MANAGING TRANSPORTATION OPERATIONS OF REUSABLE PACKAGES IN SUPPLY CHAIN SYSTEMS By Irandokht Parviziomran A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Packaging – Doctor of Philosophy 2023 ABSTRACT Containers and packages contribute to about 28% of more than 292 million tons of municipal solid waste generation. Although a shift from single-use to reuse models seems inevitable for more sustainable packaging, there are economic and environmental aspects of reusables’ operational management implications that are yet to be investigated. As more than half of the total costs of logistics operations come from transportation and transportation operations have been perceived as the main barrier in successful implementation of reusable packages, the main purpose of this research is to develop analytical frameworks for managing transportation operations of reusable packages. Transportation operations of reusable packages comprise of two tasks: delivery of full reusable packages from a central depot (product/packaging sites) to customers’ door and pickup of empty ones from customers’ door and transporting them back to the depot (cleaning facilities) where they are cleaned, refilled, and reused. The delivery of full reusables and the pickup of empty ones can be mathematically modeled as the vehicle routing problem with backhauls (VRPBs). The aim of the VRPB is to construct a set of optimal routing plans for a fleet of homogenous vehicles located at the depot such that (i) full packages are loaded at the depot and transported to customers’ door, (ii) empty packages are collected from customers’ door and returned to the depot, (iii) each vehicle starts (ends) its route from (at) the depot, (iv) each customer (with delivery and/or pickup requests) is visited exactly once and exactly by one vehicle, and (v) the total requests served by a vehicle does not exceed the vehicle’s capacity. In this research, we mathematically formulate the delivery and pickup operations of reusable packages as the VRPB in two different networks: static and dynamic. In the static network, all input data (such as customers’ request and location) is known before transportation operations start, while in the dynamic one, all or part of input data is revealed during transportation operations. To formulate the problem, we initially propose a new mixed-integer programming (MIP) model for a special class of the VRPB that serves delivery requests prior to pickup requests. In this case, we assume that customers either have delivery or pickup request and not both. We expand our model to a broader version of the VRPB where all or some customers have both delivery and pickup requests, and a pickup request can be served before a delivery request. We further develop an algorithm to manage transportation operations of reusable packages in dynamic networks. We present an analysis of the efficiency of transportation operations under a variety of factors that could provide valuable insights for decision-makers in adopting reuse models. Copyright by IRANDOKHT PARVIZIOMRAN 2023 This dissertation is dedicated to my mother and my husband. Thank you for always believing in me. v ACKNOWLEDGMENTS I would like to thank my wonderful mother, Sanam, and my beloved sister, Elmira, for always being supportive through every stage of my life, no matter how far away they are. I am thankful to my husband, Navid, for helping me to handle the burden of a doctoral student life. I am thankful for all of the times he listened to my concerns, no matter what, to make me feel better. I would also like to thank the rest of my family and friends for their inspiration and continuous support. I would like to express my sincere gratitude to my adviser, Dr. Monireh Mahmoudi, for providing me with a great research topic for my dissertation, and for giving constant help and guidance during the program. I would also like to thank the rest of my committee: Dr. Matthew Daum, Dr. Euihark Lee, and Dr. Bahare Kiumarsi for helping me out with my dissertation. I would like to thank Dr. Alireza Boloori from the Milgard School of Business at University of Washington Tacoma and Dr. Xuesong (Simon) Zhou from the School of Sustainable Engineering and the Built Environment at Arizona State University for their valuable insight and collaboration. I would also like to thank faculty, staff, and my peers in the School of Packaging at Michigan State University who have made my time at Michigan State University always engaging and exciting. vi TABLE OF CONTENTS LIST OF ABBREVIATIONS ...................................................................................................... viii CHAPTER 1. INTRODUCTION ................................................................................................... 1 CHAPTER 2. REUSABLE PACKAGING IN SUPPLY CHAINS: A REVIEW OF ENVIRONMENTAL AND ECONOMIC IMPACTS, LOGISTICS SYSTEM DESIGNS, AND OPERATIONS MANAGEMENT ................................................................................................ 11 CHAPTER 3. A LAGRANGIAN DECOMPOSITION SOLUTION APPROACH FOR THE VEHICLE ROUTING PROBLEM WITH BACKHAULS .......................................................... 56 CHAPTER 4. EVALUATING OPERATING MODELS AND URBANISM FOR TRANSPORTATION OPERATIONS OF CIRCULAR REUSE PLATFORMS ....................... 93 CHAPTER 5. TRANSPORTATION OPERATIONS OF REUSABLE PACKAGES IN REAL- TIME NETWORKS: AN APPLICATION OF DYNAMIC VEHICLE ROUTING PROBLEM WITH BACKHAULS ................................................................................................................. 121 CHAPTER 6. CONCLUSION.................................................................................................... 160 REFERENCES ........................................................................................................................... 163 APPENDIX A. ROBUSTNESS CHECKS FOR CHAPTER 4 ................................................. 181 APPENDIX B. OTHER NUMERICAL RESULTS FOR CHAPTER 4 .................................... 188 APPENDIX C. DETAILS ON VEHICLE’S CAPACITY ......................................................... 189 vii LIST OF ABBREVIATIONS VRP Vehicle routing problem DVRP Dynamic vehicle routing problem OVRP Open vehicle routing problem VRPB Vehicle routing problem with backhaul DVRPB Dynamic vehicle routing problem with backhaul MIP Mixed-integer programing AP Assignment problem CFRS Cluster first-route second TSP Traveling salesman problem OTSP Open traveling salesman problem DOD Degree of dynamism viii CHAPTER 1. INTRODUCTION Motivations According to the U.S. Environmental Protection Agency (EPA, 2020), containers and packaging make up a major portion of the municipal solid waste, amounting for 82.22 million tons of generation in 2018 (28.1% of the total 292.36 million tons of generation). The recycling and landfilled rates of municipal solid waste, generated by containers and packaging, are also reported as 53.9% and 37.1% in 2018, respectively. Despite all efforts to improve the recycling rate of containers and packaging, experts believe that the most effective resolution for reducing the containers and packaging share in generating municipal solid waste is the implementation of reuse models (EPA, 2020). Said by TIME (2021), “Reuse, as well as plain elimination of a lot of packaging we don’t need, will also have to be a crucial part of the solution”. Reuse models for the container and packaging refer to reusable packages that are produced from durable and long-lasting materials; are designed for more than one use-cycles; and are circulated through a closed-loop supply chain system. According to Ellen MacArthur Foundation (2019), “Converting 20% of plastic packaging into reuse models is a $10 billion business opportunity that benefits customers and represents a crucial element in the quest to eliminate plastic waste and pollution”. Moreover, reuse models could contribute to more sustainable packaging, which entails a closed-loop life cycle of packages and packaging materials, is economically robust, and has a minimum environmental impact (SPC, 2011; Gustavo et al., 2018). Although the pivotal shift from single-use to reuse models is an excellent idea and seems inevitable for more sustainable packaging, it is crucial to study their operational management implications to make sure that they are justifiable from both environmental and economic 1 perspectives; otherwise, disposable/recyclable packages can easily beat reusable alternatives not only from the convenience standpoint, but also from environmental and economic perspectives. The main purpose of this research is managing transportation operations of reusable packages in supply chain systems. According to Armstrong & Associates Inc., up to 58% of the total global logistics costs could come from the transportation costs (Rodrigue, 2020). Therefore, it is crucial to manage transportation operations of reusable packages as efficient as possible since it can be deemed as another barrier in proper implementation of reuse models. Overview Containers and packages can be classified into three categories based on their functionality (Palsson, 2018) as: (i) primary packages which are the first layer of packaging and are in direct contact with products, (ii) secondary packages which are an outer packaging layer of the primary packages and may be used to prevent theft or to bundle primary packages together, and (iii) tertiary or transit packages which are used for bulk handling, warehouse storage, and transport shipping. In this research, we attribute the aforementioned categories for the containers and packages to reuse models and address the transportation operations of reusable primary packages since, based on our extensive review of literature presented in chapter 2, we observe that reusable primary packages have received less attention in the extant literature compared to secondary/tertiary options. Transportation operations of reusable primary packages comprise of two tasks: delivery of full packages and pick up of empty ones. Despite the differences in the specifications of the business models for implementing reuse models by different packaging industries, it can be perceived that the utilization of a circular/closed-loop shopping platform is common among practitioners, where products that are packed in reusable packages will be delivered from product/packaging companies 2 to customers (and/or distributed to the distribution centers), and empty packages are collected from customers (and/or from collection centers) and transported back to a facility for being cleaned, refilled, and reused. In this research, we address the direct delivery/pickup business model (i.e., delivering full packages to customers’ door and picking up empty ones from customers’ door) because this model can significantly contribute to reusable packages customers’ convenience and loyalty (see, e.g., Chang et al., 2010; Harvard Business Review, 2010). After all, the issues of customer acquisition and retention are known as one of the main barriers in adopting reuse models (see, e.g., Temper Pack, 2020; Big Commerce, 2022). An example of the direct delivery-pickup operations for reusable primary packages is the circular shopping platform called the Loop that is proposed by TerraCycle in 2019 (GreenBiz, 2019). In the Loop shopping platform (see Figure 1.1), (i) customers place their orders via the Loop’s website, (ii) the Loop collects all requested products in a reusable tote and have them delivered to customers, (iii) once products are consumed and packages are empty, customers replace them in the tote and submit a pickup request on the Loop’s website, and (iv) the empty packages are collected and transported back to a facility for being inspected, cleaned, refilled, and reused. Managing the direct delivery-pickup operations of reusable primary packages is more complex than that for traditional single-use packages, mainly due to the high volume and frequency of deliveries/pickups. In this research, we develop an analytical framework to streamline such complex operations in managing reusables in modern supply chain systems. 3 Figure 1.1. The Loop shopping platform for reusable primary packages (GreenBiz, 2019). The direct delivery-pickup operations of reusable primary packages can be mathematically modeled as the vehicle routing problem with backhauls (VRPBs). The aim of the VRPB is to construct a set of optimal routing plans for a fleet of homogenous vehicles located at depot such that (i) all deliveries are transported from the depot to customers, (ii) all pickups are collected from customers and transported back to the depot, (iii) each vehicle starts (ends) its route from (at) the depot, (iv) each customer is visited exactly once and exactly by one vehicle, and (v) the total requests served by a vehicle does not exceed the vehicle’s capacity. In addition to the forgoing characteristics, there are two other assumptions that combination of them forms four classes of the VRPB (see Figure 1.2): the type of service and backhauling strategy (Parragh et al., 2008a; Battarra et al., 2014). The VRPB can be classified with respect to the type of service as the VRPB with single demand and the VRPB with simultaneous demand. While in the former one, each customer requires either a pickup or a delivery service (not both), in the later one, customers may request for a simultaneous service of pickup and delivery. Regarding backhauling strategy (prioritizing serving delivery requests prior to pickup requests), the VRPB is classified into the VRPB with backhaul solution and the VRPB with mixed solution. In this research, we mathematically 4 formulate the direct delivery-pickup operations of reusable primary packages as the VRPB and present and analytical framework for solving this problem in static and dynamic networks. Figure 1.2. A schematic illustration for the four classes of the VRPB (Parragh et al., 2008a; Battarra et al., 2014). When all input data such as customers’ request and location are known before transportation operations start, the VRPB is static (Psaraftis et al., 2016). The static VRPB aims to construct a set of optimal routing plans for a fleet of homogeneous vehicles located at the depot for serving all offline requests (deliveries and/or pickups that are placed before the start of transportation operations). In this regard, presented in chapter 3, we first propose a new mixed-integer programing (MIP) model for the first class of the VRPB (i.e., the VRPB with single demand and backhaul solution that is depicted in Figure 1.2 (a)) that allows the main problem to be decomposed into several sub- 5 problems. More specifically, we apply Lagrangian decomposition to decompose the main problem into two open vehicle routing problems (OVRPs) and one assignment problem (AP). At each iteration, we solve the forgoing sub-problems and update Lagrangian multipliers to reduce the gap between lower and upper bounds of the global optimal solution. We propose two different layouts, parallel and sequential, for solving the foregoing sub-problems and then analyze the impact of these arrangements on the solution quality as well as computational efficiency of our proposed Lagrangian decomposition algorithm by testing them on two benchmark datasets in the extant literature. Furthermore, we test our model on a real-world transportation network, Lansing network, that is geographically located in the state of Michigan, USA. To reduce the computational burden of solving the proposed problem on this dataset, we present a cluster first-route second (CFRS) algorithm and then analyze the impact of the vehicle capacity on the solution quality of our proposed algorithm. Furthermore, in chapter 4, we expand our proposed MIP model in chapter 3 to all four classes of the VRPB (see Figure 1.2) and solve them by adapting the proposed CFRS solution approach under a variety of factors that provide relevant insights for decision makers and could help them to find solutions in adopting reuse models. To address the issue of geographical accessibility of reusable packaging customers and how it may impact the operations of reusables, we showcase our analytical model on two real-world transportation networks that are located in the state of Michigan, USA: the one in Lansing (representing an urban area) and the other one in Charlotte (representing a rural area). Moreover, to reflect on the economic and environmental implications of transportation operations of reusable primary packages, we develop a simulation model, where the distance traveled is transformed to cost and emission amounts. 6 In the real-world transportation operations, with the increasing demand for rapid and high- quality services (Ninikas and Minis, 2014; Zhu et al., 2016), reuse models require to operate in the real-time where all or part of input data is revealed during transportation operations. When input data is evolved in the real-time, the VRPB is dynamic (Psaraftis et al., 2016). Then, the aim of the dynamic VRPB (DVRPB) is to (i) collect the information of online requests that are placed during transportation operations, (ii) decide on the acceptance/rejection of requests for the same-day service, and (iii) if acceptance is the case, update/reoptimize the existing routes. Managing the direct delivery-pickup operations of reusable primary packages in dynamic/real-time networks is critical because reusable primary packages are manufactured from durable high-quality materials, they are then accounted as an asset for the corresponding product/packaging company (depending on who owns the package) and should be costly to be replaced. Therefore, it is crucial to collect empties from customers’ door rapidly to reduce the chance of damage, loss, or theft. In addition, due to the limited number of such packages in the system, it is crucial that they are rapidly returned to the system for cleaning, refilling, and reusing in order to reduce the possibility of inventory shortage of such packages. In chapter 5, we develop an analytical framework for the DVRPB that contributes to streamline such complex operations in closed-loop supply chain systems. In the DVRPB, there exists an initial routing plan that assigns offline (delivery and/or pickup) requests to a fleet of homogenous vehicles located at depot. To construct the initial routing plan, we hold the routing assumptions of the VRPB with single demand and backhaul solution (see Figure 1.2 (a)) and construct it using the MIP model and the CFRS solution approach presented in chapters 3 and 4. When transportation operations start, vehicles leave the depot to serve their assigned customers as instructed in the initial routing plan. As the initial routing plan unfolds, we name it as the existing routing plan 7 which is constantly updated based on the revealed information of online requests. In this research, online requests can only be pickups. The reason is that to serve an online delivery request, the vehicle must return to the depot in the middle of the working shift to load the full packages which is not aligned with the basic assumption of the VRPB (i.e., each vehicle must start and end its route at the depot). Then, we propose a local search heuristic algorithm for the DVRPB that decides on the acceptance/rejection of online requests and updates the exiting routing plan, simultaneously. We test our proposed analytical framework for the DVRPB on two previously introduced real- world transportation networks: Lansing and Charlotte networks in order to investigate the urbanism of customers’ location on the efficiency of the direct delivery-pickup operations of reusable primary packages. We also investigate the responsiveness of our proposed framework under a variety of real-time decision-making modes that provide relevant insights for decision makers and could help them to find solutions in operating reuse models in the real-time. Objectives The main purpose of this research is to develop an analytical framework for managing transportation operations of reusable primary packages in the closed-loop supply chain systems. Granted the aforementioned virtues for reusable packages, the main objectives of this dissertation are listed as follows: 1. Developing an analytical framework for managing transportation operations of reusable packages in static networks that could be applied for different operating models (serving a mixed sequence of pickup and delivery requests versus serving all delivery requests first and pickup requests later; customers with either delivery or pickup requests but not both versus customers with simultaneous demand). 8 2. Evaluating the impact of operating models (mentioned in objective #1) and urbanism (dense versus sparse networks) on the environmental and economic costs of transportation operations of reusable packages. 3. Develop an analytical framework for managing transportation operations of reusable packages in dynamic networks where customers have the chance of receiving the same- day service. Organization This research is organized as follows: In chapter 2, we present a systematic review of the extant literature in the light of the environmental and economic costs of reusable packaging, the design of reusable packaging logistics systems, and the implications of operations management for reusable packaging. Based on our analysis of existing studies, we then deliver insights and potential opportunities for future research on reusable packaging. In chapter 3, we initially review the exiting literature on the VRPB. Then, we propose a new MIP model for the VRPB with single demand and backhaul solution. We further show the steps taken to decompose the proposed problem using Lagrangian decomposition approach followed by parallel, sequential, and CFRS solution approaches. Then, we present our numerical experiments on two benchmark datasets in the extant literature, and a real-world transportation network, Lansing network. Finally, we summarize this chapter with discussions on possible extensions. In chapter 4, we review the extant literature followed by presenting the expanded MIP model for four classes of the VRPB. Then, we present the proposed CFRS solution approach for solving these problems. Afterward, we showcase our model on two real-world transportation networks: 9 Lansing and Charlotte networks. Finally, we present future research directions and a summary of the concluding remarks. In chapter 5, we address the transportation operations of reusable packages in the real-time. To conduct this task, we initially review the extant literature on the DVRPB. Recalling the MIP model and the CFRS solution approach for the VRPB presented in chapters 3 and 4, we then present a real-time decision-making analytical framework for the DVRPB that collects the information of newly revealed customers in the real-time, decides on the acceptance/rejection of their requests, and if acceptance is the case, updates/reoptimizes the exiting routing plan. We showcase our proposed framework on previously introduced Lansing and Charlotte networks. Finally, we summarize this chapter with concluding remarks and directions for future research. In chapter 6, we summarize the concluding remarks of this dissertation, deliver insights and potential opportunities for future research. 10 CHAPTER 2. REUSABLE PACKAGING IN SUPPLY CHAINS: A REVIEW OF ENVIRONMENTAL AND ECONOMIC IMPACTS, LOGISTICS SYSTEM DESIGNS, AND OPERATIONS MANAGEMENT Introduction Background The pivotal shift from single-use to reusable packaging has recently challenged the concept of packaging ownership. This shift has made a package an asset for the product company, and hence, the company is motivated to make the package as long-lasting and durable as possible. TerraCycle is a small company that has recently compelled more than two dozen of the world’s biggest brands such as Nestlé, PepsiCo, and Procter & Gamble to begin testing reusable packaging for their products (Makower, 2019). TerraCycle has unveiled a new circular delivery service for consumers called “Loop”, which is a circular shopping platform that replaces single-use packaging with a durable, reusable one. Consumers can order goods from the Loop website (or that of a partner) and have them delivered like traditional products ordered online. Customers pay a small deposit for a package that has been designed for 100 or more use-cycles. When the container becomes empty, customers place it in a specially designed tote for pickup or, in some cases, can bring it to a retailer. They can choose whether they want that product replenished; if not, their deposit is returned or credited to their account. The empties are sent to a facility where they are washed and refilled. The focus of the Loop’s service is on the rotation of primary packages for basic products such as shampoo, toothpaste, ice cream, etc. (the concept of primary packaging will be explained in the next paragraph). Palsson (2018) classified packages based on their layer or functionality into three different categories: (1) primary packaging which is the packaging that first envelops the product and holds 11 it. This category of packaging is in direct contact with the product; (2) secondary packaging which is an outer packaging layer of the primary packaging and may be used to prevent theft or to bundle primary packages together; and (3) tertiary or transit packaging which is used for bulk handling, warehouse storage, and transport shipping. Tertiary packaging. StopWaste and Reusable Pallet & Container Coalition (2007) provided a list of virtues reusable tertiary packaging brings to the system. Reusable transport packages improve workers safety and ergonomics, because (1) their material and design reduce or eliminate injuries due to box cutting, staples, and broken containers, (2) their ergonomically designed handles and access doors improve workers safety, (3) their standardized sizes and weights reduce back injuries, and (4) they reduce the risk of slip and fall injuries by removing in-plant debris. Reusable transport packaging also provides just-in-time delivery of the finished products, because it provides standardized ordering quantities which can improve ordering procedures and inventory tracking. In addition, it provides more frequent shipments of smaller quantities and offers deliveries close to the time of consumption which can reduce the number of days that dollars and inventory are nonproductive. Secondary packaging. Reusable secondary packaging can have advantages that are common with the tertiary option (see StopWaste and Reusable Pallet & Container Coalition, 2007). Both can reduce the product damage, because the risk of packaging failure during the transportation is lower when using reusables compared to when using single-use containers. They can also improve the quality of the finished product delivered to the end user (consumer) as ventilated reusable containers increase shelf-life and freshness. Furthermore, using these packaging systems for shipping products in a supply chain can make substantial cost-savings since cost of reusable packages can be spread over several years. In addition, both packaging systems can be beneficial 12 from waste management perspective as they produce less waste to be managed for recycling or disposal. Finally, one of the main reasons of using such packaging systems is their environmental impacts. By using this type of containers, the need for building disposal facilities or recycling facility centers is dampened. Using this type of containers for delivering products may also reduce the greenhouse gas emission rates and overall energy consumption of the whole system. Primary packaging. Makower (2019) listed three virtues for this category of packaging systems: (1) it moves from disposal or recycling to reuse which is a huge environmental upgrade; (2) it moves from relatively low value packaging materials to arguably luxury or game-changing packaging materials (e.g., from multi-layered plastic film to stainless steel, glass, or engineered plastics); and (3) it brings out new features that could have never been experienced by disposable packages (e.g., a double wall stainless steel container that keeps ice cream frozen for a number of hours after removing it from the freezer). Different types of reusable packaging are observed with different terms in the literature. For example, “returnable packaging materials” and “returnable transport items” are the terms used for reusable primary and tertiary packaging, respectively (Carrasco-Gallego et al., 2012). Refillable glass bottles for beverages (Goh and Varaprasad 1986, Del Castillo and Cochran 1996), gas cylinders (Kelle and Silver 1989a, b), containers for chemicals, single-use cameras (Toktay et al. 2000), special packaging designed for transporting medical equipments, wind turbine parts, and steel coils (Rubio et al. 2009) are some examples of returnable packaging materials. Pallets, maritime containers (Crainic et al. 1993), railcars (Young et al. 2002), standardized vessels for fluid transportation, crates, tote boxes, collapsible plastic boxes, trays (Duhaime et al. 2001), roll cages (Carrasco-Gallego and Ponce-Cueto 2009), barrels, trolleys, pallet collars, racks, lids, etc. are some examples of returnable transport items being used in business-to-business settings. 13 Returnable transport items can be also used in business-to-customer settings such as supermarket trolleys, baggage trolleys in airports and train stations, and wheeled bins arranged by local councils (Breen 2006). Aims and contributions Granted the aforementioned virtues for reusable packages, a company willing to adopt such a system for their products should address the following questions before altering their current packaging system: (1) is reusable packaging environmentally and economically feasible? (2) If so, what is the proper design for their logistics system? (3) What are the implications of operations management for reusable packages? In Figure 2.1, we illustrate these steps schematically. In this research, we aim to contribute to the literature by reviewing existing studies in light of the foregoing three questions and identify potential directions/opportunities for future research in this regard. To name a few, the future research could (i) incorporate environmental factors (e.g., carbon taxes, environmental externalities, and eco-costs), consumers’ behavior, and packaging designs in measuring costs, (ii) explore the impact of ownership and third-party logistics in the operations of reusable packaging systems, (iii) analyze such systems under more complicated, and yet realistic, settings (e.g., multiple sender-recipient pairs, variations in the quality of packages, asymmetric information between third parties and senders/recipients), and (iv) consider inter-parties and product-demand-package coordination in managing operations. For a comprehensive discussion regarding these items and many others, one can refer to section 2.6. 14 Step1: Step 3: Step 2: Reusable packaging Feasibility/viability Logistics system design operations management • Measure environmental and economic costs • Design activities and responsibilities of • Inventory management • Comparison of reusable against single-use participants in a reusable packaging system. • Routing and scheduling packaging o Participants: sender, central agency, carrier, • Purchasing and repairing policies and recipient • Performance measurements Figure 2.1. An illustration for the aims of this research. It should be noted that the reusable primary packaging is a newer concept compared to secondary/tertiary options. Therefore, the existing literature have primarily focused on supply chains using reusable packaging with these options. As a result, we have observed the aforementioned research directions and opportunities with respect to these types of reusable packaging. Nevertheless, given the scope of our proposed research directions (e.g., costs, ownership, complexity of the system, quality of the package, and symmetric information, etc.), all these opportunities could also be applied for a primary reusable packaging option. To the best of our knowledge, this research is among the first studies reviewing the operations of reusable packaging systems. Glock (2017) has recently provided a review on returnable transport items, albeit our approach is different from the following standpoints: (1) as one of our classification schemes, we review the literature based on both economical/environmental factors that would impact costs (or criteria to measure these costs); (2) we consider the literature discussing various issues that might arise due to a packaging ownership; (3) we analyze the literature based on various factors in the inventory management of reusable packages, such as a planning horizon, a balance between the supply and demand of packages, and the number of usage for a reusable package; (4) we shed lights on both quantitative and qualitative studies on reusable packaging; and (5) we consider both peer-reviewed journal papers and conference proceedings in searching for relevant studies in the literature. 15 The rest of the chapter is organized as follows. Section 2.2 presents our review methodology. Section 2.3 provides a review of studies analyzing the impact of environmental and economic factors on reusable packaging supply chains. Section 2.4 provides a review of studies on various designs for reusable packaging logistics systems. Section 2.5 provides a review of studies on the operations management of reusable packaging. In section 2.6, we discuss various opportunities for future research and conclude this chapter. Review methodology To identify studies that have focused on reusable packaging in supply chains, we conducted a structured literature review based on the methodologies of Tranfield et al. (2003), Cooper (2010), Mayring (2010), and Cooper (2015), which is comprised of four steps: material collection, bibliometric analysis, content analysis, and material evaluation. While we discuss the first two steps in this section, the latter two cases will be explained in sections 2.3-5 and 2.6, respectively. Material collection The following inclusion criteria were defined before searching the literature: 1. Studies that shed lights on considerations/implications that should be made when deciding to shift from single-use to reusable packages. 2. Studies published in peer-reviewed academic journals or conference proceedings. 3. Studies written in English. Our methodology of the literature review, which resulted in the selection of 86 studies, can be summarized as follows (see Table 2.1 for more details): 1. We applied keywords such as “reusable packages”, “reusable packaging”, “reusable packaging material”, “returnable packaging material”, “returnable containers”, and “returnable transport items” to the online scholarly database Scopus. We added those 16 papers to our sample with at least one of these keywords in their title, abstract, or list of keywords. This search led to a sample of 2,857 studies. 2. Based on specific information in the title, abstract, or keywords of these papers, we shrank the sample to 128 relevant papers. 3. We then read these papers completely, and based on their relevance, 62 papers remained in the sample. 4. All references of the studies in our sample (from step 3) were checked (forward snowball search). This led to an additional 34 papers, out of which 10 were found relevant and added to the sample. 5. All studies that cited those of the sample (from step 3) were checked (backward snowball search). This led to an additional 21 papers, out of which 14 were found relevant and added to the sample. Table 2.1. Review protocol. Refine type Description No. • Studies are identified through database Scopus search along with a forward/backward snowball search. • Studies that focus on considerations that should be made when shifting from single- Inclusion use to reusable packages. --- criteria • Studies written in English. • Studies published by February 2019. • Studies published in peer-reviewed academic journals or conference proceedings. Defined “reusable packages”, “reusable packaging”, “reusable packaging material”, “returnable --- keywords packaging material”, “returnable containers”, and “returnable transport items” Keyword Online database Scopus with the defined keywords: studies that include at least one of 2,857 search these keywords in their title, abstract, or list of keywords. Filtering I Checking relevance of content by reading the title, abstract, and keywords of the paper. 128 Filtering II Checking relevance of content by reading the whole paper. 62 Forward snowball All references of the studies from filtering II were checked. 10 approach Backward snowball All works that cited studies from filtering II were checked. 14 approach Final sample 86 size 17 Bibliometric analysis After analyzing the sampled studies in detail, we classified them into three different categories: (1) works that evaluate the feasibility/viability of reusables in terms of environmental and economic factors, (2) works that provide information for the design of a logistics system adopting reusable packages for their product, and (3) works that provide information on the operations management of reusables in supply chains. Figure 2.2 illustrates the number of research studies under these categories. 25 20 Number of studies 15 By 2000 2001-2010 10 2011-2019 5 0 Environmental and economic Logistics system design Operations management impacts Figure 2.2. The number of studies under three categories. Among these categories, we further classified categories (1) and (3). Indeed, we differentiated studies on the feasibility/viability of reusable packages into two sub-categories: works that have discussed either factors affecting environmental and economic costs of reusables or criteria for measuring these costs. Furthermore, we differentiated studies on the operations management of 18 reusable packages into four sub-categories: works that have studied inventory management of reusables, scheduling and routing of reusables, reusables’ repairing and purchasing policies, and performance measurements of reusables. Regarding the inventory management of reusables, we further differentiated the studies into three sub-sub-categories: works that have studied factors affecting inventory management of reusables (e.g., time horizon), mathematical modeling for inventory management of reusables, and tracking technologies for managing the inventory of reusables. Figure 2.3 shows this classification. To this end, we also present the number of studies conducted per content category in Table 2.2. Operations management, environmental and economic feasibility/viability, and logistics system design constitute 65%, 31%, and 4% of these studies, respectively. Reusable packaging Environmental Logistics system Operations and economic design (§4) management (§5) impacts (§3) Criteria Inventory Repairing and Performance Scheduling and measuring management purchasing measurements routing (§5.2) impacts (§3.1) (§5.1) policies (§5.3) (§5.4) Factors affecting Factors impacting inventory management impacts (§3.2) (§5.1.1) Mathematical modeling of inventory management (§5.1.2) Tracking technologies for managing inventory (§5.1.3) Figure 2.3. Classification of reusable packaging studies based on three main categories. 19 Table 2.2. Number of studies on reusable packaging based on three main categories and corresponding sub-categories (see Figure 2.3 for this classification). Topics # studies References Environmental/economic In total: factors 27 ≤ 2000: Dubiel (1996), Van Doorsselaer and Lox (1999) Criteria for measuring 2001-2010: Ross and Evans (2003), Singh et al. (2006) environmental and economic 8 2011-2019: Menesatti et al. (2012), Goudenege et al. (2013), costs Goellner and Sparrow (2014), Katephap and Limnararat (2015) ≤ 2000: Kroon and Vrijens (1995), McKerrow (1996), Rosenau et al. (1996), Twede (1999), Van Doorsselaer and Lox (1999) 2001-2010: Ross and Evans (2003), Gonzalez-Torre et al. (2004), Factors affecting the Lee and Xu (2004), Twede and Clarke (2004), Mollenkopf et al. environmental and economic 21 (2005), Tsiliyannis (2005a,b), Grimes-Casey et al. (2007) costs 2011-2019: Levi et al. (2011), Palsson et al. (2013), Accorsi et al. (2014), Carrano et al. (2015), Zhang et al. (2015), Katephap and Limnararat (2017), González-Boubeta et al. (2018), Bortolini et al. (2018) ≤ 2000: Lützebauer (1993), Kroon and Vrijens (1995) In total: Logistics system design 2001-2010: Hellström and Johansson (2010) 3 2011-2019: --- In total: Operations management 56 ≤ 2000: Schrady (1967), Florez (1986), Kelle and Silver (1989a,b), Bojkow (1991), Dejax et al. (1992), Crainic et al. (1993), Rosenau et al. (1996), McKerrow (1996), Holmberg et al. (1998), Cheung and Chen (1998), Buchanan and Abad (1998), Brewer et al. (1999), Shayan and Ghotb (2000) 2001-2010: Duhaime et al. (2001), Choong et al. (2002), McFarlane and Sheffi (2003), Lampe and Strassner (2003), De Jonge (2004), Inventory management 35 Minner and Lindner (2004), Angeles (2005), Vijayaraman and Osyk (2006), Foster et al. (2006), Johansson and Hellström (2007), Thoroe et al. (2009), Ilic et al. (2009), Hellström (2009), Carrasco- Gallego and Ponce-Cueto (2009) 2011-2019: Maleki and Meiser (2011), Mason et al. (2012), Kim et al. (2014), Kim and Glock (2014), Glock and Kim (2014), Cobb (2016a), Hariga et al. (2016) ≤ 2000: --- 2001-2010: Leung and Wu (2004), Karimi et al. (2005), Di Francesco et al. (2009) Scheduling and routing 10 2011-2019: Soysal (2016), Ech-Charrat and Amechnoue (2016), Ech-Charrat et al. (2017a,b,c), Sarkar et al. (2017), Iassinovskaia et al. (2017) ≤ 2000: Kelle and Silver (1989a,b) Repairing and purchasing 2001-2010: --- 5 policies 2011-2019: Atamer et al. (2013), Limbourg and Pirotte (2018), Yang et al. (2018) ≤ 2000: --- 2001-2010: Chew et al. (2002), Chonhenchob and Singh (2003), Twede and Ckarke (2004), Breen (2006), Chonhenchob et al. Performance measurements 8 (2008) 2011-2019: Maleki and Reimche (2011), Glock and Kim (2016), Cobb (2016b) 20 Finally, we note that the majority of the literature falls into journal articles in the field of operations research, transportation, and packaging. Table 2.3 summarizes the list of top 5 journals where studies on reusable packaging systems have been published. Table 2.3. Top 5 journals published studies on reusable packaging systems. Number of No. Journal’s name papers 1 Packaging Technology and Science 10 2 International Journal of Production Economics 9 Transportation Research Part E: Logistics and Transportation 3 5 Review 4 International Journal of Production Research 3 International Journal of Physical Distribution and Logistics 5 3 Management In concluding this section, we note that, as part of our review methodology, we discussed material collection and bibliometric analysis in sections 2.2.1-2, whereas content analysis will be discussed in sections 2.3-5, and material evaluation will be discussed when delivering research opportunities in section 2.6. Reusable packaging: Environmental and economic impacts Reuse strategies are known to reduce the demand for raw materials and decrease the quantity of waste to landfill. However, such strategies have been criticized by decision makers due to their potential link to other types of resource consumption, environmental, and economic factors that are less tangible but not less important. More vehicles, added weight, reverse logistics costs, greenhouse gas emissions, and energy to clean packages and totes are the tolls of using reusable packaging. In this section, we review studies on environmental and economic impacts of reusable packages for all three types of packaging. 