WI ILLIWI/IW Wt L ; . ~,;, . 4 [f4 53/ 7337 =4 1524.34 4’ 3) ,‘V-'¢r/\-’»-rr-- {"fim 2‘ '0 , '2. ,4 1/4 "'/ '. I b. , oo lad-4699,5452 a} 0:21,; ,1 ‘1', I I f 7: NH}? -’,%”495 E 5557353! h" :13'35‘tj " 7“ - I‘i This is to certify that the dissertation entitled A FINITE SET APPROACH TO THE TANZANIA CASHENNUT PROCESSING INDUSTRY FACILITY LOCATION-ALLOCATION PLANNING presented by Hamisi Omari Dihenga has been accepted towards Tulfillment of the requirements for _ . Agricultural P h D degree in r . Englneerlng Major professor Date flflfé/féi/yffz MSUiJ an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES _—__. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. Q I FA' 1.. ' ' ‘ 'N‘W'JT': Mlco;wq« zi'YE ;' l‘rln partial tu;7il€menr ' . ~- - a .; ' " tor :ne 069:»: - I 4 00670; uf Pfif.a3..‘ - {‘13: _. "pun-lent or Agricuim- we”. a. it . 4 . 198i: A FINITE SET APPROACH TO THE TANZANIA CASHENNUT PROCESSING INDUSTRY FACILITY LOCATION-ALLOCATION PLANNING By Hamisi Omari Dihenga A DISSERTATION 7 Submitted to ‘ _ T9527 ~ Michigan State University D“f‘i : n partial fulfillment of the requirements , ,;G“ for the degree of 7 --T;_; DOCTOR OF PHILOSOPHY 'pgrtment of Agricultural Engineering ‘ 1984 THY '“N PY'HIPV II I 31-7. It?- ‘JI-‘I .. 't I" "2’ a I Y‘ I \ t 1L‘ I .n othr' ‘. .«t r', L» _ ‘4i' “ -4 "r ABSTRACT A FINITE SET APPROACH TO THE TANZANIA CASHENNUT PROCESSING INDUSTRY FACILITY LOCATION-ALLOCATION PLANNING BY Hamisi Omari Dihenga The state of the art in cashewnut handling and process- ing in Tanzania was reviewed, and, where appropriate, com- pared to other cashew producing countries. A finite set approach was used in the development of the cashew industry facility model. Subroutines were developed to interface with a LINDO code. Sensitivity analysis designed to answer some "adaptive if" issues——namely, possi- bilities of changes in cashew production levels, forcing facilities into solutions, and processing facility capacity changes——were explored. A 15% increase in predicted cashew production in the 1983-84 season resulted in no changes in facility locations, despite slight changes in routing configurations and a 7% increase in total system cost. Processing capacity utiliza- tions——50%, 75%, and 100% of rated capacity——resulted in 9, 5, and 5 open plants, respectively, for the 1982-83 season cashew production level (32,500 metric tons). A plant capacity utilization increase from 75% to 100% resulted in a 5% increase in system costs compared with 24% and 28%, respectively, for 50% to 75% and 50% to 100 percent. Hamisi Omari Dihenga _ The model solutions provide for locations and reloca- i tions of facilities, indicating open plants in optimal solution. The model highlights some planning alternatives that may be more attractive with respect to unmodeled issues. Facility configurations, location robustness index, » —v—._..___. shipment routes, and system costs are used as criteria for facility system evaluation and elaboration. The philosophy inferred is "insight" and "not numbers." APPROVED: l I fi/flM/M : Dr. Robert H. Wilkinson Major Professor 1k (5%02 Dr. Donald M. Edwards Department Chairman Dedicated to Allah. The most Gracious. the most Merciful. Cherisher and the Sustainer of the worlds. in ,,_ Master of Day of Judgment. "f' The One we worship and Thine aid we seek. sf 5r;;-rjv-~’- , ‘.,._ ’1‘. Dr \li ACKNOWLEDGMENTS I would like to thank God for guiding me in my endeavors. I would also wish to extend my sincere thanks to my brother, Mr. Kitwana 0. Mohamed, who raised me up and provided me with both financial and social support through my academic life. I would also like to give a note of appreciation to my wife Zubeda, son lssa, and the Michigan State community who helped make most of my trying days seem bearable. I also wish to express my deep appreciation to many who have contributed to this project: to the Agency for International Development for offering me the sponsorship and for its generous financial support during research implementation; to Dr. Kenneth Ebert of 1550 and Ms. Jane Tolbert and staff of the USAID office in Dar es salaam for sponsorship administrative support; to General Manager E. M. Makota and Directors B. A. Unga, Z. J. Mpogolo, and A. H. Mwenkalley of the Cashewnut Authority of Tanzania who made my data collection possible and a more rewarding experi- ence; to my Guidance Committee: Professors R. H. Wilkinson, 6. Brown, J. Gerrish, S. Yelon, M. Esmay, and R. Black who provided me with a day-to-day guidance and for their invalu- able corrections, suggestions, and criticisms in the iii TABLE OF CONTENTS LIST OF TABLES ........................................ viii LIST OF FIGURES ....................................... x CHAPTER 1 INTRODUCTION .................................. 1 1.1. Purpose of Study ........................ 1 1.2. Scope of Study .......................... 5 1.3. Study Objectives ........................ 7 2 LITERATURE REVIEW ............................. 9 2.1. Cashew Studies and Cashew Growing ....... 9 2.1.1. Distribution and Ecology ........ 9 2.1.2. Propagation by Seed ............. 10 2.1.3. Shoot, Root, Flowering, and Fruiting ........................ 16 2.2. Cashew Harvesting, Handling, and Processing .............................. 19 2.2.1. Cashew Harvesting, Handling, and Cashew Products .................. 19 2.2.2. Nut Processing ................... 23 2.3. Distribution Planning Systems ............ 32 2.4. Optimization Techniques and Facility Location Analysis ........................ 36 2.4.1. Optimal Solution: An Aid to Analysts' Intuition .............. 36 2.4.2. Facility Location Problems: Scope 40 MODEL 3.1. 2.4.3. Morphology of Location Systems ... 45 2.4.4. Location Analysis ................ 46 Multiperiod Location Analysis ............ 75 Generating Planning Alternatives ......... 78 2.6.1. The HSJ Method ................... 83 2.6.2. Estimating Differences Among Alternatives ..................... 85 DEVELOPMENT .......................... ;... 87 Methodological Choices in Facility Location Studies ......................... 87 Model Formulation ........................ 93 3.2.1. Problem Definition and Objective.. 93 3.2.2. A General Mathematical Formulation 93 3.2.3. Mathematical Formulation: Cashewnut Industry Problem ....... 96 Background Model Design Data Review ...... 100 3.3.1. Cashew Production in Tanzania .... 100 3.3.2. Tanzania Cashew Industry: Produc- tion Levels, Costs, Plant Capacities, and Distances ........ 110 CASHEN INDUSTRY LOCATIONAL STUDY MODEL: MODEL DATA INPUTS ANALYSIS, ALGORITHM, AND ASSUMPTIONS 115 4.1. Input Data Analysis ...................... 115 4.1.1. Estimating Production Levels ("Demand Pattern") ............... 115 4.1.2. Estimating the Cost Functions .... 117 4.1.3. Facility Capacities . ............. 132 4.2. The Algorithm ............................ 132 4.3. General Procedures and Model Assumptions.. 136 SENSITIVITY ANALYSIS AND RESULTS ............... 141 vi APF 6 APPENDIX A 711'"an l—KCu 5.1. General ................................. 5.2. Effects of Changes in Demand ............ 5.3. Evaluation of Facility Capacity Changes.. 5.4. Effects of Forcing Certain Facilities into Solution ........................... 5.5. Computational Experience ................ CONCLUSIONS AND RECOMMENDATIONS ............... CASHEHNUT AUTHORITY OF TANZANIA FUNCTIONAL STRUCTURE ..................................... PRODUCTION CENTER MILEAGE CHART ............... FACTORY PRODUCTION COSTS: 1980-81 ............ TYPICAL COST OF PRODUCTION ITEMS .............. MTHARA FACTORY PRODUCTION FIGURES: 1981-82 ... CASHEH PRODUCTION LEVELS: DERIVATION OF PRE- DICTION EQUATION USING ANALYSIS OF VARIANCE AND ORTHOGONAL POLYNOMIALS ........................ LAGRANGE MULTIPLIER TECHNIQUE ................. PREDICTION EQUATIONS: CASHENNUT PRODUCTS FOR EXPORT ........................................ FORTRAN INTERFACED SUBROUTINES ................ SAMPLE MODEL OUTPUTS .......................... SAMPLE SHIPMENT PLANS ......................... REFERENCES ............................................ 157 164 167 170 171 172 173 174 175 181 184 186 193 209 220 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 TABLE 2.1 LIST OF TABLES Estimates of Cashew Nut Yields from a Cashew Tree .......................................... Composition of Cashew Apple ................... Nutritive Value of Cashew Nuts (28.37 g) Roasted in Oil ................................ Cashew Kernel Classification: Grade Designa— tions ......................................... Tableau Form for Design Data and Transportation Cost Matrix, Cij .............................. Raw Cashewnut Procurement, 1973-74 to 1982-83.. Tanzania Cashewnut Export (Rawnuts and Kernels) 1975-1982 ..................................... Breakdown of Cashew Product Exports: Amounts and as a Percentage of Procurement ............ First Five-Year Plan (1976-77 to 1981-82) Esti- mates of Increases in Rawnut Production: Actual and Projected .......................... Average Annual Fixed Costs and Operating Costs and Amounts Processed for Existing Processing Plants ........................................ Recovery Rates for Lindi and Kibaha Cashew Factories: 1980-81 and 1981-82 ............... Production Center Production Level Estimates (Metric Tons) by Subreqion: 1982-83 to 1991-92 Cost Per Tonne—Year Matrix for Fixed Charge Setting ....................................... Cost Per Tonne-Year Matrix for Fixed Charge Setting ....................................... 18 22 25 31 101 102 106 107 109 111 114 118 129 130 5.1 5.} 5.8 5.8 Model Variables: Source, Description, and Assumptions ................................... 137 Sensitivity Analysis Variable Inputs: Model Verification Parameters ....................... 142 Coefficients for the Allowed Plant-Production Center Combinations - Assignments Costs ....... 143 Effects of Predicted Changes in Production Center Levels (Demand) on Plant Locations' Annual Systems Costs and Routing of Cashew and Its Products for the 1982-83 to 1986-87 Seasons 144 Demand Change Summary of Effects .............. 152 Robustness Index (RI) for the Six Sets of Solutions ..................................... 155 Effects of Cashewnut Processing Plant Capacity Changes on Plant Location, Allocation, and Total System Cost ............................. 156 Effects of Forcing Certain Key Facilities into Solutions ..................................... 160 Computational Requirements for Solving the Fixed Charge Problems ......................... 165 2.9 2.1 FIGURE 2.12 2.13 LIST OF FIGURES Cashewnut Production: India and Mozambique ... Cashewnut Production: Brazil and Tanzania .... World Aggregate Cashew Production and Regional Trends ........................................ Tanzania Cashewnut Producing Regions .......... Correct Positioning: Seed Planted at Stake, Stalk-End Upwards ............................. Incorrect Positioning: Seed Placement at Stake Effect of Planting Depth on Germination ....... The Cashew Fruit: Nut and Apple .............. Section of a Cashew Nut ....................... The Cashew Products ........................... Twelve Steps in Processing Shelled Nuts ....... Sketch of a Comprehensive Distribution System: Agricultural Product Handling ................. Planning Problem with Two Local Optima ........ An Optimal Solution Within the Inferior Region of a Multiobjective Analysis: A Case for an Incomplete Multiobjective Model ............... Fixed Charge Approximation for a Concave Cost Function ...................................... Two Piecewise Linear Fixed Variable Cost Approximations ................................ Three Linear Segments in the Piecewise Linear Approximation ................................. 13 14 15 21 21 24 26 33 38 39 50 51 52 2.16 2.17 2.18 2.23 Tangent-Chord Approximation to Facility Costs for Potential Facility 1 ...................... 55 A General Branch-and-Bound Solution Procedure for All Integer Linear Programs ............... 56 Graphical Representation of the Problem ....... 59 The Fixed Charge Cost Function and Its Approximation ................................. 60 Tableau Representation of a Generalized Assign- ment Problem (GAP) ............................ 64 GAP Tableau Representation of a p-Median Problem ....................................... 65 GAP Tableau Representation of a Median Problem with Capacity and Investment Constraints ...... 66 GAP Tableau Representation of a Simple Facility Location Problem .............................. 71 GAP Tableau Representation of Capacitated Facility Location Problem ..................... 72 Elements of a Planning Process ................ 80 Effects of Number of Facilities on Cost Ele- ments of a Distribution System ................ 89 Search Procedure Complexity Versus Complexity of Cost Functions ............................. 92 Cashewnut Production in Tanzania (1965-66 to 1982-83) and Cashew Prices .................... 104 Cashewnut Processing Plant Operating Costs (Combined-Concave Cost Function) .............. 113 Relationship Between Subregions' Annual Cashew- nut Production Level (Transformed and Real Data) and Production Year for a Ten-Year Period 116 Tanita I: Cashewnut Processing Total Cost Curve: Cost Coefficients ..................... 119 Tanita II: Cashewnut Processing Total Cost Curve ......................................... 120 Kibaha and Tunduru: Cashewnut Processing Total Cost Curve .................................... 4.10 4.11 4.12 5.1 5.2 5.3 Kilwa: Cashewnut Processing Total Cost Curve.. Lindi: Cashewnut Processing Total Cost Curve.. Mtama: Cashewnut Processing Total Cost Curve.. Nachingwea: Cashewnut Processing Total Cost Curve ......................................... Likombe, Masasi, Newala I and Newala II: Cashewnut Processing Total Cost Curve ......... Mtwara: Cashewnut Processing Total Cost Curve. Main Frame Flow Chart for Facility Location- Allocation Study .............................. Flow Chart: Bender's Cut - HJS Search ........ Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1982-83) .............. Facility Location—Allocation Plan and Raw Cashewnut Routing Plan (1983-84) .............. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1984-85) .............. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1985-86) .............. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1986-87) .............. Facility LocationaAllocation Plan and Raw Cashewnut Routing Plan (1987-88) .............. Facility Location—Allocation Plan with 50% Plant Capacity Utilization .................... Facility Location-Allocation Plan with 75% Plant Capacity Utilization .................... Facility Location-Allocation Plan: Tanita Il Forced into Solution .......................... Facility Location-Allocation Plan: Mtwara Forced into Solution .......................... Facility Location-Allocation Plan: Tanita I and II Forced into Solution ................... 122 123 124 125 126 127 Dric duct deSte CHAPTER 1 INTRODUCTION 1.1. Purpose of Study Estimates of the Food and Agriculture Organization (FAO) of the United Nations (1980) rank Tanzania fourth after India, Mozambique, and Brazil in terms of volume of cashewnut export trade. The world market price for the whole cashewnut kernel has been remarkably buoyant, with prices tending to remain firm even with increases in pro- duction (Menninger, 1977). The general trend, however, of the aggregate world cashewnut production is on the decline in all producer countries except Brazil (Figures 1.1—1.3). Reasons for the decline include droughts, poor transport and storage facilities, and, in the case of Tanzania, the destabilizing effects of the "villagization program" under which most individual producers were relocated to distances great enough that they could not effectively attend to their cashewnut plantations (Daily News, 1982). Cashewnut production in Tanzania is predominantly a smallholder enterprise. Effective the 1984—85 season, the Cashewnut Authority of Tanzania (CATA)——which until the 1983-84 season was responsible for promoting cashewnut pro- duction, collection, processing, and marketing——will only IU ‘ \ II\ C 0 5 2 ll AmOECOH 000v O .3 105 COwUUZUOLQ IOCMOU Cashew Production (000 tonnes) N O O 150 100 .5533 01 O 250 2.22:: A ?