AGGREGATE PLANNING IN MANUFACTURING OF REUSABLE CONTAINERS By Jinli Tao A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Packaging - Master of Science 202 1 AGGREGATE PLANNING IN MANUFACTURING OF REUSABLE CONTAINERS By Jinli Tao Aggregate production planning (APP) is a method to make several decisions simultaneously on production, inventory, and workforce levels over a finite time horizon , aiming to maximize the profit or minimize the cost while meeting fluctuating demands . Building mathematical models that reflect real - world problems is often difficult, as the constraints are usually intricate and may interact with each other. Decomposing the interconnected system into a number of independent phases could simplify the problem; however, it may not guarantee the optimality of the best solutions due to the missed constraints between stages . In this study , two mixed integer programming models for the manufacturin g of reusable plastic containers are presented. O ne is based on the flow of the material and the other is based on the level of the workforce at each period. The proposed models are able to (i) deal with varying demand, (ii) reflect various regulations and restrictions of public and private warehouses for storing materials, and (iii) identify the importance of subcontracting when demand increases dramatically. B oth mathematical models are implemented in the case of packaging manufacturing . A comprehensive sensitivity analysis are conducted on different parameters of the problem to test the effect of their changes . To sum up, t he general framework of the mathematical models not only can be used for the reusable con tainer manufacturing but also the manufacturing of any type of product with a similar supply chain network. iii I would like to thank my advisor, Dr. Monireh Mahmoudi, for all her help and guidance that she has given me over the past two yea rs. I would like to express my gratitude to the members of my examination committee, Dr. Alireza Boloori, and Dr. Euihark Lee. I would also like to thank the School of Packaging faculty for supporting me during these two years of my graduate study. Finall y, I would like to thank my husband who has always supported me, and my mother who helped me take care of my two children. iv LIST OF TABLES ................................ ................................ ................................ ......................... vi LIST OF FIGURES ................................ ................................ ................................ ...................... vii KEY TO ABBREVIATIONS ................................ ................................ ................................ ........ ix CHAPTER 1 - INTRODUCTION ................................ ................................ ................................ .. 1 CHAPTER 2 - LITERATURE REVIEW ................................ ................................ ....................... 3 CHAP TER 3 - FORECASTING METHODS ................................ ................................ ................ 8 Forecasting in Supply Chain Planning ................................ ................................ ......................... 8 Forecasting Techniques ................................ ................................ ................................ ................ 9 CHAPTER 4 - AGGREGATE PLANNING ................................ ................................ ................ 17 CH APTER 5 - PROBLEM STATEMENT ................................ ................................ .................. 22 Problem Statement ................................ ................................ ................................ ..................... 22 Production Planning Optimization ................................ ................................ ............................. 27 A mathematical model based on the flow of material ................................ ............................. 27 A mathematical model based on the working hours of machines. ................................ .......... 30 CHAPTER 6 - COMPUTATIONAL EXPERIMENTS ................................ ............................... 33 Forecasting Demands with the Static Method ................................ ................................ ........... 34 Forecasting Demand with Adaptive Methods ................................ ................................ ............ 38 Forecasting demands with moving average method ................................ ............................... 38 Forecasting demands with the simple exponential smoothing method ................................ ... 41 ................................ ................................ ................. 44 ................................ ................................ ............. 47 Error Measurement for the Forecasting ................................ ................................ ..................... 50 Computational Result ................................ ................................ ................................ ................. 53 CHAPTER 7 - SENSITIVITY ANALYSIS ................................ ................................ ................. 57 CHAPTER 8 - CONCLUSION ................................ ................................ ................................ .... 60 APPENDICES ................................ ................................ ................................ .............................. 62 APPENDIX A: Forecasted demand for Black and Clear Containers ................................ ........ 63 APPENDIX B: The Optimal APP Solution s ................................ ................................ ............. 64 APPENDIX C: GAMS Programming Code for the APP Model Based on the Material Flow . 66 APPENDIX D: GAMS Programming Code for the APP Model Based on the Flow of Workforce Level ................................ ................................ ................................ ................................ ........... 70 v BIBLIOGRAPHY ................................ ................................ ................................ ......................... 74 vi Table 3.1: Comparison of adaptive forecasting methods ................................ .............................. 10 Table 3.2: Definition of factors in the systematic component ................................ ...................... 10 Table 5.1: Notations used for both mathematical programming models ................................ ...... 25 Table 5.2: Variables used for the mathematical modeling of the aggregate planning based on the material flow ................................ ................................ ................................ ........................... 29 Table 5.3: Variables used for the mathematical modeling of the aggregate planning based on the working hours of the equipment ................................ ................................ ............................. 32 Table 6.1: Historical demand data ................................ ................................ ................................ 33 Table 6.2: Input range for obtaining trendline ................................ ................................ .............. 35 Table 6.3: Formulas for forecasting demands using the static method ................................ ......... 38 Table 6.4: Formulas for forecasting demands using the moving average method ....................... 40 Table 6.5: Formulas for forecasting with simple exponential smoothing method ....................... 43 ................................ ............... 46 Table 6.7: Formulas fo ................................ ........... 49 Table 6.8: Estimation of errors using adaptive methods ................................ ............................... 51 ................................ ............................... 52 Table 6.10: Parameters used in both models ................................ ................................ ................ 53 Table 7.1: Results of sensitivity an alysis. ................................ ................................ ..................... 57 Table A: Forecasted demand for black and clear containers over 18 years ................................ .. 63 Table B1: The optimal solution for the APP model based on the flow of materials .................... 64 Table B2: The optimal solution for the APP model based on workforce level ............................ 65 vii Figure 5.1: A principal scheme of an extruder ................................ ................................ ............. 23 Figure 5.2: A scheme for mechanical thermoforming press ................................ ......................... 23 Figure 5.3: The manufacturing process of plastic containers ................................ ...................... 24 Figure 6.1: Historical demand data ................................ ................................ ............................... 33 Figure 6.2: Linear regression for the deseasonalized demands of black plastics and the quarter number ................................ ................................ ................................ ................................ .......... 35 Figure 6.3: Linear regression for the deseasonalized demands of clear plastics and the quarter number ................................ ................................ ................................ ................................ .......... 35 Figure 6.4: Forecast demands for black plastic containers using the static method ..................... 37 Figure 6.5: Forecast demands for clear plastic containers using the static method ...................... 37 Figure 6.6: Forecast demands for black plastic containers using four - period moving average method ................................ ................................ ................................ ................................ ........... 39 Figure 6.7: Forecast demands for clear plastic containers using four - period moving average method ................................ ................................ ................................ ................................ ........... 40 Figure 6.8: Forecasting results for black plastic containers using the simple exponential smoothing method ................................ ................................ ................................ ......................... 42 Figure 6.9: Forecasting results for clear plastic containers using the simple exponential smoothing method ................................ ................................ ................................ ......................... 43 ........................ 45 .......................... 46 ......................... 48 .......................... 49 Figu re 6.14: Comparison of different adaptive methods for black containers ............................. 50 viii Figure 6.15: Comparison of different adaptive methods for clear containers .............................. 51 Figure 6.16: Estimated historical and forecasting future demands for clear plastic containers using ................................ ................................ ............ 52 Figure 6.17: Estimated historical and forecasting future demands for black plastic contain ers ................................ ................................ ............ 52 Figure 6.18: The optimal solution for the extruding and warehousing processes ob tained from solving the APP model based on the flow of the material ................................ ............................ 54 Figure 6.19: The optimal solution for the thermoforming proces s from solving the APP model based on the flow of the material ................................ ................................ ................................ .. 54 Figure 6.20: The optimal solution for the extruding and warehousing processes obtained from solving the APP model based on the working hour ................................ ................................ ...... 55 Figure 6.21: The optimal solutions for the the rmoforming process from solving the APP model based on the working hour ................................ ................................ ................................ . 55 Figure 7.1: Sensitivity analysis over various parameters ................................ .............................. 57 ix APP : Aggregate Production Planning MAPE : Mean Average Percentage Error MAD : Mean Average Deviation TS : Tracking Signal SKU : S tock - Keeping - Unit GAMS: General Algebraic Modeling System 1 It is generally a challenging task to manag e a multiple - stage manufacturing process to meet customer needs while keeping costs as low as possible through mathematical modeling , as each process may have its considerations (e.g., particular machinery, process capacity, pla nt capacity, and trained workforce) , and the complexity of the various constraints . A ggregating all steps such that there will not be any shortage or surplus in terms of material and/or workforce at each stage and period of manufacturing increase the size of the modeling to reflect the real - world situations . Modeling the phases separately could simplify the problems because it could red uce the number of constraints and variables involved. However, the result may not be accurate due to the lost connection between various segments of the process es . Most of the existing research ing papers mathematically model such problems based on material flow . In this work , two mixed - integer programming models are presented , one based on material flow and the other based on working hours, to determine the optimal material and working time flow between the stages of the manufacturing process and the optima l workforce assigned to each phase . The models are applied to a case of packaging manufacturing. The consistent results obtained from both models prove the feasibility of the model based on working hour s . Besides , the models reflect various considerations for public and private storage. Finally, a comprehensive analysis is performed to examine the effect of various parameters , such as the length of the planning horizon, the number of available extruders, the annual increase in the raw material price, the labor costs, and the subcontracting cost on the optimal solution. The labor costs are proved to be the most sensitive factor, and the investment of extra extruder s over the planning horizon is not so necessary in the test condition . The influence of subcontracting on optimal solutions during the 2 planning period is also signified. The general framework of the mathematical models can be used not only for the manufacturing of re us able container s but also for any type of product with a similar supply chain network. 3 APP is a method of mak ing multiple decisions about production, inventory, and workforce levels simultaneously over a finite time horizon (Pan and Kleiner, 1995; Wang and Yeh, 2014). The decisions can be made at the long - , intermediate - , and short - term levels (Sultana et al., 2014). The APP problem aims to minimize total costs while satisfying time - varying demand assuming fixed sales and production capacity (Nam and Logendran,1992; Pan and Kleiner, 1995). Although APP is more appreciated when demand is fluctuating, or resources are scarce, it is not recommended in cases of excess capacity (Gansterer, 2015). Nam and Logendran (1992) classified the existing APP methods into exact or heuristic methods based on the optimality of the solution s . T he exact solution approach es include linear programming model, linear decision rule, lot size model, goal programming, e tc . T he category of heuristic (near - optimal) solutions consists of search decision rule, production switching heuristics, management coefficient model, and simulation model. Pan and Kleiner (1995) proposed a classification of APP models based on the soluti on techniques, includ ing informal approaches, mathematical models, linear programming models, linear decision rules, heuristic techniques, management coefficients models, and search procedures using computer simulation. Various approaches have been develo ped to solve the APP problems , but very few of them have been implemented on real - world problems. Nam and Logendran (1992) point out that these approaches have more theoretical, rather than practical , value. The models have assumptions, for example, the de terministic demands and workforce with the same level of expertise , that do not reflect on the actual situations (Gilgeous, 1987; DuBois and Oliff, 1991; Pan and Kleiner, 1995; 4 García et al., 2009). Also, frequent change in the level of the workforce may be un appreciated in reality. The indirect cost, such as human resources, marketing, and finance are not integrated into the formulation of the APP models (García et al., 2009) . Lot size models incorporate scheduling issues associated with lot size indi visibilities into capacity planning decisions, but they require detailed information throughout the planning horizon, which is quite expensive to gather and process. Search decision rule methods incorporate a variety of cost functions that vary periodicall y as capacity levels changes, adapting to change s in operational conditions , and flexib ly replicat ing multiple types of planning objectives. However, the cost of the methods is quite high . Moreover , particular expertise is required to accomplish such a com plex task (Gilgeous, 1987; DuBois and Oliff, 1991; Pan and Kleiner, 1995). Dejonckheere et al. (2003) cited obstacles to applying APP methods , including formulating the model, interpretation of results, and disaggregati ng from the overall optimal results. The m anagement coefficient models help to reduce the inconsistency of manage ment decision s by eliminating . Eilon (1975) stated that simulation models can resolve some real scheduling issues and are well adapted to spec ific supply chains. However, this method is quite costly, and the results are not guaranteed to be optimal (Nam and Logendran,1992). Various models have been proposed to facilitate the use of APP in the industry. Ebert (1976) presented a method for the APP in a variable productivity setting. Apart from the administrative, initial investment, materials, and overhead cost s , the planning costs are also considered in the model. Kamien and Li (1990) introduced a multi - p eriod production planning model that integrates subcontracting as a production planning strategy. The authors also demonstrated the smoothing effect of outsourcing by reducing the fluctuation of production and inventory levels. Van Mieghem (1999) used a si ngle - period, competitive stochastic investment game model in a stochastic demand 5 setting to examine the interaction between capacity, inventory, and pricing decisions. Dejonckheere et al. (2003) utilized the filter theory to connect the dynamics of order r eplenishment to production planning strategies. Techawiboonwong and Yenradee (2003) offered a multi - product APP model where the workforce can be exchanged between different production lines. Jain and Palekar (2005) provided a configuration - based formulati on, where a product line consisting of several stages is used for manufacturing various products at different rates. Moreover, machines at each stage are allowed to combine to form various production lines. Tian and AbouRizk (2010) developed a simulation - b ased model that modeled the dynamics and constraints of the production, storage, and distribution processes of the whole process. The model was applied successfully in search ing for the best production plan for asphalt production operation s ; however, varyi ng demand made the production planning quite challenging. Sillekens et al. (2011) built a mixed - integer programming model for the APP in the automotive industry. The model is focused on the adaption of the capacity of a single production line by adjusting the workforce and working times. Chinguwa et al. (2013) explored