21 Criteria for measuring environmental and economic costs Several studies have defined criteria for measuring environmental and economic costs of reusable packaging. For example, Van Doorsselaer and Lox (1999), Ross and Evans (2003), and Singh et al. (2006) have considered the total energy consumption as a criterion in this regard. Ross and Evans (2003) showed that the energy consumed during transportation is negligible in comparison to the overall energy consumption of the system. This finding is on the contrary with this notion that transportation emissions are perceived to be the main reason for not utilizing the reusing option. The studies conducted by Dubiel (1996), Ross and Evans (2003), Singh et al. (2006), Menesatti et al. (2012), and Katephap and Limnararat (2015) measured the environmental costs of reusable packaging by quantifying the waste produced by using these packages in different industries. Goudenege et al. (2013) defined investment/transportation costs and carbon dioxide emissions as relevant criteria measuring the economic and environmental costs of reusable containers, respectively. The study conducted by Goellner and Sparrow (2014) measured the environmental impacts of single-use and reusable containers in transportation of pharmaceutical and biological materials by quantifying greenhouse gases generated from employing these packaging systems. The results of this study showed that, compared to single-use packaging containers, reusable options emit less carbon dioxide and have less potential to generate acidification, eutrophication, photochemical ozone, human toxicity, and postconsumer waste. In addition, this study confirmed that reusable containers are about half the mass of the average single-use packaging containers since a single-use container includes materials for insulation, gel packs, gel bricks, and corrugate, while a reusable container contains vacuum-insulated panels, thermal isolation chamber, phase change media, and outer corrugate. This, in turn, results in lower transportation emissions despite extra trips that a reusable container needs to make. 22 Factors affecting the environmental and economic costs There are several studies in the literature that have focused on factors affecting the environmental and economic costs of reusable packaging. For example, Rosenau et al. (1996) incorporated methods used for calculating costs (e.g., payback period, accounting rate of return, the net present value), ownership (i.e., who owns the containers?), and the percentage of the use of a reusable container. Twede (1999) showed that the capacity of the industry seeking for eco- friendly packaging in providing the storage space (for empty containers), labor and space (to sort containers), and washing and repair operations is the most important factor affecting the economic viability of using these containers. Furthermore, Van Doorsselaer and Lox (1999) showed that breakage rate of glass bottles plays an important role on justifying reusable packaging in the beer industry. The authors claimed that if the breakage rate remains below 5%, using reusable glass bottles can be environmentally and economically justifiable. The study conducted by Ross and Evans (2003) revealed that the geographical location of certain processes can significantly affect the environmental costs of a packaging system. Gonzalez-Torre et al. (2004) showed that factors such as size of the sectors, distribution system design, and demands of the foreign market result in different environmental impacts and reverse logistics policies in European bottling and packaging companies. Lee and Xu (2004) showed that factors such as weight of package, length of service life, degree of recyclability, total number of reusable parts used in the package, and total amount of products being transported per trip can affect both economic and environmental costs of reusable containers. Mollenkopf et al. (2005) showed that some factors such as the size of reusable containers, average daily volume of product to be transported, delivery distance, cycle time, total number of units per container (pack quantity), and fluctuation in peak volume can affect costs of reusable containers. 23 Mollenkopf et al. (2005) showed that reusable containers are more economically justifiable if larger containers are involved, and/or the average daily volume of product to be transported is high, while single-use containers are more economically practical when delivery distance, cycle time, pack quantity, and/or fluctuation in peak volume increases. The study by Tsiliyannis (2005a) showed that factors such as annual reuse frequency, lifetime, maximum number of reuse trips, amount of packaging present in the market, annual production and net trade imports, recycle rate, reuse rate, and consumer discard rate affect the environmental impacts of reusable packaging systems. Tsiliyannis (2005b) demonstrated that the conventional recycling rate fails to reflect environmental performance of a packaging system with recyclable or reusable containers. They also found that reuse rate, total number of trips using reusable containers, and lifetime of reusable containers are not enough to evaluate the environmental performance of a packaging system. This study introduced a new index that is the total amount of material used for packaging to deliver the product from the manufacturing plant to the consumer’s door to reflect the immediate improvements of performance of a packaging system from an environmental perspective. Among all studies that have analyzed the impacts of various factors on the environmental and economic costs of reusable packages, Grimes-Casey et al. (2007) is the only study that analyzed the impact of consumers’ behavior on the economic impacts of reusable packaging. The authors showed that, although refillable bottles seem to be more economical in the long run, it is wiser to set incentives for refillable bottles only if consumers’ return rate is high. Those consumers that keep or dispose refillable bottles, or those consumers that reduce their demand in response to the return incentive drive costs up and force the whole system to use disposable bottles instead. As a result, the authors concluded that the optimal strategy for choosing a bottle packaging is totally dependent on the level of consumers’ cooperation. Palsson et al. (2013) analyzed the effect of 24 factors such as packaging refill rate, packaging material, transport, materials handling, waste handling, and administration on the environmental and economic costs of reusable packaging in a case study at Volvo Logistics Corporation. Surprisingly, the authors showed that, in this specific industry, a single-use packaging system is more attractive from both economic and environmental perspectives, and as a result, reusables are not always superior from a sustainability standpoint. Accorsi et al. (2014) showed that several factors such as containers’ service life, washing rate, waste disposal treatment, as well as network geography can affect the environmental and economic costs of reusables. Several studies have focused on the impacts of management strategies on the environmental and economic justifiability of reusable packaging. For example, Kroon and Vrijens (1995) showed that reusable containers are justifiable only if several parties participate in the logistics system. McKerrow (1996) concluded that reusable containers work best under the following circumstances: (1) when a single party is responsible for determining the standards associated with the quality of new containers to be purchased or used containers to be maintained (while meeting all parties’ expectation); (2) the whole system is controlled by a single authority; (3) the containers are monitored from the beginning to the end of distribution process; (4) reusable containers ownership is more appreciated among parties; and (5) the authority’s objective is to minimize the collection process of empty containers, maximize the containers’ utilization, and satisfy all drop points’ requirements and preferences. Another example of the studies that have analyzed the impact of management strategies on the environmental and economic costs of reusables is the study conducted by Twede and Clarke (2004). The authors presented two case studies, the U.S. automobile manufacturing industry and the U.K. supermarket industry, and showed that in the former, reusable packaging can reduce 25 purchase and disposal costs. However, the operational cost can become a challenge if the logistics of containers are not well-managed. Inefficient allocation (getting the right number of right kinds of empty containers to the right place at the right time) and ineffective tracking increase the number of containers needed in a system, and thus, the total cost of the system. The authors showed that the best solution is to involve a third-party logistics (a transportation service provider) to manage the containers flow. Twede and Clarke (2004) also showed that reusable packaging in the supermarket industry can reduce purchase and disposal costs only if the containers remain in the custody of one company. Carrano et al. (2015) is the only study in the literature that has analyzed the environmental impacts of reusables under different management strategies over various phases of their life cycle. The management strategies considered in their research include (a) single use, (b) reusage by purchasing reusables by the user, and (c) reusage by leasing reusables by the user. The life cycle of a reusable is also divided into five phases: (1) raw material sourcing, (2) manufacturing, (3) transportation and use, (4) refurbishing, and (5) end of life disposal. Carrano et al. (2015) found that during phases (1)-(2) and (3), strategy (c) and (a) result in the lowest carbon emission, respectively. During phase (4), depending on the service condition (i.e., whether the reusables are loaded with light loads and/or subjected to a good handling environment) the emission rate may vary. For the case of reusage (strategies (b) or (c)), the emission does not vary significantly by the handling condition if the pallets are loaded with light loads. However, when servicing heavy loads, strategy (b) and (c) provides the lowest emission if reusables are subjected to a good and a rough handling environment, respectively. During phase (5), strategy (a) is the worst one. Zhang et al. (2015) is another example of studies that have assessed the economic (transportation and inventory) costs of reusable containers under different management strategies. 26 The management strategies considered in that research are dedicated and shared modes. The study showed that in the shared mode, the cost saving is negatively correlated with the number of package categories. In an extreme case, each supplier needs unique packages, and therefore, shared mode does not make sense. The paper concluded that the shared mode could be more practical for the networks with balanced demands where there is a small gap between the demand of two areas. In addition, if the ratio of the packages that are not returned to the supplier (because they are broken, lost, or stolen) is high, shared mode could be beneficial to reduce the cost. The authors also proved that, since in the shared mode the long-distance trips of empty packages are replaced by short-distance trips, time savings and subsequently, cost reduction should be expected. Finally, Katephap and Limnararat (2017) analyzed the operational, economic, and environmental costs of reusable packages under different reverse logistics managements, i.e., the single-, round-, and multi-trip arrangements. The authors found that the multi-trip arrangement is the most viable option from both operational and environmental perspectives, while the single-trip arrangement is the most favorable option from an economic perspective. This is because its payback period is the shortest among other arrangements. Table 2.4 summarizes the studies that have focused on factors influencing the environmental and economic costs of reusable packaging in various supply chain systems. 27 Table 2.4. A list of papers studied the factors affecting the environmental and economic impacts of reusable packages. Factors Reference Industry Factors in details Management Technological Process Kroon and General (any Vrijens ✓ Management strategies logistic system) (1995) McKerrow General (any ✓ Management strategies (1996) logistic system) The method being used for Rosenau et al. General (any calculating costs, ownership, ✓ ✓ (1996) logistic system) the percentage of use of a reusable container Capacity of the industry seeking for eco-friendly packaging in providing the General (any Twede (1999) ✓ ✓ storage space for empty logistic system) containers, labor and space to sort containers, and washing and repair operations Van Doorsselaer The breakage rate of glass Beverages ✓ and Lox bottles (1999) Refrigerators Ross and Geographical location of and freezers ✓ Evans (2003) various process steps manufacturing Size of the sectors, distribution Gonzalez- Food and ✓ ✓ ✓ system design, the demands of Torre (2004) beverages the foreign market Weight, number of reusable Lee and Xu parts, number of transported Food (yogurt) ✓ ✓ (2004) pottles per trip, service life, and recyclability Cost of purchasing new Automobile containers, containers’ strength, Twede and manufacturing standardization, cycle time, Ckarke ✓ ✓ ✓ and supermarket empty miles and extra handling, (2004) industry ergonomics, management strategies Size of reusable containers, average daily volume of product to be transported, delivery Mollenkopf Automobile ✓ ✓ distance, cycle time, total et al. (2005) manufacturing number of units per container (pack quantity) and fluctuation in peak volume 28 Table 2.4 (cont’d) Annual reuse frequency, lifetime, maximum number of reuse trips, amount of Tsiliyannis packaging present in the Beverages ✓ ✓ (2005a, b) market, annual production and net trade imports, recycle, reuse, and discard rate of consumer Grimes- General (any Companies’ incentives for Casey et al. ✓ logistic system) using eco-friendly packaging (2007) Silva et al. Automobile Weight of the container, total ✓ ✓ (2013) manufacturing cycle time, containers’ lifetime Packaging fill rate, packaging Palsson et al. Automobile material, transport, materials ✓ ✓ ✓ (2013) manufacturing handling, waste handling, and administration Lifespan, washing rate, waste Accorsi et al. Food and ✓ ✓ disposal treatment, network (2014) beverages geography Management strategies, number of package categories, gap Zhang et al. Automobile ✓ ✓ ✓ between the demand of two (2015) manufacturing areas, ratio of failed containers, logistics system design Carrano et al. General (any ✓ Management strategies (2015) logistic system) Katephap and General (any Limnararat ✓ Management strategies logistic system) (2017) Reusable packaging: Logistics system design Before implementing reusable packaging for packing of basic products, it is crucial to design different components of the system and define activities and responsibilities of different stakeholders. For example, if a company decides to partner with TerraCycle, then who is responsible for damages on refillable packages? What will be the role of consumers in this loop? Who will be in charge of shipment and collection of these packages in such a hyper-local delivery/pickup (i.e., requiring frequent delivery/pickup)? In this section, we review studies that have analyzed the logistics system design of reusable packaging. It should be noted that, compared to our discussion on environmental/economic factors (Section 2.3) or operations management of 29 reusables (Section 2.5), logistics system design of reusable packaging has been addressed in a handful of studies. In particular, all instances found in the literature have focused on a tertiary packaging option. Designing a return logistics system is mainly based on reusable containers’ ownership and the responsibility of managing, cleaning, controlling, maintaining, and storing these containers. Kroon and Vrijens (1995) provided a comprehensive discussion about potential designs based on the study conducted by Lützebauer (1993). In this regard, return logistic systems are categorized as switch-pool systems, systems with return logistics, and systems without return logistics. Switch-pool systems: They are referred to systems where every participant has its own portion of containers and is responsible for cleaning, controlling, maintaining, and storing. A switch-pool system can be designed as a sender-recipient or sender-carrier-recipient system. In the former, the sender is responsible for managing the return flow of containers. In the latter, an ownership switch takes place at every exchange of containers among participants, and the carrier is responsible for managing the return flow of containers. Systems with return logistics: They are defined as the third-party’s ownership in which a central agency owns the containers and is responsible for the return of the containers after they have been emptied by the recipient. In this system, the recipient bundles empty containers and stores them until a sufficient number of containers has accumulated for cost-effective collection. Regarding the role of the central agency in this supply chain, systems with return logistics can be designed as a transfer system or a depot system. In the transfer system, the central agency is only responsible for return of containers from the recipient to the sender, and the sender is fully responsible for tracking, management, cleaning, maintenance, storage, as well as stock level of containers. In the 30 depot system, the idle containers are stored at depots by the central agency. The central agency cleans the containers (if necessary) and maintain them at the depot to be used for next shipments. There are two different designs for depot systems: with booking and with deposit. In depot system with booking, the sender has an account with the central agency. When containers are delivered to the sender, the corresponding quantity is debited in the sender’s account. Similarly, when the sender sends the containers to a recipient, the corresponding quantity is credited in the sender’s account, and debited in the recipient’s account. The sender should submit the necessary data to the agency for each shipment. This allows the agency to control the flows of the containers. In the depot system with deposit, the sender pays the agency a deposit for the number of containers delivered to his site. The deposit equals at least the value of the containers. The sender debits his recipient for this deposit, who does the same with his recipient, and so on. The moment the containers are delivered to the final destination, they are collected by the agency. Then, the agency refunds the deposit to the party from which the containers were collected. The deposit finances the shrinkage of the containers. The refundable deposit encourages quick return of empty containers and prevents the empty containers being stocked in one plant for a long period of time. Systems without return logistics: Here, the central agency owns the containers, the sender rents the containers from the agency, and the sender is fully responsible for return logistics, cleaning, control, maintenance and storage. Hellström and Johansson (2010) introduced a new variation of logistics system design based on the foregoing proposed categories by Kroon and Vrijens (1995). Based on their classification, there are three types of control strategies for managing reusable containers: switch-pool system, transfer system, and depot system. In the switch-pool system, a fixed number of reusable containers are assigned to each participant, and when loaded containers are delivered to the 31 recipient, the recipient must give the sender the same number of empty containers in return. In the transfer system, the sender is fully responsible to track, manage, maintain, and store containers, while in the depot system, containers are maintained and stored in depots by a main agency. In the depot system, the sender sends fully loaded containers to the recipient, and then, the depot collects and returns the empty containers from the recipient. Depot systems can be coupled with deposits, where the sender pays the central agency a deposit for every single container used. Then, the deposit is refunded when the container is returned to the depot. Table 2.5 summarizes various logistics system designs of reusable packaging. Table 2.5. Various logistics system designs of reusable packaging. Managing, 6. Logistics 9. Cleaning 7. Participants8. Ownership maintaining, & storing References system design responsibility responsibility 15. Kroon and Vrijens 10. Switch-pool 11. Sender- 12. All 13. All (1995); Hellström 14. Sender systems recipient participants participants and Johansson (2010) 17. Sender- 16. Switch-pool 18. All 19. All 21. Kroon and Vrijens carrier- 20. Carrier systems participants participants (1995) recipient 22. Systems with 27. Kroon and Vrijens return 23. Sender-central 24. Central (1995); Hellström logistics; agency- 25. Sender 26. Sender agency and Johansson transfer recipient (2010) system 28. Systems with 33. Kroon and Vrijens 29. Sender-central return 30. Central 31. Central (1995); Hellström agency- 32. Central agency logistics; agency agency and Johansson recipient depot system (2010) 34. Systems 35. Sender-central 36. Central 39. Kroon and Vrijens without return agency- 37. Sender 38. Sender agency (1995) logistics recipient Reusable packaging: Operations management Reusable containers are loaded at the sender’s plant to transport products to the recipient’s site. Concurrently, empty containers are returned from the recipient to the sender. The exchange occurs 32 at the recipient’s site. Returned containers are inspected to decide whether they are in a condition to be used for the next shipment or not. Containers that meet the pre-defined criteria are sent for loading products. Damaged containers are transferred to the repair department to determine whether or not they are repairable. If not, they are sent for disposal. Repaired containers are sent for loading products. If the total number of containers is not enough to satisfy the demand, the sender purchases or rents new or used containers. Based on this premise, Figure 2.4 presents the flow of reusable containers between a sender and a recipient. Picking up empty Inspection No No Repairable? Disposal containers (Reusable?) Yes Yes Recipient Yes Adding parts, Safety stock level cleaning, or (enough?) sanitizing Delivering full containers No Purchasing or renting containers Figure 2.4. The flow of reusable containers between a sender and a recipient. Operations management of reusable containers is one of the main concerns of companies who are willing to adopt reusable containers for their own business. For those willing to adopt reusable packages for basic products, managing the operations of reusable packages is more critical and complicated due to the high volume and frequency of deliveries/pickups (i.e., hyper-local deliveries/pickups). The existing literature on the operations management of reusable packages can be categorized as follows: studies focusing on (1) managing the inventory of reusable packages at warehouse storages, (2) the scheduling and routing of reusable packages, (3) determining 33 optimal disposal, repairing, and replacement policies for reusable packages, and (4) measuring the performance of reusable packaging systems. In the following sections, we will review studies corresponding to each of these categories. Inventory management 2.5.1.1. Factors impacting inventory management One of the factors affecting the inventory management of reusable packages is the length of planning horizon. Despite various studies analyzing the impact of planning horizon on the production planning and control of a product, very few studies have focused on the impact of planning horizon on the inventory planning and control of reusable packages. Florez (1986) is among first studies investigating the impact of this factor on managing reusable containers. Surprisingly, this study showed that sensitivity of optimal solution of containers allocation and distribution decision to the length of planning horizon is negligible. However, the study clearly stated that this conclusion cannot be generalized to other cases as the length of planning horizon may vary upon the concentration of activities in the network. Contrary to the finding by Florez (1986), the study conducted by Crainic et al. (1993) showed that the length of planning horizon is critical in determining the optimal solution of containers allocation and distribution. The authors showed that having information about the future supply and demand of containers is necessary for determining the length of the planning horizon. Dejax et al. (1992) explicitly showed that the planning horizon should be long enough (to consider the set of arrivals and departures of coming trips) and yet short enough (to include the labor contracts, safety regulations, and other practical considerations). The study conducted by Holmberg et al. (1998) showed that the planning horizon should be longer than the longest transportation time in the system to lower the shortage level of empty freight cars. Cheung and Chen (1998) concluded 34 that a longer planning horizon does not necessarily provide better solutions, while the study by Choong et al. (2002) showed that a longer planning horizon encourages the system to use cheaper and slower transportation modes. Choong et al. (2002) also found that the effect of planning horizon length on empty containers management may vary upon the number and location of depots. One of the most recent studies by Carrasco-Gallego and Ponce-Cueto (2009) presented a dynamic regression model for forecasting the returns of reusable containers in closed-loop supply chain systems to determine the length of planning horizon. Aside from planning horizon, there are studies in the literature analyzing the impact of other factors on the inventory management of reusable packaging. For example, the study conducted by Bojkow (1991) assessed the impact of the trippage number for reusable containers on the inventory control of reusable packages. The trippage number is the total number of times a container is reused as part of its life cycle. Bojkow (1991) derived a formula for calculating the average number of trips made by a reusable container. The study developed a simulation model that generates a random data for average trippage number of reusable containers considering the total number of lost and existing containers in the system. Another example is the study conducted by Duhaime et al. (2001) in which they analyzed the impact of the balance between the inventory of supply and demand locations on the inventory control of reusables. Duhaime et al. (2001) studied the use of reusable containers between Canada Post and its large business customers to determine whether the out-of-stock issue is truly due to the shortage of containers. The authors developed a minimum- cost-flow model and showed that Canada Post has enough containers to satisfy demand, and the periodic shortage is due to the inventory imbalance between supply and demand locations. 35 2.5.1.2. Mathematical modeling of inventory management Several studies have focused on mathematical modeling of the inventory management of reusable packages. Kelle and silver (1989a) is the first study that presented a mathematical framework in this regard. Their model considered several assumptions to simplify the model (such as zero lead time). Buchanan and Abad (1998) also formulated the inventory management of reusable containers and simplified their model by assuming that the probability of a container being returned is independent of the age of the container. The age of the container is the time since the container was last issued. To the best of our knowledge, no research study has incorporated this factor into the modeling of the inventory control of reusables. Developing a mathematical model in which the probability of a container’s return would be dependent upon its age, considering this assumption that the age-mix of the containers remains stable from period to period, is one of the subjects that can be further explored. Kim et al. (2014) is the first seminal study that have explored the inventory control of reusable containers when they are used for transporting perishable products. The authors assumed that the return lead time of the containers is stochastic, and the product being shipped by these containers deteriorates if delivery is late. Their model has some assumptions that could be improved by future research. First, the containers’ capacity and return lot size could determine the production lot size of the supplier. Second, the return quantity of containers is assumed to be deterministic, while in practice, this return has a stochastic nature due to the possibility of getting lost or being damaged during transport. Finally, one could consider this fact that, once being repaired, damaged containers can be of inferior quality than new ones. This might, in turn, impose limitations on their usability. 36 Glock and Kim (2014) is among the first studies considering different logistical costs associated with the use of reusable containers and interactions between reusable containers and the distribution of finished products in an inventory control model. The model proposed by Glock and Kim (2014) has some assumptions that can be improved by future research works. The first one is that in their model, the inventory cost of a container is independent of its capacity/size, while in practice, larger containers require more storage space and consequently impose higher inventory costs. The second is that the authors assumed a supplier can freely choose the size of containers, while in practice, the supplier has limited options to choose from (e.g., the restrictions imposed by the transport service provider). The last one is about the objective of the model which is to minimize the total number of containers in the system. By choosing this objective, only as many containers are kept in stock as are needed to ship the largest batch quantity, while another strategy can be to increase the total number of containers such that the supplier would be able to ship a second batch before the containers return shipment of the first batch is received. Hariga et al. (2016) is another study that considered the interactions between reusable containers and finished products in a logistics cost model. Hariga et al. (2016) presented a model for single supplier-single retailer using reusable containers for shipping finished products, in which the supplier has the option to rent reusable containers from a transportation service provider if the return of empty containers is delayed. Cobb (2016a) is the first work that has presented an inventory control model of reusable containers in which containers are continuously returned by multiple retailers to the single manufacturing facility. In the model presented by Cobb (2016a), inspection and repairs occur at a constant, finite rate over time. Before Cobb (2016a), prior models were implemented in a single- supplier, single-retailer supply chain, and containers were collected by the retailer and returned to 37 the supplier in a batch fashion. In Cobb (2016a), return rate and the percentage of returned containers that are repairable are random variables. One of the assumptions of the study by Cobb (2016a) is that the inspection and repair runs begin simultaneously, while scheduling of these functions under various arrangements may reveal that alternate types of inventory processing schedules can be also cost-effective. In addition, in this chapter, the demand and repair rates are assumed to be deterministic. Future research on this subject can continue to focus on situations where inputs to the problem are random. 2.5.1.3. Tracking technologies for managing inventory In closed-loop systems, two parties exchange the full and empty reusable packages among themselves. Therefore, there is a slight chance of packages getting lost and probably being stolen (due to the one-to-one exchange). But, when we talk about open-loops, there is little to no control over these packages. When a party receives fully loaded reusable packages, he takes the products and put the reusables out back. The reusables are held until somebody comes and picks them up. Either somebody comes automatically to pick them up or the party holding empty packages must notify the party that is responsible for collecting empty packages that the packages are ready to pick up. In warehouse storages, there is often no security for these assets making them vulnerable to theft. Every year, several thefts of reusables are reported throughout the U.S. (e.g., Department of Justice, Central district of California, 2018). Asset tracking offers many benefits to supply chain participants, including reusable packaging and carried products location and recovery, product movement and speed, inventory management, possession, loss and damage accountability, and process improvement. There are potential benefits of asset visibility on costs associated with supply chain systems using reusable packages for shipments. Rosenau et al. (1996) and McFarlane and Sheffi (2003) 38 showed that asset visibility can optimize the containers’ configuration and fleet size. McKerrow (1996) and McFarlane and Sheffi (2003) showed that asset visibility can increase reusable containers’ availability. McFarlane and Sheffi (2003) showed that asset visibility can provide the feature of automatic handling as well as historical repair data collection, and hence, reduce the repair and maintenance costs of reusable containers. The study conducted by Angeles (2005) showed that asset visibility can decrease rental charges and deposits. Brewer et al. (1999) and Shayan and Ghotb (2000) concluded that having asset visibility can provide better performance resulting in less transportation costs. Shayan and Ghotb (2000) also showed that asset visibility can reduce transportation costs by reducing erroneous shipments. Vijayaraman and Osyk (2006) showed that asset visibility can decrease the use of warehouse space, enable automatic sorting and handling, and allow automatic cleaning procedures, and therefore, lower transportation costs. There are various technologies that have been developed to improve the automatic identification and tracking of reusable containers. In the study conducted by Maleki and Meiser (2011), five different technologies were introduced to improve this identification/tracking in supply chains: barcode, passive radio-frequency identification (RFID), active RFID, Wi-Fi, and global positioning system (GPS). The authors found that barcode systems are more economical compared to other auto-ID technologies. In addition, barcode systems have relatively low requirements for manual labor and are more compatible with the current inventory systems. However, there are still several disadvantages in using barcode technology for tracking reusable containers. For example, as each container must be manually scanned, the process can become quite tedious, time-consuming, and susceptible to errors. Also, barcodes must be visible and reachable to be scanned, and any damage to the barcode can cause it to be unreadable. Finally, barcode systems do not provide information about real-time location of the containers. For further 39 information about existing technologies for tracking reusable containers, one can refer to Maleki and Meiser (2011). De Jonge (2004) compared RFID tags to barcodes, and here are the list of RFID tag’s privileges: they can be read faster since they can be read simultaneously; they should not be necessarily visible to be read; they can deal with rough and dirty environments better since tags can be integrated into the packaging materials; they can be automatically read, and therefore, no labor cost is added to the system; and finally, information can be changed if needed, while in barcode systems, a new label is required if information is needed to be changed. In this research, we mainly focus on studies that have analyzed an RFID as a technology used to track reusable packages in supply chains. Angeles (2005) introduced an RFID technology and provided several case studies and implementation guidelines for managers based on published reports. Several research works have focused on the implementation of RFIDs in managing containers such as Lampe and Strassner (2003) for managing beer kegs, Foster et al. (2006) for managing containers in automotive industry, and Johansson and Hellström (2007) for controlling the wooden pallets and plastic containers. A number of studies have also focused on the impact of the RFID technology on inventory models. For example, Thoroe et al. (2009) studied the impact of this technology for tracking reusable containers in an inventory model with deterministic inputs. The authors adopted their inventory control model from the basic models proposed by Schrady (1967) and Minner and Lindner (2004) and analyzed the changes on the optimum inventory control policy due to the implementation of the RFID technology for tracking reusable containers. The authors extended the base model by introducing additional variable costs and examined the effect of this change on profitability of implementing the RFID technology in reusable containers’ tracking systems. Kim and Glock (2014) studied the impact of the RFID technology for tracking reusable containers 40 where the fraction of containers that are returned to the supplier is stochastic. Kim and Glock (2014) found that, if using of the RFID technology can improve the predictability of containers’ flow and encourage recipients to return empty containers quicker, the use of an RFID would be justifiable from economic perspective. The authors also found that the average return rate of containers and the reparability of containers are positively correlated with the reservation price of an RFID-tagged container. Ilic et al. (2009) studied the impact of the RFID technology on a high-volume and low-value reusable containers management model and particularly quantified the impact of this technology from a financial perspective, i.e., cost savings in the trip fee and asset investment. The authors showed that cost savings in the trip fee are mainly due to the transferal of loss costs and cost reductions in data management, while cost savings in the asset investment is mainly due to the overall performance improvement of reusable containers’ cycle time. By increasing the visibility and measurability, customers will also pay more attention to avoid unnecessary penalties caused by loss or breakage, and therefore, the whole system becomes more efficient. Despite its wide application, the RFID technology has also received pushback. To this end, De Jonge (2004) is among the first studies in the literature that have discussed the issues associated with the implementation of this technology: standardization, legislation concerning ultra-high frequencies, physical characteristics of an RFID, and maturity of available technology and lack of knowledge and experience. Furthermore, Mason et al. (2012) discussed the issues associated with the RFID technology used for the inventory control of gas cylinders in the packaged gas industry. Considering both positive and negative aspects of the RFID technology for tracking reusables, Hellström (2009) studied how and why companies should implement the RFID technology for tracking reusable containers, and under which circumstances the benefits of implementing this 41 technology outweigh the costs. We note that the model proposed by Hellström (2009) is only applicable for closed-loop supply chains and cannot be applied for open-loop systems. In open- loop supply chain systems, the implementation of an RFID requires some levels of information sharing as well as incentive alignments among supply chain parties, which makes the model more complicated. Scheduling and routing Leung and Wu (2004) is the first study that addressed the scheduling and routing of reusables by proposing a mathematical model for the maritime repositioning of empty containers. The authors proposed a multi-scenario time-extended optimization model with stochastic demands for empty containers. Later, Karimi et al. (2005) presented a linear programming model for routing and scheduling of multi-product tank containers being widely used for transporting fluid chemicals. Their model was designed to minimize the cost of transportation and cleaning of tank containers. Moreover, Di Francesco et al. (2009) conducted a research study to address the containers maritime-repositioning problem. In this research, several uncertainties associated with future supplies and demands, residual transportation capacity of the vessels, and maximum number of empty containers that can be loaded/unloaded on/from vessels were considered. Soysal (2016) is the first study that proposed a mathematical model for the routing of reusable containers in a closed-loop supply chain with respect to different factors such as fuel consumption, demand uncertainty, and shipping multiple products. The authors showed that introducing the emission factor into the model can better show the trade-off between economic and environmental benefits of reusable containers. Ech-Charrat and Amechnoue (2016) and Ech-Charrat et al. (2017a, b, c) are among the studies that also present mathematical models in which the objective is to minimize the managing cost of reusable containers with respect to emission constraints. 