$: QR w T? w 200 ‘ 2:31 ... ...;- 'r.. :1:-: .' -: ~ F '-2 :. O .':-.‘- in an 7% L: w 22 A; '.: 'j ‘..'.'_ 1'3." :' .' 1- 3'.- :- '-'-‘- 'L':'-. 1'. O '- '1 ‘ ".' .'~'.'.-' 231'. 8 ‘.1~- "5 ;,. "..'-5.; c, 150 * :=.:~. .2 :11}: ”2,31 '2': [-3 -_. $11.2: -.' I, .1; '.- - ‘ 5;“. ' :3...- : .. 1' " :-. :31 2.9.3.: 3.5:" .2 :13" ' '3 {1.5" :35" '3:- {.25 ..' 5.“: +4 :5 :2. . -' :- ' .;_'.: .2 31:1. U -_'.'; - .. . .- = 100 --= ' ::-- r.-' :.-':'. :.'.,- ‘U 1.3:. .... '_‘:.: .,- . 0-33.12: .'-: 2 If :. .‘-.'-.": :f-Z. ‘- '. 312' .; _':-': -_.;_ij= ':-'._' -I o. If: '2 5} :- 2:. ‘23:: z 2'5: tit-ES . ,t '- '- - ;- -:'_ w -'.- '..- .--. P." 5 50 1 '3‘: 5: :1 3: $3 '5' w 95 J -‘ -': "' 23 331': ?-E-: - . €21"- ' ' 355:3 1:":1 3:.‘12 3"“. :2‘ II" 0 ' ' 3L5. 74 76 78 80 82 74 76 78 80 82 Year of Production Year of Production (a) INDIA (b) MOZAMBIQUE Figure 1.1. Cashewnut Production: India and Mozambique. Source: FAO Production Yearbooks, Vols. 33, 34, 36. 2 m I «mUCEOu Dee» 6 ..‘U CO~HU2DOLQ kctmcb Cashew Production (000 tonnes) 150 _~ N O to O 01 O (.0 O 200 d U"! 0 VI 0) C C O p '- o L3 O 1.9::1—1 t: 0 HI- ”51—1_ _,-.-.-. -— 100 .:_ .- .: U .1 :.' ...I' 3 .2 . at =: sn U a PT :2 11... 11.1 O n: ,:..'_ ... ..' . ”'4 L : . .. . ... 1'2‘ . D. ....1 ';..1' {I '. . 2". 2'1 " ... .' 1' . 3 '.' '.1 .2‘. a) . ' '- .-".. Z 50 :3 :' ..'-‘2 U} '1: :1, ."-~ I'D 2': J.- y- u 1 a {-21 '5 . z 13:1 1 11.1 :J _: ,y .. 0 .3. f 74 76 78 80 82 74 76 Year of Production Year of Production (a) BRAZIL (b) TANZANIA Figure 1.2. Cashewnut Production: Brazil and Tanzania. Source: FAO Production Yearbooks, Vols. 33, 34, 36. 600 -§fi UT 0 O A h C O (.0 O O N O O Cashew Production (000 tonnes) _n O O A .' ,0 32', 74 7s 78 80 82 Year of Production Figure 1.3. World Aggregate Cashew Production and Regional Trends: (1) Africa, (2) Asia, (3) South America. Source: FAO Production Yearbooks, Vols. 33 . 34, 36. 1101 0pm 1189': C651 Hiti ,_ ..., ..’..«gawfl . "nu- --.. 5 act as a marketing-processing board (Makota, 1983). CATA currently operates thirteen processing plants scattered all over the cashewnut-producing region (Figure 1.4). The decline in production in Tanzania resulted in closure of several processing plants necessitating transshipment of raw cashews to other processing centers. It is anticipated, however, that with the government drive to encourage devel- opment of new plantations (Mwenkalley, 1983), there is a need to develop a plan for phasing in and phasing out cashewnut processing facilities and locating facilities within the region. When dealing with the location of more than one facility, there frequently exists an associated allocation problem. The term "location-allocation” as used in this study is then usually employed. 1.2. Scope of Study As Geoffrion and Powers (1980) put it, "the question: How many warehouses should we have? is deceptively simple because a proper answer requires answering, at the same time, a host of interdependent questions." A comprehensive planning model with optimization capability should be a managerial tool that can be used to deal not only with facility location but also with a variety of additional management questions. So often the preoccupation with just locating facilities tends to distract from the wider range of issues that needs to be considered. 35 KENYA 40 Kibaha .. salaa- ..§’"" - R060 / ,'..--.'" Kiln ) AD R0 o, ..'-..'...,o F KiVinja .fi,/. ./ LINDI...-'.‘":.'..l 1. "EMIBIA . ‘1’ “nun-"1.1m? , Lindi «. .- .. \yachingwea E "3‘. ~L.» - _,., ngea . "1M3 ..'“ MALA I RUVUMA “"11""3. ...... ' Eu Newala l asa Mt rara -""'H02AHBIQUE Production Centers a Main Production Areas Cashew Factory ...... Railway Regional Boundaries I1“" Main Roads TANZANIA Figure 1.4. Tanzania Cashewnut Producing Regions. It was therefore believed in this study that optimiza— tion models may be more effective and more useful to decision makers if these models were also used to generate planning alternatives that are good with respect to the modeled objective(s) and different with respect to the deci— sions they specify. The assumption here was that some of these alternatives may be better than others with respect to the unmodeled issues and, in this way, extend the utility of the optimization models. The location-allocation problem in the cashew industry in Tanzania, like many resource planning problems in Third World countries, may be regarded as a public sector problem. A prominent strategy in public sector location problems is the selection of an optimal set of locations out of a candi- date set which seeks to provide service for the maximum pos- sible population. Each community to be served has an eligible set of facility sites which are within its maximum service distance or time. The objectives of this study were to address such requirements. 1.3. Study Objectives This study addressed the following objectives: (1) Review the state of the art in cashewnut production, handling, and processing in Tanzania. (2) Develop a location-allocation model for planning opti- mal (or near optimal) plant locations and relocations for the cashewnut industry in Tanzania. 8 Generate planning alternatives and formulate evaluation and elaboration criteria. Demonstrate the model's capability in planning a facil- ity location system by answering the adaptive "what if" questions focusing on: , (a) changes in demand* (production center production level) structure; [if . (b) facility capacity changes; and, ' (c) forcing facilities into solution. 1gduction level figures in the text specified as . 'aons“ refer to metric tons (equivalent to CHAPTER 2 LITERATURE REVIEW 2.1. Cashew Studies and Cashew Growing 2.1.1. Distribution and Ecology The cashewnut (Anarcadium occidentale Linn.) is native to tropical Central and South America (mainly Brazil) and the West Indies, but cultivation has spread to other tropi- cal countries, notably Tanzania, Mozambique, Kenya, and India. Although it may be found growing at elevations up to 1200 m (4000 ft), it is best suited to lower elevations. Cashews have become naturalized in regions with average annual rainfalls ranging from 3800 mm (150 in) to as low as 500 mm (20 in). In parts of southern Tanzania, where cashews are grown on a substantial commercial scale, the average annual rainfall is commonly 760—1016 mm (30-40 in). Cashews are often grown in soils that are considered too poor or stony for most other crops, but the trees prefer loams or sandy loams to very sandy soils. Whatever the nature of the topsoil, free drainage and the absence of brackish conditions are considered to be essential (Argles, 1975). Given soils with suitable textures, the cashew trees appear to tolerate a fairly wide range of pH values. 10 Little appears to be known about the reactions of cashews to temperature apart from the fact that they are sensitive to frost (Morton, 1962) and to excessively hot, dry weather such as occurs in parts of northern India. Nor has any information been encountered on their response to variations in day length. 2.1.2. Propagation by Seed The large part of the cashewnuts that figure in expand- ing world trade are harvested from wild, self-sown seedlings or from plantations in which the trees have been raised i_ 5123 from seeds planted at stake. The preference for plant- ing seeds at stake instead of growing them in nurseries arises from difficulties experienced in transplanting young seedlings (Garner, et al., 1975). Cashew seeds vary markedly in size and weight, but only the latter is closely correlated with the kernel content because the larger nuts commonly contain air pockets between the kernel and shell or between the cotyledons and their kernels are sometimes defective. A more useful criterion of seed quality is its specific gravity. Sorting out seed on the basis of specific gravity has been done by using a sugar solution. In trials in Tanzania, four sugar solu- tions were used to separate cashewnuts into four categories with specific gravities ranging from below 1.000 to about 1.075. The speed of germination, the percentage germina- tion, and yields in the first three harvest years rose 11 progressively with each rise in the specific gravity of the seeds. This led to the general advice that only seeds which sink in a solution composed of .068 kg (1% lb) sugar to 4.5 1 (1 gal) water should be used for planting. The work on seed selection reported above is related to nuts col— lected at one time when they are fully mature. Studies in India and Tanzania suggest, however, that neither the time when the mature nuts are collected nor their stage of maturity are likely to affect germination appreciably (Turner, 1956; Rao and Hassan, 1956; Rao, et al., 1957; Morton, 1960; Mutter and Bigger, 1961; Northwood, 1967). Although there is no experimental evidence reported, it is in fact often recommended that sun drying of cashewnuts intended for seed, sometimes as long as 12 to 14 days, should be conducted. 0n the other hand, chilling (exposure to 4°C for 15 minutes) has accelerated the germination of cashew seeds. In India, seed treatment with growth promoter gibberellic acid improved seed germination. Nuts that had been stored for 8 to 10 months, then soaked in water for 24 to 48 hours before planting, gave slightly improved germina- tion percentage, and germination was hastened by one to four days. The overnight soaking of all cashew seeds, whether fresh or stored, is a generally recommended practice in Mysore, India. On trials in Italy, however, it was found that the viability of the nuts altered very little for two years after picking, provided they were kept dry (Rao and 12 Hassan, 1957; Rao, et al., 1957; Ibanez, 1968; Shanmugavelu, 1970). The planting of seed at stake is normally done as early as possible during the rainy season. The usual practice is to prepare planting pits (46x46x46 cm in size) a month or more before the date of planting. The pits are left open until about two weeks before planting, after which they are filled with topsoil to which farm yard manure or compost may be added. Burned earth and ashes may also be added. Artificial fertilizers are not normally applied, although the application of rock phosphate is sometimes advocated, particularly when transplanting seedlings. Recommendations for spacing vary widely among regions, ranging from 6x6 m (20x20 ft) to 12x12 m (40x40 ft) and up to 15x15 (50x50 ft) for windy sites. Dagg and Tadey (1967) reported that rainfall, rather than soil fertility, should be the factor given greatest attention when deciding on the spacing to adopt. As a partial insurance against pest predation and failure of some seeds to germinate, it is com- mon practice to plant two or three seeds at each pit, indi- vidual seeds per hill being about 23 cm (9 in) apart. Where seeds are planted at stake, it is preferable to place the seed with stalk end upwards but inclined at an angle (Figure 2.1). An incorrect way of planting is shown in Figure 2.2. Figure 2.3, on the other hand, shows the effect of depth of planting on the average germination (Albuquerque, et al., Figure 2.1. (a) (b) Correct Positioning: Seed Planted at Stake, Stalk—End Upwards. (a) Straight (b) Inclined G CD”) (5 Figure 2.2. Incorrect Positioning: Seed Placement at Stake. ) Stalk-end down ) (a (b Stalk-end sidewards , ...- _. ___..__,"._____.._ .— 50 40 30 L 20 Germination (days after planting) 5 1o 15 ' 20 Planting Depth (cm) Figure 2.3. Effect of Planting Depth on Germination (Days After Planting). (After Garner, et al. 1975) 16 1958; Ahmed, 1959; Saville and Bennison, 1959; Mutter and Bigger, 1961; Viswanathan, 1961; Garner, et al., 1975). 2.1.3. Shoot, Root, Flowering, and Fruiting The cashew is a spreading evergreen tree that may reach 12 m in height. The pattern of growth presented by bearing trees may perhaps be defined as one of indetermin- . ate flushing, with individual shoots emerging at different times over comparatively long periods. This suggests that pronounced physiological differences may exist between indi- vidual branches of a tree throughout the periods when the growth is occurring. The nature of these physiological dif- ferences is unknown, although in the flower flush they seem to be reflected in differences in the sturdiness and total leaf area of bearing and nonbearing shoots (Bigger, 1960; Ohler, 1967). Even in very young trees only 1% to 2% years old, a 1 very extensive lateral root system may be found, extending I to almost twice the diameter of the canopy. Inasmuch as some top growth may occur throughout the year, it is proba- ’ bly safe to assume that some root.growth is also taking 1 place during the greater part of the year (Tsakiris and ciable crops in their third to fifth year, although some t Northwood, 1967). Cashew trees usually start bearing appre- ; fruits may be borne on trees that are younger and sometimes little more than one year old (Ekrement, 1965; Northwood, a".-- «4,. h ...h. 17 1966). Table 2.1 shows the estimates of yield from a cashew tree by age (CATA estimates). The cashew inflorescence is a lax, terminal, many- flowered panicle in which male flowers outnumber hermaphro- dite flowers, usually about six to one. Inflorescences are usually borne on only one of the main growth flushes, although, occasionally, some flowers and fruits are borne on one of the other growth flushes. In India, wind is thought to be the main pollinating agent, but elsewhere (East Africa and Brazil), various insects appear to play an important role. Following pollination, there may be a sub- stantial fall of fruitlets, which is generally attributed to physiological causes but may also be accentuated by insect attack. A period of two to three months elapses between fruit set and fruit maturity. The true fruit, or nut, reaches its maximum size during the first half of this period, whereas the fleshy cashew apple, consisting of an enlarged pedicel, receptacle, and disc, makes most of its growth during the second half of the period. The harvesting period lasts from 1% to 3-4 months, depending on the region. In Tanzania, the harvesting period starts in October and extends to February. Although light showers of rain during flowering may not be harmful and may even sometimes be beneficial, heavy rain during this period or during fruit development may result in crop loss. Cashews are also subject to attack in many areas Table 2.1. Estimates of Cashew Nut Yields from a Cashew Tree (kg per tree per season versus age of tree). Age of Tree Yield of Raw Nuts Yield of Raw Nuts per Tree per Hectare* (Years) (kg) (kg) 1 - _ 2 - - 3 _ _ 4 0.5 35 5 2.5 175 6 4.0 280 7 5.5 285 8 6.5 455 9 7.5 525 10 8.5 595 11 9.5 665 *1 hectare is equivalent to 70 trees at 10x12 m spacing. m ——» -— -—-'~ .. 19 by various species of thrips and by cuspid bugs (Helopeltis spp) as well as by fungal mildew (Bigger, 1960; Ohler, 1967). 