the APP problem for a specific furniture firm. The best solution was obtained using the informal trial and error method on spreadsheets. Sadeghi et al. (2013) developed a fuzzy grey goal prog ramming model in which the g rey numbers were adopted to deal with the uncertainty of parameters. The model could provide a range of APP scenarios with flexibility for planners. In the APP model of a cable company regarding transportation dummy demand does not match with the supply (Sultana et al, 2014). Mendoza et al. (2014) developed a simulation modeling approach for the APP in a two - level inten sive supply chain by applying system dynamics. Gongbing and Kun (2014) established a data envelopment analysis - based APP 6 model that dealt with the uncertainty of demand with the normal distribution. Wang and Yeh (2014) proposed a modified particle swarm op timization method for solving an integer linear programming APP problem. Davizón et al (2015) formed a mathematical model to achieve optimal control, which includes the level of production, inventory, capacity, as well as related costs of the workforce in the same formulation. Gholamian et al. (2015) built a fuzzy multi - objective mixed - integer nonlinear programming model of the APP problems under the context of some uncertain parameters, where multiple suppliers, manufacturers , and customers are involved. M odarres and Izadpanahi (2016) proposed a multi - objective linear programming model that integrates energy saving into the APP with uncertain product demand. The objective function of their model consists of various terms: operational cost, energy, carbon em ission, and uncertainty related to demand and capacity. Rosero - Mantilla et al (2017) summarized the general process of applying the APP to solve a real - world problem. Entringer and Ferreira (2018) proposed a conceptual reference model of typical business p lanning modules that aimed to connect existing processes and aggregate planning. Yaghin (2018) presented a non - linear APP model to address the effect of varying prices and marketing expenditures in the setting of multi - site manufacturing systems and multip le demand classes. Mahmud et al. (2018) developed a multi - product and multi - period APP problem in the interactive probabilistic environment, in which some main costs, such as production, backorder, labor level, and demand are uncertain. Recently, Ruangngam and Wasusri (2019) constructed a mixed - integer linear programming model that incorporates setup time, setup cost, capacity restrictions, perishable product shelf life, and perishable supply restrictions in their formulation for a newly built fruit juice c oncentrated factory. It is noted that all models in the literature are based on material flow, however, the working hour may be more practical as it can bring convenience for work scheduling and the possibility for 7 adding constraints . Also, the fixed cost at the private warehouse is not addressed yet in the existing research about APP , wh ere contracts should be signed with the pre - determined rental area and usage period and payment. 8 F orecasting demands is fundamental to supply chain planning . Push and pull process es are two different ways of meeting customer needs . In the pull process, production actions are driven by customer actual orders ; However, the activities in the push process - a strategy used by most modern corporations - are based on a long - term prediction of customer needs before the real order arises . Forecasting has some basic characteristics . One is that i t do es n o t always match the real data . T his is why the forecast errors should be considered and measured . Another feature is that t he long - term forecast is normally less accurate than the short - term prediction , this is because the longer the time, the more factors are assumed to emerge and influence the result . At last, t he a ggregate forecast typically ha s fewer errors than the disaggregate forecast, as it tend s to have a smaller standard deviation for the errors . Before selecti ng an appropriate forecasting method, it is necessary to conduct a thorough investigation of the factors including h istorical demand, l ead time of product replenishment, p lanned promotion activitie s, e conomic situation , and s trateg ies (Chopra, 2017) . It also requires cooperation at the level of the entire supply chain. This is because the activities of each party in the supply chain are interrelated . F orecasting at an appropriate level of aggregation can effectively lower error since it is usually more precise than disaggregated forecasts . Forecasting must be monitor e d , and its error measured for further decision - making . 9 F orecasting method s are divided into two categories, qualitative and quantitative . Qualitative forecasting method s are primarily impl emented when less historical information is available that only human judgment with expertise can be used for the forecasts . Time series, causal, simulation are the main methods that fall under the category of quantitative method. Time - series forecasting methods are suitable wh ere historical demand implies its future trend well . Causal forecasting methods are established on the assumption that the demand forecast is highly correlated w ith certain environmental factors, such as policy and interest rate . Simulation forecasting is a method in which imitat ing the consumer choices that induce demand . Forecasting using a combination of several methods is deemed to have a better perform ance th an forecasting with one method . T he time - series method s are applied in th is research . Any observed demand can be considered as a combination of the systematic - and random component. The goal of forecast ing is to achieve the systematic part, instead of the random portion that is hardly predictable. Three factors are taken to define the predicting model of the systematic component: (i) l evel: the systematic element , (ii) trend: the change rate of demand for the next period , and (iii) seasonality: the predictable seasonal fluctuations in demand. Th e re are three common types of equations reflect ing the relation between the systematic component and the factors: (i) m ultiplicative: s ystematic component = level × trend × seasonal factor; (ii) a dditive: s ystematic component = level + trend + seasonal factor; and (iii) m ixed: s ystematic component = (level + trend) × seasonal factor. T he mixed equation is selected for the calculation of this project as it is considered the most accurate (Chopra, 2017) . 10 S tatic and adaptive forecasting models are based on distinct assumptions of the factors . Static methods presume the estimated level, trend, and seasonality constant, whereas adaptive models integrate the var ying effect of these parameters. Four common adaptive forecast techniques are listed and compared in Table 3.1 . Table 3 . 1 : Comparison of adaptive forecasting methods Before presenting the forecasting methods, some basic definitions , as shown in Tab le 3.2, are necessary to be introduced . Table 3 . 2 : Definition of factors in the systematic component In the static forecasting method , t he forecast demand in Period is thus given as (Chopra, 2017) : ( 1 ) , w here is the number of pieces of historic al data available. The first step is to estimate the level and trend for Period . This begins with de seasona lizing the demand data. That is, to reduce the seasonal fluctuations in the original demand data. Below are two equations for obtain ing the deseasonalized demand ( ) , one when is even and the other when is odd. In which each historic al data is given equal weight. 11 ( 2 ) , w here is the periodicity which is the number of periods after which the seasonal cycle repeats (Chopra, 2017) . T he level and trend for Period can then be retained by applying a linear equation for the period ( ) and the deseasonalized demand data ( ) . ( 3 ) The next step is to estimate seasonal factors with the following formula . The seasonal factor for Period is the ratio of actual demand to the deseasonalized demand and is given as: ( 4 ) The seasonal factors are then averaged for each season: ( 5 ) Where, , is the seasonal cycles in the data, for all periods of the form , and . In adaptive forecasting, t he forecast ing demand for Period is expressed as follows : 12 ( 6 ) Unlike in the static forecasting methods, an additional step of revising factors is required in adaptive forecasting techniques to compensate for the forecast value after minimizing the forecasting error of historic data. The e rror for Period is stated as Eq. (7). ( 7 ) Four adaptive forecasting methods are introduced in this portion . The m oving average method is used when the trend and seasonality are absent. In this method, the level for Period is estimated as the average d demand over the most recent periods. Since it is assumed that nearby observations in past are likely to be close to the future demand . The e quation for the N - period moving average is presented as follows: ( 8 ) The forecast is evaluated as: and ( 9 ) The new moving average is calculated by adding the latest observation of demand and dropping the oldest one. The revised moving average serves as the next forecast. 