42 Sarkar et al. (2017) is among the first studies that proposed a mathematical model for scheduling and routing of reusable containers in which a third-party logistics transports the containers loaded by finished products to the retailers and collects empty containers to the supplier. The results obtained from solving their model provided the optimal planning horizon, order quantity of the retailers, size of the containers, as well as optimal shipment schedule for different retailers. An extension of this seminal work would be to consider stochastic returns and random demand pattern when the third-party logistics is the sole responsible for transporting containers. Another extension of this work could be to incorporate random defectives with rework in the production line since the model proposed by Sarkar et al. (2017) assumes a perfect production system. Iassinovskaia et al. (2017) is the most recent study proposing a mixed-integer programming model for the inventory routing of reusable containers with time windows and simultaneous pickup and delivery in closed-loop supply chains. In their model, the supplier is responsible for collecting empty containers from retailers’ locations. Each retailer has a preferred delivery time window. Each participant has a storage capacity. In addition, transporting empty and/or loaded containers are performed simultaneously through a set of homogenous vehicles. All inputs in their model are assumed to be deterministic. Optimal repairing and purchasing policies There are very few studies in the literature focusing on determining optimal disposal, repairing, and purchasing policies for reusable packages. One of these studies is the study conducted by Kelle and Silver (1989a) in which they presented a stochastic model for the optimal purchasing policy of new reusable containers. The authors conducted another study (Kelle and Silver, 1989b) to find the optimal purchasing policy for reusable containers in a logistics system, where various 43 forecasting procedures are considered based upon different amounts of information. The information includes the total number of containers being issued and/or returned. The authors found that, even if the demand pattern is known, the system may need to purchase new containers from time to time, because sometimes the inventory level of containers is too low compared to the demand during the containers’ replacement lead time. Since the system does not want to face an unexpected drop in containers’ inventory level, the demand should be forecasted, and an estimate of the accuracy of the forecast should be calculated. Atamer et al. (2013) derived optimal price and production strategies for finished products when reusable containers are used for shipping products. The authors showed that the return quantity depends on two main factors: the refund paid by the manufacturer to the customers and the customer demand. They examined their model on a logistics system with un-capacitated and capacitated production setting. Their results showed that in an un-capacitated production setting, a manufacturer always utilizes reusable containers, even if they cost more than the brand-new containers. Whereas, in a capacitated production setting, the manufacturer’s decision about using reusable containers may vary depending on the system’s parameters. This seminal paper has some assumptions that can be improved by future research works. For example, the authors considered optimal pricing and production decisions for finished products in a single period setting which can be extended to a multi-period one. Another extension could be to relax the simplifying assumption of perfect correlation between the demand and the return rates. The authors also assumed that the resource utilization of returned and brand-new containers is identical, which can be relaxed as well. Finally, their model focused on transporting products by reusable containers, while adding disposable containers as an alternative option to satisfy customer demand can be interesting to explore. 44 The recent study by Limbourg and Pirotte (2018) investigates the effect of the price of the new containers on the collection rate and resale price of existing containers in the system. The authors discussed that each reusable container leaves the logistics system due to theft, loss, or irreparable damages. Depending on how the container leaves the system, it may impose different costs (e.g., disposal cost, recycling cost) or provide some revenue (resale revenue) for the company. Recently, Yang et al. (2018) proposed a myopic purchase policy for reusable containers and employed a simulation method to show the sensitivity and robustness of their results. The authors analyzed the value of recovery information for reusable containers obtained from sensors in the environment of internet of things (IoT). IoT is a network of physical objects, e.g., reusable packaging, each equipped with a unique identifier and internet connectivity that allows for the communication and transfer of data between objects and other internet-enabled devices (Reusable Packaging Association, 2019). The recovery information dynamically tracks the recovery status of containers and provides a reliable estimate of return rate. At the end, we summarize the existing papers in the literature proposing mathematical models for operations management of reusable packages in Table 2.6. Performance measurements In this section, we review studies that have measured the performance of reusable packages in supply chain systems. Chew et al. (2002) developed several performance measurements to evaluate the operation of the gas cylinders belonging to an industrial gas manufacturer in Singapore. The manufacturer produces different gas products and sells its products to different industries. A gas cylinder as a reusable container is used in each product sale and supposed to be returned to the manufacturer after consumption. After the cylinder is returned, it will be refilled with the same gas product to get ready for the next order. Inventory management of the cylinders 45 Table 2.6. A summary of existing optimization models for operations management of reusable containers. Stochastic and deterministic Reference Objective Sender-recipient Included costs inputs Stochastic inputs: demand and Kelle and Optimizing Purchase cost of new return rates; deterministic Silver replacement time and Single-single containers, inventory inputs: stock level at the (1989a) quantity cost beginning and end of each period Stochastic inputs: demand and return rates; deterministic Buchanan Optimizing stock inputs: stock level, total Cost of replacement, and Abad level and Single-single number of containers, total penalty of shortage (1998) replacement quantity number of removed containers from the system at the beginning of each period Cost of purchasing Optimizing Stochastic inputs: demand and brand-new containers, Atamer et replacement quantity return rates; deterministic Single-single cost of using brand- al. (2013) and refundable inputs: stock level at the new and used deposit beginning of each period containers, sales price Cost of recipient's Stochastic inputs: return time; order and sender's deterministic inputs: demand Kim et al. Optimizing return lot Single-single setup, inventory cost, and return lot sizes, finished (2014) size cost of production product lot size, transport shortage, sales price capacity, and production rate Optimizing cycle Cost of recipient's time, total number of Glock and order and sender's Deterministic inputs: demand, required containers, Kim Single-multiple setup, inventory cost, number of recipients, and shipment sequence (2014) replacement cost, production rate and containers' transportation cost capacity Cost of non-tagged and RFID-tagged Stochastic inputs: return and Kim and Optimizing cycle containers, inspection repair rates; deterministic Glock time, repair lot, and Single-single cost, repair cost, cost inputs: demand and stock level (2014) replacement quantity of replacement, at the beginning of each period inventory cost Optimizing the stock Inspection cost, repair Stochastic inputs: return and Cobb level considering the Single-multiple cost, replacement repair rates; deterministic (2016a) idle time for the cost, inventory cost inputs: demand inspection operation Optimizing finished Cost of recipient's Stochastic inputs: return time; products and order and sender's Hariga et deterministic inputs: demand, containers lot sizes, Single-single setup, replacement al. (2016) production rate, and as well as number of cost, inventory cost, transportation capacity trucks for shipment transportation cost Purchase cost of Stochastic inputs: demand, brand-new containers, Optimizing return, repair, and reused rates, inventory cost of Yang et al. replacement order’s supply and repair lead times; Single-single containers, holding (2018) cycle time and deterministic inputs: stock cost of products, and quantity level at the beginning of each punishment cost of period lost sales 46 is crucial as they constitute a considerable portion of the investment and storage cost of the company. Moreover, the revenue obtained by each sale is relatively small in comparison to the cylinder’s investment cost. Therefore, it is vital to own an accurate number of cylinders to support the business operations while ensuring that the payback period is not elongated. In comparison to other reusable packages such as beverage bottles, kegs, or plastic containers, gas cylinders are quite durable, and the chance of getting lost or being stolen is quite low. Therefore, the manufacturer only purchases a new cylinder if the company’s market share increases. In this regard, four performance measurements were introduced: how frequently the cylinders are used; how long they are kept by each customer; whether they are effectively utilized; and how much safety stock is considered. Chonhenchob and Singh (2003) compared the protective performance of single-use boxes (corrugated boxes) and reusable plastic crates based on measures such as bruising and heat transfer levels during shipping and handling mangoes. Twede and Ckarke (2004) discussed two ergonomic factors, i.e., hand grips and package weight, to measure the efficiency and safety of handling materials by reusable packages. Chonhenchob et al. (2008) conducted a study to compare the protective performance of packaging systems with reusable plastic containers, single-use paper corrugated containers, and plastic foam containers. The authors evaluated the impact of different packaging systems and fruit orientation on bruising and pre-cooling time of pineapples during distribution. Bruise damage, pineapples flesh decay during storage for five days, bruise volume and changes in color, firmness, and several other factors are considered to measure the protective performance. The authors found that corrugated containers have the best protective performance for pineapples, while plastic foam containers have the worst. Also, the corrugated containers with paperboard partitions showed the lowest damage levels among other packaging systems. Finally, 47 the results showed that the highest pre-cooling rate is obtained by using reusable plastic containers, while the lowest rate is related to using single-use paper containers. Glock and Kim (2016) is another research that studied safety measures of reusable containers used for shipping products from a supplier to a retailer. In their research, they assumed that the return times of the reusable containers are stochastic and may result in delays of the next shipment from the supplier to the retailer. Consequently, the retailer may face the product shortage from time to time. To prevent from this issue, the authors suggested a reusable containers’ safety return time, safety stock, and a combination of both measures as safety measures. The authors found that implementing either a reusable container’s safety stock or a combination of both measures works better than the one where no measure is adopted or only a safety return is implemented. The authors also found that using a safety stock or a combination of a safety stock and safety return time gets more important where the uncertainty level of lead time is high and/or the shortage of the finished product is costly for the retailer. The authors concluded that for the high-risk situations, a combination of both measures is recommended. In the study conducted by Cobb (2016b), cycle time and return time were used as performance measurements of a logistics system that uses reusable containers for shipping finished products. To improve the cycle time, the author provided the following suggestions: setting more frequent collection from retailers’ site and adjusting the filling schedules when cycle time is longer than the desired target. To improve the return rate of reusable containers, the author suggested that the suppliers define a reasonable deposit system and employ GPS tracking on randomly selected containers. Breen (2006) conducted an explanatory analysis into industrial practices with business-to- business and business-to-customer relationships in which reusable containers operate for shipping 48 the products. The author found that there are several options available for companies to improve the performance of reusable containers such as communication, incentives, introducing contracts, enforcement, reminding customers about their corporate as well as moral and legal responsibility, asset management, and outsourcing logistics. All options are applicable for both business-to- business and business-to-customer relationships except corporate responsibility and outsourcing which may not be practical in business-to-customer environment. Future theoretical and empirical research analyzing the effects of collection and production schedules on the cycle time and return rate would be interesting to pursue. The relationship between deposit systems and ownership contracts between manufacturers and customers on the return rate would be also worth examination. The study conducted by Maleki and Reimche (2011) proposed three managerial recommendations to improve the performance of reusable containers in the whole supply chain: improving the communication and information flow between senders and recipients by improving the communication capacity of the existing supply chain networks; implementing liability contracts between senders and recipients via incorporation of a legal statement in the bill of lading to reduce the number of lost or damaged containers; and incorporating an automatic identification technology to help with tracking the containers. To this end, performance measurements of reusables may vary from one supply chain system to another. Studying the performance, protection, and safety measurements of reusable packages in different supply chains has not received much attention and could be worth examination. Table 2.7 summarizes the measurements used in previous research to analyze the performance of reusable packaging systems. 49 Table 2.7. A summary of previous research on performance measurements of reusable packaging systems. Performance Reference Industry Performance measurements measurements in details Protection Ergonomics Efficiency Reliability Containers’ cycle time, Chew et al. Industrial gas ✓ ✓ return time, and safety (2002) manufacturing stock Chonhenchob Bruising and heat and Singh Fruits ✓ transfer levels along (2003) shipping and handling Automobile Twede and manufacturing Hand grips and Clarke and ✓ package weight (2004) supermarket industry Bruise damage, flesh decay during storage Chonhenchob for five days, bruise Fruits ✓ et al. (2008) volume, changes in color, and firmness of the fruit Containers’ safety Glock and General ✓ return time and safety Kim (2016) stock Cobb Containers’ cycle time General ✓ ✓ (2016b) and return time Breen (2006) General ✓ ✓ Containers’ return time Maleki and Reimche General ✓ ✓ Containers’ return time (2011) Future research opportunities and conclusion Reusable packaging systems replace single-use packages with reusable ones which are used for several times in supply chain systems. In this research, we have conducted a systematic literature review on works that provide information for decision making to shift from single-use to reusable packaging systems. We differentiated these works into three categories: those studying (1) the feasibility/viability of reusable packaging systems in terms of environmental and economic factors, (2) various designs of a logistics system using reusable packages, and (3) operations management of reusables. Based on our observations from the extant literature, we also delivered several relevant research opportunities for future research. 50 Research opportunities: feasibility/viability of reusables. Future research needs to have more environmental and economic orientation towards the cost of reusables. For example, emission factors have been incorporated into optimization models for reusable packaging in a handful of studies (see, e.g., Soysal, 2016; Ech-Charrat and Amechnoue, 2016; Ech-Charrat et al., 2017a,b,c; and Bortolini et al., 2018). Although environmental and economic factors could be considered simultaneously via a multi-objective optimization model (to the best of our knowledge, this route has not been addressed in the literature), converting environmental impacts into economic drivers (e.g., carbon taxes, environmental externalities, and eco-costs) would be another viable option to better understand the trade-off between such factors. Another instance of economic factors is to analyze the impact of consumers’ behavior on the cost of reusable packaging. Indeed, one can focus on the level of consumers’ cooperation on the financial success of reusable packaging (see, e.g., Grimes-Casey et al., 2007). The design (i.e., shape and dimensions, and material selection) is another aspect of reusable packaging with economic and environmental implications (as one study incorporating the design in an optimization model, one can refer to Glock and Kim, 2014). Finally, we note that research can explore social dimensions of reusables (e.g., ergonomics of material handling) and their roles in developing decision support systems (see, e.g., Twede and Clarke, 2004). This can further contribute to an efficient execution of diverse reusable packaging practices. Research opportunities: logistics system design. Future research should focus on issues that might arise due to the ownership of reusable packages, albeit in two different capacities. First, the main assumption in the majority of existing works is that such packages are owned by the sender, while in practice, they are often leased or rented (see, e.g., Ray et al., 2006; Carrano et al., 2015; Hariga et al., 2016). More importantly, the structure of renting contracts for reusable packages, 51 which could lead to various renting periods, payment schemes, and claim management is the subject that has not been investigated in the literature (to the best of our knowledge). Second, the role of third-party logistics in shipping loaded packages and collecting empty ones needs to be explored more; see Kroon and Vrijens (1995) and Elia and Gnoni (2015) as some examples in this regard. To this end, the selection of an appropriate third-party logistics is an important practical question that is worthy of further research consideration. This could, in turn, lead to another research avenue, in that the issues of bidding by third-party logistics and proposer selection problem by the central agency are dealt with. Research opportunities: operations management. Compared to the foregoing two domains, we have found more research opportunities under the operations management of reusable packaging. We acknowledge that some of these avenues can have overlaps with the earlier domains. Nevertheless, we summarize our suggestions as follows: 1. Underlying assumptions. First, to the best of our knowledge, the majority of literature have considered a supply chain with a single sender and a single recipient, while we typically observe more complex supply chains in practice (see, e.g., Glock and Kim, 2014 and Cobb, 2016a as a few instances considering a single sender and multiple recipients). Of note, it could also be worthwhile to investigate the impact of RFIDs on how they could contribute to the management of such complex systems. Second, another common assumption is the condition of reusables considered as binary parameter (i.e., whether it is usable or not). However, the condition may indeed vary from new (perfect condition) to minor scratches (not needing repair) to a condition where repair is needed. This raises another question about the quality of packages and imposes limitations on their usability (i.e., damaged packages can be of inferior quality to new ones). Therefore, appropriate models could be 52 developed to explore the question on how to select such packages. Third, third-party logistics are assumed to have full information about the cost parameters of both senders and recipients (e.g., Glock and Kim, 2014). In addition, these parties are assumed to be willing to cooperate in this process. However, lack of cooperation could result in misaligned incentives between various entities. Therefore, a viable research direction is to explore and devise incentive mechanisms that could facilitate this alignment. One last note about the assumptions is directed to the probability of a returned package, which is assumed to be independent from the package’s age (e.g., Buchanan and Abad, 1998). Developing a mathematical model, in which this probability depends on age would be another direction with real-world implications. 2. Coordination. This stream can be dealt with from two perspectives. First, coordination of a supplier’s production cycle with a third-party’s delivery schedule and recipients’ returning cycles has not been addressed in the existing decision support systems, while in practice, such coordination is a must. Also, considering the safety stock of a finished product along with that of reusables into an inventory management model would help us to protect the system against stockouts and make it more flexible in adjusting its safety measures to the cost parameters of the system. Second, future research can focus on a coordination between the lot sizes of packages, production, and consumer’s demand. To the best of our knowledge, Kim et al. (2014) seems to be the only work assuming that return lot size and capacity of reusables determine the production lot size. However, we note that such assumptions could result in unfruitful outcomes if not taking stochastic factors into consideration. Indeed, stochastic factors such as production process, return time and 53 quantity, and demand for the finished products (which often has random and seasonal components) could lead to a more realistic picture of supply chains observed in practice. 3. Industry/application. Concerning the applicability of developed optimization models under the operations management, we realized that a few papers have only developed their models based on a specific industry (see, e.g., Kim et al., 2014; Hariga et al., 2016). The operations management of reusable packaging and their characteristics may vary from industry to industry, and thus a general model may not work well for a specific industry. Future studies can also go further by reporting industry-related case studies and the lessons learned from them. 4. Deposit systems. Another research avenue can be to study deposit systems in which a sender can induce recipients to return packages earlier and more reliably. See Grimes- Casey et al. (2007) as a study analyzing how to design a deposit system for reusables. In concluding our research opportunities, it should be noted that we have observed the foregoing research directions and opportunities from the extant literature with respect to secondary and/or tertiary packaging options. As we also mentioned in Section 2.1, the reusable primary packaging is a newer concept compared to secondary/tertiary options; however, given the scope of our proposed research directions (e.g., costs, ownership, assumptions on complexity of the system, quality of the package, and symmetric information, coordination, deposit system, etc.), all these opportunities could also be exerted for a primary reusable packaging option. Our review has some limitations: (1) as our inclusion criteria, we limited the literature to those works published in English and in peer-reviewed academic journals or conference proceedings, and did not include relevant resources such as technical notes, book chapters, books, patents, etc. (granted that our choice of journal/conference papers would form a huge body of literature). (2) 54 Although we use a reasonable mix of keywords in our review via forward and backward snowball searches (e.g., “reusable packages”, “reusable packaging”, “reusable packaging material”, “returnable packaging material”, “returnable containers”, and “returnable transport items”), this review can be enhanced by including new terms such as “reusable plastic crates”, “reusable plastic containers”, and “reusable totes”. (3) Due to the nature of some studies, where extracting the focus/orientation of that study could be equivocal, their classification becomes a subjective act (as is the case in any review paper). Thus, future reviews may employ different classification schemes, and hence, their findings might be different from ours in this study. Finally, we note that this review can be enhanced by future research on reusable packaging systems (through research opportunities mentioned earlier in this section). 55 CHAPTER 3. A LAGRANGIAN DECOMPOSITION SOLUTION APPROACH FOR THE VEHICLE ROUTING PROBLEM WITH BACKHAULS Introduction According to Ellen MacArthur Foundation, “Converting 20% of plastic packaging into reuse models is a $10 billion business opportunity that benefits customers and represents a crucial element in the quest to eliminate plastic waste and pollution.” TerraCycle is a small company that has recently compelled more than two dozen of the world’s biggest brands such as Nestlé, PepsiCo, and Procter & Gamble to begin testing reusable packaging for their products (Makower, 2019). The pivotal shift from single-use to reusable packaging for basic products (e.g., ice cream, soda, shampoo) makes a package an asset for product companies, prompting them to make the package as long-lasting and durable as possible. Customers pay a small deposit for a package that has been designed for 100 or more use-cycles. When the package becomes empty, customers place it in a specially designed tote for pickup or, in some cases, can bring it to a retailer. They can choose whether they want that product replenished; if not, their deposit is returned or credited to their account. The empties are sent to a facility where they are washed and refilled. Using reusable packaging for basic products is an excellent idea; however, managing their transportation (delivery of full totes and picking up empty ones) in real-world transportation networks is a complex task (due to the high volume and high frequency of deliveries/pickups). Managing their transportation operations is also critical since using reusables should be environmentally and economically justifiable. For further information about economic and environmental impacts of reusables, one can refer to chapter 2 of this research. Delivery of full totes and picking up empty ones can be mathematically modeled by the standard VRPB (of note, the VRPB with single demand and backhaul solution is also referred as 56 the standard VRPB in the extant literature). In the standard VRPB, the customers are partitioned into linehaul and backhaul customers who require deliveries and pickups, respectively. Both linehaul and backhaul customers must be visited exactly once, all linehauls must be visited before backhauls, and all routes must contain at least one linehaul customer. All deliveries are loaded at the depot, and all pickups are transported to the depot. In this research, we aim to develop a mathematical framework for the standard VRPB that contributes to streamline such complex operations in managing reusables in modern supply chain systems. To the best of our knowledge, three exact algorithms (Toth and Vigo, 1997; Mingozzi et al., 1999; Queiroga et al. 2020) have been developed in the extant literature to solve the standard VRPB. The model proposed by Toth and Vigo (1997) is a link-based integer programming model in which all feasible subsets of linehaul/backhaul nodes are checked for the connectivity and vehicle capacity constraints in advance. The model proposed by Mingozzi et al. (1999) is a path- based set partitioning model that requires a preprocessing step in the form of connecting paths with only linehaul customers with paths with only backhaul customers. The algorithm proposed by Queiroga et al. (2020) is a branch-and-cut-and-price algorithm to solve the set partitioning model proposed by Mingozzi et al. (1999). Since the standard VRPB is an extension of the well-known capacitated vehicle routing problem (VRP), it is an NP-hard problem (Toth and Vigo, 1997; Mingozzi et al., 1999; Koç and Laporte, 2018). In this research, we propose a mathematical model that allows the main problem to be decomposed into sub-problems. More specifically, we apply Lagrangian decomposition to decompose the main problem into two OVRPs and one AP. In the OVRP, a vehicle does not return to the depot after serving the last customer on the route, or likewise the return trip to the depot is not charged (Li et al. 2007; Irnich et al., 2014). In this research, each OVRP determines optimal 57 routes for a fleet of homogeneous vehicles in order to serve linehaul/ backhaul customers, while the AP aims to find the optimal matching between linehaul and backhaul routes. At each iteration, we solve the forgoing sub-problems and update Lagrangian multipliers to reduce the gap between lower and upper bounds of the global optimal solution. We propose two different layouts, i.e., parallel and sequential, for solving the foregoing sub-problems and then analyze the impact of these arrangements on the solution quality as well as computational efficiency of our proposed Lagrangian decomposition algorithm. We test the framework on two benchmark datasets in the extant literature that were proposed by Goetschalckx and Jacobs-Blecha (1989) and Toth and Vigo (1997) and named as GJ and TV datasets, respectively. For 34 instances of the GJ dataset, compared to the best-known exact solutions, our model yields the same or better solutions in 12 instances (35%) and solutions within 2% deviation from the best-known exact solutions for the rest of instances. Our model is also capable of solving 28 out of 33 instances of the TV dataset. For this dataset, compared to the best- known exact solutions, our model yields the same or better solutions in 11 instances (33%) and solutions within 2% deviation from the best-known exact solutions in 10 instances. We also test our model on a randomly generated dataset containing 100, 250, and 500 customers, geographically distributed on the Lansing transportation network with 5,409 transportation nodes and 7,610 directed links. To reduce the computational burden of solving the VRPB on this dataset, we present a CFRS algorithm and then analyze the impact of vehicle capacity on the solution quality of our proposed algorithm. To this end, this research contributes to the literature by proposing a new link-based mixed- integer programming model for the standard VRPB that allows the main problem to be decomposed into three sub-problems: two OVRPs and one AP. The forgoing sub-problems are 58 well-known optimization problems with rich solution methodologies in the extant literature. The model presented in this paper does not require any preprocessing step such as generating feasible subsets of nodes or feasible set of paths. The remainder of the paper is organized as follows. Section 3.2 provides the existing literature on the standard VRPB. Sections 3.3 and 3.4 present our mathematical model and solution methodology, respectively. Computational experiments are presented in Section 3.5. Discussion, concluding remarks, and directions for future research form Section 3.6. Literature review Parragh et al. (2008a) classified the VRPB into four classes: In the first and second classes, customers have single demand (either delivery or pickup but not both). In the first class, all linehauls are visited before backhauls (Figure 3.1 (a)), while in the second class, any sequence of linehauls and backhauls is allowed (Figure 3.1 (b)). In Figure 3.1 (a), linehaul customer 2 is served before backhaul customers 1 and 3, while in Figure 3.1 (b), any sequence of linehaul and backhaul customers is permitted. In the third and fourth classes, at least one customer has a simultaneous demand (a delivery and a pickup). In the third class, customers demanding delivery and pickup services can be visited twice (Figure 3.1 (c)), while in the fourth class, customers demanding both services are visited exactly once (Figure 3.1 (d)). In Figure 3.1 (c), customer 1 has both delivery and pickup demand, and all deliveries occur before pickups, while in Figure 3.1 (d), any sequence of linehaul and backhaul customers is permitted (customer 1’s delivery and pickup requests are met at the same visit). 59 0 0 3 1 3 1 Depot Customer 2 2 Demand for delivery of (b) Mixed solution, single aDemand full totefor picking (a) Backhaul solution, single 0 0 up demand an empty tote demand Truck moving with one full tote Truck moving with one empty tote Empty truck 3 1 3 1 2 2 (c) Backhaul solution, simultaneous (d) Mixed solution, simultaneous demand demand Figure 3.1. Different classes of the VRPB (Parragh et al., 2008a; Battarra et al., 2014). In this research, we develop a mathematical model for the first class of the VRPB which is observed in the literature as the standard VRPB. The standard VRPB aims to construct routes for a fleet of homogeneous vehicles in order to serve a set of backhaul and linehaul customers, and the following constraints should be held: (i) each vehicle must start and end its route at the depot, (ii) the total number of linehaul customers to be served by a vehicle should not exceed the vehicle capacity, (iii) the total number of backhaul customers to be served by a vehicle should not exceed the vehicle capacity, (iv) each customer has a single demand, (v) each customer is visited exactly once, (vi) linehauls must be visited before backhauls, and (vii) all routes must contain at least one linehaul customer. To the best of our knowledge, three exact algorithms (Toth and Vigo, 1997; Mingozzi et al., 1999; Queiroga et al., 2020) have been developed in the extant literature to solve the standard 60 VRPB. Toth and Vigo (1997) proposed a link-based integer programing model and developed a branch-and-bound algorithm with Lagrangian bound to solve the problem. The model proposed by Mingozzi et al. (1999) is a path-based set partitioning model by which a valid lower bound is calculated. In their model, different heuristic methods are combined for solving the dual of the linear programming relaxation of the exact formulation. Both mathematical models have been comprehensively discussed in Toth and Vigo (2002) and Koç and Laporte (2018). Recently, Queiroga et al. (2020) has proposed a branch-and-cut-and-price algorithm to solve the set partitioning model proposed by Mingozzi et al. (1999). In addition to the existing exact algorithms, various heuristic algorithms have been developed to solve the standard VRPB which have been categorized by Koç and Laporte (2018) as follows: 1. Classical heuristics that are associated with a constructive structure of routing problems. The Clarke and wright saving algorithm, space filling, and CFRS are among the classical heuristics that have been developed to solve the standard VRPB (see, e.g., Deif and Bodin, 1984; Goetschalckx and Jacobs-Blecha, 1989; Goetschalckx and Jacobs-Blecha, 1993; Toth and Vigo, 1996). 2. Local search heuristics that improve the initial solution through searching within its neighborhood. Tabu search and ant colony are two examples of local search heuristics that have been developed to solve the standard VRPB (see, e.g., Osman and Wassan, 2002; Brandão, 2006; Ropke and Pisinger, 2006; Wassan, 2007; Gajpal and Abad, 2009; Zachariadis and Kiranoudis, 2012; Cuervo et al., 2014; Brandão, 2016; Subramanian and Queiroga, 2020). 61 3. Population search heuristics that evolve a population of candidate solutions. To the best of our knowledge, Vidal et al. (2014) is the only study that developed a genetic algorithm for different variants of the VRP, including the VRPB. 4. Neural network heuristics that employ the mapping scheme of neural network architecture. To the best of our knowledge, Ghaziri and Osman (2006) is the only study that employed neural network heuristics to solve the standard VRPB. We summarize the literature on the standard VRPB in Table 3.1. Table 3.1. Summary of the literature on the standard VRPB (partially adopted from Koç and Laporte, 2018). Solution References New model Algorithm method Deif and Bodin (1984) Heuristic Constructive heuristics Goetschalckx and Jacobs-Blecha  Heuristic Constructive heuristics (1989) Goetschalckx and Jacobs-Blecha Heuristic Generalized assignment (1993) Toth and Vigo (1997)  Exact Branch-and-bound, Lagrangian bound Set partitioning, dual problem, dual Mingozzi et al. (1999)  Exact approximation Toth and Vigo (1999) Heuristic CFRS, Lagrangian relaxation Osman and Wassan (2002) Heuristic Tabu search Brandão (2006) Heuristic Tabu search, constructive heuristics Ghaziri and Osman (2006) Heuristic Self-organizing feature maps Ropke and Pisinger (2006) Heuristic Adaptive large neighborhood search Tabu search, adaptive memory Wassan (2007) Heuristic programming Gajpal and Abad (2009) Heuristic Ant colony optimization Zachariadis and Kiranoudis (2012) Heuristic Local search Cuervo et al. (2014)  Heuristic Iterated local search Vidal et al. (2014) Heuristic Population search Yalcın and Erginel (2015)  Heuristic Fuzzy multi-objective programming Brandão (2016) Heuristic Iterated local search Queiroga et al. (2020) Exact Branch-and-cut-and-price Subramanian and Queiroga (2020) Heuristic Local neighborhood search Our proposed model for the standard VRPB Summary of notations used in this chapter is provided in Table 3.2. Let 𝐺 = (𝑉, 𝐴) be a directed graph, where 𝑉 = {0} ∪ 𝐿 ∪ 𝐵, node 0 represents the depot, 𝐿 = {1, . . . , 𝑙} corresponds to 62 𝑙 linehaul customers, 𝐵 = {𝑙 + 1, . . . , 𝑙 + 𝑏} corresponds to 𝑏 backhaul customers, and 𝐴 = {(𝑖, 𝑗): 𝑖, 𝑗 ∈ 𝑉, 𝑖 ≠ 𝑗} represents the set of all links connecting node 𝑖 to 𝑗. A non-negative traveling cost 𝑐𝑖𝑗 is associated with link (𝑖, 𝑗), and a non-negative demand 𝑑𝑖 is associated with customer 𝑖, 𝑖 ∈ 𝐿 ∪ 𝐵. 𝑘 identical vehicles with capacity 𝑄 are available at the depot. Let 𝑘𝑙 and 𝑘𝑏 denote the number of vehicles needed to serve all linehaul and backhaul customers, respectively. Then, 𝑘𝑙 = ∑𝑖∈𝐿 𝑑𝑖 ∑𝑖∈𝐵 𝑑𝑖 ⌈ ⌉, 𝑘𝑏 = ⌈ ⌉, and 𝑘 ≥ 𝑀𝑎𝑥⁡(𝑘𝑙 , 𝑘𝑏 ). 