2.2. Cashew Harvesting, Handling, and Processing 2.2.1. Cashew Harvesting, Handling, and Cashew Products Ripe cashew fruits are not plucked from trees but are left to fall to the ground and collected by hand. The fruit (Figure 2.4) varies in size from 5 to 10 cm in length and 3.8 to 5 cm in width. It is yellowish—red in color and possesses a thin, waxy skin. It is broadly conical or pear-like in shape and is usually referred to as cashew "apple." The whole nuts are removed from the apples by hand with only a small percentage of "apples“ spared for local consumption. After the nuts have been gathered, they are sun dried for two to three days. During this time, the moisture con- tent is reduced from 16% to 7% so that the nuts can be safely stored. The nuts are now either bagged and held for future processing or immediately processed. The quality of cashew apple products (juicy, astrin- gent, and nutritious, with a characteristic pleasant flavor rich in ascorbic acid, sugars, and vitamins) is largely influenced by the amount of astringency. Pantastico (1975) reports that there is a wide variation in the tannin content of juice extracted from fruits of different selections. Further, the major polyphenolic constituent in cashew apple juice has been found to be leuco-delphinidin. The juice i; 20 can yield satisfactory blends with lime juice (1.5%) and pineapple juice (50%). This is then pasteurized in bottles or cans and, if stored at 26.7°C (80°F), may have a shelf life of about 32 weeks. The juice of the cashew apple may also be fermented and made into wine. Cashew apple wine distilled into a spirit or liqueur is highly potent. The fruit and wine, rich in vitamin C, possess antiscorbutic properties. Years ago, the liqueur was valued for its diu- retic properties; it was believed to have a healthful effect on the kidneys and was prescribed in advanced cases of cholera. Other studies on biochemical and storage aspects of the cashew apple are reported by other authors; namely, Singh and Mathur (1953), Baile and Barcus (1970), Lopes (1972), and Maia, et a1. (1975). Table 2.2 shows a typical composition of cashew apple. Figure 2.5 shows a section of a cashew nut. The shell of the nut is hard, about 2.8 cm thick, and is of a honey- comb like structure on the inside. It consists of two layers with an oily liquid between them. The outer layer, which is smooth surfaced, is thin and hard. The inner layer is hard. Between the two layers is the difficult-to-handle cashew liquid (called cashewnut shell liquid, usually abbre- viated as CNSL), which has a growing commercial importance. The CNSL is comprised of aracardic acid, C22H32021, a brown crystalline substance, and cardol, C21H32021, a dark brown phenolic oil. Both are very toxic and irritating, producing blisters on the skin. CNSL has many industrial Apple Figure 2.4. The Cashew Fruit: Nut and Apple. Endocarp (Hard) L— Testa Membrane of Kernel Epiqarp (corlaceous) Mesocarp (spongy) Kernel Figure 2.5. Section of a Cashew Nut. 22 .1 Component 1 61 ,;‘Moisture 87.8 1 ;fl Proteins 0.2 ,; ,.1F‘ts 0.1 VA gaarbohydrates (90% of fermentable sugars) 11,5 hi fttalclum 0.01 S? ‘EPhosphorus 0.01 N? 0.2 mg/100 g £“‘°“‘" c 261.5 mg/100 g mlflInerals 0.2 23 uses; it is used in making resins for the manufacture of varnishes, disinfectants, special inks, brake linings, and even lubricants. Several patents have been filed for the product (Menninger, 1977). Cashews yield a number of commercial products, as sum— marized in Figure 2.6. These products are derivatives of the cashew apple and the nut. Table 2.3 shows the typical nutritive value of roasted cashew nuts (After Kuzio, 1977). 2.2.2. Nut Processing Cashewnut processing procedures in East Africa are more mechanized than those in India and Brazil. The basic proc- essing stages in all three regions, however, are the same. Figure 2.7 shows the steps involved in processing shelled nuts. Six stages can be broadly categorized in terms of the functions to be accomplished. These include: (a) Cleaning and First Calibration: This involves the removal of impurities from the outer shell. In Tanzania, this is accomplished by putting the nuts in a large tank which has jets of water and rotating brushes. As the nuts come out, they are then classi- fied into two size grades (large and small) providing the first calibration. In India and Brazil, most cleansing of nuts is done by hand. (b) Cashewnut Shell Liquid Extraction and Nut Preparation: This stage serves two purposes——the removal of CNSL and .uuNHHmua mm eofiqaa= do womucmocwa HfiaEm a xfico .mmmcccx unannoEca cu mcmwum. .muuacocm zwzmmu map .m.~ mczawm muuddus m=~»m m—uuaw au—A >=z¢~n u=u=oud uzua 24 d<==umue "maas "mu—aw Hussy: dau=m + 35:23.5 3...: =35 Table 2.3. Nutritive Value of Cashew Nuts (28.37 g) Roasted in Oil. 25 Ascorbic acid Content Amount Moisture 5.2 % Food Energy 159.0 cal Protein* 4.9 9 Fat 13.0 g Carbohydrates 8.3 9 Calcium 11.0 mg Phosphorus 106.0 mg Iron 1.1 mg Sodium 4.0 mg Potassium 132.0 mg Vitamins A 30 IU Thiamin 0.12 mg Riboflavin 0.07 mg Source: Kuzio (1977). *Notice the protein content (up to 18% of dry matter). 26 CLEANING SIZE GRADING CONDITIONING CRACKING SIZE SCREENING DRYhNG INSPECTION COLOR GRADING PACKAGING STORAGE Figure 2.7. Twelve Steps in Processing Shelled Nuts. (C) 27 contraction of the nut shell so that a space is created between the shell and the skin in order to facilitate shelling. In traditional processing units, the nuts are put into a metal drum and roasted over an open fire which is usually fed with spent shells. This traditional processing prevents any extraction of CNSL from the shell as most of it is burned off. In Tanzania, commercial factories employ another method of extraction called "CNSL Bath Extraction." In this method, nuts are put into a CNSL bath heated at about 195°-200°C for two hours. This creates explosive pres- sures within the nut and causes the "oil cells" of the pericarp to rupture, releasing CNSL into the bath for distiLHng. Another alternative involves putting the nuts in an autoclave which produces very hot steam under pressure and therefore producing the same effect as heating in a CNSL bath. This step extracts about 85% to 90% of the CNSL. The nuts are then allowed to cool for about 24 hours before shelling. Second Calibration and Shelling: The nuts are normally sorted into three size grades (small, medium, large) by using sieves (3/16", 5/16", and 5/8" sieve sizes) into separate bins. Manual shelling, which is only done in India, is accomplished by hand mostly by women workers seated on the floor using wooden sticks and carefully striking along the cleavage line of each nut, taking care not to break the kernel. As a protection, the 28 workers dip their hands in vegetable oil in order to reduce the blistering effect of CNSL. They extract the kernel and put the shell in a basket of sawdust which will absorb extra CNSL. Using the hand-shelling method, an average of 14 kg per person can be cracked in an eight-hour day, and about 90% of the kernels are whole (Anon, 1966). In Brazil, pedal-powered shellers are used. About 80% of the kernels remain whole, and in an eight-hour day, about 25 kg are shelled per per- son. The drawback of the pedal—powered shelling method is the fact that broken kernels come into con- tact with CNSL and adhere to sheller blades and shells. Such contact discolors the kernels, affects the taste, and thus decreases the market value of the nuts. In Tanzania, this step is fully automated. In one kind of shelling machine used in Tanzania, the nut is mechanically directed into a gripper which moves against a circular saw. The blade cuts a groove into the shell; then a "scooping" device is inserted into the groove to remove the kernel in a manner similar to the "shucking" method used to open an oyster. Preci- sion impact shellers are also used, where the nuts are spun at a certain angle and brought against a sharp position on the knife table for the shell to split. Once the nut has been split and separated, the prod- ucts enter the shakers (in the case of the precision impact shellers) to separate kernels from shells or (in (d) 29 the case of circular saw type shellers) kernels drop through the screen into a picking belt to be examined for shells. The kernels then enter a flat platform belt for sorting and hence through the cylindrical grader, which classifies the kernels into different size grades. The circular saw type sheller (the most modern sheller) can crack up to 96 nuts per minute. Mechanical shellers have three serious drawbacks: (1) the machines can only produce about 60% whole ker- nels, (2) they are sophisticated pieces of equipment and susceptible to failures, and (3) they are expensive. Peeling: In manual systems, the kernels are dried after shelling and then placed between finger and thumb to rub off the skin (testa). Mechanical systems include steaming the kernels until the skin is very wet, then shooting steam jets to remove the skin by impact and friction. Usually steam jets tend to over- heat and therefore scorching of kernels may occur resulting in yellow coloration of the kernels which places them in an undesirable classification. In Tanzania, the most preferred peeling method makes use of compressed air instead of steam (i.e., pneumatic peelers). After this step, the kernels are moved by conveyor belt across brushes to loosen and remove the remaining bits of skin which are picked up as the (e) (f) 30 kernels pass through a slight vacuum induced at the end of the belt. Sorting and Classification: As yet, no machine can remove 100% of the skin. All systems must have some means, usually conveyor belts, whereby the kernels are checked manually for skin. Although optic selectors are in use in most factories in Tanzania, in most cases, manual sorting is also utilized. Nith optic selectors, the kernels are fed into the machine, and those kernels having dark spots or yellow coloration on passing through a beam of light will reflect relatively more light than spotless kernels. This triggers the sensor to induce an air jet which sorts out the colored kernels. Classification of cashew kernels is now standard throughout the world. Table 2.4 shows this classification. In mechanized systems, the sheller provides some preliminary classification. Packaging: Cashew kernels have historically been packed in 5-gallon tins holding approximately 11.3 kg of kernels. Currently, the tins are vacuum packed with carbon dioxide. The vacuum packers used in 10,000-ton capacity plants in Tanzania have a packing capacity of 400-450 tins per day. The use of aluminum-foiled polyethylene packages is being contemplated as an alternative to tins (Makota, 1983). Cashew kernels are normally sold raw, mainly to avoid the duties levied on finished products by the major importing countries. 31 Table 2.4. Cashew Kernel Classification: Grade Designations. Descriptions Grade Kernels/1b Kernels/kg Nhole N210 200-210 440-462 N240 220-240 484-528 N280 260-280 572-616 N320 300-320 660-704 N400 350-400 770-880 N450 400-450 881-990 N500 450-500 991-1100 Pieces ON I Dessert Nhite I DN 11 Dessert Nhite II LNP Large Nhite Pieces SNP Small Nhite Pieces DP Dessert Pieces ScP Scorched Pieces Source: Noodroof (1978) and CATA. 32 2.3. Distribution Planning Systems The question of how many warehouses or plants should there be arises periodically at almost every company or institution offering or processing goods in the market place. This has led to numerous methods for determining facility location and has given rise to numerous computer programs addressing this problem for individual companies. The bulky and perishable nature of many agricultural products necessitates reviewing the degree of material handling to be employed, size of facility, and balancing advantages of nearness to customers against nearness to plants from a transportation viewpoint. In the instance of the cashewnut industry, issues of proper assignment of pro- duction centers to plants, proper use of cross-docking and plant direct supply, and proper pattern of inbound supply are important. Distribution planning issues should, if possible, be dealt with simultaneously since they are interdependent. Ignoring some of the interactions or dealing with issues sequentially may yield misleading conclusions and result in poor decisions. It follows that a comprehensive distribu- tion planning system along the lines of Figure 2.8 would be useful (Geoffrion and Powers, 1980). Such a model would enable calculation of performance measures (covering costs and agricultural product producers' service) for any consis- tent set of assumptions, collection of data, and design of the agricultural product handling system. Also incorporated 33 A PROOOCTION CENTERS PROCESSING PLANTS ENPORT OR CONSUMERS -Minimum and maximum -Plant capacities -Cust0mer demands for annual production level -Operating and fixed commodities for each center costs ~L0ca1 and export oBuying center costs ~Comm0dities (products) markets (fixed and variable) -Delivery policy FREIGHT RATES: Inbound, Direct, Interwarehouse, Outbound MAIN FUNCTION OF THE SOLVER: Interline °Number, location and size of plants °Plant serving zones 'All transportation flows 'Source loadings to lioilizo total cost -Processing, transportation, warehousing -Invent0ry and system reconfiguration subject to appropriate ' constraints -Processing capacity -Producer service -Single sourcing of Figure 2.8. Sketch of a Comprehensive - mans Distribution System: Agricultural Product Handling. 34 as part of Figure 2.8 is the function of the solver (solu- tion algorithm) needed to find the best (cost minimization) design for the system for any consistent set of assumptions. Geoffrion and Powers (1980) suggest that models of the type sketched in Figure 2.8 are most commonly run with non- optimizing algorithms (heuristics of one type or another). The reason is that true optimization has, until recently, been technically difficult to achieve due to the highly com- plex nature of the problems. Geoffrion and VanRoy (1979) discuss in depth the risks of using heuristics in distribu- tion planning. The importance of solution comparisons necessitates solvers that have optimization capability as opposed to error-prone solutions of heuristic solvers. A comprehensive distribution planning system along the lines of Figure 2.8 has a wide variety of potential uses. These uses may be grouped as follows: (a) Network Rationalization Issues: These are designed to answer the basic question of what is the most appropri- ate structure for the agricultural product handling system and how should products flow through the struc- ture to minimize total system operating costs given a certain required level of agricultural product producer service. (b) Adaptive "Nhat if . . .?" Issues: These allow model assumptions or data to be changed and reoptimization carried out in order to observe the change by compari- son with "base case" or "reference" run. Several (C) 35 "Nhat if . . .?" situations from environmental changes (e.g., impact of changes in demand structure, impact of higher fuel and other energy costs, impact of strikes, natural disaster, weather closure, energy shortages) and business decisions and policies (e.g., production ——plant capacity expansion proposals, plant location studies, impact of introducing new product line or dis- continuing one; marketing-—pricing policy analysis. split delivery policy analysis, expanding into new mar- ket; transportation——transportation policies, buying center capacity expansion or mechanization proposals, inventory policy comparisons; and others, including impact of combining autonomous institutional buying centers, evaluation of alternative distribution echelon structures, and implementation of priority analysis) may be incorporated. Parametric Studies: These involve systematic variation of single factors with optimization performed for sev- eral different values. The aim is to obtain a curve representing the essence of the influence of the factor being varied. These studies may help (Geoffrion, 1979) to quantify trade-offs between least system operating cost and agricultural product producers' service or the number of buying centers or plants. System sensitivity analysis with respect to any factor (e.g., inflation by cost category) as well as 36 influence of demand change over time may also be investigated. Indeed, a comprehensive optimizing distribution planning system has a surprisingly rich variety of uses beyond facility location, a must, particularly in public sector location analysis. 