13 The simple exponential smoothing method is suitable when demand demonstrates no trend or s easonality. T he initial estimate of level, , is taken to be the average of all historical data with the following equation (Chopra, 2017) . ( 10 ) , where is the total number of given demand data . The current forecast for all future periods is given as: , and ( 11 ) After observing the demand ( ) for Period , the estimate of the level is revised as follows: ( 12 ) , where is the smoothing factor for level , . Trend - : The method runs a linear regression equation between historical demands ( ) , and time ( Period ) , so that from which the initial and could be obtained. Forecast ing for Period , is expressed as (Chopra, 2017) : and ( 13 ) 14 After observing the demand for Period , the estimates for the level and trend are revised as follows: ( 14 ) ( 15 ) , where and are the smoothing factor s for the level and trend, respectively. : Initial estimates of the level , trend , and seasonal factors , ..., are obtained with the same procedure as those for static forecasting (Chopra, 2017) . In Period , given estimates of level, , trend, , and seasonal factors, , ..., . and ( 16 ) On observing demand for Period , the estimates for the level, trend, and seasonal factors are revised as follows: ( 17 ) ( 18 ) ( 19 ) 15 , w here , and are smoothing constant s for the level , trend, and the seasonal factor, . M easurement of forecast errors is essential to assess ing the accuracy of forecasting methods. There are a variety of measures to assess the error. One is mean squared error . The penalizes large errors much more significant ly than small ones as all errors are squared. Thus, it is more appropriate in situations where the cost of a large error is much larger than the gains from very accurate forecasts. It is appropriate to be exploited when forecast error has a distribution that is symmetric about zero. ( 20 ) Another measurement is the mean absolute deviation , which refers to the average of the absolute deviation over all periods . It is expressed by the following e quation : ( 21 ) The can be employed to estimate the standard deviation of the random component assuming that the random component is normally distributed. In this case , the standard deviation of the random component is: ( 22 ) 16 The mean absolute percentage error is the average absolute error as a percentage of demand and is given by: ( 23 ) The can be considered as a good choice when the underlying forecast has significant seasonality , and demand varies considerably from one period to the next. However, it is not good as if the forecast error is asymmetrically distributed. In general, one needs a method to track and control the forecasting method. One approach is to use the sum of forecast errors to evaluate the bias, where the following holds: ( 24 ) The tracking signal ( ) is the ratio of the bias and the and is given as : ( 25 ) 17 Aggregate production planning plays a n important role in industr ies . Managers want to fulfill as many customer orders as possible to make more profit; however, this is difficult because the volume of orders from customers is usually uneven, as well as there are always various resource and condition constraints. For example, lead times are typicall y long; manufacturers may need to start production before they receive orders; capacity costs often do not amount to outsourcing costs; hiring and layoff costs are often high; inventory can be expensive. Aggregate production planning , as an approach to sch edul e a capacity, production, subcontracting, inventory, stockouts, and pricing over a finite time horizon at an overall level , can help planners achieve their goal of minimiz ing the total costs or maximiz ing the profits while meeting non - constant demand s simultaneously . Specifically, it determines the levels of production, inventory, capacity (internal and outsourced), and any backlogs (unmet demand) for each period, g horizon based on the forecast demands are fully met (Chopra, 2017) . The aggregate planning acts as a broad scheme for production management and builds the boundaries within which production and distribution decisions can be made. The aggregate plan enabl es the supply chain to adapt to the capacity distributions and business agreement. It is concentrated on solving problems at the aggregate level , rather than the detailed stock - keeping - unit (SKU) level decisions. It is usually applied in advance of 3 - 18 months. In such a period, determining production levels by SKU is unrealistic as it is too early , adding production capacity may be also too late. Therefore, aggregate planning is generally limited to searching for optimal production options based on existing facilities (Chopra, 2017) . 18 It is critical to collaborate with other parties throughout the supply chain for the effective practice of aggregate planning , a s other partners are important input s for the planning (Chopra, 2017) . Moreover , many constraints lie outside these companies . S uch as the vendors or customers of their warehousing , logistics service, which are also crucial . If a manufacturing company has determined to adjust its production , its vendor, transportation, warehousing service must be informed of the plan and integrate the change into their schedules . Without engagement from upstream and downstream of the supply chain, the aggregation planning can hardly generate its complete power . The p lanning horizon should be specified before starting the aggregate planning. It indicates a time frame over which the aggregate plan produce s a solution. Another element that ought to be specified is the duration of each period within the planning hor izon , e. g., weeks, months, or quarter s . A variety of information should be gathered before employing the aggregate production planning: (i) p roduction r ate , (ii) w orkforce , (iii) o vertime , (iv) m achine c apacity l evel , (v) s ubcontracting , (vi) b acklog , (v ii) i nventory on h and . T he planner s should also identify other key information : (i) a ggregate demand forecast , , for each Period in a planning horizon that extends over periods , (ii) p roduction costs ; (iii) l abor costs, regular time ( $/hour ) , and overtime costs ( $/hour ) , (iv) c ost of subcontracting production ( $/unit or $/hour ) , (v) c ost of changing capacity , specifically, cost of hiring/laying off workforce ( $/worker ) and cost of adding or reducing machine ca pacity ( $/machine ) , (vi) l abor/machine hours required per unit (vii) inventory holding cost ( $/unit/period ) , (viii) stockout or backlog cost ( $/unit/period ) , and ( ix ) constraints on overtime, layoffs, capital, stockouts and backlogs, and from suppliers to the enterprise (Chopra, 2017) . 19 The aggregate production plan ning can determine (i) the p roduction q uantity from r egular t ime, o vertime, and s ubcontracted t ime , (ii) i nventory h eld , (iii) b acklog or s tockout q uantity , (iv) w orkforce h ired/ l aid o ff , and (v) m achine c apacity i ncrease or d ecrease (Chopra, 2017) . The quality of the a ggregate production plan ning affects profit ability because the loss can be caused not only by insufficient or late supply but also by excess inventory and capacity. There are several noteworthy principles about implement ing high - quality aggregate production planning . First, a ggregate units for production and time should be selected at a proper level . This is because the final schedule will be disaggregated at the product level, a lthough t he production planning is carried out in aggregation . Another notable point is the bottleneck of any manufacturing facility , as it is likely to be the most constraining area that may fail the aggregated planning . T he setups and maintenance should also be considered in the model since it occupies capacity but result s in no production. Otherwise, the aggregate plan will misjudge the production capacity available, resulting in a plan that cannot be achieved in practice (Chopra, 2017) . T rade - off s must be made among capacity, inventory, and backlog costs to achieve the best plan (Chopra, 2017) . The chase strategy, the flexibility strate gy, and the level strategy are three common tactics, which are generally combined or tailored in practices. The chase strategy deals with the demands with the adjustable machine or labor . The p roblem with th is approach is that there is a high expense for t he company and it hurt s the employees due to the frequent hiring or laying - off of workers ; thus, it is only useful when the inventory cost is higher than changing the level of machine and workforce. The f lexibility strategy depends on the varying utilization rate of machines and of the fluctuating demand. This tactic avoids the issues associated with the chase strategy but presents a new problem of low machine utilization . 20 In the l evel strategy , machine capacit y and workforce are kept at a constant output rate , while inventory is used as the lever. Backlogs and surpluses are the main challenges to be dealt with under this scheme. T hinking beyond t he firm to the entire supply chain may facilitate produc ing better results of aggregate production planning . This is due to many factors outside the enterprise that may have a significant impact on the optimal aggregate plan. Not only should the firm communicate with downstream partners for a better forecast of future de mand, but also, they should work with upstream partners to review th e constraints, and with other parts o f the supply chain to improve the performance of the aggregate plan (Chopra, 2017) . Another key principl e is that an aggregate planner must make the plan flexible enough as forecasts are always inaccurate. Aggregate plan ning is an overall blueprint in advance of a specified horizon before orders emerge . The firm should be prepared for the forecast error. A s e nsitivity analysis o f the inputs is a recommended solution to the issue as it can evaluate how the varying parameters impact the optimal solution (Chopra, 2017) . The third rule is that the aggregate plan should be rerun as new data becomes available . This is because the update d i nputs may have a radical influence on the previously obtained results . Therefore, it is important to use the latest input to run again the aggregate planning to check if any adjustment s should be made (Chopra, 2017) . The final point is that a firm needs to perform aggregate planning as capacity utilization increases (Chopra, 2017) . It may be un necessary when the utilization rate is low since they can arrange production as order received . However, when the utilization r ate is high and capacity is an 21 issue, it may be too late to fit the order into the busy production line. Thus, it is necessary to apply aggregate planning in production for a firm in situations of high utilization. 