𝑄 𝑄 Table 3.2. Summary of notations used in chapter 3. Notations used in section 3.3 𝑃 The standard VRPB problem 0 Representative of the depot 𝑙 Number of linehaul customers 𝑏 Number of backhaul customers 𝑖, 𝑗 Indices of nodes 𝐿 Set of linehaul nodes, 𝐿 = {1, … , 𝑙} 𝐿0 Set of linehaul nodes and the depot, 𝐿0 = {0} ∪ 𝐿 𝐵 Set of backhaul nodes, 𝐵 = {𝑙 + 1, … , 𝑙 + 𝑏} 𝐵0 Set of backhaul nodes and the depot, 𝐵0 = 𝐵 ∪ {0} 𝑉 Set of all nodes, 𝑉 = {0} ∪ 𝐿 ∪ 𝐵 𝐴 Set of all links, 𝐴 = {(𝑖, 𝑗): 𝑖, 𝑗 ∈ 𝑉, 𝑖 ≠ 𝑗} 𝐴𝑙 Set of linehaul links, 𝐴𝑙 = {(𝑖, 𝑗): 𝑖 ∈ 𝐿0 , 𝑗 ∈ 𝐿, 𝑖 ≠ 𝑗} 𝐴𝑏 Set of backhaul links, 𝐴𝑏 = {(𝑖, 𝑗): 𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵0 , 𝑖 ≠ 𝑗} 𝐴𝑐 Set of connection links, 𝐴𝑐 = {(𝑖, 𝑗): 𝑖 ∈ 𝐿, 𝑗 ∈ 𝐵0 } 𝑐𝑖𝑗 Cost of traveling from node 𝑖 to node 𝑗, (𝑖, 𝑗) ∈ 𝐴 𝑑𝑖 Demand of node 𝑖, 𝑖 ∈ 𝐿 ∪ 𝐵 𝑄 Vehicle capacity ∑ 𝑑 𝑘𝑙 Number of vehicles required to serve linehaul customers, 𝑘𝑙 = ⌈ 𝑖∈𝐿 𝑖 ⌉ 𝑄 ∑𝑖∈𝐵 𝑑𝑖 𝑘𝑏 Number of vehicles required to serve backhaul customers, 𝑘𝑏 = ⌈ ⌉ 𝑄 𝑘 Number of vehicles available at the depot, 𝑘 ≥ 𝑀𝑎𝑥⁡(𝑘𝑙 , 𝑘𝑏 ) 𝑥𝑖𝑗 Equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑙 , is selected, and 0 otherwise 𝑦𝑖𝑗 Equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑏 , is selected, and 0 otherwise 𝑧𝑖𝑗 Equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑐 , is selected, and 0 otherwise 𝑢𝑖 Arbitrary continuous variable, 𝑖 ∈ 𝐿 𝑤𝑖 Arbitrary continuous variable, 𝑖 ∈ 𝐵 Notations used in section 3.4.1 𝑃𝑋 Sub-problem related to the linehaul routing phase 𝑃𝑌 Sub-problem related to the backhaul routing phase 𝑃𝑍 Sub-problem related to the connection phase 0′ Representative of the dummy depot 𝐿0′ Set of linehaul customers and the dummy depot, 𝐿0′ = 𝐿 ∪ {0′ } 𝐵 0′ Set of backhaul customers and the dummy depot, 𝐵0′ = {0′ } ∪ 𝐵 63 Table 3.2 (cont’d) Set of linehaul links and the links connecting linehauls to the dummy depot, 𝐴′𝑙 = 𝐴𝑙 ∪ 𝐴′𝑙 {(𝑖, 0′ ): 𝑖 ∈ 𝐿} Set of backhaul links and the links connecting the dummy depot to backhauls, 𝐴′𝑏 = 𝐴𝑏 ∪ 𝐴′𝑏 {(0′ , 𝑗): 𝑗 ∈ 𝐵} 𝜆𝑖 Lagrangian multiplier corresponding to linehaul customer 𝑖, 𝑖 ∈ 𝐿 𝛾𝑗 Lagrangian multiplier corresponding to backhaul customer⁡𝑗, 𝑗 ∈ 𝐵 𝒳 Set of decision variables in the linehaul routing phase, 𝒳 = {𝑥𝑖𝑗 : 𝑥𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴′𝑙 } 𝒴 Set of decision variables in the backhaul routing phase, 𝒴 = {𝑦𝑖𝑗 : 𝑦𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴′𝑏 } 𝒵 Set of decision variables in the connection phase, 𝒵 = {𝑧𝑖𝑗 : 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑐 } ∗ 𝒳 Optimal solution of sub-problem 𝑃𝑋 𝒴∗ Optimal solution of sub-problem 𝑃𝑌 𝑃𝑋∗ (𝜆) Optimal value of sub-problem 𝑃𝑋 𝑃𝑌∗ (𝛾) Optimal value of sub-problem 𝑃𝑌 𝑃𝑍∗ (𝜆, 𝛾) Optimal value of sub-problem 𝑃𝑍 Notations used in Section 3.4.2 𝑃𝑍 ′ Sub-problem related to generating upper bound solution 𝐿𝑡 Set of tails in the linehaul routes of 𝑃𝑋∗ , 𝐿𝑡 = {𝑖: 𝑥𝑖0′ = 1, 𝑥𝑖0′ ∈ 𝒳 ∗ } 𝐵ℎ Set of heads in the backhaul routes of 𝑃𝑌∗ , 𝐵ℎ = {𝑗: 𝑦0′𝑗 = 1, 𝑦0′ 𝑗 ∈ 𝒴 ∗ } ℎ 𝐵0 Set of heads in the backhaul routes of 𝑃𝑌∗ and the depot, 𝐵0ℎ = {0} ∪ 𝐵ℎ ′ 𝐴𝑐 Set of links connecting the tails of 𝑃𝑋∗ to the heads of 𝑃𝑌∗ or to the depot, 𝐴′𝑐 = {(𝑖, 𝑗): 𝑖 ∈ 𝐿𝑡 , 𝑗 ∈ 𝐵0ℎ } ′ 𝑧𝑖𝑗 Equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴′𝑐 , is selected, and 0 otherwise ∗ ∗ ∗ 𝑃𝑍′ (𝒳 , 𝒴 ) Optimal value of sub-problem 𝑃𝑍′ Notations used in section 3.4.3 𝑅 Maximum number of iterations 𝑟 Iteration number, 𝑟 = 0, 1, … , 𝑅 𝑔 Relative gap percentage between global lower and upper bounds 𝜀 Threshold of relative gap percentage 𝜎 Step size controller Maximum number of consecutive iterations without any improvement in the lower bound 𝛿 solution 𝜃0 The constant part of step size selected from range (0,2] 𝜃𝜆𝑟 Step size for linehaul customers at iteration 𝑟 𝜃𝛾𝑟 Step size for backhaul customers at iteration 𝑟 𝜆𝑟𝑖 Lagrangian multiplier corresponding to linehaul node 𝑖 at iteration 𝑟 𝛾𝑗𝑟 Lagrangian multiplier corresponding to backhaul node 𝑗 at iteration 𝑟 𝐿𝐵 ∗ Global lower bound 𝐿𝐵𝑟 Lower bound at iteration 𝑟 𝑈𝐵 ∗ Global upper bound 𝑈𝐵𝑟 Upper bound at iteration 𝑟 Notations used in section 3.4.4 𝑞 Indices of clusters 𝜉𝑖𝑗 The measure of dissimilarity between node 𝑖 and cluster 𝑗 𝜉𝑖𝑗′ Cost of traveling from node 𝑖 to node 𝑗 𝑞 An element of the obtained incidence matrix from the clustering phase which equals 1 if node 𝑖 𝜑𝑖 exists in cluster 𝑞, and 0 otherwise 𝑠𝑖𝑗 Equals 1 if node 𝑖, 𝑖 ∈ 𝐿 ∪ 𝐵 is assigned to cluster 𝑗, 𝑗 ∈ 𝐿 ∪ 𝐵, and 0 otherwise 𝑝𝑗 Equals 1 if node 𝑗, 𝑗 ∈ 𝐿 ∪ 𝐵 is selected as the seed of a cluster, and 0 otherwise 𝑞 𝑜𝑖𝑗 Equals 1 if link (𝑖, 𝑗), 𝑖, 𝑗 ∈ 𝐿 ∪ 𝐵 ∪ {0, 0′ } exists in cluster 𝑞, and 0 otherwise 𝓊𝑖 Arbitrary continuous variable, 𝑖 ∈ 𝐿 ∪ 𝐵 64 To formulate the standard VRPB, the constraints mentioned in section 3.2 must be held. We divide the structure of the standard VRPB into three different phases: (i) serving linehaul customers, (ii) serving backhaul customers, and (iii) connecting linehaul routes to backhaul routes or to the depot. We name the first phase as the linehaul routing phase, the second as the backhaul routing phase, and the third as the connection phase. Figure 3.2 illustrates the linehaul routing phase by solid red lines, backhaul routing phase by dashed blue lines, and connection phase by dotted black lines. Figure 3.2 depicts two feasible routes: 0 → 1 → 2 → 3 → 0 (route 1) and 0 → 4 → 5 → 6 → 7 → 8 → 9 → 10 → 0 (route 2). Route 1 only serves linehaul customers, while route 2 serves both linehaul and backhaul customers. We define the head of a route as the first customer being served by the route. Similarly, the tail of a route is defined as the last customer being served by the route. For example, in Figure 3.2, customer 1 is the head of linehaul route 1, customer 6 is the tail of linehaul route 2 and the head of connection route 2, and customer 10 is the tail of backhaul route 2. Figure 3.2. The structure of our proposed model for the standard VRPB. 65 Let 𝐿0 = {0} ∪ 𝐿 and 𝐵0 = 𝐵 ∪ {0}. Suppose 𝐴𝑙 = {(𝑖, 𝑗): 𝑖 ∈ 𝐿0 , 𝑗 ∈ 𝐿, 𝑖 ≠ 𝑗}, 𝐴𝑏 = {(𝑖, 𝑗): 𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵0 , 𝑖 ≠ 𝑗}, and 𝐴𝑐 = {(𝑖, 𝑗): 𝑖 ∈ 𝐿, 𝑗 ∈ 𝐵0 } define the set of linehaul, backhaul, and connection links, respectively. Then, decision variable 𝑥𝑖𝑗 equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑙 , is selected, and 0 otherwise. Variable 𝑦𝑖𝑗 equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑏 , is selected, and 0 otherwise. Variable 𝑧𝑖𝑗 equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑐 , is selected, and 0 otherwise. We also define arbitrary continuous decision variables 𝑢𝑖 for 𝑖 ∈ 𝐿 and 𝑤𝑖 ⁡for 𝑖 ∈ 𝐿. We formulate the standard VRPB as follows: 𝑀𝑖𝑛 𝑃 = ∑(𝑖,𝑗)∈𝐴𝑙 𝑐𝑖𝑗 𝑥𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑏 𝑐𝑖𝑗 𝑦𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑐 𝑐𝑖𝑗 𝑧𝑖𝑗 (3.1) s.t. ∑𝑗∈𝐿 𝑥0𝑗 = 𝑘 (3.2) ∑𝑖∈𝐿0 𝑥𝑖𝑗 = 1 𝑗 ∈ 𝐿, (3.3) 𝑖≠𝑗 𝑢𝑖 − 𝑢𝑗 + 𝑄𝑥𝑖𝑗 ≤ 𝑄 − 𝑑𝑗 (𝑖, 𝑗) ∈ 𝐴𝑙 , 𝑖 ≠ 0, (3.4) 𝑑𝑖 ≤ 𝑢𝑖 ≤ 𝑄 𝑖 ∈ 𝐿, (3.5) ∑𝑖∈𝐵 𝑦𝑖0 + ∑𝑖∈𝐿 𝑧𝑖0 = 𝑘 (3.6) ∑𝑗∈𝐵0 𝑦𝑖𝑗 = 1 𝑖 ∈ 𝐵, (3.7) 𝑖≠𝑗 𝑤𝑖 − 𝑤𝑗 + 𝑄𝑦𝑖𝑗 ≤ 𝑄 − 𝑑𝑗 (𝑖, 𝑗) ∈ 𝐴𝑏 , 𝑗 ≠ 0, (3.8) 𝑑𝑖 ≤ 𝑤𝑖 ≤ 𝑄 𝑖 ∈ 𝐵, (3.9) ∑𝑖∈𝐿 ∑𝑗∈𝐵0 𝑧𝑖𝑗 = 𝑘 (3.10) ∑𝑗∈𝐿 𝑥𝑖𝑗 + ∑𝑗∈𝐵0 𝑧𝑖𝑗 = 1 𝑖 ∈ 𝐿, (3.11) 𝑗≠𝑖 ∑𝑖∈𝐵 𝑦𝑖𝑗 + ∑𝑖∈𝐿 𝑧𝑖𝑗 = 1 𝑗 ∈ 𝐵, (3.12) 𝑖≠𝑗 66 𝑥𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑙 ; 𝑦𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑏 ; 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑐 , (3.13) 𝑢𝑖 ∈ ℝ, 𝑖 ∈ 𝐿; 𝑤𝑖 ∈ ℝ, 𝑖 ∈ 𝐵. (3.14) Objective function (3.1) minimizes the total transportation costs of linehaul and backhaul routes, as well as routes connecting linehaul routes to backhaul routes or to the depot. Constraints (3.2)-(3.5) are associated with the routing of linehauls, constraints (3.6)-(3.9) are related to the routing of backhauls, constraints (3.10)-(3.12) correspond to the connection between linehaul and backhaul routes, and constraints (3.13)-(3.14) define the domain of decision variables. More specifically, constraint (3.2) guarantees that 𝑘 vehicles leave the depot to serve linehaul customers. Constraint (3.3) ensures that each linehaul customer is visited exactly once. Constraints (3.4)-(3.5) are the extension of Miller–Tucker–Zemlin (1960) subtour elimination constraints that were proposed by Kulkarni and Bhave (1985). By these constraints, the connectivity between nodes of a linehaul route as well as vehicle capacity constraints are guaranteed. Constraint (3.6) ensures that 𝑘 vehicles return to the depot. Constraint (3.7) guarantees that each backhaul customer is visited exactly once. Constraints (3.8)-(3.9) holds the connection between nodes of a backhaul route and ensures vehicle capacity constraints. Constraint (3.10) ensures that 𝑘 number of connections between linehaul and backhaul routes or linehaul routes and the depot exist. Constraint (3.11) ensures that a linehaul customer is connected to either (i) another linehaul customer, (ii) a backhaul customer, or (iii) the depot. Constraint (3.12) guarantees that a backhaul customer is only connected by either (i) another backhaul customer or (ii) a linehaul customer. Constraints (3.2)- (3.5) and (3.11) result in feasible linehaul routes, and constraints (3.6)-(3.9) and (3.12) result in feasible backhaul routes. Finally, by constraints (3.10)-(3.12), linehaul routes are connected to 67 either backhaul routes or to the depot. In the next section, we propose a Lagrangian relaxation- based decomposition procedure to solve this problem. Lagrangian relaxation-based solution approach In the variable splitting method (also observed as Lagrangian decomposition in the literature), the main problem is decomposed into some sub-problems. This is done by relaxing coupling constraints and introducing corresponding penalties into the objective function. To obtain more information on Lagrangian relaxation and decomposition approach, one can refer to Fisher (1981, 1985, 2004) and Fisher et al. (1997). In the mathematical model presented in section 3.3, there are three coupling constraints, i.e., constraints (3.6), (3.11), and (3.12). Without loss of generality, one can rewrite constraint (3.6) as constraint (3.15) to make the structure of the problem suitable for further decompositions (see, e.g., Guignard and Kim, 1987; Fisher et al., 1997). ∑𝑖∈𝐵 𝑦𝑖0 ≤ 𝑘 (3.15) We also relax coupling constraints (3.11) and (3.12) into objective function (3.1) to write new objective function (3.16) as follows: 𝑀𝑖𝑛 𝑃(𝜆, 𝛾) = ∑(𝑖,𝑗)∈𝐴𝑙 𝑐𝑖𝑗 𝑥𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑏 𝑐𝑖𝑗 𝑦𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑐 𝑐𝑖𝑗 𝑧𝑖𝑗 + (3.16) ∑𝑖∈𝐿 𝜆𝑖 (∑𝑗∈𝐿 𝑥𝑖𝑗 + ∑𝑗∈𝐵0 𝑧𝑖𝑗 − 1) + ∑𝑗∈𝐵 𝛾𝑗 (∑𝑖∈𝐵 𝑦𝑖𝑗 + ∑𝑖∈𝐿 𝑧𝑖𝑗 − 1) 𝑗≠𝑖 𝑖≠𝑗 Then, problem 𝑃(𝜆, 𝛾) is subject to constraints (3.2)-(3.5), (3.7)-(3.10), and (3.13)-(3.15), and 𝜆𝑖 , 𝑖 ∈ 𝐿 and 𝛾𝑗 , 𝑗 ∈ 𝐵 are Lagrangian multipliers corresponding to the linehaul and backhaul 68 nodes, respectively. We further simplify objective function (3.16) to (3.17) by assuming 𝜆0 = 0 and 𝛾0 = 0. 𝑀𝑖𝑛 𝑃(𝜆, 𝛾) = ∑(𝑖,𝑗)∈𝐴𝑙(𝑐𝑖𝑗 + 𝜆𝑖 )𝑥𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑏(𝑐𝑖𝑗 + 𝛾𝑗 )𝑦𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑐(𝑐𝑖𝑗 + (3.17) 𝜆𝑖 + 𝛾𝑗 )𝑧𝑖𝑗 − ∑𝑖∈𝐿 𝜆𝑖 − ∑𝑗∈𝐵 𝛾𝑗 Problem decomposition and generating lower bound solution Relaxing constraints (3.11) and (3.12) enables us to decompose the standard VRPB into three sub-problems: two OVRPs associated with linehaul and backhaul routing phases, and one AP corresponding to the connection phase. Sub-problem 𝑷𝑿 : The first sub-problem is related to the linehaul routing phase. To reformulate this sub-problem as an OVRP, let node 0′ represents the dummy depot, 𝐿0′ = 𝐿 ∪ {0′ }, and 𝐴′𝑙 = 𝐴𝑙 ∪ {(𝑖, 0′ ): 𝑖 ∈ 𝐿}. Also, let 𝒳 = {𝑥𝑖𝑗 : 𝑥𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴′𝑙 } be the set of decision variables in the linehaul routing phase. Without loss of generality, we reformulate the linehaul routing sub- problem as an OVRP by adding constraint (3.18) to ensure that each linehaul customer is either connected to another linehaul customer or to the dummy depot. Of note, 𝑐𝑖0′ = 0, 𝑖 ∈ 𝐿. We also add constraint (3.19) to guarantee that 𝑘 vehicles will return to the dummy depot after serving all linehaul customers. ∑𝑗∈𝐿0′ 𝑥𝑖𝑗 = 1 𝑖 ∈ 𝐿, (3.18) 𝑖≠𝑗 ∑𝑖∈𝐿 𝑥𝑖0′ = 𝑘 (3.19) Hence, the linehaul routing sub-problem is written as follows: 69 𝑀𝑖𝑛 𝑃𝑋 (𝜆) = ∑(𝑖,𝑗)∈𝐴′𝑙 (𝑐𝑖𝑗 + 𝜆𝑖 )𝑥𝑖𝑗 (3.20) Subject to constraints (3.2)-(3.5), (3.13)-(3.14), and (3.18)-(3.19). Sub-problem 𝑷𝒀 : The second sub-problem corresponds to the backhaul routing phase. Similar to sub-problem 𝑃𝑋 , we define 𝐵0′ = {0′ } ∪ 𝐵 and 𝐴′𝑏 = 𝐴𝑏 ∪ {(0′ , 𝑗): 𝑗 ∈ 𝐵}. Also, 𝒴 = {𝑦𝑖𝑗 : 𝑦𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴′𝑏 } is the set of decision variables in the backhaul routing phase. Without loss of generality, we reformulate the backhaul routing sub-problem as an OVRP by adding constraint (3.21) to guarantee that each backhaul customer is connected by another backhaul customer or by the dummy depot. Of note, 𝑐0′ 𝑗 = 0, 𝑗 ∈ 𝐵. We also add constraint (3.22) to ensure that at most 𝑘 vehicles leave the dummy depot to serve backhaul customers. ∑𝑖∈𝐵0′ 𝑦𝑖𝑗 = 1 𝑗 ∈ 𝐵, (3.21) 𝑖≠𝑗 ∑𝑗∈𝐵 𝑦0′ 𝑗 ≤ 𝑘 (3.22) Therefore, the backhaul routing sub-problem is written as follows: 𝑀𝑖𝑛 𝑃𝑌 (𝛾) = ∑(𝑖,𝑗)∈𝐴′𝑏(𝑐𝑖𝑗 + 𝛾𝑗 )𝑦𝑖𝑗 (3.23) Subject to constraints (3.7)-(3.9), (3.13)-(3.15), and (3.21)-(3.22). 70 Sub-problem 𝑷𝒁 : The third sub-problem is related to the connection phase. Let 𝒵 = {𝑧𝑖𝑗 : 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑐 } be the set of decision variables in the connection phase. Without loss of generality, we reformulate the connection sub-problem as an AP by adding constraints (3.24) and (3.25). More specifically, constraint (3.24) states that if a linehaul node is the head of a connection route, it must be connected to either a backhaul node or to the depot. Likewise, constraint (3.25) states that if a backhaul node is the tail of a connection route, it must be connected by a linehaul node. ∑𝑗∈𝐵0 𝑧𝑖𝑗 ≤ 1 𝑖 ∈ 𝐿, (3.24) ∑𝑖∈𝐿 𝑧𝑖𝑗 ≤ 1 𝑗 ∈ 𝐵. (3.25) Then, the connection sub-problem is written as follows: 𝑀𝑖𝑛 𝑃𝑍 (𝜆, 𝛾) = ∑(𝑖,𝑗)∈𝐴𝑐(𝑐𝑖𝑗 + 𝜆𝑖 + 𝛾𝑗 )𝑧𝑖𝑗 (3.26) Subject to constraints (3.10), (3.13), and (3.24)-(3.25). The forgoing sub-problems are well-known optimization problems with rich solution methodologies in the extant literature. Let 𝑃𝑋∗ (𝜆), 𝑃𝑌∗ (𝛾), and 𝑃𝑍∗ (𝜆, 𝛾) be the optimal value of sub- problems 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 , respectively. Then, the lower bound (LB) of problem 𝑃 is calculated by (3.27). 𝐿𝐵 = ⁡ 𝑃𝑋∗ (𝜆) + ⁡ 𝑃𝑌∗ (𝛾) + ⁡ 𝑃𝑍∗ (𝜆, 𝛾) − ∑𝑖∈𝐿 𝜆𝑖 − ∑𝑗∈𝐵 𝛾𝑗 (3.27) 71 In section 3.4.3, we will explain how Lagrangian multipliers are updated at each iteration. Generating upper bound solution When foregoing sub-problems are solved independently, four types of infeasibility may occur. The first one occurs when in the optimal solution of 𝑃𝑋 , node 𝑖 is the tail of a linehaul route while in the optimal solution of 𝑃𝑍 , this node is not the head of any connection route. This implies that the left-hand side of constraint (3.11) is 0. Figure 3.3 presents an example of this type of infeasibility: node 3 is the tail of a linehaul route, while this node is not the head of any connection route. The second type occurs when in the optimal solution of 𝑃𝑋 , node 𝑖 is not the tail of any linehaul route while in the optimal solution of 𝑃𝑍 , this node is the head of a connection route. This implies that the left-hand side of constraint (3.11) is equal to 2. For example, in Figure 3.3, node 2 is not the tail of any linehaul route, while this node is the head of a connection route. The third type occurs when in the optimal solution of 𝑃𝑌 , node 𝑗 is the head of a backhaul route while in the optimal solution of 𝑃𝑍 , this node is not the tail of any connection route. This implies that the left- hand side of constraint (3.12) is equal to 0. In backhaul route 7 → 8 → 9 → 10 → 0 presented in Figure 3.3, node 7 is the head of a backhaul route, while this node is not the tail of any connection route. Finally, the fourth type of infeasibility occurs when in the optimal solution of 𝑃𝑌 , node 𝑗 is not the head of any backhaul route while in the optimal solution of 𝑃𝑍 , this node is the tail of a connection route. This implies that the left-hand side of constraint (3.12) is equal to 2. In backhaul route 7 → 8 → 9 → 10 → 0, presented in Figure 3.3, node 8 is not the head of any backhaul route, while this node is the tail of a connection route. 72 Figure 3.3. Illustration of four types of infeasibility that may occur when sub-problems 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 are solved independently. To find an upper bound solution, we use the optimal solution of sub-problems 𝑃𝑋 and 𝑃𝑌 and generate an optimal set of feasible connections between linehaul and backhaul routes. Let 𝒳 ∗ and 𝒴 ∗ denote the optimal solution of sub-problems 𝑃𝑋 and 𝑃𝑌 , respectively, and 𝐿𝑡 = {𝑖: 𝑥𝑖0′ = 1, 𝑥𝑖0′ ∈ 𝒳 ∗ } represents the set of linehaul nodes that have been selected as the tail of linehaul routes in the optimal solution of sub-problem 𝑃𝑋 . Also, let 𝐵 ℎ = {𝑗: 𝑦0′ 𝑗 = 1, 𝑦0′ 𝑗 ∈ 𝒴 ∗ } represents the set of backhaul nodes that have been selected as the head of backhaul routes in the optimal solution of sub-problem 𝑃𝑌 . We also define 𝐵0ℎ = {0} ∪ 𝐵 ℎ . We obtain an upper bound solution for problem 𝑃 by solving an AP that connects 𝑘 number of 𝑖’s, 𝑖 ∈ 𝐿𝑡 , to 𝑘 number of 𝑗’s, 𝑗 ∈ 𝐵0ℎ ). Let 𝐴′𝑐 = {(𝑖, 𝑗): 𝑖 ∈ 𝐿𝑡 , 𝑗 ∈ 𝐵0ℎ } be the set of links that connects the tails of linehaul routes ′ to the heads of backhaul routes or to the depot, and 𝑧𝑖𝑗 be a binary decision variable that equals 1 if link (𝑖, 𝑗), (𝑖, 𝑗) ∈ 𝐴′𝑐 , is selected in the upper bound solution, and 0 otherwise. Then, assignment problem 𝑃𝑍 ′ is written as follows: 73 ′ 𝑀𝑖𝑛 𝑃𝑍 ′ (𝒳 ∗ , 𝒴 ∗ ) = ∑(𝑖,𝑗)∈𝐴′𝑐 𝑐𝑖𝑗 𝑧𝑖𝑗 (3.28) s.t. ′ ∑(𝑖,𝑗)∈𝐴′𝑐 𝑧𝑖𝑗 =𝑘 (3.29) ′ ∑𝑖∈𝐿𝑡 𝑧𝑖𝑗 =1 𝑗 ∈ 𝐵ℎ , (3.30) ′ ∑𝑗∈𝐵ℎ 𝑧𝑖𝑗 =1 𝑖 ∈ 𝐿𝑡 (3.31) 0 ′ 𝑧𝑖𝑗 = {0,1} (𝑖, 𝑗) ∈ 𝐴′𝑐 (3.32) Objective function (3.28) minimizes total costs of traveling from the tails of linehaul routes to the heads of backhaul routes or to the depot. Constraint (3.29) ensures that exactly 𝑘 connections exist. Constraints (3.30) and (3.31) guarantee the matching condition. Constraint (3.32) is the binary definitional constraint. Suppose 𝑃𝑍∗′ (𝒳 ∗ , 𝒴 ∗ ) be the optimal value of sub-problem 𝑃𝑍 ′ . Then, the upper bound (UB) solution of problem 𝑃 is calculated by (3.33). 𝑈𝐵 = ⁡ 𝑃𝑋∗ (𝜆) + ⁡ 𝑃𝑌∗ (𝛾) + ⁡ 𝑃𝑍∗′ (𝒳 ∗ , 𝒴 ∗ ) (3.33) Lagrangian relaxation algorithm At each iteration, a lower bound solution and its corresponding upper bound solution are generated, and the global lower and upper bound solutions are updated. Lagrangian multipliers are also updated by applying the sub-gradient method (see, e.g., Mahmoudi and Zhou, 2016; Zhou et al., 2018; Lu et al., 2019; Yao et al., 2019). If the relative gap percentage between the global lower and upper bounds becomes less than a predefined threshold (denoted by 𝜀), the algorithm stops, and the global upper bound solution is reported as the optimal solution of problem 𝑃. Of note, the 74 𝑈𝐵∗ −𝐿𝐵∗ relative gap is denoted by 𝑔 and calculated by 100 × ( ), where 𝑈𝐵 ∗ and 𝐿𝐵 ∗ denote global 𝑈𝐵∗ upper and lower bounds, respectively. If the gap does not reach to 𝜀, the algorithm stops after 𝑅 number of iterations. We take the following steps to update Lagrangian multipliers at iteration 𝑟. Equations (3.34) and (3.37) calculate the sub-gradient of constraints (3.11) and (3.12), respectively. Equations (3.35) and (3.38) calculate step size 𝜃𝜆 and 𝜃𝛾 , respectively. 𝐿𝐵 𝑟 denotes the lower bound solution at iteration 𝑟. Lagrangian multipliers 𝜆𝑖 , 𝑖 ∈ 𝐿, and 𝛾𝑗 , 𝑗 ∈ 𝐵, are updated by equations (3.36) and (3.39), respectively. Of note, in equations (3.35) and (3.38), 𝜃0 is a constant value selected from range (0,2]. ∇𝐿(𝜆𝑟𝑖 ) = ∑𝑗∈𝐿 𝑥𝑖𝑗 + ∑𝑗∈𝐵0 𝑧𝑖𝑗 − 1 𝑖∈𝐿 (3.34) 𝑗≠𝑖 𝑈𝐵∗ −𝐿𝐵𝑟 𝜃𝜆𝑟 = 𝜃0 × (∑ 2 ) (3.35) 𝑖∈𝐿(∑𝑗∈𝐿 𝑥𝑖𝑗 +∑𝑗∈𝐵0 𝑧𝑖𝑗 −1) 𝑗≠𝑖 𝜆𝑟+1 𝑖 = 𝜆𝑟𝑖 + 𝜃𝜆𝑟 ∇𝐿(𝜆𝑟𝑖 ) 𝑖∈𝐿 (3.36) ∇𝐿(𝛾𝑗𝑟 ) = ∑𝑖∈𝐵 𝑦𝑖𝑗 + ∑𝑖∈𝐿 𝑧𝑖𝑗 − 1 𝑗∈𝐵 (3.37) 𝑖≠𝑗 𝑈𝐵∗ −𝐿𝐵𝑟 𝜃𝛾𝑟 = 𝜃0 × (∑ 2 ) (3.38) 𝑗∈𝐵(∑𝑖∈𝐵 𝑦𝑖𝑗 +∑𝑖∈𝐿 𝑧𝑖𝑗 −1⁡) 𝑖≠𝑗 𝛾𝑗𝑟+1 = 𝛾𝑗𝑟 + 𝜃𝛾𝑟 ∇𝐿(𝛾𝑗𝑟 ) 𝑗∈𝐵 (3.39) 𝜃0 Finally, we design step size controller 𝜎 to reduce 𝜃0 to when the lower bound solution 2 does not improve after 𝛿 number of consecutive iterations (Sanei et al., 2017). We propose two different layouts for solving sub-problems 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 : parallel and sequential (illustrated in 75 Figure 3.4). We further assess the impact of such arrangements on the solution quality and computational time of the overall algorithm in section 3.5. (a) Parallel layout (b) Sequential layout Figure 3.4. Solving sub-problems 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 . Figure 3.4 (a) shows that at each iteration, sub-problems 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 are solved in parallel. At each iteration, Lagrangian multipliers corresponding to the linehaul and backhaul nodes as well as global lower and upper bounds are updated. Table 3.3 presents our proposed Lagrangian relaxation algorithm when the foregoing sub-problems are solved in parallel. Figure 3.4 (b) illustrates how sub-problems 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 are solved sequentially. In this arrangement, we solve 𝑃𝑋 at iteration 𝑟. We borrow 𝑃𝑌∗ and 𝐵 ℎ from iteration 𝑟 − 1 and then solve 𝑃𝑍 such that 𝑖 ∈ 𝐿 and 𝑗 ∈ 𝐵 ℎ . Having 𝑃𝑋∗ and 𝑃𝑍∗ from iteration 𝑟 and 𝑃𝑌∗ from iteration 𝑟 − 1, we calculate 𝐿𝐵 𝑟 and 𝜆𝑟+1 . To calculate 𝑈𝐵 𝑟 , we use 𝑃𝑋∗ at iteration 𝑟 and 𝑃𝑌∗ from iteration 𝑟 − 1 and solve sub-problem 𝑃𝑍 ′ (presented in section 3.4.2). At iteration 𝑟 + 1, we solve 𝑃𝑌 , borrow 𝑃𝑋∗ and 𝐿𝑡 from iteration 𝑟, and then solve 𝑃𝑍 such that 𝑖 ∈ 𝐿𝑡 and 𝑗 ∈ 𝐵. Having 𝑃𝑌∗ and 𝑃𝑍∗ at iteration 𝑟 + 1 and 𝑃𝑋∗ from iteration 𝑟, we update 𝐿𝐵 𝑟+1 and 𝛾 𝑟+2. To calculate 𝑈𝐵 𝑟+1 , we use 𝑃𝑌∗ at 76 Table 3.3. Lagrangian relaxation algorithm when 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 are solved in parallel. Step 0: initialization 1 𝑅; 𝑟: = 0; 𝜎: = 0; , 𝛿; 𝜃0 ; 𝜀; 𝐿𝐵 ∗ : = −∞; 𝑈𝐵 ∗ : = +∞; 𝑔 = +∞; 𝜆0 : = 0; 𝛾 0 : = 0; 2 while termination condition is not met repeat steps 1-3 (i.e., if 𝑟 < 𝑅 or 𝑔 > 𝜀) Step 1: generating 𝐿𝐵𝑟 solve sub-problems 𝑃𝑋 (𝜆𝑟 ), 𝑃𝑌 (𝛾 𝑟 ), and 𝑃𝑍 (𝜆𝑟 , 𝛾 𝑟 ) presented in section 3.4.1 in parallel and compute 𝒳 ∗ 3 and 𝒴 ∗ 4 compute 𝐿𝐵 𝑟 using equation (3.27) 5 if 𝐿𝐵𝑟 > 𝐿𝐵 ∗ 6 𝐿𝐵 ∗ : = 𝐿𝐵𝑟 ; 𝜎: = 0 7 else 8 𝜎: = 𝜎 + 1 Step 2: generating 𝑈𝐵𝑟 9 solve sub-problem 𝑃𝑍′ (𝒳 ∗ , 𝒴 ∗ ) presented in section 3.4.2 10 compute 𝑈𝐵 𝑟 using equation (3.33) 11 if 𝑈𝐵𝑟 < 𝑈𝐵 ∗ 12 𝑈𝐵 ∗ : = 𝑈𝐵 𝑟 Step 3: check termination condition, update 𝜃0 , 𝜆𝑟 , and 𝛾 𝑟 𝑈𝐵∗ −𝐿𝐵∗ 13 calculate 𝑔 by 100 × ( ) 𝑈𝐵∗ 14 if 𝑔 ≤ 𝜀 do 15 break 16 else if 𝜎: = 𝛿 𝜃 17 𝜃0 : = 0; 𝜎: = 0 2 18 compute 𝜆𝑟+1 using equations (3.34)-(3.36) 19 compute 𝛾 𝑟+1 using equations (3.37)-(3.39) 20 𝑟: = 𝑟 + 1 21 return 𝑈𝐵 ∗ , 𝐿𝐵 ∗ , and 𝑔 iteration 𝑟 + 1 and 𝑃𝑋∗ from iteration 𝑟 and solve 𝑃𝑍 ′ . The foregoing steps are repeated until the predefined termination condition is met. Table 3.4 presents our Lagrangian relaxation algorithm when the foregoing sub-problems are solved sequentially. CFRS algorithm for solving large datasets Clustering is defined as the task of identifying subgroups (clusters) in the dataset such that data points in the same cluster are similar while data points in different clusters are different. For more information about the general concept and methods of clustering, one can refer to Jain et al. (1999). In the VRP and its variants, clustering methods are commonly used to break a large dataset into several smaller datasets and reduce the computational burden of solving these NP hard problems (e.g., Fisher and Jaikumar (1981) for the capacitated location-allocation problem, Toth and Vigo 77 Table 3.4. Lagrangian relaxation algorithm when 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 are solved sequentially. Step 0: initialization 1 𝑅; 𝑟: = 0; 𝜎: = 0; , 𝛿; 𝜃0 ; 𝜀; 𝐿𝐵 ∗ : = −∞; 𝑈𝐵 ∗ : = +∞; 𝑔 = +∞; 𝜆0 : = 0; 𝛾 0 : = 0; 2 solve sub-problem 𝑃𝑌 (𝛾 0 ) presented in section 3.4.1 and compute 𝒴 ∗ and 𝐵ℎ = {𝑗: 𝑦0′𝑗 = 1, 𝑦0′ 𝑗 ∈ 𝒴 ∗ }; 3 while termination condition is not met repeat steps 1-3 (i.e., if 𝑟 < 𝑅 or 𝑔 > 𝜀) 4 𝑟: = 1; 𝜆1 : = 𝜆0 ; 𝛾 1 : = 𝛾 0 ; Step 1: generating 𝐿𝐵𝑟 2 5 if 𝑟 ≡ 1 6 solve sub-problem 𝑃𝑋 (𝜆𝑟 ) presented in section 3.4.1 and compute 𝒳 ∗ and 𝐿𝑡 = {𝑖: 𝑥𝑖0′ = 1, 𝑥𝑖0′ ∈ 𝒳 ∗ } 7 solve sub-problem 𝑃𝑍 (𝜆𝑟 , 𝛾 𝑟 ) presented in section 3.4.1 such that 𝐿: = 𝐿 and 𝐵: = 𝐵ℎ 8 else 9 solve sub-problem 𝑃𝑌 (𝛾 𝑟 ) presented in section 3.4.1 and compute 𝒴 ∗ ⁡and⁡𝐵ℎ = {𝑗: 𝑦0′𝑗 = 1, 𝑦0′𝑗 ∈ 𝒴 ∗ } 10 solve sub-problem 𝑃𝑍 (𝜆𝑟 , 𝛾 𝑟 ) presented in section 3.4.1 such that 𝐿: = 𝐿𝑡 and 𝐵: = 𝐵⁡ 11 compute 𝐿𝐵 𝑟 using equation (3.27) 12 if 𝐿𝐵𝑟 > 𝐿𝐵 ∗ 13 𝐿𝐵 ∗ : = 𝐿𝐵𝑟 ; 𝜎: = 0 14 else 15 𝜎: = 𝜎 + 1 Step 2: generating 𝑈𝐵𝑟 16 solve sub-problem 𝑃𝑍′ (𝒳 ∗ , 𝒴 ∗ ) presented in section 3.4.2 17 compute 𝑈𝐵 𝑟 using equation (3.33) 18 if 𝑈𝐵𝑟 < 𝑈𝐵 ∗ 19 𝑈𝐵 ∗ : = 𝑈𝐵 𝑟 Step 3: check termination condition, update 𝜃0 , 𝜆𝑟 , and 𝛾 𝑟 𝑈𝐵∗ −𝐿𝐵∗ 20 calculate 𝑔 by 100 × ( ) 𝑈𝐵∗ 21 if 𝑔 ≤ 𝜀 22 break 23 else if 𝜎: = 𝛿 𝜃 24 𝜃0 : = 0; 𝜎: = 0 2 25 compute 𝜆𝑟+1 using equations (3.34)-(3.36) 26 compute 𝛾 𝑟+1 using equations (3.37)-(3.39) 27 𝑟: = 𝑟 + 1 28 return 𝑈𝐵 ∗ , 𝐿𝐵 ∗ , and 𝑔 (1999) for the VRPB, Cakir et al. (2015) for the capacitated VRP, Mahmoudi et al. (2019) for the pickup and delivery with transfers, Wang et al. (2020) for the location-routing problem). In this paper, we conduct the task of clustering for linehaul and backhaul nodes separately. Each cluster is served by one vehicle. The vehicle capacity is considered in the clustering phase; therefore, there is no need to consider this constraint in the routing phase. As a result, for each cluster, we solve an open traveling salesman problem (OTSP) to determine the optimal route of the vehicle. Similar to the OVRP, in the OTSP, the salesman does not return to the depot after visiting the last customer 78 on the route, or likewise the return trip to the depot is not charged. Then, we solve an AP to connect linehaul routes to backhaul routes or to the depot, obtain lower and upper bound solutions, and update Lagrangian multipliers. 3.4.4.1. Clustering phase We follow Fisher and Jaikumar (1981) and Cakir et al. (2015) to develop the CFRS algorithm. Each customer node has the potential to be the seed of a cluster. We define binary decision variable 𝑠𝑖𝑗 which equals 1 if node 𝑖 is assigned to cluster 𝑗, and 0 otherwise. We also define binary variable 𝑝𝑗 which equals 1 if node 𝑗 is selected as the seed of a cluster, and 0 otherwise. Then, the measure of dissimilarity between node 𝑖 and cluster 𝑗, denoted by 𝜉𝑖𝑗 , is calculated by (3.40): 𝜉𝑖𝑗 = 𝑐𝑖𝑗 + 𝜆𝑖 ⁡{𝑖 ∈ 𝐿⁡𝐴𝑁𝐷⁡𝑗 ∈ 𝐿}⁡ (3.40) {𝑐𝑖𝑗 + 𝛾𝑖 ⁡⁡{𝑖 ∈ 𝐵⁡𝐴𝑁𝐷⁡𝑗 ∈ 𝐵} +∞⁡⁡ ⁡⁡⁡⁡{(𝑖 ∈ 𝐿⁡𝐴𝑁𝐷⁡𝑗 ∈ 𝐵)⁡𝑂𝑅⁡(𝑖 ∈ 𝐵⁡𝐴𝑁𝐷⁡𝑗 ∈ 𝐿)} The definition of 𝜉𝑖𝑗 allows us to cluster linehaul and backhaul nodes separately. Then, our mathematical model for the clustering phase is as follows: 𝑀𝑖𝑛 ∑𝑖∈𝐿∪𝐵 ∑𝑗∈𝐿∪𝐵 𝜉𝑖𝑗 𝑠𝑖𝑗 (3.41) s.t. ∑𝑖∈𝐿∪𝐵 𝑑𝑖 𝑠𝑖𝑗 ≤ 𝑄𝑝𝑗 𝑗 ∈ 𝐿 ∪ 𝐵, (3.42) ∑𝑗∈𝐿∪𝐵 𝑠𝑖𝑗 = 1 𝑖 ∈ 𝐿 ∪ 𝐵, (3.43) ∑𝑗∈𝐿 𝑝𝑗 = 𝑘, (3.44) 79 ∑𝑗∈𝐵 𝑝𝑗 ≤ 𝑘, (3.45) 𝑠𝑖𝑗 ∈ {0,1}, 𝑝𝑗 ∈ {0,1} 𝑖, 𝑗 ∈ 𝐿 ∪ 𝐵. (3.46) Objective function (41) captures the dissimilarity between each node assigned to the cluster and the node used as the seed of the cluster. Constraint (42) ensures that the total demand assigned to a cluster does not exceed the vehicle capacity. Constraint (43) ensures that each customer is assigned to exactly one cluster. Constraint (44) guarantees that the total number of linehaul clusters is equal to 𝑘. Constraint (45) ensures that at most 𝑘 number of backhaul clusters are generated. Constraint (46) defines the domain of decision variables. 3.4.4.2. Routing phase The vehicle capacity constraint for each cluster is met by constraint (42). Therefore, instead of solving an OVRP to determine optimal routing for each cluster, we solve an OTSP. Suppose after solving the problem presented in section 3.4.4.1, linehaul and backhaul customers are partitioned to 𝑞 number of clusters. From the solution obtained from the clustering phase, we define an incidence matrix whose elements, denoted by 𝜑𝑖𝑞 , represent the existence of node 𝑖 in cluster 𝑞. 𝜑𝑖𝑞 is 1 if node 𝑖 exists in cluster 𝑞, and 0 otherwise. We note that nodes 0 and 0′ exist in all ′ clusters. The traveling cost of link (𝑖, 𝑗), denoted by 𝜉𝑖𝑗 , equals 𝜉𝑖𝑗 for {(𝑖, 𝑗)|𝑖, 𝑗 ∈ 𝐿 ∪ 𝐵}, 𝑐𝑖𝑗 for {(𝑖, 𝑗)|(𝑖 = 0⁡𝐴𝑁𝐷⁡𝑗 ∈ 𝐿)⁡𝑂𝑅⁡(𝑖 ∈ 𝐵⁡𝐴𝑁𝐷⁡𝑗 = 0)}, and 0 for {(𝑖, 𝑗)|(𝑖 ∈ 𝐿⁡𝐴𝑁𝐷⁡𝑗 = 0′ )⁡𝑂𝑅⁡(𝑖 = 𝑞 0′ ⁡𝐴𝑁𝐷⁡𝑗 ∈ 𝐵}. We define binary decision variable 𝑜𝑖𝑗 which equals 1 if link (𝑖, 𝑗) exists in cluster 𝑞, and 0 otherwise. Then, our mathematical model for the routing phase is as follows: ′ 𝑞 𝑞 𝑞 𝑀𝑖𝑛⁡ ∑𝑞 ∑𝑖,𝑗∈𝐿∪𝐵∪{0,0′ } 𝜉𝑖𝑗 𝜑𝑖 𝜑𝑗 𝑜𝑖𝑗 (3.47) 80 s.t. ∑𝑖∈𝐿0 ∪𝐵0′ 𝜑𝑖𝑞 𝜑𝑗𝑞 𝑜𝑖𝑗 𝑞 =1 ∀𝑞, ∀𝑗 ∈ 𝐿0′ ∪ 𝐵0, (3.48) 𝑖≠𝑗 ∑𝑗∈𝐿0′ ∪𝐵0 𝜑𝑖𝑞 𝜑𝑗𝑞 𝑜𝑖𝑗 𝑞 =1 ∀𝑞, ∀𝑖 ∈ 𝐿0 ∪ 𝐵0′ , (3.49) 𝑖≠𝑗 ∀𝑞, ∀𝑖, 𝑗 ∈ 𝐿 ∪ 𝐵, 𝑖 ≠ 𝓊𝑖 − 𝓊𝑗 + 𝑄𝜑𝑖𝑞 𝜑𝑗𝑞 𝑜𝑖𝑗 𝑞 ≤𝑄−1 (3.50) 𝑗, 𝑑𝑖 ≤ 𝓊𝑖 ≤ 𝑄 𝑖 ∈ 𝐿 ∪ 𝐵, (3.51) 𝑞 𝑜𝑖𝑗 ∈ {0,1}, 𝑖, 𝑗 ∈ 𝐿 ∪ 𝐵 ∪ {0, 0′ }; 𝓊𝑖 ∈ ℝ, 𝑖 ∈ 𝐿 ∪ 𝐵. (3.52) Objective function (3.47) minimizes the total transportation cost. Constraints (3.48) and (3.49) ensure that each node in each cluster is visited exactly once. Constraints (3.50) and (3.51) guarantees the connectivity of the routes. Constraint (3.52) defines the domain of decision variables. Thus far, linehaul and backhaul clusters have been routed separately. At each iteration, to obtain a lower bound solution, we solve sub-problem 𝑃𝑍 (presented in section 3.4.1) and update the global lower bound. Having 𝐿𝑡 and 𝐵 ℎ from section 3.4.4.2, we solve sub-problem 𝑃𝑍 ′ (presented in section 3.4.2) to obtain the upper bound solution and update the global upper bound. At the end of each iteration, Lagrangian multipliers corresponding to the linehaul and backhaul nodes are updated. The algorithm stops if the termination condition discussed in section 3.4.3 is met. Computational experiments We test our proposed algorithms on two benchmark datasets in the extant literature, i.e., instances proposed by Goetschalckx and Jacobs-Blecha (1989) and Toth and Vigo (1997) and 81 compare our results with the best results reported by the exact algorithms in the literature. We also test our model on our randomly generated dataset containing 100, 250, and 500 customers, geographically distributed on the real-world Lansing transportation network. Our experiments executed on an HP laptop running Intel (R) Core (TM) i7-7500U CPU processors clocked at 2.70 GHz with 7 cores and 12 GB RAM running Windows Server 10×64 Edition. The algorithms were coded in Python distribution platform, Anaconda 3, and run by Spyder integrated development environment, version 4.0.1. We used Python interface with CPLEX 12.10.0 optimization solver to generate the lower and upper bond solutions. We set a time limit of 500 seconds to solve each sub-problem at each iteration by CPLEX optimization solver. The solver returns the best-founded solution if it cannot reach the optimal solution within the predefined time limit. The multithreading technique was also applied to solve 𝑃𝑋 , 𝑃𝑌 , and 𝑃𝑍 in Step 1 of Table 3.3. We also used this technique to solve 𝑃𝑋 and 𝑃𝑍 or 𝑃𝑌 and 𝑃𝑍 in Step 1 of Table 3.4. In all experiments, we set 𝑅 = 30, 𝛿 = 3, 𝜃0 = 0.4, and 𝜀 = 1%. The Python code and datasets are available at GitHub (2020). Benchmark datasets We test our proposed algorithms (presented in section 3.4.3) on two benchmark datasets in the extant literature, i.e., instances proposed by Goetschalckx and Jacobs-Blecha (1989) and Toth and Vigo (1997) and compare our results with the best results reported by the exact algorithms in the literature. In the first dataset, named as GJ dataset, the location of customers is uniformly distributed in two-dimensional Cartesian coordinate system within the range of (0, 24000) and (0, 32000), and the depot is located at (12000, 16000). Traveling cost 𝑐𝑖𝑗 is defined as the Euclidean distance between nodes 𝑖 and 𝑗 and rounded to the nearest integer number. The demand of customers is randomly generated from a normal distribution with the mean of 500 and the standard 82 deviation of 200. The ratio of linehaul customers to the total number of customers are 0.50, 0.66, and 0.80. Table 3.5 presents our computational experiments on 34 instances of the GJ dataset, containing 25 to 68 customers. In Table 3.5, the first four columns present the information related to the instance, including instance name, number of linehaul customers, number of backhaul customers, and number of vehicles. The next four columns present the upper bound and computational time in second obtained from running our algorithms. Columns 9, 10, and 11 present the results reported by Toth and Vigo (1997), Mingozzi et al. (1999), and Queiroga et al. (2020), respectively. Finally, the last column presents the percentage of deviation (denoted by 𝐷) of our best upper bounds from the best results reported by the aforementioned exact algorithms. The negative percentage of deviation implies that our algorithm provides a better solution. Table 3.5. A comparison of our best upper bounds with the best-known solutions on the GJ dataset*. Algorithm 1. Algorithm 2. Deviati Problem specification Exact algorithms Parallel Sequential on Toth Mingoz Queiro Time Time and zi ga Instance 𝑳 𝑩 𝒌 𝑼𝑩 𝑼𝑩 𝑫⁡(%) (sec) (sec) Vigo et al. et al. (1997) (1999) (2020) 2 A1 5 8 229,884 651.51 232,668 147.91 229,886 229,886 229,886 -0.001* 0 2 A2 5 5 183,482 1322.5 186,266 160.05 180,119 180,119 180,119 1.833 0 2 A3 5 4 163,403 776.33 169,051 86.13 163,405 163,405 163,405 -0.001* 0 2 A4 5 3 155,948 478.11 160,241 41.09 155,796 155,796 155,796 0.097 0 2 B1 10 7 240,304 1887.58 241,114 1288.01 239,080 239,080 239,080 0.509 0 2 B2 10 5 199,012 1332.77 201,605 142.57 198,048 198,048 198,048 0.484 0 2 B3 10 3 170,193 82.7 170,193 24.27 169,372 169,372 169,372 0.482 0 2 C1 20 7 251,843 1850.57 257,857 1103.59 249,448 249,448 250,557 0.951 0 2 C2 20 5 216,497 934.46 215,019 638.49 215,020 215,020 215,020 0.000 0 83 Table 3.5 (cont’d) 2 C3 20 5 199,344 443.21 203,745 84.38 199,346 199,346 199,346 -0.001* 0 2 C4 20 4 195,365 260.85 200,734 134.92 195,366 195,366 195,367 -0.001* 0 3 D1 8 12 324,637 4261.65 327,674 2938.68 322,530 322,530 322,530 0.649 0 3 14648.5 D2 8 11 319,507 323,010 7454.31 316,709 316,709 316,709 0.876 0 6 3 10627.0 D3 8 7 239,482 240,396 4695.46 239,479 239,479 239,479 0.001 0 7 3 D4 8 5 207,024 7642.32 213,143 5669.41 205,832 205,832 205,832 0.576 0 3 E1 15 7 242,062 5837.02 242,487 5050.87 238,880 238,880 238,880 1.315 0 3 E2 15 4 212,600 6922.61 217,600 6095.03 212,263 212,263 212,263 0.159 0 3 E3 15 4 207,924 1222.68 208,680 144.08 206,659 206,659 206,659 0.608 0 3 F1 30 6 265,792 9098.24 272,979 9092.79 263,173 263,173 263,174 0.985 0 3 F2 30 7 265,214 6445.71 265,494 5285.15 265,213 265,213 265,214 0.000 0 3 F3 30 5 241,121 1287.4 241,726 465.5 241,120 241,120 241,121 0.000 0 3 F4 30 4 237,004 804.29 237,004 684.1 233,861 233,861 233,862 1.326 0 4 G1 12 10 312,514 6539.2 312,119 3182.03 307,274 306,305 306,305 1.863 5 4 G2 12 6 246,690 8584.31 257,496 4255.91 245,441 245,441 245,441 0.506 5 4 10891.6 G3 12 5 234,377 235,785 473.66 229,507 229,507 229,507 2.078 5 1 4 13569.2 G4 12 6 237,284 239,638 444.37 233,184 232,521 232,521 2.007 5 2 4 G5 12 5 224,028 5620.12 235,718 1137.44 221,730 221,730 221,730 1.026 5 4 G6 12 4 213,457 322.94 220,980 286.23 213,457 213,457 213,457 0.000 5 4 15649.4 H1 23 6 273,344 279,091 7435.49 268,933 268,933 268,933 1.614 5 6 4 H2 23 5 253,366 5000.87 253,599 2714.9 253,365 253,365 253,366 0.000 5 4 H3 23 4 247,449 2022.34 249,199 2966.17 247,449 247,449 247,449 0.000 5 4 H4 23 5 250,221 559.83 253,395 1158.82 250,221 250,221 250,221 0.000 5 4 H5 23 4 246,354 338.01 246,121 1381.38 246,121 246,121 246,121 0.000 5 4 H6 23 5 249,280 106.21 249,280 532.13 249,135 249,135 249,135 0.058 5 Average 234,000 4353.60 237,091 2276.33 232,542 232,494 232,527 0.59 * Improved UB obtained from running our algorithms 84 From our results in Table 3.5, we observe that, compared to the best-known solutions, our model yields the same or better solutions in 12 instances (35%) and solutions within 2% deviation from the best-known exact solutions for the rest of instances. We also observe that compared to the upper bound solutions obtained from sequential layout, parallel layout provides better solutions in 31 instances, while sequential layout outperforms parallel layout from the computational efficiency standpoint in 30 instances. These observations are consistent with this fact that there is always a tradeoff between the solution quality and computational efficiency of an algorithm. The second dataset, named as TV dataset, is generated from the VRP benchmark instances in the extant literature. For each VRP instance, three VRPB instances are generated in which the ratio of linehaul customers to the total number of customers are 0.50, 0.66, and 0.80. Similar to the GJ dataset, traveling cost is defined as the Euclidean distance between nodes and rounded to the nearest integer. The number of vehicles is calculated by the maximum number of vehicles required to serve linehaul and backhaul customers independently. Table 3.6 presents our computational experiments on the 33 instances of TV datasets, containing 21 to 100 customers. Our model is capable of solving 28 out of 33 instances of the TV dataset. From our results in Table 3.6, we observe that, compared to the best-known exact solutions, our model yields the same or better solutions in 11 instances (33%) and solutions within 2% deviation from the best-known exact solutions in 10 instances. We also observe that in the majority of instances, parallel layout provides better solutions, while sequential layout outperforms parallel layout in terms of computational efficiency standpoint. 85 Table 3.6. A comparison of our best upper bounds with the best-known solutions on the TV dataset*. Sequential Devia Problem specification Parallel layout Exact algorithms layout tion Toth Mingo Queir Time Time and zzi et oga et Instance 𝑳 𝑩 𝒌 𝑼𝑩 𝑼𝑩 𝑫⁡(%) (sec) (sec) Vigo al. al. (1997) (1999) (2020) eil22_50 11 10 3 371 18.55 384 66.62 371 371 371 0.000 eil22_66 14 7 3 366 16.76 380 72.18 366 366 366 0.000 eil22_80 17 4 3 375 110.51 388 217.62 375 375 375 0.000 eil23_50 11 11 2 682 27.8 694 9.08 682 682 682 0.000 eil23_66 15 7 2 674 146.18 649 83.45 649 649 649 0.000 eil23_80 18 4 2 623 39.19 672 6.15 623 623 623 0.000 eil30_50 15 14 2 501 84.64 508 44.33 501 501 501 0.000 eil30_66 20 9 3 539 799.02 539 777.01 537 537 537 0.371 eil30_80 24 5 3 520 5329.9 520 4146.05 514 514 514 1.154 eil33_50 16 16 3 738 1885.64 738 467.76 738 738 738 0.000 eil33_66 22 10 3 758 11857.99 754 1342.53 750 750 750 0.531 eil33_80 26 6 3 745 7044.39 753 2012.56 736 736 736 1.208 eil51_50 25 25 3 559 185.74 577 41.35 559 559 559 0.000 eil51_66 34 16 4 560 1531 557 2058.94 548 548 548 1.616 eil51_80 40 10 4 565 1576.67 584 533.13 565 565 565 0.000 eilA76_50 37 38 6 761 4572.55 750 6937.16 739 739 739 1.467 eilA76_66 50 25 7 822 1004.48 872 1004.96 768 768 768 6.569 eilA76_80 60 15 8 - - - - 781 781 781 - eilB76_50 37 38 8 830 2702.39 821 4829.98 801 801 801 2.436 eilB76_66 50 25 10 937 1536.46 937 1557.99 873 873 873 6.830 eilB76_80 60 15 12 1043 1067.96 1009 1012.95 919 919 919 8.920 eilC76_50 37 38 5 720 2059.66 733 3033.33 713 713 713 0.972 eilC76_66 50 25 6 745 12550.67 764 7096.37 734 734 734 1.477 eilC76_80 60 15 7 767 8441 794 1786.6 733 733 733 4.433 eilD76_50 37 38 4 692 234.69 712 322.5 690 690 690 0.289 eilD76_66 50 25 5 715 3406.28 743 1268.31 715 715 715 0.000 eilD76_80 60 15 6 723 6905.33 735 1802.95 703 694 694 4.011 eilA101_50 50 50 4 842 7602.65 869 1541.18 843 843 831 1.306 eilA101_66 67 33 6 868 6310.89 870 2829.1 846 846 846 2.535 eilA101_80 80 20 6 - - - - 916 908 856 - eilB101_50 50 50 7 - - - - - 933 923 - eilB101_66 67 33 9 - - - - - 1056 982 - eilB101_80 80 20 11 - - - - - 1022 1008 - 1,675.0 Average 680 3,180.32 690 664 664 663 1.65 8 * Improved UB obtained from running our algorithms 86 Dataset geographically distributed on the Lansing transportation network A transportation network consists of a set of physical nodes (e.g., intersections, freeways, merge points) and a set of directed physical links (e.g., freeway segments, arterial streets, ramps). The Lansing transportation network consists of 5,409 nodes and 7,610 directed links (Figure 3.5). Figure 3.5. Lansing transportation network with 5,409 nodes and 7,610 links. In this network, each transportation node is specified by its node id, longitude, and latitude. Each transportation link is also identified by a link id, from node id (the id of the transportation node from which the link starts), to node id (the id of the transportation node at which the link ends), and link length. The Haversine distance between transportation nodes 𝑛 and 𝑚, denoted by 𝐻𝐷, is calculated by (3.53): 87 𝐻𝐷 𝜋 𝜋 𝜋 𝜋 (3.53) = 2𝑅. arcsin⁡(√𝑠𝑖𝑛2 ( ∆𝑙𝑎𝑡) + cos ( 𝑙𝑎𝑡 ) . cos ( 2 𝑙𝑎𝑡 ) . 𝑠𝑖𝑛 ( ∆𝑙𝑛𝑔))⁡ 180 180 𝑛 180 𝑚 180 𝑙𝑎𝑡𝑛 −𝑙𝑎𝑡𝑚 , where 𝑙𝑎𝑡𝑛 and 𝑙𝑛𝑔𝑚 denote the latitude and longitude of node 𝑛, ∆𝑙𝑎𝑡 = , ∆𝑙𝑛𝑔 = 2 𝑙𝑛𝑔𝑛 −𝑙𝑛𝑔𝑚 and 𝑅 = 3963.189 miles. 2 All transportation links in this network are bidirectional. We generate three datasets with 100, 250, and 500 customers whose locations are randomly selected from Lansing transportation nodes. Then, for each dataset, we generate three standard VRPB instances in which the ratio of linehaul customers to the total number of customers are 0.50, 0.66, and 0.80. The demand of all customers is one unit. For each instance, we generate three sub-instances with the vehicle capacity of 10, 30, and 50. We calculate the cost of traveling from one customer location to another by calculating the least cost path between them using Bellman-Ford algorithm (Ford, 1956; Bellman, 1958). To examine our model on this dataset, we apply the CFRS algorithm presented in section 3.4.4. Table 3.7 presents our computational experiments on the Lansing transportation network. From our results in Table 3.7, we observe an average gap of 4.88% between lower and upper bound solutions. We also observe that, in each instance, increasing the vehicle capacity decreases the foregoing gap. This would be reasonable since a higher capacity could necessitate fewer number of vehicles to serve the same number of customers. Of note, fewer vehicles translate to fewer clusters, and hence, sub-problems. Figure 3.6 illustrates the average relative gap percentage for various vehicle capacity. Figure 3.7 depicts the route of vehicles for Lansing_100_50 when 𝑄 = 10, 30, and 50. 88 Table 3.7. Our computational experiments on the Lansing transportation network. Instance 𝑳+𝑩 𝑳 𝑩 𝑸 𝒌 𝑳𝑩∗ 𝑼𝑩∗ Gap (%) Time (sec) Lansing_100_50 100 50 50 10 5 177477 185536 4.34 148.01 Lansing_100_50 100 50 50 30 2 147886 154717 4.42 464.42 Lansing_100_50 100 50 50 50 1 132809 137515 3.42 1141.34 Lansing_100_66 100 66 34 10 7 175480 195330 10.16 667.04 Lansing_100_66 100 66 34 30 3 155034 167165 7.26 383.65 Lansing_100_66 100 66 34 50 2 145200 152003 4.48 10667.63 Lansing_100_80 100 80 20 10 8 178520 193534 7.76 839.17 Lansing_100_80 100 80 20 30 3 152944 159554 4.14 1359.35 Lansing_100_80 100 80 20 50 2 142643 144977 1.61 1806.67 Lansing_250_50 250 125 125 10 13 323706 345350 6.27 15656.57 Lansing_250_50 250 125 125 30 5 267227 280738 4.81 4316.15 Lansing_250_50 250 125 125 50 3 237913 243758 2.4 17402.55 Lansing_250_66 250 165 85 10 17 348789 377983 7.72 12198.78 Lansing_250_66 250 165 85 30 6 275420 287898 4.33 5394.18 Lansing_250_66 250 165 85 50 4 252930 262297 3.57 31100.76 Lansing_250_80 250 200 50 10 20 361317 378721 4.6 15672.49 Lansing_250_80 250 200 50 30 7 267440 292128 8.45 13899.31 Lansing_250_80 250 200 50 50 4 239229 250995 4.69 28080.71 Lansing_500_50 500 250 250 10 25 517,373 528,386 2.08 15996.13 Lansing_500_50 500 250 250 30 9 383,532 398,558 3.77 24621.43 Lansing_500_50 500 250 250 50 5 344,885 359,615 4.1 34010.72 Lansing_500_66 500 330 170 10 33 560,044 603,547 7.21 16106.16 Lansing_500_66 500 330 170 30 11 394,041 406,957 3.17 34784.04 Lansing_500_66 500 330 170 50 7 362,433 375,548 3.49 47128.21 Lansing_500_80 500 400 100 10 40 599,074 638,213 6.13 16030.67 Lansing_500_80 500 400 100 30 14 395,936 415,200 4.64 25499.37 Lansing_500_80 500 400 100 50 8 354,450 363,952 2.61 68693.02 Average 292,360 307,414 4.88 16446.98 89 Figure 3.6. The average relative gap percentage for various vehicle capacity. (a) Vehicle capacity = 10 (b) Vehicle capacity = 30 (c) Vehicle capacity = 50 Figure 3.7. An illustration of vehicle routes on Lansing_100_50. Conclusion In this study, we proposed a new mathematical model for the standard VRPB. The structure of the proposed model allowed us to utilize the Lagrangian decomposition approach to break the main problem into three well-known sub-problems: two OVRPs and one AP. At each iteration, the forgoing sub-problems are solved and Lagrangian multipliers are updated to reduce the gap between the lower and upper bounds of the global optimal solution. We proposed two different 90 arrangements, i.e., parallel and sequential, for solving the foregoing sub-problems. We further analyzed the impact of these arrangements on the solution quality and computational efficiency of our proposed Lagrangian decomposition algorithm. We tested the framework on two benchmark datasets in the extant literature that were proposed by Goetschalckx and Jacobs-Blecha (1989) and Toth and Vigo (1997) and named as GJ and TV datasets, respectively. For 34 instances of the GJ dataset, compared to the best-known exact solutions, our model yielded the same or better solutions in 12 instances (35%) and solutions within 2% deviation from the best-known exact solutions for the rest of instances. Our model was also capable of solving 28 out of 33 instances of the TV dataset. For this dataset, compared to the best-known exact solutions, our model yielded the same or better solutions in 11 instances (33%) and solutions within 2% deviation from the best-known exact solutions in 10 instances. We also observed that, in the majority of instances of both datasets, solving the foregoing sub-problems in parallel provides better solution compared to when they are solved sequentially; however, sequential layout outperforms parallel layout from the computational efficiency standpoint. We also tested our model on a randomly generated dataset containing 100, 250, and 500 customers, geographically distributed on the Lansing transportation network. To reduce the computational burden of solving the VRPB on this dataset, we presented a CFRS algorithm. Our results showed that increasing vehicle capacity in each instance decreases the gap between lower and upper bounds. This would be reasonable since a higher capacity could necessitate fewer number of vehicles to serve the same number of customers. Fewer vehicles translate to fewer clusters, and hence, sub-problems. Future work may concentrate on employing exact solution methodologies such as brand-and- bound algorithm for solving the foregoing sub-problems. Since our proposed model allows us to 91 decompose linehaul and backhaul routing phases, customized side constraints can be added to each phase without altering the other phase of the problem. 