2.4. Optimization Techniques and Facility Eocation Analysis 2.4.1. Optimal Solution: An Aid to Analysts' Intuition Optimization techniques have been used successfully to obtain "optimal" solutions to mathematical models. In con- junction with parametric or sensitivity analysis and a set of noninferior solutions using multiobjective methods, optimization techniques may also be used to derive a few "second best" solutions. The last two decades have seen rapid advances in loca- tional analysis. New methods of analysis, such as optimiza- tion techniques and mathematical models, are the roots of I this expansion in capability. In spite of the vastly expanded spectrum of alternatives that one can use, the real world, with its immense complexity, tends to defy exact analogs. As Revelle, et al. (1970) observes, these methods of analysis are panacea for pouring out "optimal solu- tions." Indeed, models should be used as aids to provide insight into the sensitivity of solutions to changes in parameters, constraints, and criteria. Optimal results 37 should be regarded as an aid to the analyst's intuition and not as a replacement for it. Brill (1979), on the use of optimization models in public sector planning, concludes that both contemporary multiobjective programming formulations and the early least- cost optimization models were developed under the optimistic philosophy of obtaining "answers." But since most public sector planning problems are characterized by a multitude of local optima (which result from wavy indifference functions due to some of the important planning elements not being captured in the formulations), extensive parametric analysis is necessary to guarantee obtaining the best solution. In fact, omitted issues may imply that an optimal planning solution lies within the inferior region of a multiobjec- tive analysis instead of along the noninferior frontier. Figures 2.9 and 2.10, adapted from Brill (1979) illustrate such limitations. Differences inherent in using optimization models and in implementing solutions have been reported in many studies. Savas (1971), focusing on problems that are inherently political in nature, discusses the limitation of formal analysis techniques. Liebman (1976) observes that different members of society may not even be able to agree on a public goal, and, even when a common goal has been accepted, there is often disagreement over how the goal might be achieved. Liebman concludes that optimization methods "cannot and should not be used to resolve these OBJECTIVE TNO (Swimming) Figure 2.9. 38 4“z—‘Indiffez‘ence \4*:%Z»— CUrves B‘ \ \ \ aninferior set \\ Frontier ‘\ \ \ \ OBJECTIVE ONE (Boating) Planning Problem with Two Local Optima (Lakeside Park Problem). A and B are local optima; A is global optimum. 39 Indifference Curves OBJECTIVE TWO NOninferior set B‘ A OBJECTIVE ONE Figure 2.10. An Optimal Solution Nithin the Inferior Region of a Multiobjective Analysis: A Case for an Incomplete Multiobjective Model. A = Best solution point if only Objective One is considered. B = Best solution if Objectives One and Two are considered. ' B' = Projection of B which lies in inferior region of Objective One. 4O conflicts" but should be used instead to provide "intuition, insight, and understanding which supplements that of the decision makers." Brill (1979) further suggests the joint use of models, analytical and optimization, and tailoring an available algorithm to provide information and insights. Brill, et al. (1982) illustrate such an approach (The Hop Skip Jump Method) in which a model is tailored to generate alternatives using a land use planning problem as an example. 2.4.2. Facility Location Problems: Scope Due to philosophical and some other pragmatic differ- ences, location problems may be divided into two sector location problems. Although both problems share the objec- tive of maximization of some measure of utility while at the same time satisfying constraints on demands and other conditions, they differ in the way the objectives and con- straints are formulated. "Broadly speaking, however, they differ because of the ownership difference. Revelle, et al. (1970) defines public sector models as those that are characterized by a criterion function involv- ing a surrogate for social utility and constraint on invest- ment in facilities or in the number of facilities. Private sector models, on the other hand, are distinguished as those in which the total cost of transport and facilities'is iso- lated as the objective to be minimized. Although decisions on private sector location involve a host of issues, some of 41 which are noneconomic in nature, minimization of cost or maximization of profit to the private owners is a reasonably accurate statement of the objective of the location deci- sion. Public sector decisions, however, are made in response to a different set of owners——society as a whole ——and, here, the objective is to minimize cost or maximize benefit, which is not easily quantifiable in monetary terms. As expected, a large portion of the analytical work for location analysis has been carried out on private sector problems. Private sector location analysis poses the basic deci- sion choice between the cost of investment and operation of facilities to meet the product demand and the cost of trans- portation. The trade-offs implied are centralization and decentralization of facilities. Revelle, et al. (1970) discusses additional factors that influence private sector location as including the stochastic nature of problems (such as variations in economic and weather conditions as well as seasonal variations that influence the demand of product) and time staging of construction, opening, expan- sion, relocation, or abandonment of facilities. Other fac- tors include changes in competition, product acceptability and technology, and what has been referred to as "external effects," such as laws limiting environmental exploitation (i.e., pollution control, preservation of ecology). Location decisions in public sector models involve all the private sector problems outlined above as well as having 42 an additional dilemma that the goals, objectives, and con- straints are neither easily quantifiable nor are they easily defined. The public sector problem may be thought to fall into two classes-—ordinary services (e.g., post offices, schools, highways, water supplies, and waste disposal) and emergency services (e.g., hospitals, fire stations, and police). There are two ways in which public sector problems may be handled. The first utilizes the objective function method normally used in private sector models where an attempt is made to identify and quantify factors affecting the social cost. This approach is difficult and has rarely been successful. The second analytical approach makes use of some surrogate for social utility. The attempt, here, is to try gaining some insight about the system under analysis rather than defining exact solutions. Examples of surro- gates that can be used in public sector formulations as out- lined by Revelle, et a1. (1970) include: (a) Average distance or time traveled to facilities by the user population. The smaller this quantity, the more accessible the system is to its users. The problem becomes one of minimizing total average distance traveled subject to a constraint on the number of facilities to be established. (b) Creation of demand. Here, the user population is not considered fixed but is determined by the location, size, and number of central facilities. The greater 43 the demand created, the more efficient the system is in fulfilling the needs of the region. (c) Maximum distance or time between any facility in the set of the population area it is intended to serve. These surrogates are then optimized subject to con- straints on investment (in monetary terms or otherwise). It should be noted, here, that it is possible that the num- ber of facilities may be set in the political arena and may not reflect budgetary restrictions. Once first solutions have been obtained, using such objectives and constraints outlined above, one can start to evaluate sensitivity of the solutions to parameter variations. If these parametric changes do not seriously influence the solution, then one examines the trade-offs between investment and utility. After considering other complexities which beset the analy- sis, such as user population characteristics and length of planning horizon, final choice might be made from among the alternatives generated at different levels of funding. Savas (1978) discusses three measures of performance of public service; namely, efficiency, effectiveness, and equity. Efficiency measures the ratio of service outputs to service inputs (usually measured in monetary terms or in manpower). The service outputs are more difficult to define and measure adequately. Examples of service outputs may include the numbers of households serviced and throughput per year. These lead to efficiency measures such as cost per ton, cost per household per year, and tons collected per 44 manhour. Among the ways of improving efficiency, research has focused on changes ranging from institutional arrange- ments for providing services (Savas, 1977) to changes in vehicle routing (Betrami and Bodin, 1973; Bodner, et al., 1970), man scheduling (Altman, et al., 1971; Ignall, et al., 1972), and siting of transfer stations or disposal points (Marks and Liebman, 1971) citing examples from refuse col- lection areas. For stationary service centers, a measure of travel time or travel distance for service recipients has been found adequate (Elshafei, 1975; Abernathy and Hershey, 1972). Effectiveness measures the adequacy of service relative to need and incorporates the notion of service quality. It is difficult to measure effectiveness in public sector serv- ices; one of the ways used includes surveys of the level of citizen satisfaction. Equity refers to the fairness, impar- tiality, or equality of service. Equity is usually a con- cern of political scientists and economists. A number of different formulas, each equitable in a meaningful way, can be and are used to allocate or distribute public services. They can be subsumed under four general principles: equal payments, equal outputs, equal inputs, and equal satisfac- tion of demand (Savas, 1978). 45 2.4.3. Morphology of Location Systems Location analysis problems may be classified into two major structural categories (Revelle, et al., 1970; Eilon, et al., 1971; Robers, 1971; Scott, 1971): (1) Location on a Network: Characterized by a solution space consisting of points on the network (both nodes and points on the arc which joins the nodes) and dis- tance measurement or time measurement (e.g., dij defined as the length or time of the shortest path from node i to node j ) along the network. (2) Location on a Plane: Characterized by an infinite solution space where facilities may be located any- where on the plane and distance measured (dij) accord- ing to a specified metric. For example: (a) Euclidean metric, such that: dijz = (x1 - xj)’ + (yi - yJ.)2 [2.1] where dij (xi, yi) are coordinates of the ith point in a is the distance between i and j and rectangular system; (b) Metropolitan metric, such that: dij = |xi - le + |yi - le [2.2] 46 2.4.4. Location Analysis The field of location theory is a large and topical one, as evidenced by the lengthy bibliographies of Francis and Goldstein (1974) and Lea (1973). Historically, loca- tion analysis began with Alfred Neber, who considered loca- tion on a plane of a factory between two sources and a single market. But it was the works of Kuhn and Kuenne (1962) and Cooper (1963) that stimulated interest in loca- tion analysis. Their works described an iterative procedure for solving the generalized Neber problem. The formulations were designed to find a single point which minimized the sum of the weighted Euclidean distances to that point. Thus, the objective was: . . _ n _ 2 z 1/2 Max1mize z - 21:1 wi[(xi xp) + (yi-yp) ] [2.3] and thus: - 2 ‘ 2 1/2 dip - [(xi-xp) + (yi'yp) J [2.4] where: dip = the Euclidean distance from point i to central point P n = number of points which are served w. = the weight attached to the ith point (goods demanded, resources spent, population served, etc.) 47 the location of the ith point relative to some X ll fixed cartesian coordinate system the unknown coordinate of the central point P xp’ yp Partial differentiation with respect to x and y yields a P D pair of equations, below, which have no direct solution to these variables: w.(x. - x ) g¥L-= xi 1 g p = O [2.5] D 1!) w.(y- - y) £51 = 21 1 5 p = O [2.6] P 1P The iterative procedure suggested involves solving Equations 2.5 and 2.6 for xp and yp in terms of wi, xi, yi, and dip’ i.e.: wixi / wl [ ] x - z. 2 2.7 p 1 aip “fl—p iyi "i The value of d is then recalculated using Equation 2.4 and in the procedure repeated until successive differences between values of x and y are negligible. D D Eucledian problems may find applications in both pri- vate and public sectors such as in warehouse locations where goods are distributed from central points and where den poi on put or ist pro tiv sub; "OEr. 48 demands emanate from outlying points to the central supply point, respectively. Problems of locating central points on a network have also received a lot of attention in both public and private sector analysis. The problem of plant or warehouse location has the following general character- istics: given a number of demand areas for a certain product, each with specified demand and a number of alterna- tive sites where facilities may be built to satisfy these demands, determine where the facilities may be placed and which demand areas are to be served by a given facility. The general mathematical formulation of the warehouse or plant location problem applicable to the private sector is: . . . _ n m m Minimize z — 2j=1 21:1 dij(xij) + 21:1 Fi(yi) [2.9] subject to: z" x = y V [2 10] j=1 ij i i ' 2'" x = 0 v [211] i=1 ii i j ’ yi Z 0 Vi [2.13] where: D = the demand at area j v I. cost of shipping quantity x. ij from i to j 49 Fi(yi) cost of establishing and operating a facility at site i, where yi is being shipped from i m = number of proposed warehouse facility sites n = number of demand areas x.. = amount shipped from facility i to demand area j y1 = total amount shipped from facility i The function Fi(yi) is, unfortunately, frequently non- linear and generally exhibits a large fixed charge on investment. The function is therefore usually concave and not amenable to linear programming. The method of treating the concave cost function has been the key element of most procedures in past research. Haldi and Nhitcomb (1967), on economies of scale in production, have conveniently assumed taking the form of a concave cost function which is approxi- mated by a piecewise linear function. The differences between solution procedures are then influenced by the shape of the approximation adopted (Elshafei, 1975). Examples of such approximations are shown in Figures 2.11 to 2.13. 