22 Pol ystyrene is widely used in the packaging industry because of its various advantages. The material is economical, transparent, easy to mold, rigid, recyclable, and with good dimensional stability. This research is mainly focused on polystyrene resins that a re used for manufacturing plastic containers in the forms of black and clear. The raw material is purchased quarterly in the form of resin pellets. Extrusion and thermoforming are the main processes to convert polystyrene resin pellets into plastic contain ers. Extrusion is a high - volume manufacturing process in which the raw plastic is melted and formed into a continuous profile through a die. The raw material is fed into a preheated extruder via a hopper. The material is then compressed to the exit side b y a rotating conical screw. Heating devices surround the barrel, softening, and melting the polymer. The melted material pumping out of the die is cooled to a solid shape in the air or through a stream of water, which is finally cut into various shapes. Th e shape of the product is determined by the die at the end of the extruder. Dyes can be also added in the process to have colored products. Extrusion is generally applied to thermoplastics which refer to materials whose polymeric structure will not change drastically after multiple cycles of heating and cooling; such a character promotes its recycling. The extruded products can be further molded by other processes, such as blow molding or thermoforming, to expand their usages. 23 Figure 5 . 1 : A principal scheme of an extruder Thermoforming is a manufacturing process, where a thermoplastic material or preform is heated to a forming temperature, stretched to a specific shape in a mold, and trimmed to a finished product. Figure 5 . 2 : A scheme for mechanical thermoforming press In this case, the black and clear plastic sheets from the extrusion process are wrapped into rolls. The option of subcontracting extruded sheets is available when the customer's need s exceed the extrusion capacity. The rolls are sent either to the thermoforming presses or the warehouses for 24 future use. Of note, two types of warehouses, i.g. public and private warehouse, are available in the setting. Figur e 5 . 3 : The manufacturing process of plastic containers Suppose quarterly historical demand for plastic containers is given. Let denote the demand forecast for containers in quarter (see Table 5. 1 for the summary of notations). The r aw material is quarterly purchased at $ per 1,000 lb. to match the planned production. Extruders produce rolls of plastic sheet s . There are number of extruders in the facility. Each extruder has a processing capacity of pounds per hour. Each extruder requires workers. The amount required is passed forward to thermoforming presses, while the rest is stored at a public and/or private warehouse. There are number of thermoformi ng presses in the plant. Each thermoforming press has a processing capacity of and requires workers. Each worker is paid $ per hour for a regular - time salary and $ per hour for overtime. Workers are limited to overtime hours per quarter. T he t raining cost per person is $ . During any quarter, extruders and thermoforming presses may be idled. In this case, the associated workers should be laid off. Laying off each worker costs $ . If an idled extruder/thermoforming press is brought online, a training cost of $ per worker is required. 25 It is assum ed that the manufacturer has the option of subcontracting the production of plastic sheets to one of its supply chain partners. S ufficient production capacity is deemed always available by the subc ontractor to make up for the shortage of plastic sheets for the thermoforming process. The manufacturer spends $ per 1,000 lb. of the plastic sheet produced by a subcontractor. Surplus plastic sheets are sent for storage. Transportation is needed to bring the sheets back from the warehouse (when they are needed) to feed the thermoforming presses. Let $ denote the total transportation cost of 1,000 lb. of plastic sheet. If the option of public warehousing is selected, material handling and storag e charge the manufacturer $ per 1,000 lb. at the end of each quarter. If the option of private warehousing is selected, two types of cost incur: (i) fixed cost: as a contract of a certain area must be signed before use and the least leasing period is three years . It means the leasing area must be paid whether it is used or not. Suppose one square foot is required per 1,000 lb. of plastic sheets in storage. Then, lease rates average $ per square foot per quarter; (ii) variable cost: private wareh ousing charges the manufacturer a variable operating cost of $ per 1,000 lb. of plastic sheet stored per quarter. T he supply chain network is illustrated in Figure 5. 3 . Table 5 . 1 : Notations used for both mathematical programming models 26 Table 5.1 Based on this premise, this investigation aim s to answer the following questions: 1. How many pounds of the plastic sheet should be produced by regular time/overtime working at each quarter? i.e., how many regular/overtime hours should extruders work each quarter? 2. How many extruder work ers should be laid off/hired at each quarter? 3. How ma n y extruder s should work at each quarter? 4. How many pounds of plastic containers should be produced by regular/overtime time working at each quarter? i.e., how many regular/overtime time hours should the thermoforming presses work at each quarter? 5. How many thermoforming workers should be laid off/hired at each quarter? 6. How ma n y thermoforming presses should work at each quarter? 7. How many pounds of plastic sheets should be sent to public or private warehousi ng or should be subcontracted? How many square feet of the private warehouse should be leased if it is involved ? 27 To address these questions, two mixed - integer programming model s , based on the material flow and the level of the workforce , along each segment of the manufacturing process are presented . A mathematical model based on the flow of material The mixed - integer programming model corresponding to the aggregate planning of reusable container manufacturing based on the material flow is shown in this sector (see Table 5 . 2 for notations). Objective function (1) includes several terms: is the raw material purchasing cost, is the cost of subcontracting, is the labor cost during the regular time working at the extruding plant, is the labor cost during the regular time working at the thermoforming plant, is the labor cost during overtime working of the extruding plant, is the labor cost during overtime working of the thermoforming plant, is the cost of training workers 28 when a new extruder is brought online, is the cost of worker training when a new thermoforming press is brought online, is the cost of laying off workers when one extruder is idle, is the cost of laying off workers when a thermoforming press becomes idle, is the cost of transporting plastic sheets to a public/private warehouse, is the cost of material handling and storage cost of plastic shee ts at a public warehouse, is the fixed leasing cost of a private warehouse, and is the variable leasing cost of a private warehouse. It is noted that is related to (the first 3 - year leasing contract), is related to (the second 3 - year leasing contract), and so on. Parameter is 1 is is related to , and 0 otherwise. Consider that there are 8 working hours per day and 63 working days pe r quarter (a total of 504 hours per quarter). Constraint (2) shows the connection between the plastic sheets in pounds produced by regular time working of extruders and the total number of working extruders at each quarter. Constraint (3) shows the co nnection between the plastic sheets in pounds produced by overtime working of extruders and the total number of working extruders at each quarter. Constraint (4) shows the connection between the containers in pounds produced by regular time worki ng of the thermoforming process and the total number of working thermoforming presses at each quarter. Constraint (5) shows the connection between the containers in pounds produced by overtime working of thermoforming presses and the total number of w orking thermoforming presses at each quarter. Constraint (6) guarantees that the total number of working extruders does not exceed the total number of extruders. Constraint (7) guarantees that the total number of working thermoforming presses does not exce ed the total number of thermoforming presses. Constraint (8) guarantees that the total number of working extruders at quarter is equal to the total number of working extruders at quarter plus the total number 29 of newly hired extruders at quarter minus the total number of laid - off extruders at quarter . Similarly, constraint (9) guarantees that the total number of working thermoforming presses at quarter is equal to the total number of working thermoforming presses at quarter plus the tot al number of newly hired thermoforming presses at quarter minus the total number of laid - off thermoforming presses at quarter . Constraint (10) ensures the flow balance of materials between extruding plant, storage warehouses, and thermoforming plant. Finally, constraint (11) guarantees that the containers (in pounds) produced by thermoforming presses at quarter are equal to the demand of quarter . Constraint (12) ensures that the level of inventory at the private warehouse at each period is less than the preset inventory level leased at the beginning of the corresponding 3 - year leasing contract. The utility rate of the private warehouse is also co nsidered. Table 5 . 