92 CHAPTER 4. EVALUATING OPERATING MODELS AND URBANISM FOR TRANSPORTATION OPERATIONS OF CIRCULAR REUSE PLATFORMS Introduction According to the U.S. Environmental Protection Agency, containers and packages contribute to 28.1% of generating municipal solid waste in the U.S. in 2018 (82.22 out of 292.36 million tons) (EPA, 2020). Although 53.9% of these 82.22 million tons come from recycled containers/packages, a huge portion of these items that are supposed to be recycled will ultimately end up in the landfills, mainly due to complexities in a recycling process. Therefore, “Reuse, as well as plain elimination of a lot of packaging we don’t need, will also have to be a crucial part of the solution” (TIME, 2021). Reuse models could contribute to more sustainable packaging, which (1) entails a closed-loop life cycle of packages and packaging materials, (2) is economically robust, and (3) has a minimum environmental impact (SPC, 2011; Gustavo et al., 2018). More recently, the onset of the COVID-19 pandemic has already accelerated the interest in reusable packaging in the grocery retail industry: “The idea of transporting things in a clean, freshly sanitized container has probably never been more popular than it is right now,” says the VP of product and category development at IFCO Systems, one of two leading providers of reusable plastic containers in the U.S. (Progressive Grocer, 2020). Despite their impact, there still exist operational challenges attributed to reusable packages that have not been addressed by the extant literature (see, e.g., our extensive review in chapter 2 and Coelho et al., 2020). In particular, we refer to the transportation operations of reusable primary packages (those that are in direct contact with products, such as shampoo, ice cream, soda, etc.) in a closed-loop supply chain, where, in addition to products being delivered to customers, the reverse logistic would handle collecting empty reusable packages from customers and returning them to a 93 depot for cleaning, refill, and reuse. Of note, managing the transportation operation of reusable packages is more complex than that for traditional single-use packages, mainly due to the high volume and frequency of delivery/pickup. Needless to say, according to the Armstrong & Associates Inc., up to 58% of the total global logistics costs could come from the transportation costs (Rodrigue, 2020), which can be deemed as another barrier in proper implementation of reusables (as mentioned in chapter 2). To address these challenges, we develop an analytical framework to optimize the transportation operations of reusables. In particular, we model the delivery and pickup operations of reusable packages by VRPBs (see, e.g., Battarra et al., 2014). The aim of this model is to construct a set of routing plans for a fleet of identical vehicles located at the depot such that each vehicle starts/ends its route at the depot, each delivery/pickup customer is visited exactly once by one vehicle, and the total requests made by these customers do not exceed the capacity of the vehicle that will serve them. We then develop a MIP model to solve the VRPB for small-scale networks. Of note, the VRPB is an NP-hard problem (see, e.g., Parragh et al., 2008b; Battarra et al. 2014), and hence, for medium to large-size problems, we develop a heuristic solution approach CFRS. This heuristic partitions delivery and pickup customers into several clusters using the K-means clustering algorithm, such that each cluster corresponds to the routing of exactly one vehicle based on its capacity. Regarding the business model that we consider in this research, we note the following point: over the past few years, various companies have launched reusable packaging initiatives. For example, the TerraCycle has utilized a circular shopping platform called the Loop (GreenBiz, 2019) where (i) customers place their orders via the Loop’s website, (ii) the Loop collects all requested products in a reusable tote and have them delivered to customers, (iii) once products are 94 consumed and packages are empty, customers replace them in the tote and submit a pickup request on the Loop’s website, and (iv) the empty packages are collected and transported back to a facility for inspection, cleaning, refill, and reuse (see Figure 4.1). Since 2019, this direct-delivery model has transitioned to in-store delivery, in that customers pick up the packages from (or deliver to) U.S. retailers like Walmart and Kroger, instead of these packages being delivered to customers directly (see, e.g., CNN, 2019; TIME, 2021). Despite this shift, the direct delivery model entails less effort for customers, which, in turn, impacts their convenience and loyalty (see, e.g., Chang et al., 2010; Harvard Business Review, 2010). After all, the issues of customer acquisition and retention are known as one of the main barriers in adopting reusables (see, e.g., Temper Pack, 2020; Big Commerce, 2022). Therefore, in this research, we analyze the direct delivery/pickup problem. This would set a stepping stone towards other versions of this problem, where, in addition to customers and distributors, stakeholders like retailers will be also taken into consideration. (a) The Loop shopping platform for reusable (b) A picture of a tote along with primary packages (GreenBiz, 2019) reusable packages/containers (CNN, 2019) Figure 4.1. The Loop shopping platform. As mentioned above, the concept of reusable packaging is relatively new, and hence, there is no granular data that provides information on the customers’ demand. To address this issue, we take the following steps: 95 1. To address the issue of geographical accessibility of customers and how it may impact the operations of reusables, we showcase our analytical model on two real-world transportation networks that are located in the state of Michigan, USA: the one in Lansing (representing an urban area) and the one in Charlotte (representing a rural area). 2. We assume that a percentage of total transportation nodes in each network (that is associated with the number of customers in that network) would participate in the reusable system. 3. The consumption rate of various products that are typically placed in a reusable tote, such as shampoo, ice cream, soda, etc., are different (see, e.g., USDA, 2021). Hence, to account for limited space in their homes for keeping these items, customers typically have either a delivery or a pickup request (but not both). Despite this notion, the extant literature have also addressed problems with simultaneous pickup and delivery requests. Nevertheless, in our analytical model, we consider various scenarios where a percentage of delivery/pickup requests are simultaneous. 4. Rearranging totes inside a vehicle (differentiating the totes with full containers that are ready for delivery from the totes with empty containers that have been picked up) would pose additional challenges for the operation of reusables. As noted by Toth and Vigo (2002a), this, in turn, can be impacted by the possibility of trucks being rear-load or both rear-/front-load. To ease these challenges, customers with delivery requests may be prioritized over those with pickup requests. Nonetheless, we consider scenarios in our analytical model based on whether or not this prioritization is granted. Figure 4.2 illustrates the concepts described under items (3)-(4). 96 Figure 4.2. A schematic illustration for the four classes of the VRPB (Parragh et al., 2008a; Battarra et al., 2014). 5. The objective function of our model returns the total distance covered by the vehicles that serve customers. To reflect on the economic and environmental implications of reusables, we then develop a simulation model, where the distance traveled is transformed to cost and emission amounts. Regarding items (2)-(3), in our numerical experiments, we also account for potential variations in these parameters. Summary of Main Findings - Compared to the CPLEX optimization solver, our heuristic returns solutions with 6.61% optimality gap, while it returns these solutions 1.38 hours faster. These are average values across all problem instances. 97 - For an urban network with 800 requests, the total distance covered to fulfill these requests is on average 532.74 (387.74) miles when we prioritize (do not prioritize) delivery over pickup requests. We also show that this much distance is on average the equivalent of $1,042.19 ($769.86) daily transportation cost and 892 (654.19) kg CO2 emission when deliveries are (are not) prioritized over pickups. - In a rural network with 120 requests, 155.18 (104.44) miles are traveled on average to meet these requests when deliveries are (are not) prioritized over pickups. This much travel is converted to $314.38 ($207.72) as the daily transportation cost and 267.32 (177.44) kg as the daily CO2 emission when we prioritize (do not prioritize) delivery over pickup requests. - Contrary to the notion that the highest (lowest) distance traveled occurs when 0% (100%) of customers have simultaneous delivery/pickup demand, we find that the highest/lowest distance may be found when the rate of simultaneous demand is a value between 0% and 100%. We further explore this phenomenon under different geographical areas and the possibility of serving deliveries before pickups to shed more lights on the interplay of various factors that typically impact the operations of reusables. The remainder of the chapter is organized as follows: Section 4.2 provides the existing literature on the VRPB and the transportation operations of reusable packages. Section 4.3-4 present our mathematical model and solution methodology, respectively. Numerical experiments, along with robustness checks, are presented in section 4.5. Concluding remarks and directions for future research form section 4.6. Related Literature From a modeling standpoint with respect to VRPBs, one can refer to Toth and Vigo (1997), Mingozzi et al. (1999), and Queiroga et al. (2020) for the case when a customer has either a 98 delivery or a pickup request but not both (i.e., single demand) and when delivery customers are prioritized over pickup ones (i.e., backhaul solution), Belmecheri et al. (2013), Ting and Liao (2013), and Wu et al. (2016) for single demand and when there is no priority in serving delivery/pickup customers (i.e., mixed solution), and Dell’Amico et al. (2006), Subramanian et al. (2011), and Wang and Chen (2013) for the case when a customer can have both delivery/pickup requests and mixed solution (also see Tütüncü et al., 2009; Belloso et al., 2017) for other studies on the VRPB with backhaul and mixed solutions). Furthermore, as a seminal study that propose a large neighborhood search heuristic to solve different variants of the VRPBs, one can refer to Ropke and Pisinger (2006). For more information on the VRPBs and its variants, one can refer to Toth and Vigo (2002b), Koç and Laporte (2018), Koç et al. (2020), and references therein. Of note, this research differs from aforementioned studies, since we solve all four classes of the VRPBs (see Figure 4.2). The extant literature has also addressed the transportation operations of reusable packages in closed-loop supply chains (for the summary of these studies, see Table 4.1). Similar to these studies, we formulate our analytical model as an MIP. However, compared to this stream of research, we (1) develop an efficient heuristic solution method to alleviate computational complexities, (2) compare the performance of this heuristic with exact solutions obtained from the CPLEX optimization solver, (3) consider various performance measures such as total transportation distance, costs, and emission, and (4) implement our model on two real-world transportation networks that are representative of urban and rural areas. Our aim is to help to provide more insight and implications to practitioners who are planning to shift their packaging system from single-use to reuse models. 99 Table 4.1. Summary of literature on the transportation operations of reusables in closed-loop supply chains. Performance Max size of the Solution Reference Demand* Solution** measures problem methodology Costs of fuel # customers: 8 CPLEX Soysal consumption, # vehicles: 8 Simultaneous Mixed optimization (2016) inventory, and vehicle’s capacity: solver drivers 600 Costs of transportation, # customers: 30 Iassinovskai inventory, # vehicles: 8 a et al. Simultaneous Mixed CFRS heuristic maintenance, and vehicle’s capacity: (2017) buying new 25 reusables Costs of production, inventory, and # customers: 50 A branch-and- Fang et al. routing # vehicles: 6 Simultaneous Mixed cut guided search (2017) Emission of vehicle’s capacity: algorithm production and — routing # customers: 12 Costs of Gurobi Gong el al. # vehicles: 3 Simultaneous Mixed transportation and optimization (2020) vehicle’s capacity: inventory solver 30 # customers: 100 Iterated local Londoño et Costs of # vehicles: 12 search with Single Backhaul al. (2020) transportation vehicle’s capacity: biased — randomization # customers: 2,400 Distance traveled, # vehicles: 24 This paper Both Both cost, and emission CFRS vehicle’s capacity: of transportation 50 * Simultaneous demand: customers request for simultaneous pickup and delivery. Single demand: customers have either a delivery or a pickup request (but not both). ** Mixed solution: there is no priority in serving delivery/pickup customers. Backhaul solution: delivery customers are served before pickup ones. Mathematical Model We first introduce notations for our mathematical model. The summary of notations is provided in Table 4.2. Let 𝐺 = (𝑉, 𝐴) be a directed graph. The set of nodes (customers) is denoted by 𝑉 = {0} ∪ ⁡𝐿 ∪ ⁡𝐵, where 0, 𝐿, and 𝐵 represent the depot, the set of linehaul customers (those with delivery requests), and the set of backhaul customers (those with pickup requests), respectively. Also, 𝐴 = {(𝑖, 𝑗):⁡𝑖, 𝑗 ∈ 𝑉, 𝑖 ≠ 𝑗} represents the set of all links that connect customers 𝑖 to 𝑗. A non-negative traveling distance 𝑐𝑖𝑗 is associated with link (𝑖, 𝑗), (𝑖, 𝑗) ∈ 𝐴. Non-negative 100 delivery and pickup requests, 𝑑𝑖 and 𝑝𝑗 , are associated with customers 𝑖, 𝑗 ∈ 𝑉 where 𝑑𝑖 = 0, 𝑖 ∈ 𝑉\𝐿 and 𝑝𝑗 = 0, 𝑖 ∈ 𝑉\𝐵. Then, the aim of the VRPB is to construct a set of routes for 𝐾 identical vehicles, each with capacity 𝑄 and located at the depot, to serve all customers such that: (1) all deliveries are collected at the depot and transported to customers, (2) all pickups are collected from customers and transported back to the depot, (3) each vehicle starts and ends its route at the depot, (4) each customer (delivery and/or pickup) is visited exactly once by exactly one vehicle, and (5) the total requests served by a vehicle does not exceed the vehicle’s capacity. Table 4.2. Summary of notations used in chapter 4. Notations used for the main model (see section 4.3) 𝑃 The mathematical model for the VRPB, see equations (4.6)-(4.17) 𝐺 Directed graph representing a transportation network, 𝐺 = (𝑉, 𝐴) 𝑉 Set of all nodes (customers), 𝑉 = {0} ∪ ⁡𝐿 ∪ ⁡𝐵 (𝑖, 𝑗: indices for customers) 0 The depot 𝐿 Set of linehaul customers, 𝐿0 = {0} ∪ 𝐿 (𝐿0 : set of linehaul customers and the depot) 𝐵 Set of backhaul customers, 𝐵0 = 𝐵 ∪ {0} (𝐵0 : set of backhaul customers and the depot) 𝐴 Set of all links, 𝐴 = {(𝑖, 𝑗):⁡𝑖, 𝑗 ∈ 𝑉, 𝑖 ≠ 𝑗} and 𝐴 = 𝐴𝑙 ∪ 𝐴𝑏 ∪ 𝐴𝑐 ∪ 𝐴𝑐 ′ ∪ 𝐴𝑐 ′′ 𝐴𝑙 Set of links connecting the depot or linehaul customers to linehaul customers, 𝐴𝑙 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐿0 , 𝑗 ∈ 𝐿, 𝑖 ≠ 𝑗} 𝐴𝑏 Set of links connecting backhaul customers to backhaul customers or to the depot, 𝐴𝑏 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵0 , 𝑖 ≠ 𝑗} 𝐴𝑐 Set of links connecting linehaul customers to backhaul customers or to the depot, 𝐴𝑐 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐿, 𝑗 ∈ 𝐵0 , 𝑖 ≠ 𝑗} 𝐴𝑐 ′ Set of links connecting backhaul customers to linehaul customers, 𝐴𝑐 ′ = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐵, 𝑗 ∈ 𝐿, 𝑖 ≠ 𝑗} 𝐴𝑐 ′′ Set of links connecting the depot to backhaul customers, 𝐴𝑐 ′′ = {(0, 𝑗):⁡𝑗 ∈ 𝐵} 𝑐𝑖𝑗 Non-negative traveling distance corresponding to link (𝑖, 𝑗) ∈ 𝐴 𝑑𝑖 Non-negative delivery request corresponding to customer 𝑖 ∈ 𝑉 where 𝑑𝑖 = 0⁡for 𝑖 ∈ 𝑉\𝐿 𝑝𝑖 Non-negative pickup request corresponding to customer 𝑖 ∈ 𝑉 where 𝑝𝑖 = 0⁡for 𝑖 ∈ 𝑉\𝐵 𝐾 Number of vehicles 𝑄 Vehicle’s capacity 𝑥𝑖𝑗 Decision binary variable for each link in set 𝐴𝑙 , see equation (4.1) 𝑦𝑖𝑗 Decision binary variable for each link in set 𝐴𝑏 , see equation (4.2) 𝑧𝑖𝑗 Decision binary variable for each link in set 𝐴𝑐 , see equation (4.3) 𝑧𝑖𝑗′ Decision binary variable for each link in set 𝐴𝑐 ′ , see equation (4.4) 𝑧𝑖𝑗′′ Decision binary variable for each link in set 𝐴𝑐 ′′ , see equation (4.5) 𝑢𝑖𝑗 Auxiliary continuous decision variable for the flow of delivery requests⁡(𝑖, 𝑗) ∈ 𝐴 𝑤𝑖𝑗 Auxiliary continuous decision variable for the flow of pickup requests, (𝑖, 𝑗) ∈ 𝐴 𝛼 Equals to 1 (0) if we solve the problem 𝑃⁡for the VRPB with backhaul (mixed) solution Notations used for K-means clustering algorithm (see section 4.4.1) 𝑅 Maximum number of iterations (𝑟: index for the iteration) 𝜏 Maximum number of iterations without improvement in solution (𝑟 ′ : index for the number of iterations without improvement in solution) 101 Table 4.2 (cont’d) 𝑘 Index for vehicle/route/cluster, 𝑘 = 1, . . . , 𝐾 𝒱 Set of the cluster centers, 𝒱 = {𝓋1 , . . . , 𝓋𝐾 }, 𝓋𝑘 ∈ 𝑉\{0}, 𝑘 = 1, . . . , 𝐾 𝐶 Set of clusters, 𝐶 = {𝐶1 , . . . , 𝐶𝐾 }, 𝐶𝑘 = {𝑖: 𝑖 ∈ 𝑉\{0}}, 𝐶1 ∩. . .∩ 𝐶𝐾 = ∅ 𝑄𝑑 Set of free delivery capacity, 𝑄𝑑 = {𝑄1𝑑 , . . . , 𝑄𝐾𝑑 } 𝑝 𝑝 𝑄𝑝 Set of free pickup capacity, 𝑄𝑝 = {𝑄1 , . . . , 𝑄𝐾 } 𝑒𝑖𝑘 Sum of traveling distance from customer 𝑖 to customer 𝑗, 𝑒𝑖𝑘 = ∑𝑗∈𝐶𝑘 𝑐𝑖𝑗 , 𝑖 ∈ 𝐶𝑘 𝑗≠𝑖 𝑒 Sum of traveling distance from each customer to his assigned cluster center, 𝑒 = ∑𝐶𝑘 ∈𝐶 ∑ 𝑖∈𝐶𝑘 𝑐𝑖𝓋𝑘 𝑖≠𝓋𝑘 Notations used for modified model (see section 4.4.2) 𝑃𝑘 The modified mathematical model for the routing of cluster/vehicle 𝑘 𝐶𝑘 Set of customers assigned to cluster 𝑘, 𝐶𝑘0 = {0} ∪ 𝐶𝑘 for 𝐶𝑘 ∈ 𝐶 𝐴𝑘 Set of links connecting the depot and customers assigned to cluster 𝑘, 𝐴𝑘 = {(𝑖, 𝑗):⁡𝑖, 𝑗 ∈ 𝐶𝑘0 , 𝑖 ≠ 𝑗} for 𝐶𝑘 ∈ 𝐶 𝑃𝑘∗ Optimal distance for cluster 𝑘 and obtained by solving problem 𝑃𝑘 𝑃∗ Total distance obtained by our proposed CFRS solution approach, 𝑃 ∗ = ∑𝐾 ∗ 𝑘=1 𝑃𝑘 Notations introduced for measuring emission (see Section 5.4) 𝑣 A vehicle’s speed 𝐷 Total distance traveled by all vehicles serving customers In order to establish decision variables for our model, we first divide the set of links 𝐴 into subsets. Suppose 𝐿0 = {0} ∪ 𝐿 and 𝐵0 = 𝐵 ∪ {0}. Then, 𝐴 = 𝐴𝑙 ∪ 𝐴𝑏 ∪ 𝐴𝑐 ∪ 𝐴𝑐 ′ ∪ 𝐴𝑐 ′′ , such that: 𝐴𝑙 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐿0 , 𝑗 ∈ 𝐿, 𝑖 ≠ 𝑗} connects the depot or linehaul customers to other linehaul customers, 𝐴𝑏 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵0 , 𝑖 ≠ 𝑗} connects backhaul customers to other backhaul customers or to the depot, 𝐴𝑐 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐿, 𝑗 ∈ 𝐵0 , 𝑖 ≠ 𝑗} connects linehaul customers to backhaul customers or to the depot, 𝐴𝑐 ′ = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐵, 𝑗 ∈ 𝐿, 𝑖 ≠ 𝑗} connects backhaul customers to linehaul customers, and 𝐴𝑐 ′′ = {(0, 𝑗):⁡𝑗 ∈ 𝐵} connects the depot to backhaul customers. Of note, when we solve the problem with a backhaul solution, the main assumptions are that pickup customers must be visited after all delivery customers, and all routes must contain at least one delivery customer (Toth and Vigo, 1997; Koç and Laporte, 2018; Queiroga et al., 2020; Londoño et al., 2021). In that case, we therefore have: 𝐴𝑐 ′ = ∅ and 𝐴𝑐 ′′ = ∅. Based on our notions above, we define the following decision variables: 102 1 if⁡link⁡(𝑖, 𝑗) ∈ 𝐴𝑙 ⁡is⁡selected, 𝑥𝑖𝑗 = { (4.1) 0 otherwise,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1 if⁡link⁡(𝑖, 𝑗) ∈ 𝐴𝑏 ⁡is⁡selected, 𝑦𝑖𝑗 = { (4.2) 0 otherwise,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1 if⁡link⁡(𝑖, 𝑗) ∈ 𝐴𝑐 ⁡is⁡selected, 𝑧𝑖𝑗 = { (4.3) 0 otherwise,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ′ 1 if⁡link⁡(𝑖, 𝑗) ∈ 𝐴𝑐 ′ ⁡is⁡selected, 𝑧𝑖𝑗 ={ (4.4) 0 otherwise,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ′′ 1 if⁡link⁡(𝑖, 𝑗) ∈ 𝐴𝑐 ′′ ⁡is⁡selected, 𝑧𝑖𝑗 ={ (4.5) 0 otherwise.⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ Furthermore, we define auxiliary continuous decision variables 𝑢𝑖𝑗 and 𝑤𝑖𝑗 , (𝑖, 𝑗) ∈ 𝐴 for the flow of delivery and pickup requests, respectively. Let 𝛼 be 1 (0) if we solve the VRPB problem with a backhaul (mixed) solution. Then, we formulate the VRPB as follows: 𝑀𝑖𝑛 𝑃 = ∑(𝑖,𝑗)∈𝐴𝑙 𝑐𝑖𝑗 𝑥𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑏 𝑐𝑖𝑗 𝑦𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑐 𝑐𝑖𝑗 𝑧𝑖𝑗 + (1 − (4.6) ′ ′′ 𝛼) ∑(𝑖,𝑗)∈𝐴𝑐′ 𝑐𝑖𝑗 𝑧𝑖𝑗 + (1 − 𝛼) ∑(𝑖,𝑗)∈𝐴𝑐′′ 𝑐𝑖𝑗 𝑧𝑖𝑗 s.t. ′′ ∑𝑗∈𝐿 𝑥0𝑗 + (1 − 𝛼) ∑𝑗∈𝐵 𝑧0𝑗 =𝐾 (4.7) ∑𝑖∈𝐿 𝑧𝑖0 + ∑𝑖∈𝐵 𝑦𝑖0 = 𝐾 (4.8) ∑𝑗∈𝐿 𝑥𝑖𝑗 + ∑𝑗∈𝐵0 𝑧𝑖𝑗 = 1 𝑖 ∈ 𝐿, (4.9) 𝑖≠𝑗 ′ ∑𝑖∈𝐿0 𝑥𝑖𝑗 + (1 − 𝛼) ∑𝑖∈𝐵 𝑧𝑖𝑗 =1 𝑗 ∈ 𝐿, (4.10) 𝑖≠𝑗 ′ ∑𝑗∈𝐵0 𝑦𝑖𝑗 + (1 − 𝛼) ∑𝑗∈𝐿 𝑧𝑖𝑗 =1 𝑖 ∈ 𝐵, (4.11) 𝑖≠𝑗 ′′ ∑𝑖∈𝐵 𝑦𝑖𝑗 + ∑𝑖∈𝐿 𝑧𝑖𝑗 + (1 − 𝛼) ∑𝑖∈{0} 𝑧𝑖𝑗 =1 𝑗 ∈ 𝐵, (4.12) 𝑖≠𝑗 103 ∑𝑗∈𝑉 𝑢𝑗𝑖 − ∑𝑗∈𝑉 𝑢𝑖𝑗 = 𝑑𝑖 𝑖 ∈ 𝑉\{0}, (4.13) 𝑖≠𝑗 𝑖≠𝑗 ∑𝑗∈𝑉 𝑤𝑖𝑗 − ∑𝑗∈𝑉 𝑤𝑗𝑖 = 𝑝𝑖 𝑖 ∈ 𝑉\{0}, (4.14) 𝑖≠𝑗 𝑖≠𝑗 𝑄𝑥𝑖𝑗 if⁡(𝑖, 𝑗) ∈ 𝐴𝑙 𝑄𝑦𝑖𝑗 if⁡(𝑖, 𝑗) ∈ 𝐴𝑏 𝑢𝑖𝑗 + 𝑤𝑖𝑗 ≤ 𝑄𝑧𝑖𝑗 if⁡(𝑖, 𝑗) ∈ 𝐴𝑐 (4.15) ′ (1 − 𝛼)𝑄𝑧𝑖𝑗 ⁡if⁡(𝑖, 𝑗) ∈ 𝐴𝑐 ′ ′′ {(1 − 𝛼)𝑄𝑧𝑖𝑗 ⁡if⁡(𝑖, 𝑗) ∈ 𝐴𝑐 ′′ ′ 𝑥𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑙 ; 𝑦𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑏 ; 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑐 ; 𝑧𝑖𝑗 ∈ {0,1}, (4.16) ′′ (𝑖, 𝑗) ∈ 𝐴𝑐 ′ ; 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑐 ′′ , 𝑢𝑖𝑗 ∈ ℝ; 𝑤𝑖𝑗 ∈ ℝ, (𝑖, 𝑗) ∈ 𝐴. (4.17) Objective function in (4.6) minimizes the total distance traveled, where the first-fifth terms correspond to the distance traveled from the depot or a delivery customer to another delivery customer, the distance traveled from a pickup customer to another pickup customer or to the depot, the distance traveled from a delivery to a pickup customer or to the depot, the distance traveled from a pickup to a delivery customer, and the distance traveled from the depot to a pickup customer, respectively. Of note, in the VRPBs with backhaul solution, the vehicle cannot travel from a pickup customer to a delivery customer nor from the depot to a pickup customer (Toth and Vigo, 1997; Koç and Laporte, 2018; Queiroga et al., 2020; Londoño et al., 2021). Therefore, if 𝛼 = 1, the fourth and fifth terms are removed from the objective function. Constraint (4.7) guarantees that 𝐾 vehicles leave the depot to serve delivery customers (when 𝛼 = 1), or delivery/pickup customers (when 𝛼 = 0). Constraint (4.8) ensures that 𝐾 vehicles return to the depot after serving the last customer (either delivery or pickup). Constraints (4.9)-(4.10) and (4.11)-(4.12) guarantee that each delivery and pickup customer is visited exactly once, 104 respectively. Constraints (4.13)-(4.15) are an extension of the connectivity and capacity constraints proposed by Mosheiov (1998), where constraints (4.13)-(4.14) guarantee the flow of delivery and pickup requests, respectively, and constraint (4.15) ensures that the total flow (i.e., total amount of loaded deliveries and pickups) at each link does not exceed the vehicle capacity. Constraints (4.16)-(4.17) define the domains of decision variables. Solution Methodology In problem 𝑃 in (6)-(14), having |𝑉| customers would result in |𝑉|2 − |𝑉| binary variables, 2(|𝑉|2 − |𝑉|) continuous variables, and |𝑉|2 + 3|𝑉| + 2 constraints. Therefore, solving this problem becomes computationally complex as the number of customers grows. To solve problem 𝑃 for large-scale transportation networks, we therefore develop a heuristic solution approach called CFRS. Clustering Phase Clustering is an unsupervised learning method that partitions data points (customers) into several clusters based on a similarity metric (e.g., a spatial measure like the Euclidian distance between customers’ location in a transportation network). As a result, a large-scale transportation network is broken down into small-scale networks that would become computationally tractable. Clustering methods have been applied in the VRP and its variants. Fisher and Jaikumar (1981) proposed a generalized AP for solving a capacitated location-allocation problem. Toth and Vigo (1999) integrated a clustering method with Lagrangian relaxation to solve the VRPB. Mahmoudi et al. (2019a) utilized a clustering method to solving a pickup and delivery problem with transfers. Wang et al. (2020) utilized a Gaussian mixture clustering algorithm for solving a green logistics location-routing problem. 105 In this research, we cluster customers into 𝐾 clusters using the K-means clustering algorithm (Jain et al., 1999). It starts with an initial set of cluster centers from our data points. Then, it assigns each datapoint into its closest cluster center and updates cluster centers by minimizing the sum of squared clustering error. The algorithm repeats these steps until a termination condition is met, e.g., there is no change in the updated cluster centers, there is no improvement in clustering error, or maximum number of iterations is reached. Of note, given the capacity of each vehicle, each cluster is served by exactly one vehicle. Based on this premise, the K-means clustering algorithm consists of four steps (we summarize this algorithm in Table 4.3): Step 0: we initiate 𝐾 cluster centers that are randomly chosen from the available datapoints (i.e., the spatial location of linehaul/backhaul customers). Let 𝒱 = {𝓋1 , . . . , 𝓋𝐾 } be the set of 𝐾 cluster centers, where 𝓋𝑘 ∈ 𝑉\{0}, 𝑘 = 1, . . . , 𝐾, is randomly initialized from the set of all customers. Step 1: Let 𝐶 = {𝐶1 , . . . , 𝐶𝐾 } be the set of clusters, where 𝐶𝑘 = {𝑖: 𝑖 ∈ 𝑉\{0}}, 𝑘 = 1, . . . , 𝐾, and these clusters are mutually exclusive, i.e., 𝐶1 ∩. . .∩ 𝐶𝐾 = ∅. We assign each customer 𝑖 ∈ 𝑉\{0} to cluster 𝐶𝑘 as follows: let 𝑄 𝑑 = {𝑄1𝑑 , . . . , 𝑄𝐾𝑑 } and 𝑄 𝑝 = {𝑄1𝑝 , . . . , 𝑄𝐾𝑝 } be the set of free capacity of vehicles for deliveries and pickups, respectively. The free capacity of vehicle 𝑘 is defined as the available amount of the capacity of vehicle 𝑘 for serving the next customer. Let 𝐶 ′ ⊆ 𝐶 be a subset of clusters such that, for 𝐶𝑘 ∈ 𝐶 ′ , 𝑄𝑘𝑑 ≥ 𝑑𝑖 and 𝑄𝑘𝑝 ≥ 𝑝𝑖 . We assign customer 𝑖 ∈ 𝑉\{0} to cluster 𝐶𝑘 , 𝐶𝑘 ∈ 𝐶 ′ , which has minimum transportation cost 𝑐𝑖𝓋𝑘 . Step 2: we update the center of each cluster based on the assigned datapoints in Step 1. To conduct this task, let 𝑒𝑖𝑘 = ∑𝑗∈𝐶𝑘 𝑐𝑖𝑗 be the sum of distances for traveling between customers 𝑖 and 𝑗, where 𝑗≠𝑖 𝑖, 𝑗 ∈ 𝐶𝑘 , 𝑖 ≠ 𝑗, and 𝐶𝑘 ∈ 𝐶, 𝑘 = 1, … , 𝐾. Then, customer 𝑖 with the least 𝑒𝑖𝑘 will be selected as the updated center of cluster 𝑘 and 𝓋𝑘 = 𝑖. 106 Step 3: we check the convergence of the algorithm. If it is not converged, we repeat Steps 1-2 for 𝑅 iterations. Furthermore, let 𝜏 be the maximum number of iterations where there is no improvement in obtained clusters and 𝑒 = ∑𝐶𝑘∈𝐶 ∑ 𝑖∈𝐶𝑘 𝑐𝑖𝓋𝑘 be the sum of distance traveled from 𝑖≠𝓋𝑘 each customer to his assigned cluster center. If 𝑒 is not lowered after 𝜏 iterations, the algorithm is then converged and the set of final clusters 𝐶 is returned. Table 4.3. Summary of the K-means clustering algorithm*. Step 0: initialization 1 𝑅; 𝑟: = 1; 𝜏; 𝑟 ′ : = 0; 𝑒𝑡𝑒𝑚𝑝 : = ∞; 𝒱: = {𝓋1 , . . . , 𝓋𝐾 } for 𝓋𝑘 ∈ 𝑉/{0}, 𝑘: = 1, … , 𝐾 Step 1: assign customers to their closest cluster center with enough delivery/pickup free capacity 2 for 𝑟 = 1 to 𝑅 3 for 𝓋𝑘 ∈ 𝒱 (𝑘: = 1, … , 𝐾) 𝑝 4 𝑄𝑘𝑑 : = 𝑄; 𝑄𝑘 ≔ 𝑄 5 for 𝑖 ∈ 𝑉\{0} 6 𝑒𝑡𝑒𝑚𝑝 : = ∞; 𝑘𝑡𝑒𝑚𝑝 : = 1 7 for 𝓋𝑘 ∈ 𝒱 (𝑘: = 1, … , 𝐾) 𝑝 8 if 𝑄𝑘𝑑 ≥ 𝑑𝑖 and 𝑄𝑘 ≥ 𝑝𝑖 and 𝑐𝑖𝓋𝑘 ≤ 𝑒𝑡𝑒𝑚𝑝 9 𝑒𝑡𝑒𝑚𝑝 : = 𝑐𝑖𝓋𝑘 ; 𝑘𝑡𝑒𝑚𝑝 : = 𝑘 10 append 𝑖 to 𝐶𝑘𝑡𝑒𝑚𝑝 𝑝 𝑝 11 𝑄𝑘𝑑𝑡𝑒𝑚𝑝 : = 𝑄𝑘𝑑𝑡𝑒𝑚𝑝 − 𝑑𝑖 ; 𝑄𝑘𝑡𝑒𝑚𝑝 : = 𝑄𝑘𝑡𝑒𝑚𝑝 − 𝑝𝑖 12 compute 𝑒 = ∑𝐶𝑘 ∈𝐶 ∑ 𝑖∈𝐶𝑘 𝑐𝑖𝓋𝑘 𝑖≠𝓋𝑘 Step 2: update the center of clusters 13 for 𝐶𝑘 ∈ 𝐶 (𝑘: = 1, … , 𝐾) 14 𝑒𝑡𝑒𝑚𝑝 : = ∞ 15 for 𝑖 ∈ 𝐶𝑘 16 𝑒𝑖𝑘 = ∑𝑗∈𝐶𝑘 𝑐𝑖𝑗 𝑗≠𝑖 17 if 𝑒𝑖𝑘 ≤ 𝑒𝑡𝑒𝑚𝑝 18 𝑒𝑡𝑒𝑚𝑝 : = 𝑒𝑖𝑘 ; 𝓋𝑘 : = 𝑖 19 update 𝒱 = {𝓋1 , . . . , 𝓋𝐾 } Step 3: check the convergence of the algorithm 20 𝑒𝑡𝑒𝑚𝑝 : = ∞ 21 if 𝑒 < 𝑒𝑡𝑒𝑚𝑝 22 𝑒𝑡𝑒𝑚𝑝 : = 𝑒; 𝑟 ′ : = 0; 𝐶𝑘𝑒𝑒𝑝 : = 𝐶 23 else if 𝑒 ≥ 𝑒𝑡𝑒𝑚𝑝 24 𝑟′: = 𝑟′ + 1 25 if 𝑟 ′ ≥ 𝜏 do 26 break 27 else if 𝑟 ′ < 𝜏 28 𝑟: = 𝑟 + 1 29 return 𝐶𝑘𝑒𝑒𝑝 * See Table 4.2 for the summary of notations. 107 Routing Phase For each cluster obtained in section 4.4.1, we modify the problem 𝑃 in (4.6)-(4.17). We then solve the modified problem using CPLEX optimization solver to determine the optimal routing itinerary for each vehicle. Let 𝐶𝑘0 = {0} ∪ 𝐶𝑘 and 𝐴𝑘 = {(𝑖, 𝑗):⁡𝑖, 𝑗 ∈ 𝐶𝑘0 , 𝑖 ≠ 𝑗} for 𝐶𝑘 ∈ 𝐶. Then, for each cluster 𝑘, the modified problem is formulated as: 𝑀𝑖𝑛 𝑃𝑘 = ∑(𝑖,𝑗)∈𝐴𝑘 ∩𝐴𝑙 𝑐𝑖𝑗 𝑥𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑘∩𝐴𝑏 𝑐𝑖𝑗 𝑦𝑖𝑗 + ∑(𝑖,𝑗)∈𝐴𝑘∩𝐴𝑐 𝑐𝑖𝑗 𝑧𝑖𝑗 + (1 − (4.18) ′ ′′ 𝛼) ∑(𝑖,𝑗)∈𝐴𝑘 ∩𝐴𝑐′ 𝑐𝑖𝑗 𝑧𝑖𝑗 + (1 − 𝛼) ∑(𝑖,𝑗)∈𝐴𝑘 ∩𝐴𝑐′′ 𝑐𝑖𝑗 𝑧𝑖𝑗 s.t. ′′ ∑𝑗∈𝐶𝑘∩𝐿 𝑥0𝑗 + (1 − 𝛼) ∑𝑗∈𝐶𝑘∩𝐵 𝑧0𝑗 =1 (4.19) ∑𝑖∈𝐶𝑘 ∩𝐿 𝑧𝑖0 + ∑𝑖∈𝐶𝑘 ∩𝐵 𝑦𝑖0 = 1 (4.20) ∑𝑗∈𝐶𝑘∩𝐿 𝑥𝑖𝑗 + ∑𝑗∈𝐶 0 ∩𝐵0 𝑧𝑖𝑗 = 1 𝑖 ∈ 𝐶𝑘 ∩ 𝐿, (4.21) 𝑘 𝑖≠𝑗 ′ ∑𝑖∈𝐶 0 ∩𝐿0 𝑥𝑖𝑗 + (1 − 𝛼) ∑𝑖∈𝐶𝑘∩𝐵 𝑧𝑖𝑗 =1 𝑘 𝑗 ∈ 𝐶𝑘 ∩ 𝐿, (4.22) 𝑖≠𝑗 ′ ∑𝑗∈𝐶 0 ∩𝐵0 𝑦𝑖𝑗 + (1 − 𝛼) ∑𝑗∈𝐶𝑘∩𝐿 𝑧𝑖𝑗 =1 𝑘 𝑖 ∈ 𝐶𝑘 ∩ 𝐵, (4.23) 𝑖≠𝑗 ′′ ∑𝑖∈𝐶𝑘 ∩𝐵 𝑦𝑖𝑗 + ∑𝑖∈𝐶𝑘 ∩𝐿 𝑧𝑖𝑗 + (1 − 𝛼) ∑𝑖∈{0} 𝑧𝑖𝑗 =1 𝑗 ∈ 𝐶𝑘 ∩ 𝐵, (4.24) 𝑖≠𝑗 ∑𝑗∈𝐶 0 𝑢𝑗𝑖 − ∑𝑗∈𝐶 0 𝑢𝑖𝑗 = 𝑑𝑖 𝑘 𝑘 𝑖 ∈ 𝐶𝑘 , (4.25) 𝑖≠𝑗 𝑖≠𝑗 ∑𝑗∈𝐶 0 𝑤𝑖𝑗 − ∑𝑗∈𝐶 0 𝑤𝑗𝑖 = 𝑝𝑖 𝑘 𝑘 𝑖 ∈ 𝐶𝑘 , (4.26) 𝑖≠𝑗 𝑖≠𝑗 108 𝑄𝑥𝑖𝑗 (𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑙 𝑄𝑦𝑖𝑗 (𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑏 𝑢𝑖𝑗 + 𝑤𝑖𝑗 ≤ 𝑄𝑧𝑖𝑗 ⁡(𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑐 (4.27) ′ (1 − 𝛼)𝑄𝑧𝑖𝑗 ⁡⁡(𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑐 ′ ′′ {(1 − 𝛼)𝑄𝑧𝑖𝑗 ⁡⁡(𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑐 ′′ 𝑥𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑙 ; 𝑦𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑏 ; 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ (4.28) ′ ′′ 𝐴𝑘 ∩ 𝐴𝑐 ; 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑐 ′ ; 𝑧𝑖𝑗 ∈ {0,1}, (𝑖, 𝑗) ∈ 𝐴𝑘 ∩ 𝐴𝑐 ′′ , 𝑢𝑖𝑗 ∈ ℝ; 𝑤𝑖𝑗 ∈ ℝ, (𝑖, 𝑗) ∈ 𝐴𝑘 . (4.29) Let 𝑃𝑘∗ be the optimal distance traveled for vehicle 𝑘⁡and obtained by solving problem 𝑃𝑘 in (4.18)-(4.29). Then, the total distance traveled can be computed as: 𝑃 ∗ = ∑𝐾𝑘=1 𝑃𝑘 ∗ (4.30) Computational Experiments We conduct our experiments on a computer running Intel (R) Core (TM) i7-4770T CPU processors clocked at 2.50 GHz with 7 cores and 8 GB RAM running Windows Server 10Ö64 Edition. The algorithms are coded in Python distribution platform, Anaconda 3, and run by Spyder integrated development environment, version 4.2.5. We use Python interface with CPLEX 20.1.0 optimization solver to generate optimal routes for each cluster, setting a time limit of 3,600 seconds. The solver returns the best-founded solution if it cannot reach the optimal solution within the predefined time limit. In the K-means clustering algorithm and for all experiments, we set 𝑅⁡ = ⁡100 and 𝜏⁡ = ⁡10. The Python code and datasets are available at GitHub (2022). 109 Problem Setting 4.5.1.1. Configuration: Networks. We implement each problem instance on two different real- world transportation networks that are representative of urban and rural areas. This will help us to shed more lights on the geographical accessibility/convenience that are deemed as one of crucial factors in launching reusable packaging systems (see, e.g., Closed Loop Partners, 2021). To this end, we consider Lansing and Charlotte networks both located in the state of Michigan, USA. These networks consist of 8,091 and 1,184 nodes, respectively, where each node is represented with an ID and unique latitude/longitude (see Figure 4.3). In our study, we assume that each node can represent a customer’s location, and all links are bidirectional. Moreover, to measure the distance, we calculate the shortest path and the corresponding distance traveled using the Dijkstra’s algorithm (Johnson, 1973). To this end, we measure the length of a given link (in miles) by the Harversian distance equation (Mahmoudi et al., 2019a). (a) Lansing network (b) Charlotte network Figure 4.3. Two real-world transportation networks. Notes. Lansing and Charlotte networks shown here have 8,091 and 1,184 nodes, and represent urban and rural geographical settings, respectively. In each network, larger black nodes are those used for our baseline experiments (section 4.5.3). These account for 800 and 120 nodes for Lansing and Charlotte areas, respectively (roughly 10% of the total nodes in each network). 110 4.5.1.2. Configuration: Customers, Requests, and Vehicles. For our baseline experiment, we consider about 10% of potential customers to have delivery and/or pickup requests. This will result in 800 and 120 requests for Lansing and Charlotte networks, respectively (see also Figure 4.3). We also account for the rate of simultaneous requests (i.e., when a customer has both delivery and pickup requests). To this end, we consider the rates {0%, 10%, . . ., 100%}. For example, in the case of Lansing, when the rate is 0%, we have 800 different customers each having either a delivery or a pickup request (but not both). However, when the rate is 100%, we have 800/2 = 400 customers each having both requests. Furthermore, we consider the capacity of each vehicle as 50 totes (see Appendix C). As illustrated in Figure 4.1, each tote is a single box that is filled with all reusable primary packages/containers (these containers are full (empty) for delivery (pickup)). Of note, under each parameters setting, we separately solve our proposed model for the backhaul solution (where deliveries must be served before pickups) and the mixed solution (where there is no priority in serving deliveries and pickups). In the following, we first compare our proposed framework with the benchmarks obtained from the CPLEX optimization solver (section 4.5.2). In section 4.5.3, we implement our framework on medium to large scale transportation networks. In section 4.5.5, we conduct various robustness checks on our input parameters to evaluate the validity of our results and findings against variations in these parameters. Comparison with the Benchmark Solution We compare the performance of our proposed CFRS approach with the exact solutions obtained from the CPLEX optimization solver. The performance measures are (1) the value of the objective function (total distance traveled) and (2) the computational time. Of note, to accommodate the exact solutions via the CPLEX, we focus on a transportation problem at a smaller 111 scale with 80 requests and vehicle’s capacity of 30. Also, for brevity, we select the rates of simultaneous delivery/pickup requests from {0%, 20%, . . ., 100%}. We notice from our results in Figure 4.4 (and those in Appendix A.1) that, although our CFRS approach yields total distance traveled that is on average 6.61% more than that obtained from the CPLEX, our approach enables us to solve the model considerably faster (i.e., on average, our approach would take 1.38 hours less to solve each problem set compared to the CPLEX). Overall, this show that our proposed method could have potentials when taking into consideration both the quality of solution and computational complexity of the transportation problem. (a) Solution: Backhaul (left: Lansing network, right: Charlotte network) (b) Solution: Mixed (left: Lansing network, right: Charlotte network) Figure 4.4. Comparison between our heuristic approach and the CPLEX optimization solver. Notes. For each solution method and network: # requests = 80 and vehicle’s capacity = 30. 112 Performance of the Proposed Solution Approach on Large-scale Transportation Networks Figure 4.5 summarizes the results. In each plot, we demonstrate the total distance traveled under different solutions (backhaul vs. mixed) and various rates of simultaneous delivery/pickup requests. We observe the following points from our results: (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 4.5. Distance traveled under different solutions and rates of simultaneous pickup/delivery requests. Notes. For each solution method and network: vehicle’s capacity = 50. The computational times are reported in Appendix B. 1. Compared to the mixed approach, following the backhaul solution (where full totes must be delivered before empty containers are picked up) results in a longer distance traveled by trucks. Although this would be expected, we also note that following a mixed approach requires handling of full/empty totes at the same time, which, in turn, can pose more challenges for truck operators (see, e.g., Toth and Vigo, 2002a). 2. Intuitively, one expects the total distance traveled to decrease as a larger proportion of customers happen to have both delivery and pickup requests. This is due to the fact that, under such circumstance, the number of unique customers (and hence the distance covered between them) would go down. Although we observe this pattern in our results, we also notice that the highest (lowest) distance traveled may not be attained under 0% (100%) 113 simultaneous demands. For example, for Lansing and Charlotte, the total distance traveled is the highest one when 10% and 40% of customers have simultaneous demands, respectively. Recall that, under 0% simultaneous demand, there are 800 and 120 unique customers in these networks, respectively. However, when 10% and 40% of customers have simultaneous demands in these networks, it results in 760 and 96 unique customers, respectively. 3. The total distance traveled is higher for urban (compared to rural) areas, or when the number of requests increase. In contrast, we also find that the distance per request would reduce when we serve an urban (relative to rural) area, or when the number of requests go up. This implies that launching reusable packaging system could bring economies of scale in more crowed areas (where the number of customers/requests is typically higher per, say, square miles) better. That said, such a finding should not discourage adopting platforms like reusable packaging in rural areas, especially since Federal agencies like the U.S. Department of Agriculture have urged for the expansion of circular economy platforms in such areas (see, e.g., National Association of Counties, 2022). Overall, our observations here could provide insights for reusable packaging companies that would like to target customers based on the interplay of their delivery/pickup demand status, their geographical accessibility, and the way the companies intend to serve these customers. Economic and Environmental Impacts Recall that the objective function in (4.6) returns the total distance traveled by vehicles that serve customers. In order to convert this distance to the total costs and emission amounts, we conduct a back-of-the-envelope simulation analysis. For emission, we consider carbon dioxide (CO2), which constitutes about 80% of greenhouse gases (EPA, 2022). We measure the emission 114 by multiplying the total distance traveled by the CO2 emission per mile traveled. We note that the average CO2 emission by a truck is reported to be 1,700 grams/mile (EDF, 2015). To account for potential variations in this number, we consider the Normal distribution 𝑁(1700, 200). Furthermore, we measure the cost by multiplying the total distance traveled by the transportation cost per mile. For the latter, ATRI (2022) reports the average marginal cost per mile for 2021 ($1.855). Although impacted by factors, such as driver’s wage, maintenance, and truck purchase payments, fuel cost remains the main element, which on average accounts for 27% of this cost (see ATRI (2022) for more details). We also account for the spike in the fuel price in 2022, as this hike has been on average 46.33% (eia.gov, 2022). Taking all the aforementioned factors into consideration, we project the average cost per mile for 2022 as 73% ∗ $1.855 + 27% ∗ $1.855 ∗ 1.4633 = $2.087. Similar to our notion for the emission, here we account for potential variations in this cost. In our baseline experiment, we consider the Normal distribution 𝑁(2, 0.2) (cost unit: $/mile). In our robustness checks, we will analyze other distributions for the emission and cost as well. Of note, to estimate the emission rate more precisely, more granular information would be required, such as travel-related factors like vehicle speed and vehicle acceleration (Lin and Niemeier, 2003; Samaras et al., 2018), highway network characteristics (e.g., road grade), vehicle- related factors like vehicle type like trucks, fuel/engine technologies, vehicle age, vehicle model, engine size, and vehicle mass (Shi et al., 2016), and other factors (see, e.g., Ligterink et al., 2012; Zhou et al., 2015; Mahmoudi et al., 2019b). However, addressing such granular information would be beyond the scope of the current research. Figure 4.6 shows the results. When we place no priority in fulfilling delivery/pickup requests in the Lansing network (with 800 requests), the average daily cost and CO2 emission are $769.86 and 654.19 kg, respectively. However, when we prioritize delivery over pickup requests, the 115 corresponding numbers go up to $1,042.19 and 892 kg, respectively. While this would be expected (as discussed in Figure 4.5, the total distance traveled under the backhaul solution is typically higher than that under the mixed solution, and more distance means more cost/emission), our results in Figure 4.6 also reveals the impact of geographical accessibility on cost and emission. In particular, although we have the similar pattern in the Charlotte network (with 120 requests), we observe that the jump in cost and emission in Charlotte occurs at a faster pace when prioritizing delivery over pickup customers; e.g., in Charlotte, this jump is about 51%, whereas it is about 35% in Lansing even if the number of requests is much higher in Lansing. This is due to the fact that, in rural areas, the distance between customers’ locations is typically higher than that in urban areas. As discussed before, granted that such findings might indicate the intrinsic impediments of rural areas in adopting reusable packages, policy makers could alleviate such effects by devising incentives that contribute to a better utilization of reusables. Robustness Checks The factors and parameters that are subject to change in our numerical experiments are: (1) geographical setting (urban vs. rural), (2) type of solution (backhaul vs. mixed), (3) rate of customers with simultaneous delivery/pickup requests, (4) vehicle’s capacity, (5) number of requests, and (6) distributions for cost/emission. While we address some of these factors in our baseline experiments (section 4.2-4), we continue to explore the sensitivity of our reported outcomes to variations in factors (4)-(6). Table 4.4 summarizes the parameters scenarios for the robustness checks. 