56 (1969) and Ellwein and Gray (1971) adopted the approximation shown in Figure 2.11 and both developed branch and bound algorithms to solve the problem. Kuehn and Hamburger (1963), in their heuristic approach for locating warehouses, assumed an approximation that transport costs are linear in the amount shipped and that facility costs are of the following form: 50 COST .1). O ------ - ------- --- QUANTITY Figure 2.11. Fixed Charge Approximation for a Concave Cost Function X. fixed charge for operating costs fl.‘ HO‘ fixed cost '1) Figu 51 I-.- --. 9.----- .-'OOOOOOOO'OA 0.-.-.0---0-'--"----O.'-.-Il -.'..-- -..---- -.'- "----'-.l hmoo QWMNTTY 2" Two Piecewise Linear Fixed Variable Cost Approximations. Figure 2.12. this“ 0057 Tar—1 H——"h—-¢1 H.~o 52 poooooooooooo o..- a... .0 o. o ......o..... a... o. 0.... -0 ......o..... .. .-- --. 0.0.0.... ....00000 J Figure 2.13. :N QUANTITY C a... 0 Three Linear Segments in the Piecewise Linear Approximation. fi = fixed annual charge at location xi = facility i's variable cost per unit cc ta DC 53 F1(Y1) 31 + biyi if facility exists [2,14] = 0 otherwise [2.15] That is, Fi(yi) consists of a fixed charge (ai), which is independent of the storage or production, and a linear cost, which depends on storage or production (in case of plants) if a facility exists. The expansion cost, b is assumed i’ constant for each facility. The solution algorithm begins with a single facility, and one additional facility is added every time until it appears that another facility cannot be added without increasing total cost. The assumption is that the best N facilities are contained in the set of the best N+1 facilities. On the other hand, Feldman, et al. (1966) assumed a more general form for facility cost——that of a continuous concave cost function. They also assumed that the transpor- tations costs were linear in the amount shipped between points. Since Fi(yi) did not have an expansion cost compon- ent in this case, the assignment of demand areas to existing facilities was much more difficult to accomplish. The authors resorted to an approximation method. Hence, the algorithm by Feldman, et al., started with all facilities and dropped out individual facilities one at a time. Solu- tion was achieved when no further savings were realized by further facility elimination. 54 Two other approaches to incorporating economies of scale of facility systems have been the tangent line approx- imation reported by Khumawala (1974) and the chord approxi- mation method used by Soland (1974). An obvious shortcoming of tangent line approximation is that using tangents only focuses on the variable costs so that fixed costs do not influence the ensuing facility assignments. Soland's approach overcomes this drawback by use of the chord (convex envelope) cost approximations. The drawback with the chord approximation approach, however, is that resulting lower bounds are generally weak and a very large solution tree must be generated (due to typically large numbers of nodes). A combined tangent-chord approximation of costs as shown in Figure 2.14 (Kelly and Khumawala, 1982) combines the advan- tages of the two approaches. Efroymson and Ray (1966) presented a solution procedure to a related but more constrained problem than that of Feldman, et al.(1966) utilizing an implicit enumeration technique——the branch-and-bound algorithm. This selective enumeration (Figure 2.15) was guided at each stage by a bound on the value of the objective function obtained at that stage. Computational experience on problems with 50 possible facility locations and 200 demand areas up to 100 facilities and 150 demand areas have been reported by Efroymson and Ray and by Spielburg (1969), respectively. The branch-and-bound algorithm used by Spielburg had some added features to speed up computation. 55 Tangent+ f F.+f.(') ; l l i i p. i 3 1 I / 3 i ./ [gs-wharf: E ./ / E i Chord* / I, 3 E \z‘ [I i 2 ./ l ; ’1 I f l /' M i i Uhused 1 n 2 Capacity 0 Q“i = 351*”- Qui TH‘ROUGHPUT Figure 2.14. Tangent-Chord Apprdximation to Facility Costs for Potential Facility i. 'TMarginal increases in facility throughput cause movement upward to the right along the tangent. - ‘Marginal decreases in facility throughput cause movement downward to the left along the chord. aChords pass through the origin when there is no unused facility capacity. Changes in throughput changes the cost reference point. 56 Solwo tho LP Roloxotloo of tho “P at Mo 1. Sot II 1: Voloo of LP oolotloo. mo o fooolhlo lotogor oolotloo. Sot LI 3 Noloo of fooolhlo iotogor oolotloo. Tho optlool solotloo ls tho foaoihlo oolotloo with Valoo = LI lrooch fro- tho too with grootost LP woloo. fioo ooriohlo (a ) that is forthost fro- holog lotogrol. dooto ti- hroochos ood tn doscoodoot nodes. ooo with: '1 5 ['1] ood ooothor with: I {[1191 J J Soloo tho LP Ioloaotloo at ooch of tho dooooodoot oodos ooo rocordo its LP woloo. Nocoqoto tho ll hy floologtho oaalooo oooroll oodoo fr. ulch thoro oro oo hraochoo. “to tho Ll as tho onl- woloo of all fooolhlo lotogor Solotloos . foooll to Cato. Figure 2.15. A General Branch-and-Bound Solution Procedure for All Integer Linear Programs (after Anderson, et al.. 1982). 57 Revelle, et al. (1970), on work by Marks (1969), reported a formulation for a fixed charge, transshipment, facility location problem. The model allowed the facilities to be constrained in their capacity, and the warehouses were considered as intermediate points between sources of product and demand centers. If either the sources or the demand areas of product were dropped, the problem reduced to a gen- eral warehouse problem, as outlined previously. The problem was then to determine which facilities should be established and which supply and demand areas each facility should serve in order to minimize the total costs of facilities and transshipment. The mathematical formulation of the problem was: . . . m m n * m p * Minimize 21:1 Fiyi + 21:1 2 C. X? + 2. z :1 cgi Xfi? [2. 16] 3‘1 lj ij 1=1 k subject to: 21.1 X121: 5.. ii, [2.17] 29:1 SIj = £E=1 xii V, [2.18] 2L1 XII i lel #1 [2.19] Di 3 2T=1 Xij 3.”; VJ [2.20] ij, Xi? Z O and Y1 = (O, 1) [2.21] where: Ci”? 1] O U U U') ‘1'! Q. fluo : x Ho p 58 CijTRj = unit cost associated with a transfer of material from facility i to sink j unit shipping cost from facility i to sink j unit variable cost associated with sink j 1, if the ith facility is built; = 0 otherwise flow of material from facility 1 to sink j flow of material from source k to intermediate point i CkiTTk+Vi = unit cost associated with the trans- fer of material from source k to facility i unit shipping cost from source k to facility 1 unit variable cost associated with using source k unit variable cost associated with using facility i fixed charge for establishing facility i amount supplied at source k upper bound on amount demanded at sink j lower bound on amount demanded at sink j capacity of the ith facility number of proposed facility sites number of demand areas number of supply points The solution technique was based on a network algorithm. Figures 2.16 and 2.17 show the network representation and the fixed charge cost function and its approximation, respectively. A capacitated node for each facility was 59 .sm_noca mg» mo cowpmucwmmcamm _mu_camcw .o_.N mc:m_m umou «fie: .ucaom .830; .ueaom Lona: n A . . v mmmc< nemewo mm_ug__uma uwmoaoga macwoa »_aa=m a How ~ pom x pom o. 1 km ..‘ em +m, .0 +2 f «MN.» .0 .8» 2s 5 :48 wwnmwum> mmumcu nonwm may“ oucz neumuwumqmu M C Cost to Build and Operate the Facility J 1 60 .,-”’IV’T”:::r.——fl:;7? /- 1 . . g x i x 3 / i / / 3 /"L Linear Approximation 3 /' of Cost Function : 1 /' E Ho Flow Through Facility Figure 2.17. The Fixed Charge Cost Function and Its Approximation. 61 added so that capacity constraints and a linear cost func- tion could be defined for each facility location. It should be noted that the approximate cost function underestimated the true cost function except when the flow through the facility wmszero or equal to 01’ the capacity of the facil- ity. It was this approximation that formed the basis for the branch-and-bound solution scheme. Assuming that all facilities were open and had an approximate cost function, the initial problem was solved using an out-of—kilter algo- rithm. The optimal solution to the fixed charge problem was found if the resulting solution was such that the flow through each facility was zero or the capacity of the facility. If not, branching took place, and a facility with its fixed charge was included or excluded from the solution until optimal solution was realized and verified. Hakimi (1964, 1965) proved for the case of a specified number of facilities that there was an optimal solution to such problems which consisted of all facilities located at nodes. Levy (1967) extended these by reporting his theorems to cases of concave transportation and facility establishment costs. Maranzana (1964) utilized an heuristic method to locate a specified number of warehouses to serve a region of specified demands. The solution criterion was a minimization of transportation costs. Although the work was directed toward the private sector problem, it fit, however, in public sector models, since the criterion was isolated 62 and distinct from the constraint on the number of facilities which normally characterized the private sector location problems. A number of facility location and location-allocation problems can be formulated and solved as generalized assign- ment problems (GAPS). The GAP is a 0-1 programming model in which it is desired to minimize the cost of assigning n "tasks" to a subset of m "agents.“ Each task must be assigned to one agent, but each agent is limited only by the amount of resource, e.g., available time (Ross and Soland, 1977). Ross and Soland (1975) outlined an efficient algo- rithm (branch-and-bound) for GAPs. A mathematical formula- tion of the GAP is: . . . 2 z Minimize 161 jeJ Cijxij [2.22] subject to: 2 A1 5 jeJ rinij S Di Vi [2.23] ’3 x =1 11L [2 24] 161 ij j ' X”. = 0 or 1 «Lid. [2.25] where: I 5 (1,2,...,m) is a set of agent indices J 5 (1,2,...,n) is a set of task indices C = cost incurred if task j is assigned agent i iJ 63 rij > 0 is amount of resource required by agent i to perform task j A. 3 O and bi > O are the minimum and maximum amounts of the resources that may be expected by agent i X ij 1 or O decision variable if task j is assigned to agent i or otherwise In facility location problems, the "task" generally represents centers of demand for a good or service and the "agents" represent the supply centers which supplies the goods or services. However, it should be noted, here, that goods or services may actually move from the supply centers to demand centers (Toregas, et al., 1971; Geoffrion and Graves, 1974) or vice versa (Meier and Vander Neide, 1974). The objective function should therefore provide the appro- priate measure of total cost of the supply system. Figure 2.18 shows a convenient way of utilizing a tableau represen- tation of the data and special features of a GAP (Ross and Soland, 1977). Figure 2.19 and Figure 2.20 give summaries in tableau form of a p-median problem and a median problem with capacity and investment constraints, respectively. Nhen capacity restrictions are included in the p-median problem, prohibition of the assignment of an arbitrary num- ber of demand centers to a given supply center is implied. For example, if b; is the maximum demand (total population to be served or facility capacity) that a supply center located at demand center i can serve, then the following constraint must be included: 64 .wczpuchm umfiwmpmu map Any .mcauuacpm ~_mcm>o one Amy 4.23 Em~noca pcmscmwmm< uwNfifimcmcmw a mo cowpmpcmmmcamm =m¢~nmh 11 En . . n m . cw. _T||I l E 2: 2: mu .m_.m wL:m_m N < F°l°|°l°l Figure 2.19. NOTES: (i) (ii) (iii) 65 “C c c ‘ ' .111, 12 13 3] 3 1 1 1 3 , c21 c22 23 _Q_I 3 1 1 1 3 , A C 0 .33 32. ifza _J 3 1 1 1 3 . , __l 0 o 0 [...] ...] .___J 2 1 1 1 GAP Tableau Representation of a p- Median Problem. (“:39 9:2) Each demand center is a potential location for a supply center. Requirement: Choose p of N demand center locations (0 O = 0 if Y 5 O The problem may then be formulated as: 06 FL an ab! DH SDE SUE 69 w. x .+£N t (w )] [2.30] N Minimize z? =11E1“ H(Zj_1 W Xi ji+V 2 =1 j ij j: =1 ij wj xij N J 1]: subject to: x.. = 1 j = 1,2,...,N [2.31] and: x.. > O [2.32] Assuming that ti is concave, resulting in a concave func- lJ tion, it is observed (Langwill, 1968) that there exists an optimal solution in which all xij are 0 or 1, i.e., each demand center has a unique supply center associated with it. Further, one can use the definitions: x 1 if demand center j is assigned to supply center ii 1 0 otherwise Cij = Viwj + t J.(w J.) and the objective function then becomes: 1w.x..) + 2'." C.‘.xi.] [2.33] N "WU H12 23. In the formulation of this as a GAP, the xij are as defined above where i=1,2,...,M and j=1,2,...,N. As in the p-median problem described above, an additional "task" is used to specify whether or not each of the M sites is designated a supply center. One additional "agent" is also required; 70 therefore, the GAP is of size m = M+1 by n = N+1. For i 5.M, additional variables need to be defined. Thus: XM+1,N+i 1 if site i is designated a supply center = 0 otherwise xi,N=i ‘ 1 ' xM+1,N+i All other x. 's for j > N must be zero, and all x. 's for ij 1] i = N+1 must be zero. The remainder of the GAP formulation requires complete specification of the r. A B., and C1 ij’ i’ i j' Figure 2.21 illustrates the structural representation of the problem. Generalization of the above simple facility location problem involves additional constraints usually on the allowable supply center configurations and on the alloca- tions. Nhere a maximum number of units that may be fur- nished by supply center i is b;, in general, there will not exist an optimal solution in which the total requirement of each demand center is met by only one supply center. This is often a very desirable solution property (Geoffrion, 1975). Figure 2.22 is the representation of the capacitated facility location problem with two configuration con- straints formulated as GAP. The last row of the GAP tab- leau may be used to impose configuration constraints other than one restricting the number of supply centers. In this example, the last row stipulates a restriction that at Figure 2.21. ”TE: 71 ‘11 ‘12 521 it] .11 H1 V2 H3 H4 7 c21 £2.21 52.3] Ed ...] w,, '2 w3 w4 E21. 522] 633 M __l '1 '2 '3 '4 . Ii] :2] $3.! 1 Facility Location Problem. ai = O; ”n+1 = 0‘ C..