2 : Variables used for the mathematical modeling of the aggregate planning based on the material flow 30 A mathematical model based on the working hours of machines. T he mixed - integer programming model corresponding to the aggregate planning of plastic container manufacturing based on the working hours of the equipment is presented (see Table 5. 3 for notations). Objective function (13) includes several terms: is the raw material purchasing cost, is the cost of subcontracting, is the labor cost during the regular time working at the extruding plant, i s the labor cost during the regular time working at the thermoforming plant, is the labor cost during overtime working of the extruding plant, is the labor cost during overtime working of the thermoforming plant, is t he cost of training workers when a new extruder is brought online, is the cost of training workers when a new thermoforming press is brought online, is the cost of laying off workers when an extruder becomes idle, is the cost of laying off workers when a thermoforming press becomes idle, is the cost of transporting plastic sheets to a public/private warehouse, is the cost of material handling and storage cost of plastic sheets at a public 31 warehouse, is the fixed leasing cost of a private warehouse, and the variable leasing cost of a private warehouse. Constraint (14) shows the connection between the regular time working hours of extruders and the total number of working extruders at each quarter. Constraint (15) shows the connection between the overtime working hours of extruders and the total number of working extruders at each quarter. Constraint (16) shows the connection between the regular time worki ng hours of the thermoforming process and the total number of working thermoforming presses at each quarter. Constraint (17) shows the connection between the overtime working hours of thermoforming presses and the total number of working thermoforming pres ses at each quarter. Constraint (18) guarantees that the total number of working extruders does not exceed the total number of extruders. Constraint (19) guarantees that the total number of working thermoforming presses does not exceed the total number of thermoforming presses. Constraint (20) guarantees that the total number of working extruders at quarter is equal to the total number of working extruders at quarter plus the total number of newly hired extruders at quarter minus the total number of laid - off extruders at quarter . Similarly, constraint (21) guarantees that the total number of working thermoforming presses at quarter is equal to the total number of working thermoforming presses at quarter plus the total number of newly hi red thermoforming presses at quarter minus the total number of laid - off thermoforming presses at quarter . Constraint (22) ensures the flow balance of materials between extruding plant, storage warehouses, and thermoforming plant. Finally, constraint (23) guarantees that the containers (in pounds) produced by thermoforming presses at quarter are equal to the demand of quarter . Constraint (24) ensures that the level of inventory at the private warehouse at each period is less than the preset inventory level leased at 32 the beginning of the corresponding 3 - year leasing contract. The utility rate of the private warehouse is also considered. Table 5 . 3 : Variables used for the mathematical modeling of the aggregate planning based on the working hours of the equipment Finally, it is noted that one can evaluate various storage strategies, i.e., usi ng either public or private storage (but not both at the same time), and a combination of both warehous e s . The following constraints provide such flexibility to the model: , where and are binary variables and is a large number. Constraints (25), (26), and (27) with guarantee that using either public or private storage is used and not both. Constraints (25), (26), and (27) with guarantee that using a combination of each storage is acceptable. 33 Historical demand data and the value of some parameters are adopted from Chopra (2017). Historical demands for black and clear plastic containers from the year 2005 to 2009 are presented in Figure 6.1 and Table 6.1. Figure 6 . 1 : Historical demand data Table 6 . 1 : Historical demand data 0 5,000 10,000 15,000 20,000 2005 2006 2007 2008 2009 Demand ('000 lb.) Year clear black 34 In Figure 6.1 , the trend and seasonality can be observed for the demands of both plastics . The demands for clear plastic container s peak every summer which is assumed to relevant to the demand for cold drinks in the season , and the demands for black one s reach the highest - level during winters . The demand s for both containers ha ve an increasing trend . It is assume d the demands will continue to increase in the following three years at historical rates . which consider s trend and seasonality, is expected to give the best forecast among all other forecasting methods . In the static forecasting method, level, trend , and seasonal factor s are assumed to be constant . The first step of forecasting with the static method is to de seasonali ze the historical data series. It is observed that periodicity which is an even number . Therefore, the first row of Eq. ( 2 ) is taken to calculat e the deseasonalized demand for quarter 3 to quarter 19 to ). The second step is to obtain a linear equation between deseasonalized demands to ) and quarter numbe r ( to ) Which can be obtained by running a linear regression analysis i n E xcel or adding a linear trendline to the data series . The equation s of the obtained linear trendline s are shown in Figure 6. 2 and 6.3 . 35 Figure 6 . 2 : Linear regression for the deseasonalized demands of black plastics and the quarter number Figure 6 . 3 : Linear regression for the deseasonal i zed demands of clear plastics and the quarter n umber Table 6 . 2 : Input range for obtaining trendline As t he coefficients of the regression equation are round , the equation s for deseasonalized demand data of both containers are as follows . ( 3A ) ( 3B ) = 226.86t + 2592.7 R² = 0.9128 0 2000 4000 6000 8000 0 4 8 12 16 20 De - seasonlized demand ('000 lbs.) Quarter Deseasonalized demand Linear (Deseasonalized demand) = 263.94t + 3612 R² = 0.8669 0 2000 4000 6000 8000 10000 0 4 8 12 16 20 De - seasonlized demand ('000 lbs.) Quarter Deseasonalized demand Linear (Deseasonalized demand) 36 T he deseasonalized demands for both plastics during all quarter s can be calculated with the above equations . All s easonal factors are calculated by Eq . ( 4 ) (in Column F) and then averaged by Eq. (5) (in Column G) . Originally , there are 20 values of seasonal factors , where they are considered five cycles ( each year is considered a cycle ) . S easonal factors at the quarter I, II, III, IV, are averaged respectively because the four seasonal factors are assumed to be repeat ed every year . For example, , etc. At last, the forecast is calculated by Eq . ( 1 ) . Results for both black and clear containers are presented in Figure 6. 4 and Figure 6. 5 . The formulas involved are also listed in Table 6. 3 . Note that the symbol in the middle of the cell number is used to lock the referenced cell so that it will not be changed automatically . 37 Figure 6 . 4 : Forecast demands for black plastic containers using the static method Figure 6 . 5 : Forecast demands for clear plastic containers using the static method 38 Table 6 . 3 : Formulas for forecasting demands using the static method In this section, various adaptive methods which have been mentioned in Chapter 3, are used to forecast the demands for black and clear plastic s in the next three years . T he values of forecasting error, , Squared , , and are accordingly calculated . V alues of and are compared among different methods to evaluate the accuracy of each method . Forecasting demands with movi ng average method The m oving average method does not consider trend or seasonality . Eq . ( 8 ) is used to calculate the level from period to by taking the average of the previous four periods . Eq . ( 9 ) is then used to calculate the forecasting demand which is equal to the level of the previous quarter . The forecasting results for the demand for black and clear plastic s are presented in Figure 6 . 6 and Figure 6. 7 , respectively . Formulas used to forecast demands are presented in Table 6 . 4 . 39 Figure 6 . 6 : Forecast demands for black plastic containers using four - period moving average method 40 Figure 6 . 7 : Forecast demands for clear plastic containers using four - period moving average method Table 6 . 4 : Formulas for forecasting demands using the moving average method 41 Forecasting demands with the simple exponential smoothing method The s imple e xponential s moothing forecasting method does not take trend or seasonality into consideration . Initial level ( ) is calculated by averaging all the historical demands by Eq . ( 10 ) . The levels over the periods from 1 to 20 ( to ) are calculated by Eq . ( 12 ) . Demands for the historical period s are then calculated by Eq . ( 11 ) , and forecasting demands for the whole forecasting horizon is equal to the value of the last observed level ( ) . Value of smoothing constant in Cell M2 is obtained by minimizing the . The results are presented in Figure 6. 8 and Figure 6. 9 . The formulas involved in the calculation are listed in Table 6. 5 . 42 Figure 6 . 8 : Forecasting results for black plastic containers using the simple exponential smoothing method 43 Figure 6 . 9 : Forecasting results for clear plastic containers using the simple exponential smoothing method Table 6 . 5 : Formulas fo r forecasting with simple exponential smoothing method 44 Forecasting demands with H Holts Model includes the element of trend. The initial level ( ) and trend ( ) is acquired through running a linear regression between historical demands ( from to ) and quarter No . ( to ) . The results for the black plastic container s are , , while , for the clear ones . The l evel and trend for Quarter s 1 to 20 are calculated by Eq . ( 14 ) and Eq . ( 15 ) , respectively . Finally, f orecasting demands during historical periods and future time are calculated . Two smoothing constants are set as and in Cell N2 and Cell O2 by minimizing the . Formulas applied in these calculations are listed in Table 6. 