116 (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 4.6. Simulation: daily transportation cost and emission amount under different solutions. Notes. For each solution method and network: vehicle’s capacity = 50. Numbers at the top of each histogram are the average (s.d.) of the outcome reported. Table 4.4. Summary of parameters for robustness checks. Parameter Values/distributions* Comparing solutions between our approach and the CPLEX optimization Vehicle’s capacity solver: {10, 30} Performance of our solution approach: {30, 50} Comparing solutions between our approach and the CPLEX optimization solver: {40, 80} # requests Performance of our solution approach: Lansing network: {800, 1600, 2400} Charlotte network: {120, 240, 360} CO2 emission (gram/mile) {𝑵(𝟏𝟕𝟎𝟎, 𝟐𝟎𝟎), 𝑈[1300, 2100], 𝑊(11, 1900)} Cost ($)** {𝑵(𝟐, 𝟎. 𝟐), 𝑈[1.5, 2.5]} * In each set, the value/distribution used for the baseline experiment is shown in bold. ** Transportation cost per mile. 117 Regarding the number of requests for the Lansing and Charlotte networks, recall that we consider 800 and 120 requests under the baseline experiment for these networks, respectively. This account for about 10% of all nodes in these networks (see Figure 4.3). In our robustness checks, we assume 20% and 30% of the representative population would participate in the reusable packaging platform, which account for 1,600 and 2,400 requests in Lansing and 240 and 360 requests in Charlotte, respectively. Furthermore, recall that we consider the Normal distribution 𝑁(1700, 200) (unit: grams of CO2/mile) in our baseline experiment. Here, we also consider the Uniform distribution 𝑈[1300, 2100] and the Weibull distribution 𝑊(11, 1900). The latter is considered to account for potential asymmetry/skewness in the emission distribution. In addition, regarding the distribution of transportation cost, while we consider the Normal distribution 𝑁(2, 0.2) (cost unit: $/mile) in the baseline experiment, here we consider the Uniform distribution 𝑈[1.5, 2.5]. Based on our results in Appendix A.1, we realize that our approach matches the solution returned by the CPLEX in numerous instances (this holds whether we alternate the number of requests or a vehicle’s capacity). Also, from our results in Appendix A.2, we notice that, when a larger proportion of population participate in the reusable packaging platform (this results in higher number of requests), the total distance traveled increases, while the distance per request served would drop. This is also in line with our findings in section 4.5.3. Furthermore, from our results in Appendix A.3, we find that, when the CO2 emission per mile traveled follows Normal or Uniform distributions, the total emission amounts are comparable. However, when we consider an asymmetric left-skewed distribution, like Weibull, the emission amount increase. In addition, the total transportation cost is comparable when we consider Uniform instead of Normal distributions. Of note, these observations hold for any geographical area or solution type. 118 Overall, our robustness checks show that our proposed framework is robust against variations in the input parameters. Conclusion We model the transportation operations of adopting reusable packages as a VRPB. While we develop a mixed integer programming model for this problem, we also devise an efficient heuristic solution approach to solve the problem for large-scale transportation networks. In particular, we implement our framework on two real-world networks that are representative of urban and rural areas. We also analyze our framework under further considerations; e.g., the possibility of serving delivery customers prior to pickup customers, or when a fraction of customers have simultaneous pickup and delivery requests. In our results, we report on various measures, such as the total distance traveled, the transportation cost, and CO2 emission. The current study sets a stepping stone towards a more comprehensive analysis of transportation operations of reusable packaging. To this end, we lay out three avenues for future research. (1) In this paper, we address a static problem, in that the pickup/delivery requests are known prior to trucks leaving the depot. The future research can adopt our proposed framework for a dynamic problem, where a part of pickup requests become known while trucks are on their routes. This feature would make the reusable packaging systems more adaptable towards customers’ needs. (2) We consider a single time period in our study. Future works can expand this to multiple periods (customers are served over a time horizon). That said, targeting a multi-period problem would pose additional challenges. To name a few, one can refer to the underlying computational complexity of methodologies that are suited for this purpose (e.g., dynamic programming), or existence of data that reveals information on customers’ consumption behaviors over time. (3) Regarding the last point, in the absence of data, future research can also focus on 119 modeling customers’ demand based on some distributions. This could also provide an opportunity for developing novel analytical models based on the combination of vehicle routing and newsvendor problems. 120 CHAPTER 5. TRANSPORTATION OPERATIONS OF REUSABLE PACKAGES IN REAL- TIME NETWORKS: AN APPLICATION OF DYNAMIC VEHICLE ROUTING PROBLEM WITH BACKHAULS Introduction Reusable packages are durable and long-lasting packages designed for more than one use-cycle in a closed-loop supply chain system. In such a system, transportation operations comprise of two tasks: (i) delivery of full packages from a depot to customers’ door and (ii) picking up empty ones from customers’ door and returning them to the depot for cleaning and reusing. Although the utilization of reusable packages has substantial environmental upgrades, the environmental and economic costs related to their transportation operations are still debatable in the quest of shifting from single-use to reusable packaging systems (see chapter 2). Therefore, it is crucial to manage the transportation operations of reusable packages as efficient as possible. Delivery of full reusable packages and picking up empty ones is a variation of pickup and delivery problems which is also known as one-to-many-to-one VRPs or VRPBs in the extant literature (see, e.g., Battarra et al., 2014; Koç and Laporte, 2018). In the VRPB, vehicles deliver some commodities from a depot to the door of a set of customers (named linehaul customers) and pick up some other commodities from the door of another set of customers (named backhaul customers) and return them back to the depot. When all input data such as customers’ request and location are known before transportation operations start, the VRPB is perceived as static (Psaraftis et al., 2016). The static VRPB aims to construct a set of optimal routing plans for a fleet of homogeneous vehicles that are all located at the depot and ready for serving offline/static customers/requests (i.e., customers who placed their requests before transportation operations start). However, in real-world transportation operations, with the increasing demand for quick 121 responses from service providers (Ninikas and Minis, 2014; Zhu et al., 2016), the system requires to operate in real-time where all or part of input data is revealed during transportation operations. When input data is evolved in real-time, the VRPB is perceived as dynamic (Psaraftis et al., 2016). Then, the aim of the DVRPB is to (i) collect the information related to online/dynamic customers/requests (i.e., customers who place their requests when transportation operations are running), (ii) decide on the acceptance/rejection of requests, and (iii) if acceptance is the case, update/reoptimize the routing plan. Managing transportation operations of reusable packages in dynamic/real-time networks is a complex task due to the high volume and frequency of requests. Furthermore, since a reusable package is manufactured from durable high-quality materials, it is accounted as an asset for the corresponding product/packaging company (depending on who owns the package) and is costly to be replaced. Therefore, it is crucial to collect empties from customers’ door quickly to reduce the chance of damage, loss, or theft. In addition, due to the limited number of such packages in the system and the potential of inventory shortage for them, it is crucial that they are quickly returned to the system for cleaning, refilling, and reusing. This research aims to develop an analytical framework for transportation operations of reusable packages in dynamic networks that contributes to streamline such complex operations in closed-loop supply chain systems. In the DVRPB, before transportation operations start, there exists an initial routing plan that assigns offline customers to the available vehicles located at the depot. In this research, offline customers consist of linehaul and offline backhaul customers. To construct the initial routing plan, we hold the routing assumptions of the VRPB with single demand and backhaul solution (i.e., standard VRPB depicted in Figure 1.2 (a)). Then, we adopt the CFRS solution approach presented in chapter 4 to construct the initial routing plan. In the CFRS solution approach, we initially 122 partition the set of linehaul and offline backhaul customers into several clusters using the K-mean clustering algorithm such that each cluster is associated with the routing of exactly one vehicle (of note, the amount of customers’ request and vehicles’ capacity are incorporated in the k-mean clustering algorithm). Then, we present the MIP model and solve it by using CPLEX optimization solver to construct the optimal routing plan for each cluster/vehicle. Finally, we integrate the optimal routing plans obtained for all vehicles to form the initial routing plan. When transportation operations start, vehicles leave the depot to serve their assigned customers as instructed in the initial routing plan. As the initial routing plan unfolds, we name it the existing routing plan which is constantly updated based on the revealed information of online customers. In this research, online customers can only be backhauls (named online backhaul customers). The reason behind this assumption is that to serve an online linehaul customer, the vehicle must return to the depot in the middle of the working shift to load the full packages which is not aligned with the basic assumption of the VRPB (i.e., each vehicle must start and end its route at the depot). Although the acceptance of all online requests for the same-day service is not practical given the limited vehicles’ capacity, limited working shifts, etc., the ultimate goal is to accept as many online requests as possible. That is why a real-time decision-making approach is introduced to handle dynamic systems, effectively. We propose a local search heuristic algorithm for the DVRPB that decides on the acceptance/rejection of online requests and updates the existing routing plan, simultaneously. To this end, We present an analytical framework for the DVRPB by taking the following steps: 1. We test our proposed analytical framework on two real-world transportation networks which are geographically located in the state of Michigan, USA. The first network is the Lansing network which represents an urban area, and the second one is the Charlotte 123 network which is representative of a rural area. Focusing on these networks as representatives of urban and rural areas enables us to investigate the dynamics of transportation operations of circular reuse platforms in different geographical settings. For both networks, in our baseline experiments, we assume that the total number of customers is roughly equivalent to 10% of the size of the network (i.e., the total number of transportation nodes). We extend our experiments to cases where the total number of customers is about 20% and 30% of the size of the network. In addition, we investigate the efficiency of real-time operations for two different vehicles’ capacity in both networks. 2. We investigate the degree of dynamism (DOD) on the efficiency and responsiveness of our model. The DOD refers to the ratio/percentage of the number of online customers to the total number of customers (Larsen et al., 2002). We change the value of DOD from 0% (the case that all backhaul customers are offline, or equivalently, pickup operations are all static) to 100% (the case that all backhaul customers are online, or equivalently, pickup operations are all dynamic) with 10% step size. 3. We investigate the effect of real-time decision-making policies on the service quality in real-time. To conduct this task, we consider: (i) Time frames at which online customers can place their requests for the same-day service. It is a common practice among service providers to define specific policies/time frames during which online customers have the chance to receive the same-day service. For example, Amazon provides a cutoff time for its prime members when placing their requests in the shopping cart, saying that if they complete their order by this time, they will be eligible for the same-day service (Amazon, 2022). It is not unrealistic to assume that if an online customer places his order before noon (the middle of the working shift), 124 he has higher chance to receive the same-day service compared to the one that places his order in the afternoon. In the baseline experiments, there is no restriction on the time at which customers place their orders, while in our other experiments, we investigate the cases in which customers are constrained to place their orders in specific times of the day (i.e., before noon). (ii) Time stamps at which the existing routing plan are updated. The existing routing plan can be updated either at the emergence of an online request or at fixed time intervals (Ninikas and Minis, 2014). In the former option, the DVRPB will be constantly run throughout the working shift. In the latter one, the working shift is partitioned into several decision epochs. At the end of each decision epoch (we name it as a decision time stamp), the information about online customers who placed their requests within that epoch is collected, the decision on acceptance/rejection of the request is made, and the existing routing plan is updated. In our baseline experiments, we assume the decision time stamp be in the middle of the working shift. We also investigate other cases where decisions are made minute-by-minute and hour-by-hour. 4. The objective function of our proposed model is the total distance traveled by all vehicles comprising of four components: (i) the distance traveled for serving offline customers, (ii) the distance traveled for serving online backhaul customers that are in the accepted list, (iii) the distance that should have been traveled for serving online backhaul customers that are in the waiting list (penalty type I), and (iv) the distance that should have been traveled for serving online backhaul customers that are in the rejected list (penalty type II). The accepted (rejected) list is the list of online backhaul customers that are decided to receive 125 (not receive) the same-day service. The waiting list is the list of online backhaul customers whose requests have not been decided yet to receive the same-day service. Summary of main findings - As the DOD increases the total distance traveled increases in both Lansing and Charlotte networks. In particular, the total distance traveled (in miles) for Lansing (Charlotte) network (i) is 522.91 (155.93) when pickup operations are completely static (0% DOD), (ii) increases with a ratio of 243.88 (44.94) miles as pickup operations incrementally shift to dynamic with a step size of 10%, and (iii) is 802.01 (193.02) when pickup operations are completely dynamic (100% DOD). - We realize that the total distance traveled increases as the network gets more crowded .This is mainly related to penalty types I and II. In addition, the distance per request and rate of accepted online requests decrease as the network becomes more crowded. Therefore, there is a trade-off between normalized distance traveled (i.e., distance per request) and the service quality ( the ratio of accepted customers to the total number of online customers). - We notice that varying vehicles’ capacity have not significantly changed the total distance traveled. We perceive that implementing less-than-truckload might be more effective in order to save space in vehicles for potential online customers. - We notice that limiting the time frame for receiving online requests up to the middle of the working shift (of note, here, the decision time stamp is set at the middle of the working shift) decreases the penalty type I that is caused by the number of online requests in the waiting list (i.e., number of online requests that no decision is made for the same-day service). The reason is that, in this scenario, the information of all online requests is collected at the decision time stamp, hence, the acceptance/rejection decision will be made 126 for these requests. Therefore, if these requests be rejected, it will be due to the limited logistical resources such as vehicles availability or duration of the working shift. In addition, when varying the decision time stamps to every minute and hour (of note, here, the time frame is set throughout the working shift) decreases penalty type I for the same reason. Adding more vehicles, dispatching vehicles from the depot with less than their actual capacity, or adjusting waiting strategies on specific locations/customers’ site could increase the chance of serving more online requests. The remainder of this chapter is organized as follows. In section 5.2, we review the extant literature on the DVRPB and present the problem statement investigated in this research. Section 5.3 presents our solution approach to construct the initial routing plan. Then, section 5.4 presents our proposed solution framework for the DVRPB. Computational experiments are presented in section 5.5. Discussion, concluding remarks, and directions for future research are presented in section 5.6. Review of literature and problem statement To develop an analytical framework for the DVRPB, we observe two directions that are required to be addressed. The first one is related to the routing assumptions that will be required for constructing the initial routing plan and/or updating the existing routing plan. The second one corresponds to the solution approach for the DVRPB. In this section, we review the extant literature on the DVRPB based on these forgoing directions and later on we present the problem statement. Extant literature on the DVRPB We recall from chapter 1 that, in the VRPB, each vehicle starts and ends its route at the depot, each linehaul/backhaul customer is visited exactly once and exactly by one vehicle, the total 127 requests of linehaul/backhaul customers to be served by a vehicle must not exceed the vehicles’ capacity, and all routes must contain at least one linehaul customer (Battarra et al., 2014). In addition to the foregoing characteristics, there are two other assumptions whose combination forms four classes of the VRPB (depicted in Figure 1.2): the type of service and backhauling strategy (Parragh et al., 2008a; Battarra et al., 2014). The VRPB can be classified with respect to the type of service as the VRPB with single demand and the VRPB with simultaneous demand. While in the former one, each customer requires either a pickup or a delivery service (not both), in the latter one, customers may request simultaneous service of delivery and pickup. Regarding backhauling strategy (whether the sequence of visiting linehaul and backhaul customers is constrained or not), the VRPB is classified into the VRPB with backhaul solution and the VRPB with mixed solution. Among the scant literature on the DVRPB, the routing assumptions proposed by Chang et al. (2003) is the VRPB with simultaneous demand and mixed solution. Studies conducted by Haghani and Jung (2005), Ninikas and Minis (2014), and Zhu et al. (2016) addressed the VRPB with mixed solution. In this research, we hold the routing assumptions of the VRPB with single demand and backhaul solution which is also referred to as the standard VRPB in the extant literature (see, e.g., Toth and Vigo, 1997; Mingozzi et al., 1999; Koç and Laporte, 2018). In the standard VRPB, in addition to the assumptions mentioned above, it is assumed that (i) each customer has a single request (delivery or pickup, but not both), and (ii) linehaul customers must be served before backhauls. Regarding the first assumption, we note that the model proposed in this study is capable to be extended to the case of the VRPB with simultaneous demand. Regarding the second assumption, we note that, in the context of reusable packages, this assumption is quite practical since the essence of the problem provides a priority to the customers with delivery requests, prevents any extra rework by logistics staff resulting from mixing full and empty packages in the 128 vehicle during the loading/unloading operations, and prevents placing dirty packages beside clean ones (Battarra et al, 2014). For more information about the extant mathematical models and the exact solution approaches for the standard VRPB, one can refer to Toth and Vigo (1997), Mingozzi et al. (1999), Queiroga et al. (2020), and the review paper conducted by Koç and Laporte (2018). To the best of our knowledge, routing assumptions considered by Zeng (2006) and Wang and Cao (2008) for the DVRPB are quite close to the one investigated in this research. In particular, Zeng (2006) studied the DVRPB with time windows and backhaul solution for the city pickup-delivery problem in the less-than-truckload couriers. The less-than-truckload courier is a specific operations system in the freight transportation industry in which a shared space of trailer is used for the transportation of small freights. It may also be used when freights do not require the use of an entire trailer. Wang and Cao (2008) addressed the DVRPB with time windows and backhaul solution. The authors identified what changes in customers’ request disrupt the initial routing plan and proposed a disruption recovery model based on a local search algorithm. Ninikas and Minis (2014) observed three directions of solution approaches for the dynamic counterpart of VRPs in the extant literature. The first solution approach employs policy-based techniques integrated with local update procedures in order to incorporate online requests into the existing routing plan. In this approach, when a new online request becomes available, it is initially stored in a dynamic list. Then, the online request is inserted into the existing routing plan. Finally, route improvement techniques such as 2-opt exchange (see Croes, 1958) and 3-opt exchange (see Lin, 1965) are employed to reoptimize the routing plan and update the list. For more information on the application of this solution approach for the general case of dynamic VRP (DVRP), one can refer to Shieh and May (1998), Larsen et al. (2002), and Mitrovi´c-Mini´c et al. (2004). 129 The second solution approach utilizes exact, heuristics, or metaheuristics algorithms to solve the static version of the problem considering all up-to-date input data. In this approach, an algorithm is initially designed to solve the static version of the problem. Then, at each decision epoch, the proposed algorithm updates corresponding input data and solves the entire problem (e.g., Gendreau et al. (1999) and Ichoua et al. (2000) proposed a Tabu search heuristic, Montenammi et al. (2005) presented an ant colony system, Chen and Xu (2006) proposed a column generation-based approach, Hong (2012) and proposed a large neighborhood search algorithm to solve the DVRP by dividing the overall planning horizon into decision epochs and solving the static version of the problem incorporating the up-to-date input data). The third solution approach exploits the dynamic nature of routing problems. Waiting-relocation strategy is an example that falls into this category (Psaraftis et al., 2016). Waiting-relocation strategy considers the possibility of positioning vehicles at strategic locations or at customer locations to wait for the arrival of potential online requests (see, e.g., Mitrovi´c-Mini´c et al.; 2004; Mitrovi´c-Mini´c and Laporte, 2004). To the best of our knowledge, studies conducted by Zeng (2006) and Wang and Cao (2008) incorporate the first solution approach, and studies conducted by Chang et al. (2003), Haghani and Jung (2005), Ninikas and Minis (2014), and Zhu et al. (2016) employ the second solution approach for the DVRPB. In this research, we construct the initial routing plan by adopting the CFRS solution approach presented in chapter 4. When transportation operations start, the initial routing plan turns into an existing routing plan. Then, we decompose the working shift into several decision epochs such that at the end of each epoch (at each decision time stamp), the information of online customers who placed their requests within that epoch is collected, a real-time decision for the acceptance/rejection of their requests is taken with respect to the limited logistical resources 130 (e.g., there must be a vehicle with enough available capacity that is not yet returned to the depot), and if acceptance is the case, the existing routing plan is updated for serving these customers using a local search heuristic algorithm. Table 5.1 summarizes the extant literature on the DVRPB. Table 5.1. Summary of the extant literature on the DVRPB. Dynamic Reference DVRPB with Solution approach Application elements Chang et al. Simultaneous demand Customers’ Tabu search - (2003) and mixed solution requests Customers’ Haghani and Mixed solution requests, Genetic algorithm - Jung (2005) Traveling time Parallel insertion heuristics Customers’ Less-than- Zeng (2006) Backhaul solution with reactive Tabu search requests truckload tour improvement Wang and Cao Customers’ Demand disruption Backhaul solution Local search heuristics (2008) requests recovery Ninikas and Customers’ Column generation-based Mixed solution - Minis (2014) requests heuristics Multi-objective evolutionary Zhu et al. Customers’ algorithm and locality- Mixed solution - (2016) requests sensitive hashing based local search Single demand and Customers’ CFRS and local search This paper Reusable packages backhaul solution requests heuristics Problem statement Table 5.2 presents a complete description of notations used in this chapter. Let 𝐿 be the set of linehaul customers and 𝐵 be the set of backhaul customers. As mentioned before, in the DVRPB, we are dealing with two types of backhaul customers: offline and online. Let 𝐵 = 𝐵 𝑜𝑓𝑓 ∪ 𝐵 𝑜𝑛 , where 𝐵 𝑜𝑓𝑓 and 𝐵 𝑜𝑛 denote the set of offline and online backhaul customers, respectively. Then, we define the DVRPB on a directed graph 𝐺 = (𝑉, 𝐴), where 𝑉 = {0} ∪ 𝐿 ∪ 𝐵 denotes the set of all nodes including the depot (denoted as {0}), and 𝐴 = {(𝑖, 𝑗): 𝑖, 𝑗 ∈ 𝑉, 𝑖 ≠ 𝑗} represents the set of all links connecting node 𝑖 to 𝑗. We define 𝑑𝑖 and 𝑝𝑖 be non-negative amount of delivery and pickup requests associated with customer 𝑖 ∈ 𝑉, where 𝑑𝑖 = 0, 𝑖 ∈ 𝑉\𝐿 and 𝑝𝑖 = 0, 𝑖 ∈ 𝑉\𝐵. We 131 also define non-negative traveling time 𝑡𝑖𝑗 associated with link (𝑖, 𝑗) ∈ 𝐴. Then, we observe two horizons for the DVRPB: before and during transportation operations. Before transportation operations start, we construct an initial routing plan for 𝐾 identical vehicles with capacity 𝑄 all located at the depot, and ready to serve offline customers (i.e., 𝑖 ∈ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 ) such that (i) all delivery requests are transported from depot to linehaul customers, (ii) all pickup requests are collected from backhaul customers and transported back to the depot, (iii) each vehicle starts and ends its route at the depot, (iv) each linehaul/backhaul customer is visited exactly once by exactly one vehicle, (v) the total amount of delivery/pickup requests served by a vehicle does not exceed the vehicle’s capacity, (vi) each vehicle must serve at least one linehaul customer, and (vii) linehaul customers must be served before any backhauls. From assumptions (i) to (vii), it is clear that offline and online backhaul customers are treated the same as these assumptions will be held when the constructed routing plan is updated. In the CFRS solution approach, we initially partition the set of offline customers 𝐿 ∪ 𝐵 𝑜𝑓𝑓 into 𝐾 clusters using K-mean clustering algorithm. We incorporate 𝑑𝑖 , 𝑝𝑖 , and 𝑄 into the K-mean clustering algorithm. That is, each cluster 𝑘, 𝑘 = 1. . . , 𝐾 determines a sub-set of offline customers that will be served exactly by one vehicle (i.e., each cluster corresponds to the routing of exactly one vehicle). After that, we solve an MIP model using CPLEX optimization solver to obtain the optimal routing plan for vehicle 𝑘, 𝑘 = 1. . . 𝐾. When transportation operations start, vehicles leave the depot as instructed in their routing plan. Let 𝑇 represents a working shift associated with a typical day of service. Throughout 𝑇, online backhaul customers 𝐵 𝑜𝑛 place their request in the real-time. To fulfill these requests, we partition 𝑇 into several decision epochs such that at the end of each decision epoch (decision time stamp), we collect the information of all up-to-date online backhaul customers. Then, based on the 132 current location of vehicles and their corresponding free capacity, online backhaul customers are assigned to vehicle 𝑘 such that (1) this assignment results in the least detour deviation, and (2) the total traveling time does not exceed 𝑇. To hold assumptions (i)-(vii), the online backhaul customer that is assigned to vehicle 𝑘 can only be inserted after the last linehaul customer or between any offline or already assigned online backhaul customers that have not been served yet at the corresponding decision time stamp. We also note that if an online backhaul customer is assigned to vehicle 𝑘, then, this assignment is fixed, which means that his request cannot be later rejected nor assigned to another vehicle. Table 5.2. Summary of notations used in chapter 5. Notations used in section 5.2 𝐺 Directed graph representing the transportation network, 𝐺 = (𝑉, 𝐴) 𝑉 Set of all nodes (customers and the depot), 𝑉 = {0} ∪ 𝐿 ∪ 𝐵 (𝑖, 𝑗: indices for nodes) 𝐴 Set of all links, 𝐴 = {(𝑖, 𝑗): 𝑖, 𝑗 ∈ 𝑉, 𝑖 ≠ 𝑗} 0 The depot 𝐿 Set of linehaul customers, 𝐿0 = {0} ∪ 𝐿 𝐵 Set of backhaul customers including both offline and online requests 𝐵 = 𝐵𝑜𝑓𝑓 ∪ 𝐵𝑜𝑛 𝐵 𝑜𝑓𝑓 Set of offline backhaul customers, 𝐵0𝑆 = {0} ∪ 𝐵 𝑆 𝑜𝑛 𝐵 Set of online backhaul customers 𝑡𝑖𝑗 Non-negative traveling time corresponding to link (𝑖, 𝑗) ∈ 𝐴 Non-negative amount of delivery request corresponding to customer 𝑖 ∈ 𝑉 where 𝑑𝑖 = 0⁡for 𝑖 ∈ 𝑑𝑖 𝑉\𝐿 Non-negative amount of pickup request corresponding to customer 𝑖 ∈ 𝑉 where 𝑝𝑖 = 0⁡for 𝑖 ∈ 𝑝𝑖 𝑉\𝐵 𝐾 Number of vehicles (𝑘: index for vehicles, clusters, or routes, 𝑘 = 1, . . . , 𝐾) 𝑄 Vehicle’s capacity 𝑇 Working shift corresponding to the time duration of one day of service Notations used in section 5.3.1 𝑅 Maximum number of iterations for the K-mean clustering algorithm Maximum number of iterations without improvement in solution for the K-mean clustering 𝜏 algorithm 𝒮 Set of cluster centers, 𝒮 = {𝓈1 , . . . , 𝓈𝐾 }, 𝓈𝑘 ∈ 𝐿 ∪ 𝐵𝑜𝑓𝑓 , 𝑘 = 1, . . . , 𝐾 𝐶𝑘 Set of customers assigned to cluster 𝑘, 𝐶𝑘 ⊆ 𝐿 ∪ 𝐵𝑜𝑓𝑓 , 𝐶1 ∩. . .∩ 𝐶𝐾 = ∅, 𝑘 = 1, . . . , 𝐾 𝑑 𝑄𝑘 Free delivery capacity for cluster 𝑘, 𝑘 = 1, . . . , 𝐾 𝑝 𝑄𝑘 Free pickup capacity for cluster 𝑘, 𝑘 = 1, . . . , 𝐾 𝑒𝑘 Sum of traveling time from customer 𝑖 to all customers in 𝐶𝑘 , 𝑒𝑖𝑘 = ∑𝑗∈𝐶𝑘 𝑡𝑖𝑗 , 𝑖 ∈ 𝐶𝑘 𝑖 𝑗≠𝑖 𝑒 Sum of traveling time from each customer to his assigned cluster center, 𝑒 = ∑𝐾 𝑘=1 ∑𝑖∈𝐶𝑘 𝑡𝑖𝓈𝑘 𝑖≠𝓈𝑘 Notations used in section 5.3.2 𝑜𝑓𝑓 𝑇𝑇𝑘 The MIP model to construct the routing plan for cluster 𝑘, 𝑘 = 1, . . . , 𝐾, see Eq. (5.1)-(5.11). 133 Table 5.2 (cont’d) 𝐶𝑘0 Set of customers assigned to cluster 𝑘 and the depot, 𝐶𝑘0 = {0} ∪ 𝐶𝑘 , 𝑘 = 1, . . . , 𝐾 𝐴𝑘 Set of all feasible links for cluster 𝑘, 𝐴𝑘 = 𝐴′𝑘 ∪ 𝐴′′𝑘 ∪ 𝐴′′′ 𝑘 , 𝑘 = 1, . . . , 𝐾 ′ Set of links connecting the depot or linehaul customers in cluster 𝑘 to other linehaul customers in 𝐴𝑘 cluster 𝑘, 𝐴′𝑘 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐶𝑘0 ∩ 𝐿0 , 𝑗 ∈ 𝐶𝑘 ∩ 𝐿, 𝑖 ≠ 𝑗}, 𝑘 = 1, . . . , 𝐾 Set of links connecting linehaul customers in cluster 𝑘 to the depot or to offline backhaul 𝐴′′𝑘 𝑜𝑓𝑓 customers in cluster 𝑘, 𝐴′′𝑘 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐶𝑘 ∩ 𝐿, 𝑗 ∈ 𝐶𝑘0 ∩ 𝐵0 , 𝑖 ≠ 𝑗}, 𝑘 = 1, . . . , 𝐾 Set of links that connects offline backhaul customers in cluster 𝑘 to the depot or to other offline 𝐴′′′ 𝑘 𝑜𝑓𝑓 backhaul customers in cluster 𝑘, 𝐴′′′ 𝑘 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐶𝑘 ∩ 𝐵 𝑜𝑓𝑓 , 𝑗 ∈ 𝐶𝑘0 ∩ 𝐵0 , 𝑖 ≠ 𝑗}, 𝑘 = 1, . . . , 𝐾 𝑥𝑖𝑗 Decision variable equals 1 if link (𝑖, 𝑗),⁡(𝑖, 𝑗) ∈ 𝐴𝑘 , is selected, and 0 otherwise 𝑢𝑖𝑗 Auxiliary continuous decision variable for the flow of delivery requests, (𝑖, 𝑗) ∈ 𝐴𝑘 𝑤𝑖𝑗 Auxiliary continuous decision variable for the flow of pickup requests, (𝑖, 𝑗) ∈ 𝐴𝑘 𝑜𝑓𝑓∗ 𝑜𝑓𝑓 𝑇𝑇𝑘 Optimal traveling time for cluster/vehicle 𝑘 and obtained by solving problem 𝑇𝑇𝑘 , 𝑘 = 1, . . . , 𝐾 𝑜𝑓𝑓∗ 𝑇𝑇 𝑜𝑓𝑓 Total traveling time for serving all offline customers, 𝑇𝑇 𝑜𝑓𝑓 = ∑𝐾 𝑘=1 𝑇𝑇𝑘 Notations used in section 5.4 𝑘 Vector representative of the spatial itinerary of vehicle 𝑘, 𝒳 𝑘 = (𝓍1𝑘 , . . . , 𝓍𝑗𝑘 , 𝓍𝑗+1 𝑘 , . . . , 𝓍𝑑𝑖𝑚(𝒳 𝑘 ) ), 𝒳𝑘 𝑘 𝑘 𝑘 1 𝐾 𝓍1 = 0, 𝓍𝑑𝑖𝑚(𝒳 𝑘) = 0, 𝓍𝑗 ∈ 𝑉\{0}, 𝒳 = {𝒳 , . . . , 𝒳 } 𝐿𝑘𝑡𝑎𝑖𝑙 Last linehaul customer to be served by vehicle 𝑘, 𝐿𝑡𝑎𝑖𝑙 = {𝐿1𝑡𝑎𝑖𝑙 , . . . , 𝐿𝐾𝑡𝑎𝑖𝑙 }. 𝑘 Vector representative of the temporal itinerary of vehicle 𝑘, 𝒯 𝑘 = (𝓉1𝑘 , . . . , 𝓉𝑗𝑘 , 𝓉𝑗+1 𝑘 , . . . , 𝓉𝑑𝑖𝑚(𝒯 𝑘 ) ), 𝒯𝑘 𝒯 = {𝒯 1 , . . . , 𝒯 𝐾 } 𝑇𝑇 𝑘 Total traveling time of vehicle 𝑘, 𝑇𝑇 = {𝑇𝑇 1 , . . . , 𝑇𝑇 𝐾 } 𝑘 1 𝐾 𝑄𝑓𝑟 Free/available capacity of vehicle 𝑘, 𝑄𝑓𝑟 = {𝑄𝑓𝑟 , . . . , 𝑄𝑓𝑟 } 𝑜𝑛 Service status of online backhaul customer 𝑖 ∈ 𝐵 , 𝑠𝑖 ∈ {0,1,2}; the meaning of 0, 1, and 2 are 𝑠𝑖 explained later in this table. 𝑜𝑖 Time stamp at which the order of online backhaul customer 𝑖 ∈ 𝐵𝑜𝑛 is received ℓ Length of decision epochs 𝑇𝑙 Decision time stamp 𝑙, 𝑇0 = 0, 𝑇𝑙 < 𝑇 (𝑙: index for decision epoch, 𝑙 > 1) Online backhaul customers who placed their requests in decision epoch 𝑙, 𝐵𝑙𝑜𝑛 = {𝑖: 𝑖 ∈ 𝐵𝑙𝑜𝑛 𝐵𝑜𝑛 , 𝑇𝑙−1 ≤ 𝑜𝑖 < 𝑇𝑙 } Detour deviation resulted from assigning online backhaul customer 𝑖 ∈ 𝐵𝑙𝐷 ⁡between two ∆𝑘𝑖 consecutive customers: (𝓍𝑗𝑘 , 𝓍𝑗+1𝑘 ), 𝑗 ∈ {1, 2, … , 𝑑𝑖𝑚(𝒳 𝑘 ) − 1}, 𝓍𝑗𝑘 ∈ {𝐿𝑘𝑡𝑎𝑖𝑙 } ∪ 𝐵𝑜𝑓𝑓 ∪ 𝐵𝑜𝑛 , 𝓉𝑗𝑘 ≥ 𝑇𝑙 , ∆𝑘𝑖 : = 𝑡𝓍 𝑘𝑖 + 𝑡𝑖𝓍 𝑘 𝑗 𝑗+1 ∆𝑘 Detour deviation resulted from assigning online backhaul customer 𝑖 ∈ 𝐵𝑙𝐷 to vehicle 𝑘 𝑇𝑇 𝑜𝑛 Total time traveled for serving online backhaul customers Set of online backhaul customers whose requests have not been decided for the same-day service 𝑁𝐼 yet, 𝑁 𝐼 = {𝑖: 𝑖 ∈ 𝐵𝑜𝑛 , 𝑠𝑖 = 0} Set of online backhaul customers whose requests have been accepted for the same-day service, 𝑁 𝑁 = {𝑖: 𝑖 ∈ 𝐵𝑜𝑛 , 𝑠𝑖 = 1} Set of online backhaul customers whose requests have been rejected for the same-day service, 𝑁 𝐼𝐼 𝑁 𝐼𝐼 = {𝑖: 𝑖 ∈ 𝐵𝑜𝑛 , 𝑠𝑖 = 2} 𝑃𝐼 Penalty type I for customers of 𝑁 𝐼 (𝛼: the corresponding penalty multiplier) 𝑃𝐼𝐼 Penalty type II for customers of 𝑁 𝐼 (𝛽: the corresponding penalty multiplier) Total travel time that is traversed and/or must be traversed for serving all offline and online 𝑇𝑇𝑇 customers 𝑖 ∈ 𝑉\{0} 134 Constructing the initial routing plan In this section, we present our CFRS solution approach for constructing the initial routing plan. The CFRS solution approach comprises of two phases: (1) clustering phase in which offline customers 𝐿 ∪ 𝐵 𝑜𝑓𝑓 are clustered into 𝐾 clusters using K-mean clustering algorithm, and (2) routing phase in which the optimal routing plan for each cluster/vehicle is constructed. In the following, we explain both phases in detail. Clustering phase The VRP and its variants are NP-hard problems which cannot be solved in a short time and with exact algorithms as the size of these problems increases. That is, heuristics algorithms such as clustering are widely used to reduce the computational burden of such problems. Clustering is a partitioning technique (Jain et al., 1999) that is used to partition/break down these complex problems into several small-scale sub-problems. For more information on the utilization of clustering methods in solving VRP and its variants, one can refer to Fisher and Jaikumar (1981), Cakir et al. (2015), Toth and Vigo (1999), Mahmoudi et al. (2019a), and Wang et al. (2020). Since the VRPB is a variant of the VRP, it is a NP-hard problem. Therefore, to deal with its computational complexity, we propose a K-mean clustering algorithm that partitions the set of offline customers 𝐿 ∪ 𝐵 𝑜𝑓𝑓 into 𝐾 clusters. The K-mean clustering algorithm is an iterative partitioning clustering technique (Jain et al., 1999) that starts with an initialized set of cluster centers from a given dataset. Then, it assigns each datapoint into its closest cluster center and updates the set of cluster centers with the aim of minimizing the clustering error. The clustering error is commonly defined as the sum of the squared of the distance from each datapoint to its assigned cluster center. The algorithm repeats these steps until a termination condition is met, e.g., 135 there is no change in the set of cluster centers, there is no improvement in minimizing the clustering error, or when the maximum number of iterations is reached. We recall that 𝑘 = 1, . . . , 𝐾 represents the index of clusters, vehicles, or routes. Let 𝒮 = {𝓈1 , . . . , 𝓈𝐾 } denotes the set of 𝐾 cluster centers that are randomly selected from the set of offline customers (i.e., 𝓈𝑘 ∈ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 ). We define 𝐶𝑘 ⊆ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 that represents a sub-set of offline customers that are/will be assigned to cluster 𝑘. It is clear that 𝐶1 ∩. . .∩ 𝐶𝐾 = ∅ (offline customer 𝑖 ∈ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 must be assigned to exactly one cluster). Also, let 𝑄𝑘𝑑 and 𝑄𝑘𝑝 be the free delivery and pickup capacity of cluster 𝑘, respectively. For the maximum number of iterations 𝑅, the following steps form the K-mean clustering algorithm: Step 1. We assign all offline customers 𝑖 ∈ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 to their closest cluster 𝑘 (i.e., add 𝑖 to 𝐶𝑘 ) that has enough free delivery/pickup capacity. To conduct this task, we set 𝐶𝑘 = ∅, 𝑄𝑘𝑑 = 𝑄, and 𝑄𝑘𝑝 = 𝑄 for 𝑘 = 1, . . . , 𝐾. Then, for each 𝑖 ∈ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 , we repeat the following sub-steps until all offline customers be assigned to their closest cluster: - Step 1-1: we determine the clusters that have enough free delivery/pickup capacity for fulfilling the request of offline customer 𝑖, i.e., 𝑄𝑘𝑑 ≥ 𝑑𝑖 and 𝑄𝑘𝑝 ≥ 𝑝𝑖 . - Step 1-2: for the clusters that are determined in step1-1, we assign offline customer 𝑖 to cluster 𝑘 that has the least 𝑡𝑖𝓈𝑘 . - Step 1-3: for the selected cluster in step 1-2, we update the corresponding free delivery/pickup capacity by setting 𝑄𝑘𝑑 = 𝑄𝑘𝑑 − 𝑑𝑖 and 𝑄𝑘𝑝 = 𝑄𝑘𝑝 − 𝑝𝑖 . Step 2. We update the center of clusters (i.e., set 𝒮). To conduct this task, for 𝑘 = 1, . . . , 𝐾, we take the following sub-steps: 136 - Step 2-1: for 𝑖 ∈ 𝐶𝑘 , we compute the sum of traveling time from customer 𝑖 to customer⁡𝑗 as 𝑒𝑖𝑘 = ∑𝑗∈𝐶𝑘 𝑡𝑖𝑗 . 