= 1] 1] c1.11.1 ‘ and C. . = r. 1] i j = 0 otherwise 1,2,... GAP Tableau Representation of a Simple 72 c11 c12 c1:1 c14 0 l a 1 1 1 w1 '2 w3 w4 61 - C C C C 0 a2 21 1 2;) , 23 , 25) _JD , , '1 '2 '3 '4 2 C31 632 3;, “2| o| a3 '-_'w '__'w ‘ 'w ‘ w b 1 2 f 3 f 4 3 C11 C42, C13 £55) .9.) a4 ""'w '__'w ' w ‘ w ‘ b 1 2 3 4 4 1y._41 , Hi Is] 1 1 1 1 1 j :3] 11 2 2 Figure 2.22. GAP Tableau Representation of Capacitated Facility Location Problem. (M=N=4 and Two Configuration Constraints) NOTES: (a) Changes to simple facility parameter specifications are: set ri,N+i = bi = [111‘ 1 = 1,2,...,N (b) Forcing supply center at i to furnish at least ai units: set ai = a; i = 1,2,...,N (c) Number of supply centers may be limited to the interval (p1,p2) by specifying 3'5”: p1 and b"+1 = pz. (0) Proper treatment of transportation costs requires the assumption that function t is linear. ii ODi Cd! prc to to ADI Gar met out SOC (19 COD to pro 73 least one of the first two sites be used and that at most one of the last two sites be used. The GAP size in this case is m = M+2 by n = N+1 (Ross and Soland, 1977). A number of algorithms have been developed for the problem of locating facilities (medians) on a network so as to minimize the sum of all the distances from each vertex to its nearest facility. Linear programming-based approaches have been outlined by Spielberg (1969), Garfinkel, et a1. (1974), and Schrage (1975). Heuristic methods similar to one by Maranzana (1964) have also been outlined by Teitz and Bart (1968). Tree search procedures, such as those by Efroymson and Ray (1966), have also been utilized by Khumawala (1972) and Christofides and Beasley (1982). The latter utilizes the lagrangean relaxation in conjunction with subgradient optimization of lower bounds to be used in tree search procedures. The lagrangean relaxation method for solving integer programming problems puts into use the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side con- straints. Dualizing the side constraints produces a lagrangean problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The lagrangean problem can thus be used in place of a linear programming relaxation to provide the bounds in the branch-and-bound algorithm (Fisher, 1981). ang 74 Held and Karp (1970) used a lagrangean problem based on minimum spanning trees to devise a dramatically successful algorithm for the traveling salesman problem. Fisher (1973) used lagrange multipliers in solving scheduling problems, and Geoffrion (1974), who coined the name "lagrangean relax- ation," demonstrated the use of the procedure in integer programming and later applied the procedure to capacitated facility location problems (Geoffrion and McBride, 1978). For most of these problems, lagrangean relaxation has pro- vided the best existing algorithm for the problem and has enabled the solution of problems of practical size (Fisher, 1981). Appendix G outlines the general description and basic structure of lagrange multiplier techniques and an example demonstrating its use. It should be noted, however, that it is often essentially impossible to solve the set of nonlinear equations involved in the lagrange multiplier method to obtain the critical (local and global minimum, maximum, and inflection) points. Even when they can be obtained, the number of critical points may be so large (often infinite) that it is impractical to attempt to iden- tify global minimum or maximum except for certain types of small problems (Hillier and Lieberman, 1980). Most of the work outlined above dealt with static solu- tion to location problems. As a result of the stochastic nature of most practical location problems, a multiperiod focus on such problems has, in some cases, been found useful and desirable. 75 2.5. Multiperiod Location Analysis Several papers have dealt with multiperiod or dynamic aspects of the discrete space location-allocation problem. The goal has, in most cases, been to devise a plan of opti- mal locations and relocations in response to predicted changes in the demand volume originating at demand points over a planning horizon. The intention is to avoid "myopic" relocations which, being based on current demands only, may not be optimal. Ballou (1968) and Nesolowsky (1973) have dealt with single facility multiperiod location problems. Klein and Klimpel (1967) applied dynamic programming to a fixed charge problem under economies of scale. Scott (1971) treated a case where facilities can enter the system one at a time with relocations allowed. Tapiero (1971) dealt with a con- tinuous space, capacitated problem under a continuous time horizon. Gunawardane (1981) considered several multiperiod public facility planning decision problems such as multi- period service coverage, multiperiod service phaseout, and multiperiod maximum coverage. Roodman and Shwarz (1975, 1977) looked at the multiperiod facility phaseout and phasein models. Nesolowsky and Truscott (1975) extended the usefulness of single period models by Revelle and Swain (1970) by introducing dynamic considerations. Shifts in the pattern of demand over a planning horizon may warrant a multiperiod analysis which investigates the tradeoffs between static 76 distribution costs and expenditures for relocating facili- ties. If positive relocation costs are associated with both the vacating of a site and the entering of a site, then the dynamic model may be formulated as follows: subject to: and: where: K N M K M . .. u Minimize 2k=121=12J=1 Aijkxijk + 2k=22j-1(CjkY'jk + Cjijk) [2.34] M _ . _ . _ £j=1xijk — 1 for l-1,2,..,N,|(-1,2,..,K [2.35] 2Mx SEARCH COMPLEXITY—’ Figure 3.2. Search Procedure Complexity Versus Complexity of Cost Functions. For exampde,a simulation model usually has a more com- plex cost function but with a more straight forward search procedure. 93 A cost minimization finite location set approach is sug- gested in this study. 3.2. Model Formulation 3.2.1. Problem Definition and Objective A finite number of potential cashewnut processing plant sites was available. Each had associated with it known fixed costs and known functional forms of operating costs, should a plant be in operation at that site. A maxi- mum capacity for a plant at each site and the production level at each production center were known or assumed pre- dictable, respectively. The distribution costs consisted of the sum of the fixed costs and operating costs of all processing plants that were actually Opened and the trans- portation costs. The Objective of the model was to devise plant location plans that minimized the system distribution COST. . 3.2.2. A General Mathematical Formulation The mathematical formulation of the problem can be presented as follows: 0 .)+ g? t .. . _ m Minimize Z - 21:1[FJYi + fi(}:J=1xJJ J=1 ] [3.1] ijxij subject to facility capacity restrictions (hitherto referred to as "supply restrictions"): 94 facility at site i (YJ=1) or absence of a facility at site i (YJ=O) n - - 2j=1 xij i Yia1 for 1 - 1,2,...,m [3.2] and subject to customer demand requirements (in this case referring to production center levels): m - - 21:1 xJJ 2 DJ for j — 1,2,...,n [3.3] and Yi = O or 1 for i = 1,2, ,m [3.4] xi" 3 O for i = 1,2, ,m J [3.5] j = 1,2,. ,n where: m = number Of potential facility sites n = number of customer demand centers tJJ = total linear cost per unit of shipping from demand center j to facility i bJ = demand at customer demand center j ai = maximum capacity of facility i xJJ = amount shipped from demand center j to facility i Fi = fixed cost associated with operating a facility at site i Y = variable denoting the establishment of a 95 fJ(-) = nonlinear operating cost, exclusive of fixed cost, associated with a facility at site 1 Special cases of this problem (already referred to in the literature review) have been treated extensively. Some of the cases can be summarized as follows: (a) If Fi = fJ(-) = O and Y1 = 1 for all i's, the above formulation reduces to a simple transportation problem. (b) If fJ(-) = O and a1 is infinite for all i's, the prob- lem becomes an uncapacitated facility location problem which has been solved using heuristics (Kuehn and Hamburger, 1963; Feldman, et al., 1966; Khumawala, 1973) and optimizing solution techniques (Erlenkotter, 1978; Khumawala, 1972; S6, 1969; Soland, 1974). Small values of ai which result in a capacitated facility location problem have been accommodated in principle by existing programming techniques, but such methods are not yet computationally efficient for most practi- cal problems (56, 1969; Davis and Ray, 1969; Ellwein and Gray, 1971; Akine and Khumawala, 1977). (c) Cases where Fi = O, a. 1 continuous, piecewise linear for all i's, have been is infinite, and fJ(-) is semi- tackled using both optimizing and approximation solu- tion methods (Feldman, et al., 1966). (d) Cases where fJ(-) is concave and continuous or semi- continuous or has a finite number of discontinuities have been addressed successfully theoretically, but 96 optimal solution techniques have not been demonstrated to be practical. (e) Heuristic approaches to the case where fi(°) has a functional form: n q Ki(zj=1xij) where O $5 '1 s; gs: “2% so . 7 u; :S" - T" 7 60 - '1 —--I 40 k — T 0 . . . . - . - - - - . - - ., -7- 65 69 73 77 81 66 7O 74 78 82 to 1 CROPPING SEASON (YEARS) 982-83) and Cashew Prices. Cashewnut Production in Tanzania (1965-66 These are based on CATA figures which are dif- ferent from FAO figures due to a different season reference. KERNEL AVERAGE PRICE “($/KG) 105 Appendix A shows the CATA functional structure. At the vil- lage level, CATA advances money to the Village Planning Com- mittee to buy the rawnuts on behalf of the Authority. A levy of 1¢ (about TAS 0.10) per kg purchased during the .season is paid to the village at the end of the season. The Authority also makes arrangements for transporting the raw- nuts to processing factories and later to port of export for the eventual sale of nuts and cashew products to buyers overseas. CATA sells raw cashewnuts to India and China and cashew kernels——the main product of cashewnut processing—— to the United States, Federal Republic of Germany, West Germany, Israel, Japan, and the Netherlands. Table 3.3 shows the cashew exports between 1975 and 1982. Table 3.4 provides the breakdown of cashew product exports between the same years. Extension services and research are the responsibility of the Directorate of Crop Development and Production, which acts as a liaison with the Extension Services and Crop Research Organization of the Ministry of Agriculture. Naliendele Research Station (Mtwara) is the main research center and conducts research on cashews in order to improve the quality of cashewnut trees, increase their resistance to disease, and reduce vulnerability to adverse weather changes. The most important research in progress worth men- tioning here is that related to development of mildew- resistant clones (of which, so far, two out of twelve clones 106 .A¢.m m_nmhv was: ummmwuogacz mcw_fiwm cusp Locum; w_:m~:mp c_ mus: “fin mo mcflmmmuoca wuwfiqsoo mmnwmmoa m ucmzop ucmgu fiucmcwm may wu_poz« min 3.3 mmmémmdp .Nmm NNK 3.53.3 mmmmp «mu—m 35 mmdm omndnmém ooem 3.: 39893. ommpm .mém mp.“ 5.8 SQNSJF gen 36 oomeod. momom omnmm mo.m NTON mmudmog: 33m SJ 29558 20mm 2.3 mm.m RJN oom.$~.m «mum mo.m oemdmimm 83:. 3.: m~.m _m.om om¢.mmm.m. Fomm mm.m o~¢[_m~.em ooomm .NnumN oo.m 3.3 mmmé—NK we: 3.. om~.mwm.nm mmmum onumn WAS a a » “mmwmwr mx\» w3~m> mmccoh mmwwmm< m:_w> mmcco» Low» mgmzmmx mhazz-m>_a cmccc .umuumnocc ucc ~c=~u< «neocuuzcoca .m.m w_nmh 110 Agriculture. The current arrangement, in which CATA pur- chases rawnuts from individual villages, will cease, and the Cooperative Unions will be responsible for the purchase and transportation of rawnuts from member villages to production centers. CATA will be responsible for the purchase and transportation of rawnuts from the production centers to the processing plants. 3.3.2. Tanzania Cashew Industry: Production Levels, COsts, Plant Capacities, and Distances Data on individual villages' rawnut production levels were available. To a large extent, location of production centers was determined by existing road networks and dis- trict boundaries. It was found realistic, on the grounds of justifying a reasonable number of truck loads and maximum distances to the production center, to cluster villages into groups, each group being served by one production center (a 120-mile radius was assumed). Appendix 8 gives the mileage chart for distances between production centers. Table 3.6 shows the capacities of existing cashewnut processing plants and their code numbers as used in this study. As a result of the increase in fuel prices, transporta- tion costs per kilometer have varied between 14¢ (TAS 1.80 per tonne per mile) per tonne in 1977 to 26¢ per tonne (TAS 3.50 per tonne per mile) in 1982 and were approximately the same for different parts of the cashewnut production region. 111 Table 3.6. Average Annual Fixed Costs and Operating Costs and Amounts Processed for Existing Processing Plants+. N536; (>51?th CaRSacfijty Changlgf d(F1) Pragusnsted “'06:?” (000 tonnes) ($000) (tonnes) ($000) 5 Lindi 5 280 2226 1045.0 6 Mtama 5 ' 312 3348 1360.2 7 Nachingwea 5 221 277 180.9 4 Kilwa 5 250 __. __. 12 Mtwara 8 375 -—- -—- 11 Likombe 10 390 -—- -—- 8 Masasi 10 390 1800 897.9 9 Newala I 10 390 4320 1680.0 10 Newala II 10 390 7680 2160.0 13 Tunduru 10 430** -—— -—- 3 Kibaha 10 430 6055 1985.9 1 Tanita I 12 650 9120 2240.0 2 Tanita II 12 680 10400 2300.0 +Based on 1980-81 CATA figures. *Includes direct labor, fixed factory overhead, and administrative expenses. *ngA estimates (factory under construction, to be commissioned in 1 4-85 . 112 According to the above trends, it is reasonable to assume that transportation costs will double every five years. Factory production costs, on the other hand, have varied among factories. Appendix C gives an example of a summary of production costs for some of the factories. Typical items included in the packing materials, variable factory overheads, and typical depreciation rates on capi- tal items are given in Appendix 0. Due to differences in cashew quality resulting from, among others, variable recov- ery rates from different factories and below-capacity opera- tion of factories, the variable costs in relation to factory throughputs are difficult to visualize. Hence, Figure 3.4 shows the combined concave cost function depicting the vari- able cost trend with increasing factory throughputs derived from cost figures in Table 3.6. Table 3.7 shows examples of factory recovery rates for the kernels and CNSL, the main products of cashew processing, for two representative fac- tories. Appendix E gives typical factory production figures showing kernel grade outputs for one year (1981-82) and regional range of kernel grade outputs that may be expected from a factory. .-"-'-"“-"'---'-‘-l-|-'--"I--'--l 1: Jay-.3: J 8 3.1-..'... ' S-.. ..'-...u'ul'q'.’ J J .’8&%% 8 8 ‘ I all-.- 113 O -------------‘------‘-.‘-l - r -.---- -.'--- -8--- .-------- FDCO- .. .-. E4 I P L I wmmw mw mm m mm» mm m w 1 1 Accccc cmcc‘cz_cwo upmv umm.m ”mpca acm>oumm ecmzu wmccm>< Amm.m« >mo uvmv ae.o~ "mama xcm>oumm cmccmx wmmcm>< ”ccmcm>o cc.ch cc.Nn cc..w c_.~H cc._n mm.m« m_.cH .c.cH cmc ccm c.c_ m._~ c.“ c.c~ m.c_ c.- c.“ m.m. ccccc>< xcouucm cc. N03 cg 55 I I cc ..3 28.538 cc cg: cg cc. I I I I 8.82 c.c m.cm c.c c.cc :1 in .11 1| c:c m.c_ 33.. ..c N? I I I I 8% ~.N_ c.- N.c c.c_ N.“ c.c~ c.c c..~ cs‘ ~.c. N.c_ c.“ c._~ _.__ c.cc c.c c.c. cra< 1| :1. c.“ c.c~ c.c_ c.cN c.c c.c_ cues‘ In 11 ..N. c.c_ c.c c..~ c.c c.m. cessacc I I I I c.~_ 4.8 cg c.c~ 552.2. I I I I m.: m.- cg cc. 28588 I I I I m... ..cm c; mg. 28.532 I I I I I I cg 93 .3538 cmzc _cccc¥ cmzc cmcccx cmzc cmcccx cmzc ccccmx cpl m a .m “r In hr, 1&1 cccc_¥ ccccc cccc_¥ ccccc ccccz Nccc-_mc_ .cc_-ccc. .Nc-ccc. ccc _m-ccc_ ”mmccccucc zccmcc cccc_¥ ccc ccc_c com mcccc ccmcccmc .c.m mcccc CHAPTER 4 CASHEN INDUSTRY LOCATIONAL STUDY MODEL: MODEL DATA INPUTS ANALYSIS, ALGORITHM, AND ASSUMPTIONS 4.1. Input Data Analysis 4.1.1. Estimating Production Levels ("Demand Pattern") Estimates of increases in production center production levels (total and zonal) creating a processing demand based on the Five-Year Plan figures seem very optimistic, as may be calculated by adding estimated yearly increases (Table 3.5) to the preceding actual production figures (Table 3.2). For example: Actual: Total production, 1982-83 season (Table 3.2): 32,743 tonnes Cahnnatad: Actual preceding year (1981-82) production (Table 3.2): 44,326 tonnes Estimated yearly increase (Table 3.5): 9,343 tonnes Estimated Total (optimistic) 53,669 tonnes This overestimation resulted from the overestimation of cashew yields per tree as well as underestimating the cashew yield decline by older trees. Figure 4.1 shows the relation between production year (beginning in the 1982-83 season) to the production level in the next five years, i.e., a production level prediction curve. The assumption made in the prediction equation derivation is that increases 115 TRANSFORMED PRODUCTION 116 2.10 ' 233.2 2'00 _ 9 Production Year Mean 185‘2 1.80 P ' 116.9 Prediction Curve 1.60 - a 4 73.7 Y a 1.4245 - 0.0089): + 0.0074x’ '0 1.40 F 4 46.5 4 I I I I I 0 2 4 6 8 10 Figure 4.1. CASHEN PRODUCTION YEARS (Beginning 1981-82 as year zero) Relationship Between Subregions' Annual Cashewnut Production Level (Transformed and Real Data) and Production Year for a Ten-Year Period. PREDICTED PRODUCTION (000 TONNES) 117 in production levels will follow a similar pattern as the decline between 1973-74 and 1981-82 seasons. The prediction equation is derived by use of orthogonal polynomials. Appendix F gives the details of the derivation procedures. Figure 4.1 also shows the transformed production year, production level means. Table 4.1 gives the production cen- ter production levels derived from the use of the prediction equation. 