6 . The results are shown in Figure 6. 1 0 and Figure 6 . 1 1 45 Figure 6 . 10 : Forecasting results for black plastic containers using H 46 Figure 6 . 11 : Forecast demands for clear plastic containers using H Table 6 . 6 : Formulas for forecasting demands using H 47 Forecasting demands with W not only trend but also seasonality into consideration . Initializing values obtained from the static method are used in this method. T he coefficients obtained from Eq . ( 3A ) and (3B) represent the initial level and trend . F or black plastic , , ; for clear plasti c, , . Four seasonal factors obtained by Eq. ( 4 ) and Eq. ( 5 ) for each plastic, are used as initializing values of seasonal factors. For black plastic , , , ; and for clear plastic , , , . Eq . ( 17 ) , Eq . ( 18 ) are used to estimate the level, trend, forecast of demand in Cells D 3, E 3 , and F 7 for the historical periods. Eq. ( 19 ) is applied to calculate seasonal factors for period to as the initial seasonal factors for the first cycle have already been obtained . Therefore, the forecasting values of demand for historical periods can be calculated by Eq. ( 16 ) . The smoothing constants , and are decided by minimizing the . The results are shown in Figure 6.1 2 and Figure 6.1 3 . The detailed formulas to build the worksheet are shown in Table 6. 7 . 48 Figure 6 . 12 : Forecast results for black plastic containers using W 49 Figure 6 . 13 : Forecast results for clear plastic containers using W Table 6 . 7 : F ormulas for forecasting demands using W 50 Table 6.7 The mean absolute percentage error ( ) , which represents the average absolute error as a percentage of demand, is selected to evaluate the forecast error . This is because the demand data have seasonality and relatively high variations from one quarter to the next . The forecasting demand for black and clear containers over the past five years using the aforementioned methods is presented in Figure 6.14 and 6.15. Smoothing constants in the simple are determined by minimizing the value of . Figure 6 . 14 : Comparison of different adaptive methods for black containers 0 5000 10000 15000 0 4 8 12 16 20 Demand forcast ('000 lb.) Quarter Historic demand Winter's model Holt's model Simple exponential smoothing method Moving average 51 Figure 6 . 15 : Comparison of different adaptive methods for clear containers From Figure 6.14 and 6.15, it is clear that fits best with the historical data comparing to the results obtained through other methods . Table 6.8 also presents and of different adaptive methods and confirm s the aforementioned claim. Table 6 . 8 : Estimation of errors using adaptive methods years are presented in Figure 6.16 and 6.17 , as well as in Table 6.9 . Of note, the forecast ing horizon is extended to 18 years, since three scenarios with various plan ning horizon s as long as 18 years will be tested in the segment of sensitivity analysis. The results are presented in APPENDIX A . 0 5000 10000 15000 20000 0 4 8 12 16 20 Demand forcast ('000 lb.) Quarter Historic demand Winter's model Holt's model Simple exponential smoothing method Moving average 52 Figure 6 . 16 : Estimat ed historical and forecasting future demands for clear plastic container s (for the coming 3 years) Figure 6 . 17 : Estimat ed historical and forecasting future demands for black plastic containers Table. 6 . 9 : 0 5000 10000 15000 20000 25000 0 4 8 12 16 20 24 28 32 Demand forcast ('000 lb.) Quarter Historical demand Winter's model 0 5000 10000 15000 20000 25000 0 5 10 15 20 25 30 Demand forcast ('000 lb.) Quarter Historical demand Winter's model 53 T he parameters adopted from Chopra (2017) , and are shown in Table 6.10 . Of note, the demand s for both colors of plastic containers are combined since they are compatible with each other at the same production line . The u tilization rate of the area at the private warehouse s is taken 80%, which is a common value. Table 6 . 10 : Parameters used in both model s A high - level modeling system for mathematical programming and optimization , e. g. General Algebraic Modeling System ( GAMS ) , of distribution 30.3.0 , is used to solve the problem due to its large size . The code is enclosed in APPENDIX C and D . Given the forecasted demands for two types of plastic containers over the coming three years, the optimal sol utions of the proposed APP models (one based on the flow of materials and the other based on working hours) are obtained. The results from both models are consistent , which confirms the achievability of the model based on the working hour . T he details of planning outcomes are included in APPENDIX B and a n illustration of each in Figure 6.18 - 6.21 . The first column at each quarter represents the forecasted demand s , and the second column represents the p roduction amount in pounds (or equivalently working hours). Regular and overtime working hours are distinguished by two types of filled 54 pattern s . Subcontracting is also demonstrated in the second column. Finally, the information related to the storage (public and private) is illustrated in the third column. Figure 6 . 18 : The o ptimal solution for the extruding and warehousing processes obtained from solving the APP model based on the flow of the material Figure 6 . 19 : The o ptimal solution for the thermoforming process from solving the APP model based on the flow of the materia l 12 +1 +1 0 5000 10000 15000 20000 25000 30000 1 2 3 4 5 6 7 8 9 10 11 12 Production('000 lb.) Quarter Demand d 13 +7 - 6 +3 - 2 +8 - 7 +3 - 3 +9 - 8 +5 0 5000 10000 15000 20000 25000 30000 1 2 3 4 5 6 7 8 9 10 11 12 Production('000 lb.) Quarter Demand d 55 Figure 6 . 20 : The o ptimal solution for the extruding and warehousing processes obtained from solving the APP model based on the working hour Figure 6 . 21 : The o ptimal solutions for the thermoforming process from solving the AP P model based on the working hour In the experiment s , the regular time production can meet the demand for most of the periods, while for several periods with relatively - high - demand that cannot be met by regular time capacity and storage from the previous quarter , overtime working become essentia l. S ubcontracting is the last option to fill the gap between capacity and customers' needs due to its relatively high cost. Note that it is only used in P eriod when the elevated demand cannot be satisfied by the full running of the current machines, i.e., the full capacity of regular and overtime working, plus the 12 +1 +1 0 2000 4000 6000 8000 10000 12000 1 2 3 4 5 6 7 8 9 10 11 12 Hour Quarter 13 +7 - 6 +3 - 2 +8 - 7 +3 - 3 +9 - 8 +5 0 2000 4000 6000 8000 10000 12000 14000 16000 1 2 3 4 5 6 7 8 9 10 11 12 Hour Quarter 56 inventory from period 9. Overtime and outsourcing are great options to provide high leveled flexibility to me s ( Mendoza et al., 2014 ). It is also noteworthy that the sheets extruded during overtime hours have neither been stored in the public n or the private warehouses . This is because a cheaper alternative, i.e., subcontracting, is assumed always available in the problem setting . The sheets produced during the regular working time are still appreciated to be stored for their competitive price (compar ed to subcontracting). The occurrence is because of the cost diff erence of product s from the sources. The p rivate warehouse is more preferred than the public warehouse because of its low price , despite the fixed leasing area needs to be confirmed ahead of a leasing period. However, the unstable demand s lead to low utilization of the public warehouse during low - demand seasons . The situation promote s adopting both types of warehousing to deal with the fluctuating demand for storage . T aking advantage of the fle xibility from the public warehouse can reduce the storage cost during high inventory seasons, as it requires no fixed cost, despite it has a higher unit price. In other words , t he surplus sheets should be sent to the private warehouse in priority unless it becomes full. The workforce level varies from one period to another to adapt to the varying demand. The workforce level in the thermoforming process fluctuates more drastically than in the extruding process. This is because o nly one worker will be laid off or trained when idling /initiating a thermoforming press , while the cost becomes six times when the same activit ies occur for the extruding process because each extruder requires six operators. Therefore, it is less appreciated to lay off workers i n the extruding department. 57 In this part , the effect of a range of parameter variations on the optimal solution is analyzed . The factors being examined are demonstrated in Figure 7.1 : (i) forecasting horizon (i.e., 3 years, 6 years, 9 years, and 18 years), (ii) number of extruders (i.e., 14, 16, 18, and 20), an annual increase of (iii) raw material price, (iv) labor costs, and (v) subcontracting cost with the following rate s : 0%, 5%, 10%, 20%, 50% and 100%. The baseline scenario is a 3 - year planning horizon, 14 number of extruders, 0% annual increase rate of raw material price, labor cost, and subcontracting price. Figure 7 . 1 : Sensitivity analysis over various parameters The results of the sensitivity analysis are summarized in Table 7.1 . Of note, to obtain the best possible solutions, the number of thermoforming presses is modified to be 32, 39, and 59 , respectively, in scenarios where the length of horizons is 6 - , 9 - , and 18 - year as the demands are assumed to keep increasing and exceed t he offered capacity . It is not considered that purchas ing or any other costs regarding the new machine in the objective function . Table 7 . 1 : Results of sensitivity analysis. 