𝑗≠𝑖 - Step 2-2: we set 𝓈𝑘 = 𝑖, 𝑖 ∈ 𝐶𝑘 with the least 𝑒𝑖𝑘 . Step 3. We repeat steps 1-2 until the termination conditions are met. Let 𝑒 = ∑𝐾 𝑘=1 ∑𝑖∈𝐶𝑘 𝑡𝑖𝓈𝑘 be 𝑖≠𝓈𝑘 the sum of traveling time from each customer to his assigned cluster center. Also let 𝜏 be a threshold for the number of iterations without improvement in 𝑒. Then, the K-mean clustering algorithm terminates if (i) there is no improvement in 𝑒 after 𝜏 iterations, or (ii) the maximum number of iterations 𝑅 is reached. Then, for each 𝐶𝑘 , 𝑘 = 1, . . . , 𝐾, we obtain the optimal routing plan by solving the MIP model presented in section 5.3.2 with CPLEX optimization solver. Routing phase Let 𝐶𝑘0 = {0} ∪ 𝐶𝑘 . We also define (i) 𝐴′𝑘 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐶𝑘0 ∩ 𝐿0 , 𝑗 ∈ 𝐶𝑘 ∩ 𝐿, 𝑖 ≠ 𝑗} be the set of links that connect the depot or linehaul customers to other linehaul customers in cluster 𝑘, (ii) 𝑜𝑓𝑓 𝐴′′𝑘 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝐶𝑘 ∩ 𝐿, 𝑗 ∈ 𝐶𝑘0 ∩ 𝐵0 , 𝑖 ≠ 𝑗} be the set of links that connect the linehaul customers to the depot or to the offline backhaul customers in cluster 𝑘, and (iii) 𝐴′′′ 𝑘 = {(𝑖, 𝑗):⁡𝑖 ∈ 𝑜𝑓𝑓 𝐶𝑘 ∩ 𝐵 𝑜𝑓𝑓 , 𝑗 ∈ 𝐶𝑘0 ∩ 𝐵0 , 𝑖 ≠ 𝑗} be the set of links that connects offline backhaul customers to the depot or to other offline backhaul customers in cluster 𝑘. Then, 𝐴𝑘 = 𝐴′𝑘 ∪ 𝐴′′𝑘 ∪ 𝐴′′′𝑘 represents the set of all feasible links that connects customer 𝑖 to customer 𝑗, 𝑖, 𝑗 ∈ 𝐶𝑘0 . Furthermore, we define 𝑥𝑖𝑗 as a binary decision variable that equals 1 if link (𝑖, 𝑗) ∈ 𝐴𝑘 , is selected, and 0 otherwise. We also define auxiliary continuous decision variables 𝑢𝑖𝑗 and 𝑤𝑖𝑗 , (𝑖, 𝑗) ∈ 𝐴𝑘 for the flow of delivery and pickup requests, respectively. Then, for each cluster 𝑘, the MIP model can be formulated as: 137 𝑜𝑓𝑓 𝑀𝑖𝑛 𝑇𝑇𝑘 = ∑(𝑖,𝑗)∈𝐴𝑘 𝑡𝑖𝑗 𝑥𝑖𝑗 (5.1) s.t. ∑𝑗∈𝐶𝑘∩𝐿 𝑥0𝑗 = 1 (5.2) ∑𝑖∈𝐶𝑘 𝑥𝑖0 = 1 (5.3) ∑𝑗∈𝐶 0 𝑥𝑖𝑗 = 1 𝑘 𝑖 ∈ 𝐶𝑘 ∩ 𝐿, (5.4) 𝑖≠𝑗 ∑𝑖∈𝐶 0 ∩𝐿0 𝑥𝑖𝑗 = 1 𝑘 𝑗 ∈ 𝐶𝑘 ∩ 𝐿, (5.5) 𝑖≠𝑗 ∑𝑗∈𝐶 0 ∩𝐵𝑜𝑓𝑓 𝑥𝑖𝑗 = 1 𝑘 0 𝑖 ∈ 𝐶𝑘 ∩ 𝐵 𝑜𝑓𝑓 , (5.6) 𝑖≠𝑗 ∑𝑖∈𝐶𝑘 𝑥𝑖𝑗 = 1 𝑗 ∈ 𝐶𝑘 ∩ 𝐵 𝑜𝑓𝑓 , (5.7) 𝑖≠𝑗 ∑𝑗∈𝐶 0 𝑢𝑗𝑖 − ∑𝑗∈𝐶 0 𝑢𝑖𝑗 = 𝑑𝑖 𝑘 𝑘 𝑖 ∈ 𝐶𝑘 , (5.8) 𝑖≠𝑗 𝑖≠𝑗 ∑𝑗∈𝐶 0 𝑤𝑖𝑗 − ∑𝑗∈𝐶 0 𝑤𝑗𝑖 = 𝑝𝑖 𝑘 𝑘 𝑖 ∈ 𝐶𝑘 , (5.9) 𝑖≠𝑗 𝑖≠𝑗 𝑢𝑖𝑗 + 𝑤𝑖𝑗 ≤ 𝑄𝑥𝑖𝑗 (𝑖, 𝑗) ∈ 𝐴𝑘 , (5.10) 𝑥𝑖𝑗 ∈ {0,1}, 𝑢𝑖𝑗 ∈ ℝ, 𝑤𝑖𝑗 ∈ ℝ (𝑖, 𝑗) ∈ 𝐴𝑘 . (5.11) The objective function (5.1) minimizes the time traveled by vehicle 𝑘 for serving offline customers that are assigned to cluster 𝑘. For linehaul and offline backhaul customers in 𝐶𝑘 , constraint (5.2) guarantees that the itinerary of vehicle 𝑘 starts by serving a linehaul customer; constraint (5.3) guarantees that vehicle 𝑘 returns to the depot after serving either a linehaul or an offline backhaul customer; constraint (5.4) guarantees that each linehaul customer is connected to another linehaul customer, to an offline backhaul customer, or to the depot; constraint (5.5) ensures 138 that each linehaul customer is connected by another linehaul customer or by the depot; constraint (5.6) ensures that each offline backhaul customer is connected to another offline backhaul customer or to the depot; constraint (5.7) ensures that each offline backhaul customer is connected by another offline backhaul customer or by a linehaul customer. We note that constraints (5.2) and (5.5)-(5.6) guarantee that all linehaul customers be served before any offline backhaul customers. Constraints (5.8)-(5.10) are an extension of the connectivity and capacity constraints proposed by Mosheiov (1998), where constraints (5.8)-(5.9) guarantee the flow of delivery and pickup requests, respectively, and constraint (5.10) ensures that the total flow (i.e., total amount of deliveries and pickups) at each link does not exceed the vehicle’s capacity. Constraint (5.11) defines the domains of decision variables. 𝑜𝑓𝑓∗ 𝑜𝑓𝑓 Let 𝑇𝑇𝑘 be the optimal solution of problem 𝑇𝑇𝑘 , 𝑘 = 1, . . . , 𝐾. Then, we define 𝑇𝑇 𝑜𝑓𝑓 be the total time traveled for serving all offline customers and compute it as: 𝑜𝑓𝑓∗ 𝑇𝑇 𝑜𝑓𝑓 = ∑𝐾 𝑘=1 𝑇𝑇𝑘 (5.12) Updating the existing routing plan 𝑘 Let 𝒳 𝑘 = (𝓍1𝑘 , . . . , 𝓍𝑗𝑘 , 𝓍𝑗+1 𝑘 , . . . , 𝓍𝑑𝑖𝑚(𝒳 𝑘 ) ) be a vector representative of the spatial itinerary of 𝑘 𝑘 vehicle 𝑘, where 𝓍1𝑘 = 0, 𝓍𝑑𝑖𝑚(𝒳 1 𝐾 𝑘 ) = 0, and 𝓍𝑗 ∈ 𝑉/{0}. Suppose 𝒳 = {𝒳 , . . . , 𝒳 }. Before transportation operations start, 𝒳 is the initial routing plan obtained from section 5.3. However, during the working shift 𝑇, 𝒳 represents the existing routing plan that is updated based on the revealed information of online backhaul customers 𝐵 𝑜𝑛 using the DVRPB algorithms presented in Tables 5.3 and 5.4. Also, let 𝐿𝑘𝑡𝑎𝑖𝑙 be the last linehaul customer that will be served by vehicle 𝑘. Then, 𝐿𝑡𝑎𝑖𝑙 = {𝐿1𝑡𝑎𝑖𝑙 , . . . , 𝐿𝐾𝑡𝑎𝑖𝑙 }. 139 𝑘 Furthermore, let 𝒯 𝑘 = (𝓉1𝑘 , . . . , 𝓉𝑗𝑘 , 𝓉𝑗+1 𝑘 , . . . , 𝓉𝑑𝑖𝑚(𝒯 𝑘 ) ) be a vector representative of the temporal itinerary of vehicle 𝑘. The temporal itinerary refers to the time stamps at which vehicle 𝑘 arrives at a customer’s location or departs from/arrives at the depot. Therefore, 𝓉1𝑘 is the time 𝑘 stamp at which vehicle 𝑘 leaves the depot and 𝓉𝑑𝑖𝑚(𝒯 𝑘 ) represents the time stamp at which vehicle 𝑘 𝑘 arrives at the depot. Then, 𝓉𝑗+1 can be computed as: 𝑘 𝓉𝑗+1 = 𝓉𝑗𝑘 + 𝑡𝓍 𝑘𝓍 𝑘 (5.13) 𝑗 𝑗+1 𝑘 𝑘 Let 𝑇𝑇 𝑘 represents the total time traveled by vehicle 𝑘. It is clear that 𝑇𝑇 𝑘 = 𝓉𝑑𝑖𝑚(𝒯 𝑘 ) − 𝓉1 . 𝑘 We also define 𝑄𝑓𝑟 be the free capacity of vehicle 𝑘 to serve the next online backhaul customer and compute it as: 𝑘 𝑄𝑓𝑟 = 𝑄 − ∑𝓍 𝑘∈𝒳 𝑘 𝑝𝓍 𝑘 (5.14) 𝑗 𝑗 1 𝐾 Let 𝒯 = {𝒯 1 , . . . , 𝒯 𝐾 }, 𝑇𝑇 = {𝑇𝑇1 , . . . , 𝑇𝑇 𝐾 }, and 𝑄𝑓𝑟 = {𝑄𝑓𝑟 , . . . , 𝑄𝑓𝑟 }. Furthermore, we define 𝑠𝑖 ∈ {0,1,2} be the service status of online backhaul customer 𝑖 ∈ 𝐵 𝑜𝑛 , where (i) 𝑠𝑖 equals 0 if no decision has been made yet, (ii) 𝑠𝑖 equals 1 if the request is accepted, and (iii) 𝑠𝑖 equals 2 if the request is rejected for the same-day service. We also define 𝑜𝑖 be the time stamp at which the order of online backhaul customer 𝑖 ∈ 𝐵 𝑜𝑛 is placed. To collect the information of online backhaul customers and further update the existing routing plan, we partition the working shift 𝑇 into 𝑙 decision epochs with the length of ℓ. Recall from section 5.2.2 that the decision epoch is a time window at which the information of online backhaul 140 customers who placed their requests within that time window are collected. Then, 𝑇𝑙 represents the decision time stamp 𝑙 (i.e., the time stamp at which decision epoch 𝑙 ends), where 𝑇𝑙 = 𝑇𝑙−1 + ℓ, 𝑇0 = 0, 𝑇𝑙 < 𝑇, and 𝑙 ≥ 1. At each 𝑇𝑙 , let 𝐵𝑙𝑜𝑛 = {𝑖: 𝑖 ∈ 𝐵 𝑜𝑛 , 𝑇𝑙−1 ≤ 𝑜𝑖 < 𝑇𝑙 } be the set of online backhaul customers who placed their requests between 𝑇𝑙−1 and 𝑇𝑙 . We note that, at the beginning of the decision epoch 𝑙, 𝑠𝑖 = 0 for all 𝑖 ∈ 𝐵𝑙𝐷 . However, throughout working shift 𝑇, the acceptance/rejection of the request is decided. If acceptance is the case, then online backhaul customer 𝑖 ∈ 𝐵𝑙𝐷 is assigned to vehicle 𝑘 that has the least detour diversion from its existing routing plan and 𝒳, 𝒯, 𝑇𝑇, 𝑄𝑓𝑟 , and 𝑠𝑖 are updated. Table 5.3. Main algorithm for the DVRPB*. Step 0: initialization 1 𝐾 𝒳: = {𝒳 1 , . . . , 𝒳 𝐾 }; 𝒯: = {𝒯 1 , . . . , 𝒯 𝐾 }; 𝑄𝑓𝑟 = {𝑄𝑓𝑟 , . . . , 𝑄𝑓𝑟 }; 𝑇𝑇: = {𝑇𝑇 1 , . . . , 𝑇𝑇 𝐾 }; 𝐿𝑡𝑎𝑖𝑙 = 1 1 𝐾 {𝐿𝑡𝑎𝑖𝑙 , . . . , 𝐿𝑡𝑎𝑖𝑙 }; ℓ⁡and 𝑇𝑙 Step 1: collect the information of online customers at each decision epoch 2 for 𝑇𝑙 ⁡ 3 update 𝐵𝑜𝑛 AND set 𝐵𝑙𝑜𝑛 : = {𝑖: 𝑖 ∈ 𝐵𝑜𝑛 , 𝑇𝑙−1 ≤ 𝑜𝑖 < 𝑇𝑙 } 4 if 𝐵𝑙𝑜𝑛 ≠ ∅ Step 2: decide on the acceptance/rejection of online requests 5 for 𝑖 ∈ 𝐵𝑙𝑜𝑛 6 𝑘𝑡𝑒𝑚𝑝 : = 0; ∆𝑡𝑒𝑚𝑝 : = ∞ 7 𝒳 𝑡𝑒𝑚𝑝 : = 𝒳; 𝒯𝑡𝑒𝑚𝑝 : = 𝒯; 𝑄𝑡𝑒𝑚𝑝 : = 𝑄𝑓𝑟 ; 𝑇𝑇𝑡𝑒𝑚𝑝 : = 𝑇𝑇 8 for 𝑘 = 1 to 𝐾 9 call the internal algorithm for the DVRPB from Table 5.4 𝑘 10 return updated 𝒳 𝑘 ⁡, 𝒯 𝑘 , 𝑄𝑓𝑟 , 𝑇𝑇 𝑘 , 𝑠𝑖 , ∆𝑘 𝑘 11 if 𝑠𝑖 = 1 AND ∆ ≤ ∆𝑡𝑒𝑚𝑝 12 𝑘𝑡𝑒𝑚𝑝 : = 𝑘 AND ∆𝑡𝑒𝑚𝑝 : = ∆𝑘 Step 3: update the exiting routing plan for serving accepted online request 13 if 𝑘𝑡𝑒𝑚𝑝 ≠ 0 ⁡𝑘𝑡𝑒𝑚𝑝 ⁡𝑘𝑡𝑒𝑚𝑝 ⁡𝑘𝑡𝑒𝑚𝑝 ⁡𝑘 ⁡𝑘 14 𝒳𝑡𝑒𝑚𝑝 : = 𝒳 ⁡𝑘𝑡𝑒𝑚𝑝 ; 𝒯𝑡𝑒𝑚𝑝 : = 𝒯 ⁡𝑘𝑡𝑒𝑚𝑝 ; 𝑄𝑡𝑒𝑚𝑝 : = 𝑄𝑓𝑟𝑡𝑒𝑚𝑝 ; 𝑇𝑇𝑡𝑒𝑚𝑝 𝑡𝑒𝑚𝑝 : = 𝑇𝑇 ⁡𝑘𝑡𝑒𝑚𝑝 15 𝒳: = 𝒳𝑡𝑒𝑚𝑝 ; 𝒯: = 𝒯𝑡𝑒𝑚𝑝 ; 𝑄𝑓𝑟 : = 𝑄𝑡𝑒𝑚𝑝 ; 𝑇𝑇: = 𝑇𝑇𝑡𝑒𝑚𝑝 16 return 𝑇𝑇, 𝑠𝑖 for 𝑖 ∈ 𝐵𝑜𝑛 * See Table 5.2 for the summary of notations. At decision time stamp 𝑇𝑙 , let (𝓍𝑗𝑘 , 𝓍𝑗+1 𝑘 ) be two consecutive customers that will be but not served yet by vehicle 𝑘, where 𝑗 ∈ {1, 2, … , 𝑑𝑖𝑚(𝒳 𝑘 ) − 1}. We note that 𝓍𝑗𝑘 must be the last 141 linehaul customer, offline backhaul customers, or accepted online backhaul customers that are assigned to vehicle 𝑘 (i.e., 𝓍𝑗𝑘 ∈ {𝐿𝑘𝑡𝑎𝑖𝑙 } ∪ 𝐵 𝑜𝑓𝑓 ∪ 𝐵 𝑜𝑛 ) but have not been served yet at 𝑇𝑙 (i.e., 𝓉𝑗𝑘 ≥ 𝑇𝑙 ). Let ∆𝑘𝑖 be the detour deviation resulted from inserting online backhaul customer 𝑖 ∈ 𝐵𝑙𝐷 between 𝓍𝑗𝑘 and 𝓍𝑗+1 𝑘 and computed as: ∆𝑘𝑖 = 𝑡𝓍 𝑘𝑖 + 𝑡𝑖𝓍 𝑘 (5.15) 𝑗 𝑗+1 Table 5.4. Internal algorithm for the DVRPB*. Step 0: initialization 𝑘 1 𝑇𝑙 ; 𝒳 𝑘 ; 𝒯 𝑘 ; 𝑄𝑓𝑟 ; 𝑇𝑇 𝑘 ; 𝐿𝑘𝑡𝑎𝑖𝑙 ; 𝑖 ∈ 𝐵𝑙𝑜𝑛 ; 𝑝𝑖 ; 𝑠𝑖 : = 0; ∆𝑘 : = ∞ 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 2 𝒳𝑡𝑒𝑚𝑝 : = 𝒳 ; 𝒯𝑡𝑒𝑚𝑝 : = 𝒯 𝑘 ; 𝑄𝑡𝑒𝑚𝑝 : = 𝑄𝑓𝑟 ; 𝑇𝑇𝑡𝑒𝑚𝑝 : = 𝑇𝑇 𝑘 Step 1: check the total traveling time and capacity of vehicle 𝑘 3 𝑘 𝑘 if 𝑄𝑓𝑟 < 𝑝𝑖 OR 𝑇𝑙 ≥ 𝒯𝑑𝑖𝑚(𝒯 𝑘 )−1 4 𝑠𝑖 : = 2 Step 2: find the insertion point that has the least detour deviation for vehicle 𝑘 5 else 6 for (𝓍𝑗𝑘 , 𝓍𝑗+1𝑘 ), 𝑗 ∈ {1, 2, … , 𝑑𝑖𝑚(𝒳 𝑘 ) − 1} AND 𝓍𝑗𝑘 ∈ {𝐿𝑘𝑡𝑎𝑖𝑙 } ∪ 𝐵𝑜𝑓𝑓 ∪ 𝐵𝑜𝑛 AND 𝓉𝑗𝑘 ≥ 𝑇𝑙 𝑘 7 ∆𝑖 : = 𝑡𝓍 𝑘𝑖 + 𝑡𝑖𝓍 𝑘 𝑗 𝑗+1 8 if ∆𝑘𝑖 ≤ ∆𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 9 𝒳 𝑘 : = 𝒳𝑡𝑒𝑚𝑝 ; 𝒯 𝑘 : = 𝒯𝑡𝑒𝑚𝑝 ; 𝑄𝑓𝑟 : = 𝑄𝑡𝑒𝑚𝑝 ; 𝑇𝑇 𝑘 : = 𝑇𝑇𝑡𝑒𝑚𝑝 𝑘 𝑘 10 ∆ : = ∆𝑖 𝑘 11 update 𝒳 𝑘 , 𝒯 𝑘 , 𝑄𝑓𝑟 , 𝑇𝑇 𝑘 Step 3: check if the total traveling time of vehicle 𝑘 does not exceed the working shift after insertion 12 if ⁡𝑇𝑇 𝑘 ≤ 𝑇 13 𝑠𝑖 : = 1 𝑘 𝑘 𝑘 𝑘 𝑘 14 𝒳𝑡𝑒𝑚𝑝 : = 𝒳 𝑘 ; 𝒯𝑡𝑒𝑚𝑝 : = 𝒯 𝑘 ; 𝑄𝑡𝑒𝑚𝑝 : = 𝑄𝑓𝑟 ; 𝑇𝑇𝑡𝑒𝑚𝑝 : = 𝑇𝑇 𝑘 14 else 16 𝑠𝑖 : = 2 Step 4: update the routing plan of vehicle 𝑘 17 𝒳 𝑘 : = 𝒳𝑡𝑒𝑚𝑝 𝑘 𝑘 𝑘 𝑘 𝑘 ; 𝒯 𝑘 : = 𝒯𝑡𝑒𝑚𝑝 ; 𝑄𝑓𝑟 : = 𝑄𝑡𝑒𝑚𝑝 ; 𝑇𝑇 𝑘 : = 𝑇𝑇𝑡𝑒𝑚𝑝 18 return 𝒳 𝑘 , 𝒯 𝑘 , 𝑄𝑓𝑟 𝑘 , 𝑇𝑇 𝑘 , 𝑠𝑖 , ∆𝑘 * See Table 5.2 for the summary of notations. Let ∆𝑘 be the least ∆𝑘𝑖 obtained for (𝓍𝑗𝑘 , 𝓍𝑗+1 𝑘 ). Then, the request of customer 𝑖 ∈ 𝐵𝑙𝐷 is accepted 𝑘 (𝑠𝑖 = 1) if there exist vehicle 𝑘 such that 𝑄𝑓𝑟 − 𝑝𝑖 ≥ 0 and 𝑇𝑇 𝑘 + ∆𝑘 − 𝑡𝓍 𝑘𝓍 𝑘 ≤ 𝑇. Finally, we 𝑗 𝑗+1 142 assign customer 𝑖 ∈ 𝐵𝑙𝐷 to vehicle 𝑘 that has the least detour deviation (i.e., least ∆𝑘 ) and update 𝑘 corresponding 𝒳 𝑘 , 𝒯 𝑘 , 𝑇𝑇 𝑘 , and 𝑄𝑓𝑟 . Tables 5.3 and 5.4 present our proposed analytical framework for the DVRPB. Let 𝑇𝑇𝑇 represent the total time that is traveled and/or must be traveled for serving all customers 𝑖 ∈ 𝑉/{0}. Then, 𝑇𝑇𝑇 comprises of four components: 1. Total time traveled for serving offline customers 𝑖 ∈ 𝐿 ∪ 𝐵 𝑜𝑓𝑓 that is denoted as 𝑇𝑇 𝑜𝑓𝑓 and is computed by Eq. 5.12 in section 5.3. 2. Total time traveled for serving online backhaul customers 𝑖 ∈ 𝐵 𝑜𝑛 whose requests have been accepted for the same-day service (𝑁 = {𝑖: 𝑖 ∈ 𝐵 𝑜𝑛 , 𝑠𝑖 = 1}), denoted as 𝑇𝑇 𝑜𝑛 and can be computed as 𝑜𝑓𝑓∗ 𝑇𝑇 𝑜𝑛 = ∑𝐾 𝑘 𝑘=1 𝑇𝑇 − 𝑇𝑇𝑘 (5.16) 3. Total time that must be traveled for serving online backhaul customers 𝑖 ∈ 𝐵 𝑜𝑛 whose requests have not been decided for the same-day service (i.e., 𝑠𝑖 = 0). We name this component penalty type I (denoted as 𝑃𝐼 ). It is clear that online requests that are received after 𝑇𝑙 , cannot be decided for the sameday service. Let 𝑁 𝐼 = {𝑖: 𝑖 ∈ 𝐵 𝑜𝑛 , 𝑠𝑖 = 0} be the set of online backhaul customers whose requests have not been decided for the same-day service, and 𝛼 represents the corresponding penalty multiplier. Then, penalty type I can be computed as 𝑃𝐼 = 𝛼|𝑁 𝐼 | (5.17) 143 4. Total time that must be traveled for serving online backhaul customers whose requests have been rejected for the same-day service (i.e., 𝑠𝑖 = 2). We name this component penalty type II (denoted as 𝑃𝐼𝐼 ). It is clear that not all online requests can be accepted for the same-day service because of limited resources. For example, at the corresponding decision time stamp, there may not exist available en-route vehicles with enough free capacity to serve the online backhaul customer 𝑖 ∈ 𝐵 𝑜𝑛 . Another example is the case that the corresponding detour devotion from assigning the online backhaul customer 𝑖 ∈ 𝐵 𝑜𝑛 to vehicle 𝑘 may require the vehicle to travel beyond its working shift. Under these conditions, the request cannot be accepted for the same-day service. Let 𝑁 𝐼𝐼 = {𝑖: 𝑖 ∈ 𝐵 𝑜𝑛 , 𝑠𝑖 = 2} be the set of online backhaul customers whose requests have been rejected for the same- day service, and 𝛽 represents the corresponding penalty multiplier. Then, penalty type II can be computed as 𝑃𝐼𝐼 = 𝛽|𝑁 𝐼𝐼 | (5.18) Then, 𝑇𝑇𝑇 is 𝑇𝑇𝑇 = 𝑇𝑇 𝑜𝑓𝑓 + 𝑇𝑇 𝑜𝑛 + 𝑃𝐼 + 𝑃𝐼𝐼 (5.19) Computational experiments We conduct our experiments on a computer running Intel (R) Core (TM) i7-4770T CPU processors clocked at 2.50 GHz with 7 cores and 8 GB RAM running Windows Server 10×64 Edition. The algorithms are coded in Python distribution platform, Anaconda 3, and run by Spyder 144 integrated development environment, version 4.2.5. The Python code and datasets are available at GitHub (2022b). Problem setting We test our proposed framework for the DVRPB on two real-world transportation networks, Lansing and Charlotte networks (both located in the state of Michigan, USA) that have been introduced in chapter 4. Recalling from chapter 4, these networks consist of 8,091 and 1,184 nodes, respectively. Each node is represented with an ID and unique latitude/longitude (depicted in Figure 4.3) We assume that each node can represent a customer’s location, and all links are bidirectional. The majority of benchmark datasets in the extant literature assume there exists a direct link between every two nodes, while we know that this is not a valid assumption in real-world transportation networks. To calculate the distance between each pair of nodes, we calculate the shortest path between them using the Dijkstra’s algorithm (Johnson 1973). We note that the length of a given link (in miles) is measured by the Harversian distance equation (Mahmoudi et al. 2019a). We further compute the traveling time from one node to another as a linear function of the distance and speed (miles per hour). For simplicity, we assume a constant speed of 25 miles per hour for this calculation. We also assume 𝑝𝑖 and 𝑑𝑖 does not exceed one unit as each request is a pickup/delivery of a single reusable tote. We also assume that a customer has a single request (either delivery or pickup, but not both). This type of the VRPB is called the VRPB with single demand. Baseline experiments: First and foremost, we note that Lansing and Charlotte networks are representative of urban and rural areas, respectively. This characteristic enables us to investigate the effect of the urbanism on the efficiency of transportation operations of reusables. Furthermore, we randomly generate the set of customers from both networks by roughly considering 10% of the 145 total nodes in each network as the location of customers. This results in 800 and 120 customers for Lansing and Charlotte networks, respectively (shown by black dots in Figure 4.3). Then, we randomly select 50% of these customers as backhauls. For example, in the case of Lansing, 400 (out of 800) customers are backhauls. In addition, we consider the ratio of DOD be 0%, 10%, …, 100%. For example, in the Lansing network, (i) 0% DOD means all 400 backhaul customers have placed their requests before transportation operations start (pickup operations are completely static), (ii) 10% DOD implies that 40 (out of 400) backhaul customers are placing their requests when transportation operations are running (10% of pickup operations are dynamic), and (iii) the 100% DOD means that all 400 backhaul customers are placing their requests when transportation operations are running (pickup operations are fully dynamic). Moreover, for the reason explained in Appendix C, we set the capacity of vehicles as 50. Furthermore, we consider minute as the unit of time to track transportation operations more precisely. We also set the working shift be one typical day of service starting at time 0 and ending at time 480 (i.e., 8 hours or 480 minutes is the duration of one day of operations). We randomly generate time stamps at which online backhaul customers place their requests as an integer number from 0 to 480 minutes using Uniform distribution (i.e., 𝒰(0,480) excluding 0 and 480). We also set the length of decision epoch as 240 minutes. Other scenarios: We further extend our experiments by (i) generating the time frames at which online backhaul customers place their requests as 𝒰(0,240), (ii) setting the length of decision epoch as 1 and 60 minutes, (iii) setting the number of customers as 1600 and 2400 in Lansing network and 240 and 360 in Charlotte network (accounting for about 20% and 30% of total nodes in Lansing and Charlotte networks), and (iv) setting the capacity of vehicles as 30. Table 5.5 presents a summary of parameters that we consider for generating our experimental instances. 146 Table 5.5. Summary of parameters for generating experimental instances. Parameters Values/distributions* Network {Lansing, Charlotte} Lansing network: {800, 1600, 2400} # customers Charlotte network: {120, 240, 360} Vehicles’ capacity {30, 50} DOD {0%, 10%, …, 100%} Time stamps at which online customers {𝓤(𝟎, 𝟒𝟖𝟎), 𝒰(0,240)} can place their requests Length of decision epoch {1, 60, 240} * The baseline experiment is shown in bold. All experiments: We construct the initial routing plan using the CFRS solution approach explained in section 5.3. In the clustering phase, we set 𝑅⁡ = ⁡100 and 𝜏⁡ = ⁡10. In the routing phase, we use Python interface with CPLEX 20.1.0 optimization solver to generate optimal routes for each cluster, setting a time limit of 3,600 seconds. Therefore, the solver returns the best- founded route if it cannot reach the optimal route within the predefined time limit. From the results, we observe that vehicles finish their tasks in a couple of hours or so. This is because the request of majority of online backhaul customers are rejected for the same-day service due to the unavailable en-route vehicles. To resolve this issue in our experiments, we merge routing plans by solving a bin-packing problem such that (i) each route must be assigned to exactly one bin, (ii) for routes assigned to a bin, there is a 60-minutes reloading time from finishing a route to start another one, (iii) the total traveling time of routing plans assigned to a bin along with associated reloading time should not exceed the working shift. Then, for the above-mentioned merged routing plans, we slightly modify the DVRPB algorithms presented in section 5.4. Our proposed modifications are two-fold: (i) for each bin, the time that a route starts, is the total traveling time of previous route plus the 60-minutes reloading time. With this arrangement, the corresponding temporal 147 itinerary is computed by Eq. 5.13; (ii) an online backhaul customer can be inserted to a route if, after insertion, the total traveling time of the corresponding bin does not exceed the working shift. We present our experimental results in the form of the total distance traveled considering that distance is a linear function of traveling time and traveling speed (we consider 25 mile/hour as 𝑇𝑇 𝑜𝑓𝑓 traveling speed). To compute penalty type I and II using Eq. 5.17-5.18, we set 𝛼 = |𝐿∪𝐵𝑜𝑓𝑓 | and 𝑇𝑇 𝑜𝑛 𝛽= , respectively (of note, to avoid further notations, 𝑇𝑇 𝑜𝑓𝑓 and 𝑇𝑇 𝑜𝑛 are also representative |𝑁| of their distance counterparts). Impact of geographical accessibility of customers’ location Figure 5.1 (a) and (b) depict the total distance traveled for Lansing and Charlotte networks, respectively. As it is shown, the total distance traveled (in miles) for Lansing (Charlotte) network (i) is 522.91 (155.93) when pickup operations are completely static (0% DOD), (ii) increases with a ratio of 243.88 (44.94) miles as pickup operations incrementally shift to dynamic with a step size of 10%, and (iii) is 802.01 (193.02) when pickup operations are completely dynamic (100% DOD). (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 5.1. Total distance traveled for Lansing and Charlotte networks. Note. For each network: vehicles’ capacity = 50, the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. 148 We further decompose the total distance traveled into four components (mentioned in section 5.3.2) of the objective function of the DVRPB (see Figure 5.2). It can be observed that in Lansing network, the increase of the total distance traveled is mainly due to penalty types I and II (i.e., the request of majority of online customers have not been decided or have been rejected for the same- day service). The reason can be the limitation of logistical resources in Lansing network that avoids accepting more online requests. In despite, in the Charlotte network, the increase of the total distance traveled is mainly due to serving more online customers. (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 5.2. Decomposition of the total distance traveled based on four components defined for the objective function of the DVRPB. Note. For each network: vehicles’ capacity = 50, the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. Intuitively, one can expect that the total distance traveled in urban areas be higher than rural areas as it is showcased for Lansing and Charlotte networks in Figure 5.1. We also find that (i) the distance per request would reduce when serving an urban area (compared in Figure 5.3 (a)). The average distance per request is 0.82 and 1.60 miles for Lansing and Charlotte networks, respectively; and (ii) more online requests are accepted for the same-day service in rural area 149 (compared in Figure 5.3 (b)). The average number of accepted to the total number of online requests is 0.24 and 0.52 for Lansing and Charlotte networks, respectively. Although finding (i) may imply that launching reusable packages could bring economies of scale in urban areas but finding (ii) demonstrates that the service of operating in the real-time for urban areas will not be as efficient as rural areas. That is, there is a trade-off between transportation cost and service quality for urban verses rural areas if operating in the real-time. (a) Distance / # requests (b) # accepted / # total online requests Figure 5.3. Comparison of the cost and service quality of transportation operations for urban verses rural areas. Note. For each network: vehicles capacity = 50, the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. In Lansing network, # requests = 800. In Charlotte network, # requests = 120. We further investigate the impact of geographical accessibility of customers’ location for scenarios when about 20% and 30% of the representative population would participate in the reusable packaging platform, which account for 1,600 and 2,400 requests in Lansing and 240 and 360 requests in Charlotte, respectively. Figure 5.4 illustrates the total distance traveled and distance per requests for forgoing scenarios. We notice that, when a larger proportion of population participate in the reusable packaging platform, the total distance traveled increases, while the distance per request served would drop. Then, we decompose the total distance traveled into four 150 components of the objective function of the DVRPB for each forgoing scenario depicted in Figure 5.5. We realize that the increase in the total distance traveled verses the increase in the participation in the reusable packaging platform in both, Lansing and Charlotte networks, is mainly due to penalty types I and II. This implies that service quality is adversely affected as the size of networks grows no matter it is urban or rural area. The reason is two-fold: (i) the limitation of logistical resources such as vehicles’ availability at right time, in right location, and with enough free capacity; and (ii) the setting of the DVRPB (i.e., the impact of the real-time decision -making policies’ modes). For example, the result could be different if the DVRPB be implemented at every minute or online customers be restricted to place their requests up to the middle of the working shift. We investigate this effect in section 5.5.3. (a) Lansing network (b) Charlotte network Figure 5.4. Comparison of the roughly 10%, 20%, and 30% of nodes participations in the reusable packaging platform. Note. For each experiment: vehicles capacity = 50, the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. 151 Lansing network Charlotte network (a) 10% participation (Lansing: # requests = 800, Charlotte: # requests = 120) (b) 20% participation (Lansing: # requests = 1600, Charlotte: # requests = 240) (c) 30% participation (Lansing: # requests = 2400, Charlotte: # requests = 360) Figure 5.5. Decomposition of the total distance traveled based on four components defined for the objective function of the DVRPB and for roughly 10%, 20%, and 30% of nodes participations in the reusable packaging platform. Note. Left and right sides present experiments for Lansing and Charlotte networks, respectively. For each experiment: vehicles’ capacity = 50, the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. 152 In the baseline and forgoing scenarios, we set vehicles’ capacity as 50. Therefore, we expand our baseline experiment to the scenario that vehicles’ capacity is 30. Lower vehicles’ capacity enforces a greater number of vehicles en-route. Intuitively, it is expected that total distance traveled be mainly due to greater number of accepted online customers. Figure 5.6 presents the total distance traveled and Figure 5.7 depicts the decomposition of the total distance traveled into four components of the objective function of the DVRPB based on different values of vehicles’ capacity. We realize that decreasing the value of vehicles’ capacity does not significantly impact the total distance traveled in both networks (see Figure 5.6). However, in contrast from our general intuition, the number of accepted online customers has not been improved (see Figure 5.7). That is, the less-than-truckload may be more effective where we do consider lower than actual value for the vehicles’ capacity when solving the static problem in order to reserve more room for online requests in real-time. (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 5.6. Comparison of total distance traveled based on vehicles’ capacity. Note. For each experiment: the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. 153 Lansing network Charlotte network (a) Vehicles’ capacity = 50 (b) Vehicles’ capacity = 30 Figure 5.7. Decomposition of the total distance traveled based on four components defined for the objective function of the DVRPB and different values of vehicles’ capacity. Note. Left and right sides present experiments for Lansing and Charlotte networks, respectively. For each experiment: the distribution of time frame for receiving online requests = 𝒰(0,480), and the length of decision epoch = 240 minutes. In Lansing network, # requests = 800. In Charlotte network, # requests = 120. Impact of the real-time decision-making modes Figure 5.8 presents the total distance traveled and Figure 5.9 depicts the decomposition of the total distance traveled into four components of the objective function of the DVRPB based on different distributions of time frame for receiving online requests. Although the total distance traveled has not significantly changed but we notice that the penalty type I (caused by the number 154 of not-decided online requests) is negligible when 𝒰(0,240). The reason is that when online customers are restricted to place their requests within specific time frame (here, at the middle of working shift when we have also set our baseline decision time stamp), the DVRPB could decide on acceptance/rejection of requests (more likely a decision is made for an online request). We also conclude that the rejection of requests is due to limited logistical resources such as vehicles availability or duration of working shift. Adding more vehicles or adjusting waiting strategies on specific locations or on customers’ site could increase the chance of serving more online requests. (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 5.8. Comparison of total distance traveled based on the distribution of time frame for receiving online requests. Note. For each experiment: vehicles’ capacity = 50, and the length of decision epoch = 240 minutes. 155 Lansing network Charlotte network (a) Distribution of time frame for receiving online requests = 𝒰(0,480) (b) Distribution of time frame for receiving online requests = 𝒰(0,240) Figure 5.9. Decomposition of the total distance traveled based distribution of time frame for receiving online requests. Note. Left and right sides present experiments for Lansing and Charlotte networks, respectively. For each experiment: vehicles’ capacity = 50, and the length of decision epoch = 240 minutes. In Lansing network, # requests = 800. In Charlotte network, # requests = 120. Finally, we investigate the impact of the length of decision epochs (or the decision time stamps) on the efficiency if the DVRPB. Figure 5.10 and 5.11 present total distance traveled, and the decomposition of the total distance traveled into four components of the objective function of the DVRPB based on the length of decision epoch 240, 60, 1 (minutes), respectively. Although the total distance traveled has not changed dramatically, we notice that it is resulted from either serving more online customers or penalty type II. 156 (a) Lansing network (# requests = 800) (b) Charlotte network (# requests = 120) Figure 5.10. Comparison of total distance traveled based on different length of decision epochs. Notes. For each experiment: vehicles’ capacity = 50, and the distribution of time frame for receiving online requests = 𝒰(0,480). Conclusion We model the transportation operations of reusable packages in real-time networks as DVRPBs. We construct an initial routing plan for a fleet of vehicles located at depot to serve all offline (linehaul and offline backhaul) customers by proposing a CFRS solution approach. When transportation operations start, vehicles leave the depot to serve customers that are assigned to them. As the initial routing plan unfolds, online (online backhaul) customers place their requests in real-time. We present an analytical framework that collects the information of online customers, decides on the acceptance/rejection of their requests, and if acceptance is the case, updates/reoptimizes the pre-planned routes. We showcase our proposed framework on two real- world transportation networks that are representative of urban and rural areas. We also analyze our framework under further considerations related to (i) the DOD of transportation networks, and (ii) the real-time decision-making modes. 157 Lansing network Charlotte network (a) Length of decision epoch = 240 (minutes) (b) Length of decision epoch = 60 (minutes) (c) Length of decision epoch = 1 (minute) Figure 5.11. Decomposition of the total distance traveled based on four components defined for the objective function of the DVRPB and different length of decision epochs. Notes. Left and right sides present experiments for Lansing and Charlotte networks, respectively. For each experiment: vehicles’ capacity = 50, and the distribution of time frame for receiving online requests = 𝒰(0,480). In Lansing network, # requests = 800. In Charlotte network, # requests = 120. 158 We find that the limited logistical resources such as vehicles’ capacity and the duration of working shift could affect the number of accepted online requests for the same-day service. To enhance the quality of serving online customers, future research could incorporate (i) waiting strategy at special locations such as customers’ location, and (ii) the concept of less-than-truckload couriers (meaning to account for a portion of vehicles’ capacity when constructing the initial routing plan) into our proposed framework in order to reserve more room for serving potential online customers in real-time. Considering stochastic distribution of online customers could also be another avenue for future research. Furthermore, in this research, we project traveling distance into traveling time by assuming a constant traveling speed. The future research can adopt our proposed framework to incorporate a time-dependent traveling time for traveling from one customer to another one. Finally, we model the DVRPB as a two-index commodity flow problem. However, in our computational experiments we solve a bin packing problem to merge short routes. Therefore, future research could propose a two-echelon model that minimizes the number of bins (vehicles) and traveling distances, simultaneously. 159 CHAPTER 6. CONCLUSION In chapter 1, we extensively discussed the motivation, overview, and objectives of this research. In chapter 2, we present a systematic review of the literature in the light of the environmental and economic costs of reusable packaging, the design of reusable packaging logistics systems, and the implications of operations management for reusable packaging. Based on our analysis of existing studies, we then deliver insights and potential opportunities for future research on reusable packaging. To name a few, the future research could (i) incorporate environmental factors (e.g., carbon taxes, environmental externalities, and eco-costs), consumers’ behavior, and packaging designs in measuring costs, (ii) explore the impact of ownership and third-party logistics in the operations of reusable packaging systems, (iii) analyze such systems under more complicated, and yet realistic, settings (e.g., multiple sender-recipient pairs, variations in the quality of packages, asymmetric information between third parties and senders/recipients), and (iv) consider inter- parties and product-demand-package coordination in managing operations. Chapter 3 proposes a new mathematical model for the VRPB with single demand and backhaul solution. The structure of the proposed model allows us to utilize the Lagrangian decomposition approach to break the main problem into three well-known sub-problems: two OVRPs and one AP. At each iteration, the forgoing sub-problems are solved and Lagrangian multipliers are updated to reduce the gap between the lower and upper bounds of the global optimal solution. In this chapter, we present two different arrangements, i.e., parallel and sequential, for solving the foregoing sub-problems. We further analyze the impact of these arrangements on the solution quality and computational efficiency of our proposed Lagrangian decomposition algorithm by testing them on two benchmark datasets in the extant literature. We observe that, in the majority 160 of instances of both datasets, solving the foregoing sub-problems in parallel layout provides better solution compared to when they are solved sequentially; however, sequential layout outperforms parallel layout from the computational efficiency standpoint. We also tested our model on a real- world transportation network that is geographically distributed on Lansing in the state of Michigan, USA. To reduce the computational burden of solving the proposed VRPB on this dataset, we presented a CFRS algorithm. Our results showed that increasing vehicle capacity in each instance of this dataset decreases the gap between lower and upper bounds. This would be reasonable since a higher capacity could necessitate fewer number of vehicles to serve the same number of customers. Fewer vehicles translate to fewer clusters, and hence, sub-problems. Then, we conclude that future work may concentrate on employing exact solution methodologies such as brand-and- bound algorithm for solving the foregoing sub-problems. Since our proposed model allows us to decompose linehaul and backhaul routing phases, customized side constraints can be added to each phase without altering the other phase of the problem. Chapter 4 expands the mathematical model and the CFRS solution approach presented in chapter 3 for four classes of the VRPB. In this chapter, we implement our framework on two real- world networks that are representative of urban and rural areas. We also analyze our framework under further considerations, e.g., the possibility of serving delivery requests prior to pickup requests, or when a fraction of customers have simultaneous pickup and delivery requests. In our results, we report on various measures, such as the total distance traveled, the transportation cost, and CO2 emission. This chapter sets a stepping stone towards a more comprehensive analysis of transportation operations of reusable packaging. In chapter 5, we model the transportation operations of reusable packages in real-time networks as DVRPB. We construct an initial routing plan for a fleet of vehicles located at depot to serve all 161 offline (linehaul and offline backhaul) customers by proposing a CFRS solution approach. When transportation operations start, vehicles leave the depot to serve customers that are assigned to them. As the initial routing plan unfolds, online (online backhaul) customers place their requests in real-time. We present an analytical framework that collects the information of online customers, decides on the acceptance/rejection of their requests, and if acceptance is the case, updates/reoptimizes the pre-planned routes. We showcase our proposed framework on two real- world transportation networks that are representative of urban and rural areas. We also analyze our framework under further considerations related to (i) the DOD of transportation networks, and (ii) the real-time decision-making modes. To this end, we lay out three avenues for future research. The future research can expand our proposed framework for VRPB and DVRPB to multiple periods problems (customers are served over a time horizon). That said, targeting a multi-period problem would pose additional challenges. To name a few, one can refer to the underlying computational complexity of methodologies that are suited for this purpose (e.g., dynamic programming). The future research can also focus on modeling customers’ demand based on some distributions. This could also provide an opportunity for developing novel analytical models based on the combination of vehicle routing and newsvendor problems. For the DVRPB, the future research can devise (i) waiting strategies at customers or some strategic locations, and (ii) less-than-truckload concepts in order to increase the rate of served online requests in the real-time. 162 REFERENCES Accorsi, R., Cascini, A., Cholette, S., Manzini, R., & Mora, C. (2014). Economic and environmental assessment of reusable plastic containers: a food catering supply chain case study. International Journal of Production Economics, 152, 88–101. https://doi.org/10.1016/j.ijpe.2013.12.014 Amazon. (2022). Order with prime free same-day delivery. Retrieved from https://www.amazon.com/gp/help/customer/display.html?nodeId=GFUT24ALVAVC6VFD. Accessed October 14, 2022. American Transportation Research Institute (ATRI). (2022). Bringing Reusable Packaging Systems to Life. Retrieved from https://truckingresearch.org/atri-research/operational-costs- of-trucking/. Accessed August 27, 2022. Angeles, R. (2005). RFID technologies: supply chain applications and implementation issues. Information Systems Management, 22 (1), 51-65. https://doi.org/10.1201/1078/44912.22.1.20051201/85739.7 Atamer, B., Bakal, I.S., & Bayındır, Z.P. (2013). Optimal pricing and production decisions in utilizing reusable containers. International Journal of Production Economics, 143, 222–232. https://doi.org/10.1016/j.ijpe.2011.08.007 Automotive Guider. (2022). What size trucks does UPS have? Retrieved from https://www.automotiveguider.com/automotive-guides/what-size-trucks-does-ups-have/. Accessed June 15, 202. Battarra, M., Cordeau, J. F., & Iori, M. (2014). Pickup and delivery problems for goods transportation. In P. Toth & D. Vigo (Eds.), Vehicle routing: Problems, methods, and applications (pp. 161-192). Society for Industrial and Applied Mathematics. Bellman, R. (1958). On a routing problem. Quarterly of applied mathematics, 16(1), 87-90. https://doi.org/10.1090/qam/102435 Belloso, J., Juan, A.A., Martinez, E., & Faulin, J. (2017). A biased‐randomized metaheuristic for the vehicle routing problem with clustered and mixed backhauls. Networks, 69(3), 241-255. https://doi.org/10.1002/net.21734 Belmecheri, F., Prins, C., Yalaoui, F., & Amodeo, L. (2013). Particle swarm optimization algorithm for a vehicle routing problem with heterogeneous fleet, mixed backhauls, and time windows. Journal of intelligent manufacturing, 24(4), 775-789. https://doi.org/10.1007/s10845-012-0627-8 Big Commerce. (2022). Why You Need to Offer Sustainable Packaging – and How to Do It Right. Retrieved from https://www.bigcommerce.com/blog/sustainable-packaging/#what-is- sustainable-packaging. Accessed June 09, 2022. 