4.1.2. Estimating the Cost Functions Three costs are considered in this study: an annual fixed charge, processing plant operating cost coefficient, and unit shipment cost. The annual fixed charge includes both investment costs and construction of facilities and other fixed indirect charges (administrative expenses, direct labor, and fixed factory overheads) which are incurred when the facility is or was constructed, regardless of its output. Figures listed in Table 3.6 form the basis of fixed charge estimates for each existing factory. The operating cost coefficient estimates are based on the com- bined concave cost function (Figure 3.4). Combined individ- ual fixed cost and operating cost functions shown in Figures 4.2 to 4.10 are used to determine the cost coeffi- cients. A "pseudo" tangent-chord approximation, with one and also two linear segments in the pseudo, piecewise-linear approximation, are shown in Figures 4.2 to 4.10 as two examples of finding approximations to the fixed costs (Fi*) 118 .xcccc.c + xcccc.c - mcmc._ u w ”cc_ccccc cc_cc_cccc cc ccmcc+ . .caccc_c m4 :o_pu:uoca cmucmu co_pu=ooca ._.v m_nmc TOTAL COST ($000) 119 3200 — 2800 - I / i 600 E 1 2 2000 1 : 1 3200 ‘ . 2 1’ s i 1600 I r, = $650.000* 3 A, x $333/ton/year i 1200 r; - $1,750,000 1 A' - SSO/ton/year I 800 Fixed A, = $188/ton/year' 1 l i 400 - E i .2. as] o f v W fi ' V ‘ ' ' 2400 4800 7200 9600 12000 FACILITY THROUGHPUT (TONNES/YEAR) Figure 4.2. Tanita l: Cashewnut Processing Total Cost Curve: Cost Coefficients. 120 .L___--...---. - ---..- ----...-.---..----.....--.- .- .--- . --.. --.--.. 120000 3200 r / z / / ’fiyr” 2800 r .230?" 100 / ’ g / 2 1 / 2400 - I / l E 300 I l 2000 r /’ 1 8 3’ S D ’ a 8 z: :: / .= g 1600 ; r2 . $680,000* 3 1 A2 a: $321/ton/year g g r; . $1,800,000 1200 E A; = $50/ton/year 800 1 Fixed A2 = $188/ton/year* i 400 - i i A A i A A O 2400 4800 7200 9600 FACILITY THROUGHPUTS (TONNES/YEAR) Figure 4.3. Tanita ll: Cashewnut Processing Total Cost Curve. 121 2800 ’ 2400 2000 1600 TOTAL COST ($000) 1200 F = $430,000* A = s375/ton/year .F' = $1,280,000 A' = $94/ton/year --‘* ...--.-...- -...- . -..-. c .- --.--..———----o--.§-. -.. 1 800 E IFixed A - $275/ton/year’ 400 I i A A l I i A J A A 0 2400 4800 7200 9600 FACILITY THROUGHPUT (TONNES/YEAR) Figure 4.4. Kibaha and Tunduru: Cost Curve. Cashewnut Processing Total 122 2400,... 2000 - , 1600 F’ 400 S c: c> 1’3 63 1200 I- I o 0 U 0 < 1 :5 l p. : F 250,000* 800 g 4 $ : A4 = $438/ton/year i r; = $440,000 E A; = $267/ton/year 400 ; 1 Fixed A = $375/ton/year* 0 2400 4800 FACTORY THROUGHPUTS (TONNES/YEAR) Figure 4.5. Kilwa: Cashewnut Processing Total Cost Curve: Cost Coefficients. 123 2400 ’ Tangent —-9 / / 2000 ' 1600 r 8 C) O 69 g 1200 . i U _1 5 F5 $280,000* ._ 800 15 = $417/ton/year r5 = $750,000 As' = SZZZ/ton/year 400 I Fixed A = s333/cofi/yearw i 1 1 0 2400 4800 7200 FACILITY THROUGHPUT (TONNES/YEAR) Figure 4.6. Lindi: Cashewnut Processing Total Cost Curve. 124 I 2000 - 1600 . 8 o1 c: 3 l O 1— . i g 1200 I: .4 l 175 1 F 312 000* E g 6 $ ’ 800 1 A6 2 $458/ton/year 550 I Fé - $600,000 1 Ag 2 $250/ton/year 400 1 1 Fixed A = $375/ton/year* 1 _ 1 ' 4 O 2400 4800 FACTORY TRHOUGHPUT (TONNES/YEAR) Figure 4.7. Mtama: Cashewnut Processing Total Cost Curve. 125 2400 . 2000 - 3 , 1600 b 8 c: O 22 5 1200 - 8 ...l ,5 $221,000* :3 7 F- 800 n...— 7 a $458/ton/year 800 F} = $500,000 550 A} a: $250/ton/year 400 Fixed A = $375/ton/year* A l 0 2400 4800 FACTORY THROUGHPUT (TONNES/YEAR) Figure 4.8. Nachingwea: Cashewnut Processing Total Cost Curve. 126 2800 r I /' l 2400 r E 2000 - 1 E l ’3‘ E s O 9 1 r: 1600 r ,’ i : L. 2’ 0’ I m /! 1 8 / // a : A 0 ' g 1200 ’ I / :5 = $390.00” 5 ’— /—T600-7_/‘ i a: $375/ton/year . é / 2000 : F' I: $7,260,000 5 800 E A' = $83/ton/year E i i 5 Fixed 1 = $225/ton/year i 400 I 3 E E 1 E I I 1 1 4 4 A 44 A a n j l O 2400 4800 7200 9600 FACILITY THROUGHPUT (TONNES/YEAR) Figure 4.9. Likombe, Masasi, Newala I and Newala Il: Cashewnut Processing Total Cost Curve. 28OOT-—- 2400 2000 2600 1200 TOTAL COST ($000) 800 400 127 :6 N 1 - N ..'?J N 11 - N . Fixed b.-..-0... ... - $375,000* $375/ton/year $900,000 SISO/ton/year = $250/ton/year ‘0‘------- ----o ------- ----0..-.---.--COOC-QCCC-oocoooo “0---.--- Figure 4.10. 2400 4800 7200 FACILITY THROUGHPUT (TONNES/YEAR) Mtwara: Cashewnut Pr Total Cost Curve. ocessing 128 and operating cost coefficients (A?) using piecewise linear approximation methods. The word "pseudo" is used in this case because the fixed charge is found from the extrapola- tion of the tangent at the minimum capacity in that linear segment rather than the extrapolation of the chord itself, which usually tends to overestimate fixed costs. This approximation of the fixed cost estimates is a compromise between piecewise-linear and the tangent-chord approxima- tion methods (see Figures 2.12 and 2.14 for detail). In this study, however, due to financial reasons, operating costs are based upon a fixed charge cost rather than two segments in a piecewise linear approximation and are included in the unit shipment cost data inputs. This means that each unit of cashewnut sent from a production center to a processing plant brings with it both shipment and operating charges to the assigned processing plant site. Conceptually, processing plants in the feasible set (free facilities) are competing to drop their production centers in order to avoid incurring their fixed costs and shipment costs. On the other hand, processing plants in the feasible set earmarked open are competing to serve production centers in an effort to reduce the variable system's cost. Tables 4.2 and 4.3 give the cost per tonne per year matrix, C , where, for the first stage (Production Center - 13 Processing Plants): .2618? 3 31ch coccioeeouac<3~ .3. 25.: .339 9:: 2.. 2232.3 «5:. cog—2:. 129 wmmmmmmmmmmmmmwmmmw b“.."u~‘u“”utw.wn!l O 5 (CI... 00 I... M S .M I I 6 v a 3 I n -.c .cc.c cc ....c 11 cccccc .ccccc .ccccccccc c...c cc...c ccccccc c.~cc c.ccc c.mcc c.~cc c.ccc c.ccc c.ccc c.ccc c.cmc c.c~c ~.c~ c.cc~ c.c- m..- c.m~c ..ccc -.ccc c.cnc c.cc~ ccc.cc ccc.cc «- ...ccx c.cc. -.ccc c.~.m c.ccc c.cc~ c.cc~ -.ccc c.ccc ..ccc -.ccc c.cc~ ~.mcc c.ccc c.ccc c.cc c.cm ..ccc c.cc ~.c~ ccc.mc ccc.c c. .cucccc c.-cc c.ccm ..ccm c.9cc c.cc~ c.cc~ c.c.c c.ccc c.c~c c.ccc ~.cc~ c.cmc c.ccc -.ccc c.cc c.cm c.cc- c.cc ~.c~ ccc.cc ccc.c. cc cc .c.,.. c.ccc c.ccc c.~cm c.ccc c.c~c c.mc~ c.ccc c.ccc ..mcc c.cc~ c.ccc c.mc~ c.c~c m.cc c.cc c.mc- c.cc c.c~ c.mc ccc.cc ccc.cc c. c .cczcg c.c¢c “.mc. c.~cm c.ccc c.c~c c.mc~ c.mcc c.ccc c.mcc c.cc~ c.mnc c.mc~ -.c- m.co ..'. c.mcp P.0n c.- c.mc occ.cc ccc.o- c c....: c.~cc c.ccc -.cmm c.ccc c..c~ c.cc~ c.ccc c..cc ..ccc c.c- -.ccc c.cc- ~.cc c.-c c... c.cc c.c~ -.cc c.ccc ccc.cc ccc.cc c ...ccccccc c.ccc c.ccc c.ccm c.c~c c.ccc c.-cc c.ccc c.cmc c.cmc c.cc~ c..~c c.cc- c.cc c... c.cn ..cc c.c~ c.cc ~.ccc ccc.c~ ccc.c c ...c: ..ccc c.-cc c.ccm c.cc~ c.mcc c.ccc c.ccc c.c~c c.ccc c.cc~ c.ccc c.cc- ~.cc- c.cc -.~c c.cc c.~c c.~c c.cc ccc.~cc ccc.c c cccc. c.cmc c.ccc ~.cc. c.cc~ ..mcc c.ccc c.ccc c.mc~ c.cc~ c.ccc ~.ccc ~.ccc c.ccc c.-c c.cc c.c~ c.cc “.mcc ...“ ccc.cc ccc.c m ..cc. c.cc~ c.ccc ..ccc c.cc- c.ccc ~.-c -.cc~ c.ccc c.-c- c.ccc -.cc~ c.cm c.cc~ c.cc- c.c~c ~.ccc c.ccc c.m- c.cmc ccc.cm ccc.c c .c.ccc -.mc c.cc. c.cc. ~.mc c.cc c.c~c ~.cc c.- «.m. c.ccc c.ccc c.ccc c.c~c c.cc~ c.c~c c.mc~ c..cc c.ccc c.ccc ccc.cc ccc.cc c cc .ccc.c c.cc- c.ccc c.c- c.~c m.cc c.cc c.c. c.cc ~.cc c.ccc c.~cc c..cc ~.-cc c.c~c c.cc~ c.cm~ c.ccc c.ccc m.~cc ccc.c ccc.~c N c .ccc.c c.cc- c.~cc c.cc~ c.- c.cc c.cc- ~.c. c.m~ m.c~ c.ccc c.ccc c.ccc c.ccc c.c~c ..ccc ~.cc~ c.ccc c.ccc c.ccc ccc.cmc ccc.~c c cesc ceac cc cc cc o. m. c. n- ~c cc c- c c c c a c n N p ccc cup . ccc ccc .ccccc .ccccccccc ccc cccccccc-c..cc .accuumm mocezu umxcc so; «xccuo: cum»-m=:op com umou .~.v ocnoc 130 .A~wuwoar ca oousoorv caucuoceoc-c.\u~m co. .....t .ncnou to... u:- nocccoonau «ca—a couscoec. m m u m m m M u. n a m. u n m a n H a. a u .a .w a nu u. u. u .w w. m. u u‘ u ”V n a a u .. .. u .. . u w u u m ... u x n u. u u u u a a .m u. C .- 33: 3.: ~62: c5; «.2: c.8c 3; 3.3 2: mat. in: was c..c~c c.8c 93.. 3: 9cm :2 5mm cut... or cc cc .c=.c c.ccc c.ccnc c.m~c c.~cc n.9cm c.0co o.~cc c.~cc c.ccc c.cocc c.cccc ~.ccc o.~c~c n.caec ..c~9c c.cmc o.~ccc c.cocc m.ococ on. ac cc acaccac ~.mc¢c c.ccc c.cccc “.moc m.coc n.cco c.cmc c.c~o o.ccc c.cc~ ~.~m c.cmc c.ccc ...c~ c.cc~ c.omc o.acc c.cc~ ~.0cc on. cc cc ...,c: c.coc c.mccc c.ccoc m.ceo c.ccm c.c~c c.oco o.moc m.cmc c.moo c.0cc c.oOc c.ccc c.coc ..ccc ..coc c.cc~ c.mcc ~.cm mcc c ~c 38.: 28 cmmccc “.32 :3 3.? :2 925 .38 ES 33 3: c.§ 3: 5.2 mic ...Sc “.3 ..mcc cém 8c ccc cc cc :3... 3.8 3.8 20cc :2. :3 3R. 3: 38 3c: 3% c.ch 2c. 33 98c ...: 5.3 92. ..cc ..mcc 8c 2 2 c2...- 28 c6: c.2cc 95. 0.30 ficmm ~.c3 38 92:. 3% c..ch :2 9.9. Sac ...: c6: 92. ..cc ...“: 8c 2 c. c».c.x m.mco o.ccc o.~ccc c.moo ..com c.ccc c.cmc c.c~c_ o.ccc m.ccc c.ccc a.omc ~.~cc o.~c ..c. c..mc c.mm o.~c m.cc~ can cc . .3228. 93. :8 «.92 :3 3R. ~43 3:. 3: .22. 3x... cécc 9c: mic wk of ..ccc “.3 cc.c~c c.8~ c- m c. ...... £2. .22: :2: :8 ....R .33 ..os 33 :2 c2? :3 «...: c.8~ 9.3 ”.3 2c 92 ...: mic Ncc m o cuecc m.c=c o.ccoc c.cmc o.ccm c.c~c c.c~c m.coc c.ccm c.ccm c.coo c.cmc °.c°~ c.cc~ ..ccc ~.mc m.cn c.¢mc ~.oc~ m.c¢c coc m m ..cc. c.coa m.cc~c c.cmc ~.goc ~.oc~ c.-c o.co. c.ccc c.~cc c.coo ..cmm n.~¢c o.ccc m.ccc c..c~ c.oo~ c.omc c.o~c ~.ooc om~ m c 2.2:. eacc .79. :2 Ex c.2c ..2 ~52 c2. 35 :2 :2. 3: cut. 3:. c..c$ 3% 2a 38 .6: 3.. cc c cc 3:... ..ch 92.. 3: c.m~ ..Sc ...cccc ..3 0.3 mi :2. 93. as: 3: 2.3 38 ER :3 3.2. 38 as ~c c c .cce.c c.coc m.m.. o.~cc c.m~ c.ccc o.ccc c.e. c.°m c.oc c.c~c o.ccc m.c~c ..mcc c.omo c.ccm ~.c~m c.ccc c.cmc c.ccc one ~c c ceoec coooc cc cc cc cc cc cc cc ~c cc 2 c. c c. o a c c N c c: c: c c: 33;: .2322. ccccccccctcccc: .accccmm oucuzu caxcc Lou cxcgcat Loocucccoc can cmou .m.e ocnoc 131 CH J) ___ Return Journey Distance) (Unit Shipmen5 Between Facilities Costs + Unit Operating Costs Coefficient [4.1:] and for the second stage (Processing Plant - Export Depot): C( I ’J) = Return Journey Distance) (Unit Shipment) [4.2] Between Facilities Costs Currently, only Dar es salaam and Tanga are designated ports of export. A solution of two transportation problems is preferred to the solution of a transshipment problem, mainly because the procedure allows for decentralization of facilities regionalwise. The export amount (%) for each cashew product (?) is predicted using prediction equations based on figures in Table 3.4 (derivation summary is given in Appendix J): Rawnuts: ? = 75.01 - 5.96x r2 = .89 [4.3] Kernels: ? = 8.53 + 7.74x r’ = .85 [4.4] CNSL: Y = 0.78 + 0.67x r’ = .85 [4.5] where x is the production year (x=1,2,...,10). But since export depots can be assumed to have no capacity restrictions, shipment of all the output of each source to that one destination with the least associated marginal cost results in the least-cost allocation (Ademosun and Noble, 1982). It follows, therefore, that each process- ing unit will be allocated to the nearest export depot. 132 This excludes the necessity for solving the second trans- shipment problem. Nhere data are not complete and for new factories to be established, average estimates for existing factories are used to estimate costs of similar capacity facilities. 4.1.3. Facility Capacities For the establishment of new factories, available figures for the modern 10,000-tonne capacity factory (such as Kibaha or Tunduru) are used. Apparently, the 10,000-tonne capacity factories are preferred by CATA authorities, and, as the operating cost function curve indicates, it is with good reason. For established factories, capacities in Table 3.6 are used. 4.2. The Algorithm The algorithm used to solve the cashew industry facility location-allocation problem was constructed accord-- ing to the flow charts in Figures 4.11 and 4.12. A FORTRAN interface was used to link the linear, interactive, and discrete optimizer (LINDO) package. The major features of LINDO (Schrage, 1982) are: (a) Flexibility: Over forty commands for data input, editing, optimization, display, file handling, and sensitivity analysis. (b) Power: Depends upon the size of the computer, but current capabilities of approximately BOO rows and 133 c 3 l IIIEREACE VIII lIIOO IICORPORRIE TIE HAIRIX GENERATOR I- if 2 RERO ORIR SE1 OR IRE IIIIIAl EORHOLRIIOI “31.5 I" RIO IEIOERS COT SOLVE TOE ' IRAISPORIAIIOI PROBLEM URIIE SOLOTIOI SIORE SOLOIIO RCIORL RIO X C ) Figure 4.11. Main Frame Flow Chart for Facility Location-Allocation Study. 134 SEI OP IIIIIAL EORHOLAIIOR 1 stain so (LINED, IOIDR) LOOP DISPLAY IRE SOLOIIOR REAO IRE OBJECIIVE VALUE EORIOLAIE ARO IICORPORAIE OEROER'S COISIRAIII Figure 4.12. Flow Chart: Bender's Cut- HSJ Search. 135 4,000 columns. (c) Ease of Use: HELP is available interactively. (d) Multifaceted: LINDO has interfaces for MP5 (industry standard) files, user-written FORTRAN subroutines, and general purpose text editors. (e) Portability: Versions of LINDO running all major com- puter types (DEC 10, DEC 20, DEC VAX, IBM 370, Amdahl. Honeywell-Bull, Prime, Burroughs, CDC/Cyber, HP 3000, Harris and Sigma computers). LINDO solves the integer programming problems by a branch-and-bound search procedure. In order to add a new constraint, a Bender's decomposition strategy (here referred to as Bender's cut) is used. This captures the advantage of the fact that each time an integer solution has been found, LINDO calls a subroutine NENIP (actual, bound), but just before NENIP is called, LINDO sets ACTUAL to the value of the objective function. A user-supplied version of NENIP (Appendix I) is used to generate new constraints on the problem. The strategy is to add a constraint after an optimal solution has been found. This allows the incor- poration of the hop-skip-jump routine (Section 2.6.1) for generation of alternatives provided by budgetary relaxation of the objective function value. 136 4.3. General Procedures and Model Assumptions The cashew industry facility location problem defined in the previous sections naturally breaks down into two con- nected problems: the main problem (solved by interfacing a user-supplied subroutine USER) and the subproblem (solved by interfacing user-supplied subroutines USER(INPROB) and NENIP(ACTUAL, BOUND)). The locational-allocational part forms the main problem. A solution to the main problem is a configuration of open facilities, at some sites, capable of processing all cashewnuts from the production centers and an associated allocation process in the form of a transporta- tion problem. The subproblem is designed to answer some other adaptive "if" questions. Several assumptions have been made in formulating the cashew industry processing and handling model. Table 4.4 outlines the source, description, and assumptions of model variables. The total costs that will be derived will not reflect the costs resulting from Closing of plants. Since closing of plants rather than expansion of numbers of plants is being contemplated, capital costs are considered irrelevant and therefore are not part of system costs. 