58 Tabl e 7.1 From Table 7.1, a combination of public and private storage is indicated as the most favorable option in most scenarios. However, there are some exceptions. In the second 3 - year of the 6 /9 - year planning horizon, public storage is solely recommended. This is because the demands are more volatile over the 2 nd or 3 rd leasing period within which the demand is assumed to keep increasing. P rivate storage then becomes less attractive because it requires a minimum 3 - year lease, which causes more losses due to the unused area dur ing low - demand seasons than it would have saved by not using the public warehouse. D uring the third leasing period (only under the scenario of the 9 - year planning horizon ) where the demand is relatively high , subcontracting becomes an 59 appealing strategy du e to the significant insufficiency of the capacity . T here would be no need for storage, neither in public nor in private storage. Increases in raw material prices or labor costs also make subcontracting very appealing , because it is assume d that the subco ntracting price remains the same. In other words, the extruding facility will be idle, and the supply completely rel ies on the subcontrac ting , so neither public nor private storage will be needed. Once subcontracting becomes costly, manufacturing the produ ct at the facility becomes more appealing, that is why both private and public warehouses are important in such cases. Labor cost is the most sensitive parameter in the examination, as it is the dominating component of the production costs. 60 This study addresses an important optimization problem of aggregate production planning for the case of manufacturing reusable plastic containers. Such a problem aims to coordinate various segments of the supply chain such as production, inventory, and wor kforce levels together. Operations planning for these segments separately would be much less complex since fewer variables and constraints that connect these segments will be dealt with; however, it does not guarantee that resources (e.g., raw materials, s torage space, machines, workforce) are used optimally. The manufacturing of reusable containers involves two main processes: extruding plastic sheets and thermoforming. Besides these processes, one can decide to store extra sheets extruded from the first p hase (extrusion) to use for the second phase (thermoforming) in future periods. This can significantly reduce the concern of shortage when demand increases during a season. In the experiment, there are two options for storage: public and private warehouses . Each has its regulations. In the meantime, the option of subcontracting with unlimited production capacity exists to make up the limited production capacity. This complicated production planning problem is mathematically modeled in two different ways: o ne based on the flow of materials and the other based on the level of the workforce. Both models produce the same results. The problem is coded with GAMS, distribution 30.3.0, and a comprehensive sensitivity analysis under various scenarios is carried out. In the sensitivity analysis, the impacts of various factors are examined, including the length of the planning horizon, the number of extruders, the annual increase rate of raw material price, annual labor costs, and subcontracting cost on the optimal sol ution. The proposed framework can be used not only for 61 reusable container manufacturing but also for the manufacturing of any type of product with a similar supply chain network. Future exploration can be directed toward case studies in which various cons traints for manufacturing phases, as well as limitations and regulations on subcontracting/third - party logistics and warehousing, be reflected on the APP models. This investigation provides a better understanding of the complicated APP models and presents a great tool for practitioners who would like to apply such decision support systems for their production lines. 62 63 Table A : Forecast ed demand for black and clear containers over 18 years 64 Table B 1 : The o ptimal solution for the APP model based on the flow of materials 65 Table B 2 : The o ptimal solution for the APP model based on workforce level 66 67 68 69 70 71 72 73 74 75 Aziz, R. A., Paul, H. K., Karim, T. M., Ahmed, I., & Azeem, A. (2018). Modeling and optimization of multi - layer aggregate production planning. Journal of Operations and Supply Chain Management (JOSCM) , 11 (2), 1 - 15. Chinguwa, S. , Madanhire, I., & Musoma, T. (2013). A decision framework based on aggregate production planning strategies in a multi - product factory: A furniture industry case study. International Journal of Science and Research (IJSR) , 2 (2), 370 - 383. Chopra, S., Meind l, P., & Kalra, D. V. (2017). Supply chain management: strategy, planning, and operation (Vol. 232). Boston, MA: Pearson. Davizón, Y. A., Martínez - Olvera, C., Soto, R., Hinojosa, C., & Espino - Román, P. (2015). Optimal control approaches to the aggregate pr oduction planning problem. Sustainability , 7 (12), 16324 - 16339. Dejonckheere, J., Disney, S. M., Lambrecht, M., & Towill, D. R. (2003). The dynamics of aggregate planning. Production Planning & Control , 14(6), 497 - 516. Dubois, F.L. & Oliff, M.D. (1991). Agg regate production planning in practice. Production and Inventory Management Journal , 32(3), 26 - 30. Ebert, R. J. (1976). Aggregate planning with learning curve productivity. Management Science , 23 (2), 171 - 182. Eilon, S. (1975). Five approaches to aggregate production planning. AIIE Transactions , 7 (2), 118 - 131. Entringer, T. C., & da Silva Ferreira, A. (2018). Proposal for a reference model for sales operations planning and aggregate planning. International Journal of Advanced Engineering Research and Science , 5 (8). Gansterer, M. (2015). Aggregate planning and forecasting in make - to - order production systems. International Journal of Production Economics , 170 , 521 - 528. Garcia - Sabater, J. P., Maheut, J., & Garcia - Sabater, J. J. (2009, July). A decision s upport system for aggregate production planning based on MILP: A case study from the automotive industry. In 2009 International Conference on Computers & Industrial Engineering (pp. 366 - 371). IEEE. 76 Gholamian, N., Mahdavi, I., Tavakkoli - Moghaddam, R., & Mah davi - Amiri, N. (2015). Comprehensive fuzzy multi - objective multi - product multi - site aggregate production planning decisions in a supply chain under uncertainty. Applied soft computing , 37 , 585 - 607. Gilgeous, V. (1987). Aggregate planning in UK manufacturin g companies. International Journal of Operations & Production Management , 7 (1), 50 - 61. Gongbing, B. I., & Xiang, K. (2014). Aggregate planning based on stochastic demand DEA model with an application in production planning. Management Science and Engineer ing , 8 (4), 82 - 87. Jain, A., & Palekar, U. S. (2005). Aggregate production planning for a continuous reconfigurable manufacturing process. Computers & operations research , 32(5), 1213 - 1236. Kamien, M. I., & Li, L. (1990). Subcontracting, coordination, flexi bility, and production smoothing in aggregate planning. Management Science , 36 (11), 1352 - 1363. Mahmud, S., Hossain, M. S., & Hossain, M. M. (2018). Application of multi - objective genetic algorithm to aggregate production planning in a possibilistic enviro nment. International Journal of Industrial and Systems Engineering , 30 (1), 40 - 59. Mendoza, J. D., Mula, J., & Campuzano - Bolarin, F. (2014). Using systems dynamics to evaluate the tradeoff among supply chain aggregate production planning policies. Internati onal Journal of Operations & Production Management . Modarres, M., & Izadpanahi, E. (2016). Aggregate production planning by focusing on energy saving: A robust optimization approach. Journal of Cleaner Production , 133 , 1074 - 1085. Nam, S. J., & Logendran, R. (1992). Aggregate production planning: A survey of models and methodologies. European Journal of Operational Research , 61 (3), 255 - 272. Pan, L., & Kleiner, B. H. (1995). Aggregate planning today. Work Study . Rosero - Mantilla, C., Sánchez - Sailem a, M., Sánchez - Rosero, C., & Galleguillos - Pozo, R. (2017, June). Aggregate production planning, case study in a medium - sized industry of the rubber production line in Ecuador. In IOP Conference Series Mater Science Engineering (Vol. 212). Ruangngam, I., & Wasusri, T. (2019). Aggregate planning using mixed integer programing: A fruit juice concentrated factory case study. In ECMS (pp. 249 - 253). 77 Sadeghi, M., Hajiagha, S. H. R., & Hashemi, S. S. (2013). A fuzzy grey goal programming approach for aggregate prod uction planning. The International Journal of Advanced Manufacturing Technology , 64 (9 - 12), 1715 - 1727. Sillekens, T., Koberstein, A., & Suhl, L. (2011). Aggregate production planning in the automotive industry with special consideration of workforce flexibi lity. International Journal of Production Research , 49 (17), 5055 - 5078. Sultana, M., Shohan, S., & Sufian, F. (2014). Aggregate planning using transportation method: a case study in cable industry. International Journal of Managing Value and Supply Chains , 5 (3), 19. Techawiboonwong, A., & Yenradee, P. (2003). Aggregate production planning with workforce transferring plan for multiple product types. Production planning & control , 14 (5), 447 - 458. Tian, X., Mohamed, Y., & AbouRizk, S. (2010). Simulation - based aggregate planning of batch plant operations. Canadian Journal of Civil Engineering , 37 (10), 1277 - 1288. Van Mieghem, J. A. (1999). Coordinating investment, production, and subcontracting. Management Science , 45 (7), 954 - 971. Wang, S. C., & Yeh, M. F. (2014) . A modified particle swarm optimization for aggregate production planning. Expert Systems with Applications , 41 (6), 3069 - 3077. Yaghin, R. G. (2018). Integrated multi - site aggregate production - pricing planning in a two - echelon supply chain with multiple demand classes. Applied Mathematical Modelling , 53 , 276 - 295.