163 Bojkow, E. (1991). Basic considerations on the calculation of the trippage number for returnable containers. Packaging Technology and Science, 4 (6), 315–331. https://doi.org/10.1002/pts.2770040605 Bortolini, M., Galizia, F. G., Mora, C., Botti, L., & Rosano, M. (2018). Bi-objective design of fresh food supply chain networks with reusable and disposable packaging containers. Journal of Cleaner Production, 184, 375-388. https://doi.org/10.1016/j.jclepro.2018.02.231 Brandão, J. (2006). A new tabu search algorithm for the vehicle routing problem with backhauls. European Journal of Operational Research, 173(2), 540-555. https://doi.org/10.1016/j.ejor.2005.01.042 Brandão, J. (2016). A deterministic iterated local search algorithm for the vehicle routing problem with backhauls. Top, 24, 445-465. https://doi.org/10.1007/s11750-015-0404-x Breen, L. (2006). Give me back my empties or else! A preliminary analysis of customer compliance in reverse logistics practices (UK). Management Research News, 29, 532–551. https://doi.org/10.1108/01409170610708989 Brewer, A., Sloan, N., & Landers, T.L. (1999). Intelligent tracking in manufacturing. Journal of Intelligent Manufacturing, 10 (4/3), 245-50. https://doi.org/10.1023/A:1008995707211 Buchanan, D.J. & Abad, P.L. (1989). Optimal policy for a periodic review returnable inventory system. IIE Transactions, 30 (11), 1049–1055. https://doi.org/10.1023/A:1007507729962 Cakir, F., Street, W. N., & Thomas, B. W. (2015). Revisiting cluster first, route second for the vehicle routing problem. no. August. Carrano, A.L., Pazour, J.A., Roy, D., & Thorn, B.K. (2015). Selection of pallet management strategies based on carbon emissions impact. International Journal of Production Economics, 164 (6), 258–270. https://doi.org/10.1016/j.ijpe.2014.09.037 Carrasco-Gallego, R. & Ponce-Cueto, E. (2009). Forecasting the returns in reusable containers closed-loop supply chains. A case in the LPG industry. In XIII Congreso de Ingeniería de Organización, (311–320). Carrasco-Gallego, R., Ponce-Cueto, E., & Dekker, R. (2012). Closed-loop supply chains of reusable articles: A typology grounded on case studies. International Journal of Production Research, 50(19), 5582–5596. https://doi.org/10.1080/00207543.2011.649861 Chang, K.C., Chen, M.C., Hsu, C.L. & Kuo, N.T. (2010). The effect of service convenience on post‐purchasing behaviours. Industrial Management and Data Systems, 110(9), 1420–1443. https://doi.org/10.1108/02635571011087464 164 Chang, M.S., Chen, S.R., & Hsueh, C.F. (2003). Real-time vehicle routing problem with time windows and simultaneous delivery/pickup demands. Journal of the Eastern Asia Society for Transportation Studies, 5, 2273-2286. Chen, Z.L. & Xu, H. (2006). Dynamic column generation for dynamic vehicle routing with time windows. Transportation Science, 40(1), 74-88. https://doi.org/10.1287/trsc.1050.0133 Cheung, R.K. & Chen, C. (1998). A two-stage stochastic network model and solution methods for the dynamic empty container allocation problem. Transportation Science, 32 (2), 142–162. https://doi.org/10.1287/trsc.32.2.142 Chew, E.P., Huang, H.C., & Horiana. (2002). Performance measures for returnable inventory: a case study. Production Planning and Control, 13 (5), 462–469. https://doi.org/10.1080/09537280210142790 Chonhenchob, V., Kamhangwong, D., & Singh, S.P. (2008). Comparison of reusable and single- use plastic and paper shipping containers for distribution of fresh pineapples. Packaging Technology and Science, 21(2), 73–83. https://doi.org/10.1002/pts.780 Chonhenchob, V. & Singh, S.P. (2003). A comparison of corrugated boxes and re-usable plastic containers for mango distribution. Packaging Technology and Science, 16, 231–237. https://doi.org/10.1002/pts.630 Choong, S.T., Cole, M.H., & Kutanoglu, E. (2002). Empty container management for intermodal transportation networks. Transportation Research Part E: Logistics and Transportation Review, 38 (6), 423–438. https://doi.org/10.1016/S1366-5545(02)00018-2 Closed Loop Partners. (2021). Bringing Reusable Packaging Systems to Life. Retrieved from https://www.closedlooppartners.com/wp-content/uploads/2021/01/CLP_Bringing-Reusable- Packaging-Systems-to-Life.pdf. Accessed August 20, 2022. CNN. (2019). How to solve the world’s plastics problem: Bring back the milk man. Retrieved from https://www.cnn.com/interactive/2019/01/business/loop-reusable-packaging-mission- ahead/index.html. Accessed June 02, 2022. Cobb, B.R. (2016a). Inventory control for returnable transport items in a closed-loop supply chain. Transportation Research Part E: Logistics and Transportation Review, 86: 53-68. https://doi.org/10.1016/j.tre.2015.12.010 Cobb, B.R. (2016b). Estimating cycle time and return rate distributions for returnable transport items. International Journal of Production Research, 54(14) 4356-4367. https://doi.org/10.1080/00207543.2016.1162920 Coelho, P.M., Corona, B., ten Klooster, R., & Worrell, E. (2020). Sustainability of reusable packaging–Current situation and trends. Resources, Conservation and Recycling: X, 6, 100037. https://doi.org/10.1016/j.rcrx.2020.100037 165 Cooper, H. (2015). Research Synthesis and Meta-Analysis: A Step-by-Step Approach. Vol. 2. Thousand Oaks, CA: SAGE Publications, Inc. Cooper, H.M. (2010). Research Synthesis and Meta-Analysis: A Step-By-Step Approach, 4th ed. Sage Publications, Thousand Oaks. Crainic, T.G., Gendreau, M., & Dejax, P. (1993). Dynamic and stochastic models for the allocation of empty containers. Operations Research, 41 (1), 102–126. https://doi.org/10.1287/opre.41.1.102 Croes, G.A. (1958). A method for solving traveling-salesman problems. Operations research, 6(6), 791-812. https://doi.org/10.1287/opre.6.6.791 Cuervo, D.P., Goos, P., Sörensen, K., & Arráiz, E. (2014). An iterated local search algorithm for the vehicle routing problem with backhauls. European Journal of Operational Research, 237(2), 454-464. https://doi.org/10.1016/j.ejor.2014.02.011 De Jonge, P.S. (2004). Making waves: RFID adoption in returnable packaging. LogicaCMG. Deif, I., & Bodin, L.D. (1984). Extension of the clarke and wright algorithm for solving the vehicle routing problem with backhauling. In Proceedings of the Babson Conference on Software Uses in Transportation and Logistics Management, Babson Park, MA. Dejax, P., Benamar, F., Crainic, T.G., & Gendreau, M. (1992). Short term container fleet management: issues, models and tools. In: Proceedings of the 6th World Conference on Transport Research, Lyon, France, 29 June–3 July. Del Castillo, E. & Cochran, J.K. (1996). Optimal short horizon distribution operations in reusable container systems. Journal of the Operational Research Society, 47 (1), 48–60. https://doi.org/10.1057/jors.1996.5 Dell’Amico, M., Righini, G., & Salani, M. (2006). A branch-and-price approach to the vehicle routing problem with simultaneous distribution and collection. Transportation science, 40(2), 235-247. https://doi.org/10.1287/trsc.1050.0118 Department of Justice, Central district of California, 2018. Retrieved from https://www.justice.gov/usao-cdca/pr/recycling-executives-spearheading-scheme-involving- stolen-postal-service-equipment. Di Francesco, M., Crainic, T.G., & Zuddas, P. (2009). The effect of multi-scenario policies on empty container repositioning. Transportation Research Part E: Logistics and Transportation Review, 45(5), 758–770. https://doi.org/10.1016/j.tre.2009.03.001 Dubiel, M. (1996). Costing structures of reusable packaging systems. Packaging Technology and Science, 9, 237–254. https://doi.org/10.1002/(SICI)1099-1522(199609)9:5%3C237::AID- PTS368%3E3.0.CO;2-7 166 Duhaime, R., Riopel, D., & Langevin, A. (2001). Value analysis and optimization of reusable containers at Canada Post. Interfaces, 31 (3), 3–15. https://doi.org/10.1287/inte.31.3.3.9636 Ech-Charrat, M.R. & Amechnoue, K. (2016). Dynamic hybrid approach for reusable containers management in a close-loop supply chain. In Multimedia Computing and Systems (ICMCS), 2016 5th International Conference on (548-553). Ech-Charrat M.R., Amechnoue K., & Zouadi, T. (2017a). Dynamic planning of reusable containers in a close-loop supply chain under carbon emission constraint. International Journal of Supply and Operations Management, 4 (4), 279-297. Ech-Charrat, M.R., Amechnoue, K., & Zouadi, T. (2017b). Genetic algorithm for reusable containers management problem. In International Conference on Advanced Information Technology, Services and Systems (50-58). Springer, Cham. Ech-Charrat, M.R., Amechnoue, K., & Zouadi, T. (2017c). Hybrid resolution approaches for dynamic assignment problem of reusable containers. In MATEC Web of Conferences (105, 00009). EDP Sciences. Elia, V., & Gnoni, M.G. (2015). Designing an effective closed loop system for pallet management. International Journal of Production Economics, 170 (C), 730–740. Ellen MacArthur Foundation. (2019). Reuse – Rethinking Packaging. Retrieved from https://www.ellenmacarthurfoundation.org/publications/reuse. Accessed December 20, 2020. Environmental Defense Fund (EDF). (2015). Green Freight Math: How to Calculate Emissions for a Truck Move. Retrieved from https://business.edf.org/insights/green-freight-math-how-to- calculate-emissions-for-a-truck-move/. Accessed August 27, 2022. Fang, X., Du, Y., & Qiu, Y. (2017). Reducing carbon emissions in a closed-loop production routing problem with simultaneous pickups and deliveries under carbon cap-and-trade. Sustainability, 9(12), 2198. https://doi.org/10.3390/su9122198 Fisher, M.L. & Jaikumar, R. (1981). A generalized assignment heuristic for vehicle routing. Networks, 11, 109-124. https://doi.org/10.1002/net.3230110205 Fisher, M.L. (1981). The Lagrangian relaxation method for solving integer programming problems. Management Science, 27(1), 1-18. https://doi.org/10.1287/mnsc.27.1.1 Fisher, M.L. (1985). An applications oriented guide to Lagrangian relaxation. Interfaces, 15(2), 10-21. https://doi.org/10.1287/inte.15.2.10 Fisher, M.L. (2004). The Lagrangian relaxation method for solving integer programming problems. Management Science, 50, 1861-1871. https://doi.org/10.1287/mnsc.1040.0263 167 Fisher, M.L., Jörnsten, K.O., & Madsen, O.B. (1997). Vehicle routing with time windows: Two optimization algorithms. Operations research, 45(3), 488-492. https://doi.org/10.1287/opre.45.3.488 Florez, H. (1986). Empty container repositioning and leasing: An optimization model. Ph.D. Dissertation, Polytechnic Institute of New York, New York. Ford Jr, L.R. (1956). Network flow theory (No. P-923). Rand Corp Santa Monica Ca. Foster, P., Sindhu, A., & Blundell, D. (2006). A case study to track high value stillages using RFID for an automobile OEM and its supply chain in the manufacturing industry, Proceedings of the 2006 IEEE International Conference on Industrial Informatics, 56–60. Gajpal, Y. & Abad, P.L. (2009). Multi-ant colony system (MACS) for a vehicle routing problem with backhauls. European Journal of Operational Research, 196(1), 102-117. https://doi.org/10.1016/j.ejor.2008.02.025 Gendreau, M., Guertin, F., Potvin, J.Y., & Taillard, É. (1999). Parallel tabu search for real-time vehicle routing and dispatching. Transportation Science, 33(4), 381-390. https://doi.org/10.1287/trsc.33.4.381 Ghaziri, H. & Osman, I.H. (2006). Self-organizing feature maps for the vehicle routing problem with backhauls. Journal of Scheduling, 9(2), 97-114. https://doi.org/10.1007/s10951-006- 6774-z GitHub. (2020). Vehicle-Routing-Problem-with-Backhaul. Retrieved from https://github.com/IrandokhtPVZ/Vehicle-Routing-Problem-with-Backhaul. Accessed August 20, 2022. GitHub. (2022a). Vehicle-Routing-Problem-with-Backhaul. Retrieved from https://github.com/IrandokhtPVZ/Vehicle-Routing-Problem-with-Backhaul. Accessed August 20, 2022. GitHub. (2022b). Vehicle-Routing-Problem-with-Backhaul. Retrieved from https://github.com/IrandokhtPVZ/Vehicle-Routing-Problem-with-Backhaul. Accessed August 20, 2022. Glock, C. H. (2017). Decision support models for managing returnable transport items in supply chains: A systematic literature review. International Journal of Production Economics 183 (Part B): 561–569. Glock, C.H., & Kim, T. (2014). Container management in a single-vendor–multiple-buyer supply chain. Logistics Research, 7, 112. Glock, C.H. & Kim, T. (2016). Safety measures in the joint economic lot size model with returnable transport items. International Journal of Production Economics, 181: 24-33. 168 Goellner, K. N. & Sparrow, E. (2014). An environmental impact comparison of single-use and reusable thermally controlled shipping containers. The International Journal of Life Cycle Assessment, 19 (3): 611–619. Goetschalckx, M. & Jacobs-Blecha, C. (1989). The vehicle routing problem with backhauls. European Journal of Operational Research, 42(1), 39-51. https://doi.org/10.1016/0377- 2217(89)90057-X Goetschalckx, M. & Jacobs-Blecha, C. (1993). The vehicle routing problem with backhauls: Properties and solution algorithms. Georgia Institute of Technology, Atlanta. Technical report. Goh, T.N. & Varaprasad, N. (1986). A statistical methodology for the analysis of the life-cycle of reusable containers. IIE Transactions, 18, 42–47. Gong, I., Lee, K., Kim, J., Min, Y., & Shin, K. (2020). Optimizing vehicle routing for simultaneous delivery and pick-up considering reusable transporting containers: Case of convenience stores. Applied Sciences, 10(12), 4162. https://doi.org/10.3390/app10124162 González-Boubeta, I., Fernández-Vázquez, M., Domínguez-Caamaño, P., & Prado-Prado, J.C. (2018). Economic and environmental packaging sustainability: A case study. Journal of Industrial Engineering and Management, 11(2), 229-238. Gonzalez-Torre, P.L., Adenso-Diaz, B., & Artiba, H. (2004). Environmental and reverse logistics policies in European bottling and packaging firms. International Journal of Production Economics, 88 (1), 95–104. Goudenege, G., Chu, C., & Jemai, Z. (2013). Reusable containers management: from a generic model to an industrial case study. Supply Chain Forum, 14 (2), 26–38. GreenBiz. (2019). Loop’s launch brings reusable packaging to the world’s biggest brands. Retrieved from https://www.greenbiz.com/article/loops-launch-brings-reusable-packaging- worlds-biggest-brands. Accessed April 28, 2022. Grimes-Casey, H.G., Seager, T.P., Theis, T.L., & Powers, S.E. (2007). A game theory framework for cooperative management of refillable and disposable bottle lifecycles. Journal of Cleaner Production, 15 (17), 1618-1627. Guignard, M. & Kim, S. (1987). Lagrangean decomposition: A model yielding stronger Lagrangean bounds. Mathematical Programming, 39(2), 215-228. https://doi.org/10.1007/BF02592954 Gustavo, J.U., Pereira, G.M., Bond, A.J., Viegas, C.V., & Borchardt, M. (2018). Drivers, opportunities and barriers for a retailer in the pursuit of more sustainable packaging redesign. Journal of Cleaner Production, 187, 18-28. https://doi.org/10.1016/j.jclepro.2018.03.197 169 Haghani, A. & Jung, S. (2005). A dynamic vehicle routing problem with time-dependent travel times. Computers and operations research, 32(11), 2959-2986. https://doi.org/10.1016/j.cor.2004.04.013 Hariga, M.A., Glock, C.H., & Kim, T. (2016). Integrated product and container inventory model for a single-vendor single-buyer supply chain with owned and rented returnable transport items. International Journal of Production Research, 54 (7), 1964–1979. Harvard Business Review. (2010). Stop Trying to Delight Your Customers. Retrieved from https://hbr.org/2010/07/stop-trying-to-delight-your-customers. Accessed June 09, 2022. Hellström, D. (2009). The cost and process of implementing RFID technology to manage and control returnable transport items. International Journal of Logistics Research and Applications, 12 (1), 1–21. Hellström, D. & Johansson, O. (2010). The impact of control strategies on the management of returnable transport items. Transportation Research Part E: Logistics and Transportation Review, 46 (6), 1128–1139. Holmberg, K., Joborn, M., & Lundgren, J.T. (1998). Improved empty freight car distribution. Transportation Science, 32 (2), 163–173. Hong, L. (2012). An improved LNS algorithm for real-time vehicle routing problem with time windows. Computers & Operations Research, 39(2), 151-163. https://doi.org/10.1016/j.cor.2011.03.006 Iassinovskaia, G., Limbourg, S., & Riane, F. (2017). The inventory-routing problem of returnable transport items with time windows and simultaneous pickup and delivery in closed-loop supply chains. International Journal of Production Economics, 183, 570-582. https://doi.org/10.1016/j.ijpe.2016.06.024 Ichoua, S., Gendreau, M., & Potvin, J.Y. (2000). Diversion issues in real-time vehicle dispatching. Transportation Science, 34(4), 426-438. https://doi.org/10.1287/trsc.34.4.426.12325 Ilic, A., Ng, J.W.P., Bowman, P., & Staake, T. (2009). The value of RFID for RTI management. Electronic Markets, 19 (2–3), 125–135. Irnich, S., Toth, P., & Vigo, D. (2014). The family of vehicle routing problems. In P. Toth & D. Vigo (Eds.), Vehicle routing: Problems, methods, and applications (pp. 1-33). Society for Industrial and Applied Mathematics. Jain, A., Murty, M., & Flynn, P. (1999). Data clustering: A review. ACM Computing Surveys (CSUR), 31(3), 264-323. https://doi.org/10.1145/331499.331504 170 Johansson, O. & Hellström, D. (2007). The effect of asset visibility on managing returnable transport items. International Journal of Physical Distribution and Logistics Management, 37(10), 799–815. Johnson, D. B. (1973). A note on Dijkstra’s shortest path algorithm. Journal of the ACM (JACM), 20(3), 385–388. https://doi.org/10.1145/321765.321768 Karimi, I.A., Sharafali, M., & Mahalingam, H. (2005). Scheduling tank container movements for chemical logistics. American Institute of Chemical Engineers, 51 (1), 178–197. Katephap, N. & Limnararat, S. (2015). Waste reduction of returnable packaging: A case study of reverse logistics in an auto parts company. Proceeding of the 2015 IEEE International Conference on Industrial Engineering and Engineering Management, 1598 – 1602. https://doi.org/10.1109/IEEM.2015.7385917. Katephap, N. & Limnararat, S. (2017). The operational, economic and environmental benefits of returnable packaging under various reverse logistics arrangements. International Journal of Intelligent Engineering and Systems, 10(5), 210-219. Kelle, P. & Silver, E.A. (1989a). Purchasing policy of new containers considering the random returns of previously issued containers. IIE Transactions, 21(4), 349–354. Kelle, P. & Silver, E.A. (1989b). Forecasting the returns of reusable containers. Journal of Operations Management, 8, 17–35. Kim, T. & Glock, C.H. (2014). On the use of RFID in the management of reusable containers in closed-loop supply chains under stochastic container return quantities. Transportation Research Part E: Logistics and Transportation Review, 64, 12–27. Kim, T., Glock, C.H., & Kwon, Y. (2014). A closed-loop supply chain for deteriorating products under stochastic container return times. Omega, 43, 30–40. Koç, Ç. & Laporte, G. (2018). Vehicle routing with backhauls: Review and research perspectives. Computers & Operations Research, 91, 79-91. https://doi.org/10.1016/j.cor.2017.11.003 Koç, Ç., Laporte, G., & Tükenmez, İ. (2020). A review of vehicle routing with simultaneous pickup and delivery. Computers & Operations Research, 122, 104987. https://doi.org/10.1016/j.cor.2020.104987 Kroon, L. & Vrijens, G. (1995). Returnable containers: an example of reverse logistics. International Journal of Physical Distribution and Logistics Management, 25 (2), 56–68. Kulkarni, R.V. & Bhave, P.R. (1985). Integer programming formulations of vehicle routing problems. European Journal of Operational Research, 20(1), 58-67. https://doi.org/10.1016/0377-2217(85)90284-X 171 Lampe, M. & Strassner, M. (2003). The potential of RFID for moveable asset management, Proceedings of the 5th International Conference on Ubiquitous Computing. Larsen, A., Madsen, O.B.G.D., & Solomon, M. (2002). Partially dynamic vehicle routing—models and algorithms. Journal of the Operational Research Society, 53(6), 637-646. https://doi.org/10.1057/palgrave.jors.2601352 Lee, S.G. & Xu, X. (2004). A simplified life cycle assessment of re-usable and single-use bulk transit packaging. Packaging Technology and Science, 17, 67–83. Leung, S.C.H. & Wu, Y. (2004). A robust optimization model for dynamic empty container allocation problems in an uncertain environment. International Journal of Operations and Quantitative Management, 10 (4), 1–20. Levi, M., Cortesi, S., Vezzoli, C., & Salvia, G. (2011). A Comparative life cycle assessment of disposable and reusable packaging for the distribution of Italian fruit and vegetables. Packaging Technology and Science, 24, 387-400. Li, F., Golden, B., & Wasil, E. (2007). The open vehicle routing problem: Algorithms, large-scale test problems, and computational results. Computers and Operations Research, 34(10), 2918- 2930. https://doi.org/10.1016/j.cor.2005.11.018 Ligterink, N.E., Tavasszy, L.A., & Lange, R. (2012). A velocity and payload dependent emission model for heavy-duty road freight transportation. Transportation Research Part D: Transport and Environment, 17, 487–491. https://doi.org/10.1016/j.trd.2012.05.009 Limbourg, S. & Pirotte, M. (2018). How to include the durability, resale and losses of returnable transport items in their management? Proceedings, Logistics Operations Management, Université du Havre, le Havre, France, http://hdl.handle.net/2268/223968. Lin, J., & Niemeier, D. A. (2003). Regional driving characteristics, regional driving cycles. Transportation Research Part D: Transport and Environment, 8(5), 361–381. https://doi.org/10.1016/S1361-9209(03)00022-1 Lin, S. (1965). Computer solutions of the traveling salesman problem. Bell System Technical Journal, 44(10), 2245-2269. https://doi.org/10.1002/j.1538-7305.1965.tb04146.x Londoño, J.C., Tordecilla, R.D., Martins, L.D.C., & Juan, A.A. (2021). A biased-randomized iterated local search for the vehicle routing problem with optional backhauls. Top, 29(2), 387- 416. https://doi.org/10.1007/s11750-020-00558-x Loop & TerraCycle (Kelsey Moffitt – Manager, Industrial Design) (2022). Personal communication. Lu, G., Zhou, X., Mahmoudi, M., Shi, T., & Peng, Q. (2019). Optimizing resource recharging location-routing plans: A resource-space-time network modeling framework for railway 172 locomotive refueling applications. Computers & Industrial Engineering, 127, 1241-1258. https://doi.org/10.1016/j.cie.2018.03.015 Lützebauer, M. (1993). Systems for returnable transport packaging (in German: Mehrwegsysteme für Transportverpackungen), Deutscher Fachverlag GmbH, Frankfurt am Main. Mahmoudi, M., Chen, J., Shi, T., Zhang, Y., & Zhou, X. (2019a). A cumulative service state representation for the pickup and delivery problem with transfers. Transportation Research Part B: Methodological, 129, 351–380. https://doi.org/10.1016/j.trb.2019.09.015 Mahmoudi, M., Song, Y., Miller, H. J., & Zhou, X. (2019b). Accessibility with time and resource constraints: Computing hyper-prisms for sustainable transportation planning. Computers, Environment and Urban Systems, 73, 171–183. https://doi.org/10.1016/j.compenvurbsys.2018.10.002 Mahmoudi, M. & Zhou, X. (2016). Finding optimal solutions for vehicle routing problem with pickup and delivery services with time windows: A dynamic programming approach based on state–space–time network representations. Transportation Research Part B: Methodological, 89, 19-42. https://doi.org/10.1016/j.trb.2016.03.009 Makower, J. (2019). Two steps forward: Loop’s launch brings reusable packaging to the world’s biggest brands. Retrieved from https://www.greenbiz.com/article/loops-launch-brings- reusable-packaging-worlds-biggest-brands. Maleki, R. A. & Meiser, G. (2011). Managing returnable containers logistics-A case study: Part II-Improving visibility through using automatic identification technologies. International Journal of Engineering Business Management, 3(2), 45–54. Maleki, R.A. & Reimche, J. (2011). Managing returnable containers logistics-a case study part I- physical and information flow analysis. International Journal of Engineering Business Management, 3 (2), 1–8. Mason, A., Shaw, A., & Al-Shamma’a, A. (2012). Peer-to-peer inventory management of returnable transport items: a design science approach. Computers in Industry, 63 (3), 265– 274. Mayring, P. (2010). Qualitative Inhaltsanalyse. Handbuch Qualitative Forschung in Der Psychologie, 601–613. Berlin: Springer. McFarlane, D. & Sheffi, Y. (2003). The impact of automatic identification on supply chain operations, International Journal of Logistics Management, 14(1), 1-17. McKerrow, D. (1996). What makes reusable packaging systems work. Logistics Information Management, 9 (4), 39–42. 173 Menesatti, P., Canali, E., Sperandio, G., Burchi, G., Devlin, G., & Costa, C. (2012). Cost and waste comparison of reusable and disposable shipping containers for cut flowers. Packaging Technology and Science, 25 (4), 203–215. Miller, C.E., Tucker, A.W., & Zemlin, R.A. (1960). Integer programming formulation of traveling salesman problems. Journal of the Association for Computing Machinery, 7(4), 326-329. https://doi.org/10.1145/321043.321046 Mingozzi, A., Giorgi, S., & Baldacci, R. (1999). An exact method for the vehicle routing problem with backhauls. Transportation Science, 33(3), 315-329. https://doi.org/10.1287/trsc.33.3.315 Minner, S. & Lindner, G. (2004). Lot sizing decisions in product recovery management in reverse logistics: Quantitative models for closed-loop supply chains, R. Dekker, M. Fleischmann, K. In-derfurth, L.N. Van Wassenhove (eds.), Springer-Verlag. Mitrovi´c-Mini´c, S., Krishnamurti, R., & Laporte, G. (2004). Double-horizon based heuristics for the dynamic pickup and delivery problem with time windows. Transportation Research Part B: Methodological, 38(8), 669-685. https://doi.org/10.1016/j.trb.2003.09.001 Mitrović-Minić, S. & Laporte, G. (2004). Waiting strategies for the dynamic pickup and delivery problem with time windows. Transportation Research Part B: Methodological, 38(7), 635- 655. https://doi.org/10.1016/j.trb.2003.09.002 Mollenkopf, D., Closs, D., Twede, D., Lee, S., & Burgess, G. (2005). Assessing the viability of reusable packaging: a relative cost approach. Journal of Business Logistics, 26 (1), 169–197. Montemanni, R., Gambardella, L.M., Rizzoli, A.E., & Donati, A.V. (2005). Ant colony system for a dynamic vehicle routing problem. Journal of Combinatorial Optimization, 10(4), 327-343. https://doi.org/10.1007/s10878-005-4922-6 Mosheiov, G. (1998). Vehicle routing with pick-up and delivery: tour-partitioning heuristics. Computers & Industrial Engineering, 34(3), 669–684. https://doi.org/10.1016/S0360- 8352(97)00275-1 National Association of Counties. (2022). USDA seeks to grow circular economy in rural areas. Retrieved from https://www.naco.org/articles/usda-seeks-grow-circular-economy-rural- areas. Accessed August 23, 2022. Ninikas, G. & Minis, I. (2014). Reoptimization strategies for a dynamic vehicle routing problem with mixed backhauls. Networks, 64(3), 214-231. https://doi.org/10.1002/net.21567 Osman, I.H. & Wassan, N.A. (2002). A reactive tabu search meta‐heuristic for the vehicle routing problem with backhauls. Journal of Scheduling, 5(4), 263-285. https://doi.org/10.1002/jos.122 174 Palsson, H. (2018). Packaging logistics: Understanding and managing the economic and environmental impacts of packaging in supply chains. Kogan Page Publishers. Palsson, H., Finnsgard, C., & Wänström, C. (2013). Selection of packaging systems in supply chains from a sustainability perspective: The case of Volvo. Packaging Technology and Science, 26 (5), 289–310. Parragh, S.N., Doerner, K.F., & Hartl, R.F. (2008a). A survey on pickup and delivery problems. part I: Transportation between customers and depot. Journal für Betriebswirtschaft, 58(2), 81-117. Parragh, S.N., Doerner, K.F., & Hartl, R.F. (2008b). A survey on pickup and delivery problems. Journal für Betriebswirtschaft, 58, 21–51. https://doi.org/10.1007/s11301-008-0033-7 Progressive Grocer. (2020). Tackling Supply Chain Challenges with Reusable Transport Packaging. Retrieved from https://progressivegrocer.com/tackling-supply-chain-challenges- reusable-transport-packaging. Accessed June 01, 2022. Psaraftis, H.N., Wen, M., & Kontovas, C.A. (2016). Dynamic vehicle routing problems: Three decades and counting. Networks, 67(1), 3-31. https://doi.org/10.1002/net.21628 Queiroga, E., Frota, Y., Sadykov, R., Subramanian, A., Uchoa, E., & Vidal, T. (2020). On the exact solution of vehicle routing problems with backhauls. European Journal of Operational Research, 287(1), 76-89. https://doi.org/10.1016/j.ejor.2020.04.047 Ray, C.D., Michael, J.H., & Scholnick, B.N. (2006). Supply-chain system costs of alternative grocery industry pallet systems. Forest Products Journal, 56 (10), 52–57. Reusable Packaging Association. (2019). A Smarter, technology-driven supply chain with reusable packaging systems, Retrieved from https://reusables.org/download-asset- technologies-white-paper/. Rodrigue, J.P. (2020). The geography of transport systems. Routledge. Ropke, S. & Pisinger, D. (2006). A unified heuristic for a large class of vehicle routing problems with backhauls. European Journal of Operational Research, 171(3), 750-775. https://doi.org/10.1016/j.ejor.2004.09.004 Rosenau, W.V., Twede, D., Mazzeo, M.A., & Singh, S.P. (1996). Returnable/reusable logistical packaging: a capital budgeting investment decision framework. Journal of Business Logistics, 17(2), 139-165. Ross, S. & Evans, D. (2003). The environmental effect of reusing and recycling a plastic based packaging system. Journal of Cleaner Production, 11, 561–571. 175 Rubio, S., et al. (2009). Implementing a reverse logistics system: a case study. International Journal of Procurement Management, 2 (4), 346–357. Samaras, C., Tsokolis, D., Toffolo, S., Magra, G., Ntziachristos, L., & Samaras, Z. (2018). Improving fuel consumption and CO2 emissions calculations in urban areas by coupling a dynamic micro traffic model with an instantaneous emissions model. Transportation Research Part D: Transport and Environment, 65, 772–783. https://doi.org/10.1016/j.trd.2017.10.016 Sanei, M., Mahmoodirad, A., Niroomand, S., Jamalian, A., & Gelareh, S. (2017). Step fixed- charge solid transportation problem: A Lagrangian relaxation heuristic approach. Computational and Applied Mathematics, 36(3), 1217-1237. https://doi.org/10.1007/s40314-015-0293-5 Sarkar, B., Ullah, M., & Kim, N. (2017). Environmental and economic assessment of closed-loop supply chain with remanufacturing and returnable transport items. Computers and Industrial Engineering, 111, 148-163. Schrady, D.A. (1967). A deterministic inventory model for repairable items. Naval Research Logistics Quarterly, 14 (3), 391–398. Shayan, M.K. & Ghotb, F. (2000). The importance of information technology in port terminal operations, International Journal of Physical Distribution and Logistics Management, 30(3/4), 331-44. Shi, S., Lin, N., Zhang, Y., Cheng, J., Huang, C., Liu, L., & Lu, B. (2016). Research on Markov property analysis of driving cycles and its application. Transportation Research Part D: Transport and Environment, 47, 171–181. https://doi.org/10.1016/j.trd.2016.05.013 Shieh, H.M. & May, M.D. (1998). On-line vehicle routing with time windows: Optimization-based heuristics approach for freight demands requested in real-time. Transportation Research Record, 1617(1), 171-178. https://doi.org/10.3141%2F1617-24 Silva, D.A.L., Renó, G.W.S., Sevegnani, G., Sevegnani, T.B., & Truzzi, O.M.S. (2013). Comparison of disposable and returnable packaging: A case study of reverse logistics in Brazil. Journal of Cleaner Production, 47, 377–387. Singh, S.P., Chonhenchob, V., & Singh, J. (2006). Life cycle inventory and analysis of re‐usable plastic containers and display‐ready corrugated containers used for packaging fresh fruits and vegetables. Packaging Technology and Science, 19(5), 279–293. Soysal, M. (2016). Closed-loop Inventory Routing Problem for returnable transport items. Transportation Research Part D: Transport and Environment, 48, 31-45. https://doi.org/10.1016/j.trd.2016.07.001 176 StopWaste, Reusable Pallet & Container Coalition (RPCC). (2007). Reusables 101: Think outside the box: think reusable. Retrieved from http://usereusables.org/sites/default/files/reusables101.pdf. Subramanian, A. & Queiroga, E. (2020). Solution strategies for the vehicle routing problem with backhauls. Optimization Letters, 1-13. Subramanian, A., Uchoa, E., Pessoa, A.A., & Ochi, L.S. (2011). Branch-and-cut with lazy separation for the vehicle routing problem with simultaneous pickup and delivery. Operations Research Letters, 39(5), 338-341. https://doi.org/10.1016/j.orl.2011.06.012 Sustainable Packaging Coalition (SPC). (2011). Definition of sustainable packaging. Retrieved from https://sustainablepackaging.org/wp-content/uploads/2017/09/Definition-of- Sustainable-Packaging.pdf. Accessed May 10, 2022. Temper Pack. (2020). Reusable Packaging: Great Potential, But Challenges Ahead. Retrieved from https://www.temperpack.com/reusable-packaging-great-potential-but-challenges-ahead/. Accessed June 09, 2022. Thoroe, L., Melski, A., & Schumann, M. (2009). The impact of RFID on management of returnable containers. Electronic Markets, 19 (2–3), 115–124. TIME. (2021). Reusable Packaging Is the Latest Eco-Friendly Trend. But Does It Actually Make a Difference? Retrieved from https://time.com/6101846/is-reusable-packaging-sustainable/. Accessed June 02, 2022. Ting, C.K. & Liao, X.L. (2013). The selective pickup and delivery problem: Formulation and a memetic algorithm. International Journal of Production Economics, 141(1), 199-211. https://doi.org/10.1016/j.ijpe.2012.06.009 Toktay, L.B., Wein, L.M., & Zenios, S.A. (2000). Inventory management of remanufacturable products. Management Science, 46 (11), 1412–1426. Toth, P. & Vigo, D. (1996). Heuristic algorithms for the vehicle routing problem with backhauls. In L. Bianco & P. Toth (Eds.), Advanced methods in transportation analysis (pp. 585–608). Springer-Verlag, Berlin. Toth, P. & Vigo, D. (1997). An exact algorithm for the vehicle routing problem with backhauls. Transportation Science, 31(4), 372-385. https://doi.org/10.1287/trsc.31.4.372 Toth, P. & Vigo, D. (1999). A heuristic algorithm for the symmetric and asymmetric vehicle routing problems with backhauls. European Journal of Operational Research, 113(3), 528- 543. https://doi.org/10.1016/S0377-2217(98)00086-1 Toth, P. & Vigo, D. (2002a). VRP with backhauls. In P. Toth & D. Vigo (Eds.), The vehicle routing problem (pp.1-26). Society for Industrial and Applied Mathematics. 177 Toth, P. & Vigo, D. (2002b). VRP with backhauls. In P. Toth & D. Vigo (Eds.), The vehicle routing problem (pp.195-224). Society for Industrial and Applied Mathematics. Tranfield, D., Denyer, D., & Smart, P. (2003). Towards a methodology for developing evidence- informed management knowledge by means of systematic review. Br. J. Manag, 14 (3), 207- 222. Tsiliyannis, C.A. (2005a). Parametric analysis of environmental performance of reused/recycled packaging. Environmental Science and Technology, 39, 9770–9777. Tsiliyannis, C.A. (2005b). A new rate index for environmental monitoring of combined reuse/recycle packaging systems. Waste Management and Research, 23, 304–313. Tütüncü, G.Y., Carreto, C.A., & Baker, B.M. (2009). A visual interactive approach to classical and mixed vehicle routing problems with backhauls. Omega, 37(1), 138-154. https://doi.org/10.1016/j.omega.2006.11.001 Twede, D. (1999). Can you justify returnables? Transportation and Distribution, 40 (4), 85–88. Twede, D. & Clarke, R. (2004). Supply chain issues in reusable packaging. Journal of Marketing Channels, 12 (1), 7–26. U.S. Environmental Protection Agency (EPA). (2020). Advancing sustainable materials management: Facts and figures report, 2018 facts and figures fact sheet. Retrieved from https://www.epa.gov/facts-and-figures-about-materials-waste-and-recycling/advancing- sustainable-materials-management. Accessed April 27, 2022. U.S. Environmental Protection Agency (EPA). (2022). Overview of Greenhouse Gases. Retrieved from https://www.epa.gov/ghgemissions/overview-greenhouse-gases. Accessed August 27, 2022. Van Doorsselaer, K. & Lox, F. (1999). Estimation of the energy needs in the life cycle analysis of one-way and returnable glass packaging. Packaging Technology and Science, 12 (5), 235e239. Vidal, T., Crainic, T.G., Gendreau, M., & Prins, C. (2014). A unified solution framework for multi- attribute vehicle routing problems. European Journal of Operational Research, 234(3), 658- 673. https://doi.org/10.1016/j.ejor.2013.09.045 Vijayaraman, B.S. & Osyk, B.A. (2006). An empirical study of RFID implementation in the warehousing industry. International Journal of Logistics Management, 17(1), 6-20. Wang, H.F. & Chen, Y.Y. (2013). A coevolutionary algorithm for the flexible delivery and pickup problem with time windows. International Journal of Production Economics, 141(1), 4-13. https://doi.org/10.1016/j.ijpe.2012.04.011 178 Wang, X. & Cao, H. (2008). A dynamic vehicle routing problem with backhaul and time window. In 2008 IEEE International Conference on Service Operations and Logistics, and Informatics, 1, 1256-1261. https://doi.org/10.1109/SOLI.2008.4686592 Wang, Y., Peng, S., Zhou, X., Mahmoudi, M., & Zhen, L. (2020). Green logistics location-routing problem with eco-packages. Transportation Research Part E: Logistics and Transportation Review, 143, 102118. https://doi.org/10.1016/j.tre.2020.102118 Wassan, N. (2007). Reactive tabu adaptive memory programming search for the vehicle routing problem with backhauls. Journal of the Operational Research Society, 58(12), 1630-1641. https://doi.org/10.1057/palgrave.jors.2602313 Wu, W., Tian, Y., & Jin, T. (2016). A label based ant colony algorithm for heterogeneous vehicle routing with mixed backhaul. Applied Soft Computing, 47, 224-234. https://doi.org/10.1016/j.asoc.2016.05.011 Yalcın, G.D. & Erginel, N. (2015). Fuzzy multi-objective programming algorithm for vehicle routing problems with backhauls. Expert Systems with Applications, 42(13), 5632-5644. https://doi.org/10.1016/j.eswa.2015.02.060 Yang, T., Fu, C., Liu, X., Pei, J., Liu, L., & Pardalos, P. (2018). Closed-loop supply chain inventory management with recovery information of reusable containers. Journal of Combinatorial Optimization, 35, 266-292. Yao, Y., Zhu, X., Dong, H., Wu, S., Wu, H., Tong, L.C., & Zhou, X. (2019). ADMM-based problem decomposition scheme for vehicle routing problem with time windows. Transportation Research Part B: Methodological, 129, 156-174. https://doi.org/10.1016/j.trb.2019.09.009 Young, R.R., Swan, P.F., & Burn, R.H. (2002). Private railcar fleet operations: the problem of excessive customer holding time in the chemicals and plastics industries. Transportation Journal, Fall, 51–59. Yue, C., Hall, C.R., Behe, B.K., Campbell, B.L., Lopez, R.G., & Dennis, J.H. (2010a_. Investigating consumer preference for biodegradable containers. Journal of Environmental Horticulture, 28(4), 234–243. Yue, C., Hall, C.R., Behe, B.K., Campbell, B., Dennis, J.H., & Lopez, R.G. (2010b). Are consumers willing to pay more for biodegradable containers than for plastic ones? Evidence from hypothetical conjoint analysis and non-hypothetical experimental auctions. Journal of Agricultural and Applied Economics, 42, 757–772. Zachariadis, E.E. & Kiranoudis, C.T. (2012). An effective local search approach for the vehicle routing problem with backhauls. Expert Systems with Applications, 39(3), 3174-3184. https://doi.org/10.1016/j.eswa.2011.09.004 179 Zeng, K. (2006). Dynamic vehicle routing problem with backhaul and time window and its application in the less-than-truckload (LTL) trucking industry (Doctoral dissertation, University of Cincinnati). Zhang, Q., Segerstedt, A., Tsao, Y.-C., & Liu, B. (2015). Returnable packaging management in automotive parts logistics: dedicated mode and shared mode. International Journal of Production Economics, 168 (10), 234–244. Zhou, X., Tanvir, S., Lei, H., Taylor, J., Liu, B., Rouphail, N. M., & Frey, H. C. (2015). Integrating a simplified emission estimation model and mesoscopic dynamic traffic simulator to efficiently evaluate emission impacts of traffic management strategies. Transportation Research Part D: Transport and Environment, 37, 123–136. https://doi.org/10.1016/j.trd.2015.04.013 Zhou, X., Tong, L., Mahmoudi, M., Zhuge, L., Yao, Y., Zhang, Y., Shang, P., Liu, J., & Shi, T. (2018). Open-source VRPLite package for vehicle routing with pickup and delivery: a path finding engine for scheduled transportation systems. Urban Rail Transit, 4(2), 68-85. https://doi.org/10.1007/s40864-018-0083-7 Zhu, Z., Xiao, J., He, S., Ji, Z., & Sun, Y. (2016). A multi-objective memetic algorithm based on locality-sensitive hashing for one-to-many-to-one dynamic pickup-and-delivery problem. Information Sciences, 329, 73-89. https://doi.org/10.1016/j.ins.2015.09.006 180 APPENDIX A. ROBUSTNESS CHECKS FOR CHAPTER 4 Appendix A.1. Comparison with the Benchmark (CPLEX optimization solver) Lansing network Charlotte network (a) Backhaul solution, # requests = 40, vehicle’s capacity = 3 (b) Mixed solution, # requests = 40, vehicle’s capacity = 30 (c) Backhaul solution, # requests = 40, vehicle’s capacity = 10 (d) Mixed solution, # requests = 40, vehicle’s capacity = 10 Figure A.1. Comparison between our heuristic approach and the CPLEX optimization solver. Notes. Backhaul solution: delivery customers are served before pickup ones. Mixed solution: there is no priority in serving delivery/pickup customers. 181 Appendix A.2. Performance of the Proposed Solution Approach Lansing network Charlotte network (a) # requests = 800, vehicle’s capacity = 30 (b) # requests = 120, vehicle’s capacity = 30 (c) # requests = 800, vehicle’s capacity = 50 (d) # requests = 120, vehicle’s capacity = 50 (e) # requests = 1600, vehicle’s capacity = 50 (f) # requests = 240, vehicle’s capacity = 50 (g) # requests = 2400, vehicle’s capacity = 50 (h) # requests = 3600, vehicle’s capacity = 50 Figure A.2. Distance traveled under different solutions and percentage of simultaneous pickup/delivery requests. Notes. Backhaul solution: delivery customers are served before pickup ones. Mixed solution: there is no priority in serving delivery/pickup customers. 182 Appendix A.3. Economic and Environmental Impacts Lansing network Charlotte network # requests = 800, vehicle’s capacity = 30 # requests = 120, vehicle’s capacity = 30 # requests = 800, vehicle’s capacity = 50 # requests = 120, vehicle’s capacity = 50 # requests = 1600, vehicle’s capacity = 30 # requests = 240, vehicle’s capacity = 30 # requests = 2400, vehicle’s capacity = 30 # requests = 360, vehicle’s capacity = 30 Figure A.3. Simulation: daily transportation cost and emission amount under different solutions. Notes. Distribution for transportation cost = 𝑁(2,0.2) (unit: $/mile). Backhaul solution: delivery customers are served before pickup ones. Mixed solution: there is no priority in serving delivery/pickup customers. 183 Figure A.3. (cont’d) Lansing network Charlotte network # requests = 800, vehicle’s capacity = 30 # requests = 120, vehicle’s capacity = 30 # requests = 800, vehicle’s capacity = 50 # requests = 120, vehicle’s capacity = 50 # requests = 1600, vehicle’s capacity = 30 # requests = 240, vehicle’s capacity = 30 # requests = 2400, vehicle’s capacity = 30 # requests = 360, vehicle’s capacity = 30 Notes. Distribution for transportation cost = 𝑈[1.5,2.5] (unit: $/mile). 184 Figure A.3. (cont’d) Lansing network Charlotte network # requests = 800, vehicle’s capacity = 30 # requests = 120, vehicle’s capacity = 30 # requests = 800, vehicle’s capacity = 50 # requests = 120, vehicle’s capacity = 50 # requests = 1600, vehicle’s capacity = 30 # requests = 240, vehicle’s capacity = 30 # requests = 2400, vehicle’s capacity = 30 # requests = 360, vehicle’s capacity = 30 Notes. Distribution for CO2 emission = 𝑁(1700,200) (unit: gram/mile). 185 Figure A.3. (cont’d) Lansing network Charlotte network # requests = 800, vehicle’s capacity = 30 # requests = 120, vehicle’s capacity = 30 # requests = 800, vehicle’s capacity = 50 # requests = 120, vehicle’s capacity = 50 # requests = 1600, vehicle’s capacity = 30 # requests = 240, vehicle’s capacity = 30 # requests = 2400, vehicle’s capacity = 30 # requests = 360, vehicle’s capacity = 30 Notes. Distribution for CO2 emission = 𝑈[1300,2100] (unit: gram/mile). 186 Figure A.3. (cont’d) Lansing network Charlotte network # requests = 800, vehicle’s capacity = 30 # requests = 120, vehicle’s capacity = 30 # requests = 800, vehicle’s capacity = 50 # requests = 120, vehicle’s capacity = 50 # requests = 1600, vehicle’s capacity = 30 # requests = 240, vehicle’s capacity = 30 # requests = 2400, vehicle’s capacity = 30 # requests = 360, vehicle’s capacity = 30 Notes. Distribution for CO2 emission = 𝑊(11,1900) (unit: gram/mile). 187 APPENDIX B. OTHER NUMERICAL RESULTS FOR CHAPTER 4 Lansing network Charlotte network # requests = 800, vehicle’s capacity = 30 # requests = 120, vehicle’s capacity = 30 # requests = 800, vehicle’s capacity = 50 # requests = 120, vehicle’s capacity = 50 # requests = 1600, vehicle’s capacity = 50 # requests = 240, vehicle’s capacity = 50 # requests = 2400, vehicle’s capacity = 50 # requests = 360, vehicle’s capacity = 50 Figure B.1. Computational time under different solutions and percentage of simultaneous pickup/delivery requests. Notes. Backhaul solution: delivery customers are served before pickup ones. Mixed solution: there is no priority in serving delivery/pickup customers. 188 APPENDIX C. DETAILS ON VEHICLE’S CAPACITY According to AutomotiveGuider (2022), the dimension of a typical delivery truck (like the one used by the UPS) is 𝐿 ∗ 𝑊 ∗ 𝐻 = 26 ∗ 10 ∗ 7 (unit: ft), which is equal to 312 ∗ 120 ∗ 84 (unit: inch). Also, according to Loop & TerraCycle (2022), the dimension of original totes utilized by the Loop is 𝐿 ∗ 𝑊 ∗ 𝐻 = 15 ∗ 16 ∗ 19 (unit: inch). Accounting for the maneuverability of a truck operator and the fact that there are typically 2 shelves with 2-3 decks on each shelf, any number between 8 ∗ 3 ∗ 2 = 48 to 10 ∗ 3 ∗ 2 = 60 totes can be handled by a truck. To reflect this, we consider the capacity of 50 totes in our baseline experiment. 189