137 Table 4.4. Model Variables: Source, Description, and Assumptions. . Source/ . . 2:7 Variable Equivalence Description and Assumptions ACTUAL User-supplied The value of objective function after APPCOL BEN(I) CAP(I) C(I,J) CIJMIN subroutine NENIP. Dummy argument. LINDO subroutine User-supplied subroutine NEWIP. Subroutine USER. User-supplied subroutines USER, USER(INPROB) , and NENIP. Subroutine NENIP solution has been achieved. e.g., APPCOL (NAME, NONZ, VAL, IRO, TRUBLE). Appends a column whose name is stored in the vector NAME which has NONZ coefficients stored in vector VAL with associated row numbers stored in vector IRO. Bender's constraint coefficient. Initialw ized zero. Relates to savings associated with opening of a preferred site. Processing plant capacities array. Rated capacities (PCmax) in Table 3.6 apply to existing factories and 10,000-tonne capacity factory is used for to-be- established factories. Assumed expansion is from 50% to 100% of rated capacity. Allowed production center-processing plant assignment costs. Assignment costs include unit shipment cost, unit process- ing cost (A). Appendix B (distance charts-—di 's), transportation costs per km ton of 6¢ for first five years of prediction and 52¢ for second five years are used. One linear approximation oper- ating cost charge (fixed charge) is used. It should be noted that use of two or three segments in piecewise linear approx- imations, respectively, doubles or triples the number of processing plants to be considered in the formulation since segments break down to pseudo facilities. Figures 4.2-4.10 apply. Assignment costs defining the closest (lowest) production center-processing plant combination. Table 4.4 (continued). 138 . ’Sourcel . . . Variable Eguivalence Description and Assumptions DEFROw LINDO subroutine. e.g., DEFRON (IDIR, RHS, IDRON, TRUBLE). Defines a row to the current formulation where IDIR=1 if < or minimization for row 1, 0 if =, and -1 if > or maximiza- tion for row 1; the right-hand side value of the row (+RHS), the row number assigned to this row (=IDROW) and the logical TRUBLE returned - TRUE. D(I) Subroutine USER Production center production level (D(I)=b- referred in Table 3.1). Levels are predicted in Table 4.1. DP Subroutine NENIP Summation of dual prices on the con- straints for open plants. FMND Subroutine NENIP Minimum number of processing plants allowed open. Variable. FMXO Subroutine NENIP Maximum number of processing plants allowed open. Variable. FXD Subroutine USER Fixed costs (also referred to as F, or F; referred also as in the text) for the processing plants. F(I) in subrou- Table 3.6 and Figures 4.2-4.10 apply. tine USER(INPROB). GIANT Subroutine USER Control argument. Aborts solution if (INPROB) limit exceeded. GLOBAL Subroutine NENIP Accumulates cost of current optimal solution. GO LINDO subroutine e.g., G0 (LIMGO, KONDN). Command to go and solve the problem. LIMGO defines the limit on number of pivots (LIMGO=O or default specifies limits set by LINDO: KONDN specifies on return solution status (4=optima1, 5=unbounded, 2=infeasible). All subroutines. FPotential processing plant sites (1:1,2. ...,NP or M). Feasible set approach is assumed and all potential production cen- ter sites in Table 4.1 can be potential plant sites. Processing plants are regarded as supply points in this study supplying theirgprocessing capacities. 139 Table 4.4 (continued). . Source? . .= := =. Variable Equivalence Description and Assumptions INIT LINDO subroutine Reinitializes storage in preparation for lnew problem. INSERT LINDO subroutine e.g., INSERT(I, J, AMT, NOADD). Replaces element in row I, column J of the matrix. If NOADD=1, then AMT replaces the old value at I,J; otherwise AMT is added to old value. J All subroutines Potential production center sites (J=1,2,...,NC or N). Refer to Table 4.1. In this study, production centers are regarded as demand points; their produc- tion levels create a demand for a processing capacity. M All subroutines Number of supply points (processing plants). N All subroutines Number of demand points (production centers). NC Subroutine USER Number of customers (demand points). NDIM1 Subroutine USER Limit on the number of processing plants for which the subroutine is designed. NDIM2 Subroutine USER Limit on the number of production centers for which the subroutine is designed. NP Subroutine USER Number of processing plants. OUTSOL LINDO subroutine Prints a standard solution report. REPVAR LINDO subroutine e.g., REPVAR (J, PRIMAL, DUAL). Returns the value of variable J in PRIMAL and its reduced cost in DUAL. SBCOST Subroutine NENIP Accumulates the subproblem or transporta- tion costs. X(I,J) Subroutine USER Amounts of cashewnut shipment from pro- duction center J to processing plant I. Table 4.4 (continued). 140 . Source/ . . .: Variable Equivalence Description and Assumptions Y(I) Subroutine USER Equal to 1 if potential plant site I has been designated open; equal to 0 otherwise. Z Subroutine Defines the objective function to be USER(INPROB) [finimized. CHAPTER 5 SENSITIVITY ANALYSIS 5.1. General In an effort to answer some of the "adaptive if" ques- tions, the model was run for several model inputs designed to highlight the issue under study. Table 5.1 shows the setting of the model inputs for each run for the respective "adaptive if" issue. The program was run on the CDC Cyber 750 under the Michigan State University Hustler System. 5.2. Effects of Changes in Demand Table 5.2 shows in tableau form the allowed combina— tion coefficients that were used in arriving at the location- allocation solutions. Routinely, the omitted coefficients, C 's (designating the nonpreferred production center- 13 processing plant combinations),were assigned a very large number, thus rendering them unacceptable routes. Table 5.3 shows the optimal or near optimal solutions to the cashew industry in relation to changes in predicted production cen- ter production levels (demand levels). Figures 5.1 to 5.6 show the allocation of production centers to processing plants for six years (1982-83 to 1987-88). As summarized in Table 5.4, based on the 1982-83 season figures, a fifteen 141 142 x x x x m.m cocpacom x x x x N.m opcc x x x x c.m mwcccccumu accucoc x x x x ~.N ,.mwacmco x x x x c.~ cucumamu x x x x m.c cmcm>wc x x x x m.c coccuzuocav x x x x ¢.c x x x .x m.c x x x x N.— x x x x c.c namewo accmcmcu N c o ooc mu om mm.o m~.o m m e m N c * mammc cocczcom cc flay csx\wccoc\wv cam =cc m>ccamu<= mocccccumc umucoc cccomamu cmou cmmc coccuacocm OLwE .mcmcmsmcma coccmuccccm> cwuoz coccmccoamcmc u c cmcsacc mcnmccw> mcmccmc< xcc>cccmcwm .c.m chMF 143 W .... w m m I u u... u. m. ...... n I m n m a u m: m ... m c m c. . ... 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' J z) ,/~«J {_n '!- ' .lrvll 'lf"~m~‘“<" a~’ o /~'° (V ’ "' i8, \ ! . l .1 -’ S‘Rvnu”" I”, x '.V\ O .fi’x ) ' ,‘\.. C)—— ‘W ." A. If ‘B‘ . ' . 0 0 Figure 5.1. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1982-83). . Production level - 32,473 metric tons. 2. Plants in solution: Kibaha, Nachingwea, Lindi, Newala I, and Newala II. 3. 80% of cashew production comes from the 'southern region, 20% from the northern region (cf., plant ratio 4:1). -. Note: 147 KEY 0 Production Center Location © Production Center Plant Location 0 Processing Plant Location 0 dP d R ' “" _..Cashew an ro ucts outing 'r.r~. A O O \ 12:23.“. . ‘f‘ueii; occ. ' J I /~«J {.. 1., / ,/' ' , ' /’ {/f‘~~‘<.h O rm“ (V " " 5‘ \. }L C) i .c.) j S 'va".V/ . H . \H. o, Figure 5.2. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1983-84). 148 Production Center Location Production Center Plant Location lo©05 O Processmg Plant Location , Cashew and Products Routing ”Orv. . 7 0 ~‘ gdhhfiz‘, '83 m8. ' J I /~«J i’~. ‘. / ," /N\. v: (v/ f ‘4" . C, \ I O \ ,J -’ ' .4. ,/ ‘i‘bxcf‘\ /' (J T’\5 ‘I 1". fi.\ C) 0‘ ’0 ’3: o. "- . o v’ O—‘fiégo Figure 5.3. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1984-85). 149 REV 0 Production Center Location 6) Production Center Plant Location 0 Processing Plant Location —r— Cashew and Products Routing or.r~. 7 ‘~. . f. - ..., - ' J I .h'J 1P. ;/ lrz’ ./”\c_‘ ’0'. (V I \l. H. \ ! \ ,J -’ ' A ./ ‘5 V~cf'\"(’ . ..‘. . O \ C O'\'\r_' V Figure 5.4. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1985-86). 150 Production Center Location 05 © Production Center Plant Location 0 Processing Plant Location Cashew and Products Routing Figure 5.5. Facility Location—Allocation Plan and Raw Cashewnut Routing Plan (1986-97). 151 IE! 0 Production Center Location © Production Center Plant Location 0 Processing Plant Location 0 —.- Cashew and Products Routing .r,r~. e7 . 0 . \cgfifi'O /' " O“ J ' f” /) ,z’fl" I '!v . /. f \.\c 0 .r"‘" (V I 0’. O H. > I! \ . ' ..-. ,/ ‘1 'va“.V / H \ 0 Figure 5.6. Facility Location-Allocation Plan and Raw Cashewnut Routing Plan (1987-88). 152 Table 5.4. Demand Change Summary of Effects. ‘T::rease 16:?i jNJEE;:_Tj$1nC¥§§§?_77:_ Year De(m%a)nd Plgrits Systegfost 1982-83* _ 5 _ 1983-84 51 7 37 1984-85 55 7 39 1985-86 55 7 44 1986-87 83 8 61 1987-88** 109 10 144 *Reference base (32,473 metric ton production level) **Based on 52¢/tonne-km transportation cost Increase in total system cost does not necessarily reflect the resulting profits to CATA or prices the farmers get. 153 percent change in cashew production did not lead to a change in location or number of processing plants, and there was only a seven percent change in total system cost. This is an indication of the robustness of the solutions between the 1983-84 and 1985-86 seasons. A configuration of processing plants shows preference for the southern region in all solutions, and this is con- sistent with the fact that 80% of the cashew production ccmes from the south. Appendix K shows the shipments for each corresponding routing plan. Routing from a production cen- ter with a plant located at that site to another plant may seem more a "shuffle," but this feature may be utilized to an advantage such that the production in the domain of one production center but close to the designated processing plant in another center's domain can be shipped to the assigned plant directly instead of shipping to its center first. This may result in further reduction in transporta- tion costs. On average, the contribution to the total sys- tem cost is 54% assignment costs (including transportation and processing costs-—variable costs) and 46% fixed costs. This implies that the use of either cost category alone in solving the facility location-allocation problem may be inadequate and probably result in misleading solutions. Robustness of locations to possible futures may be highlighted by calculating what is referred to in this study as "robustness index (RI)." This index does not necessarily have the same meaning as that used by statisticians. The RI 154 of each location can be calculated by the ratio: RI _ Number of sets in which location appears ' Total number offsets The assumption is that the location with the highest robust- ness index should be chosen as the first location to be utilized. Nhere two sites are close together, they become alternatives. For sites with similarly high RI scores, con- sideration of short-run costs incurred by each possible decision and the stability of the locations should be made. Table 5.5 shows an example summary of the above procedure based on the near optimum solutions for the 1982-83 to 1987-88 cashew production seasons (see Table 5.3). 5.3. Evaluation of Facility Capacity Changes Three levels of capacities were assumed and incorpor- ated into the model. Table 5.6 shows the effect of these three capacity levels on plant location, allocation of pro- duction centers to processing plants, and system total cost based on the 1982-83 season demand. There is little loca- tional difference for plants in the solution between 75% and 100% processing capacity, but a large difference exists between 50% and 75% capacity. This is mainly a result of a more concentrated production in the southern part of the cashew growing region, forcing more plants to be open in that area due to drops in plant capacities. A shift from 50% to 75% and 100% capacity utilization resulted in, 155 Table 5.5. Robustness Index (RI) for the Six Sets of Solutions.* Plant Location RI Kibaha Nachingwea Newala I 1'00 Newala II Masasi 0.83 Kilwa 0.67 Mtama Mtwara 0'33 Lindi Likombe Tunduru 0.17 Tanga Utete *Only processing plants in the solution are reported. A high RI location should be considered first in opening of plants and considered last in a closing strategy. The con- verse is true for low RI locations. 156 Table 5.6. Effects of Cashewnut Processing Plant Capacity Changes on Plant Location, Allocation, and Total System Cost.* Plant Capacity Utilized (%) Item 50 75 100 Number of Plants 9 5 5 Plant Location (2) Tanita II (1) Tanita I (3) Kibaha (4) Kilwa (5) Lindi (5) Lindi (5) Lindi (8) Masasi (7) Nachingwea (6) Mtama (9) Newala I (9) Newala I (7) Nachingwea (10) Newala II (10) Newala II (B) Masasi (9) Newala I (10)Ihwahill (13) Tunduru Total System Cost (3) 4.585.980 3,477,514 3,332,088 Allocation of Centers Figure 5.7 Figure 5.8 Figure 5.1 *Based on 1982-83 production figures (demand). 157 respectively, 24% and 27% increases in total system cost, whereas a shift from 75% to 100% capacity utilization resulted in only a 4% increase. Figures 5.7 and 5.8 show facility location-allocation plans with 50% and 75% plant capacity utilization, respectively. 5.4. Effects of Forcing Certain Facilities into Solution Most decisions in public sector problems are made in a political arena, probably with little bearing on budgetary constraints. A procedure for forcing certain facilities may therefore provide a device for analyzing such decisions. In this study, the facility location is forced into solution by assigning a very small Ci.value (e.gc.one is used in this study), in effect, makinthhe route the least expensive. Table 5.7 shows three examples of forced facility systems; Figures 5.9-5.11 depict routing plans. Subsystem costs do not reflect realistic costs of the system due to the imposed (I '5. Actual subsystem costs are given and compared to the 13' unforced solution system. One should note that a higher cost penalty tends to be associated with locations of low robustness. Forcing Mtwara (RI=.33) resulted in a 30% subsys- tem cost increase; forcing Tanita II (thow) alone resulted in a 39% increase and even a larger increase (130%) if both Tanita I amill were forced into solution. In this case, it is not easy to directly assess the effect of these increases on profits to CATA or their contribution to prices farmers receive for their cashewnuts. 158 <:) Production Center Location (:> Production Center and Plant Location 0 Processing Plant Location Cashew and Products Routing ”‘1 .83 J / .cc- \1 o r (\ l .2 5. ~ «x. x vx Figure 5.7. Facility Location-Allocation Plan with 50% Plant Capacity Utilization. Based on 1982-83 production figures. 159 Production Center Location Production Center and Plant Location 0 1 Processing Plant Location .rorVQ ‘ 1-©O Cashew and Products Routing . Figure 5.8. Facility Location-Allocation Plan with 75% Plant Capacity Utilization. Based on 1982-83 production figures. .mmm.8mm.m8 8c 8888 588888888 88ccac88 8888888=+ .888: 888 888:8cc 88c8888888 388888 mmimmoc8 .cocpmuoc zucccuwm umugom«* 160 888.8 cc 8c8382 c8c 88c 888.888.c 888.c88.c cc.8 8888cc 888.8c 88cc 8cc88c c8c 8.8 888.8. «.c 88c88c ccc 888.8 ..888382 c8cc 888.. M8soxcc “cwc . . . . . 888.8 88; 8c888z c . 88 888 8c8 8 888 888 8 8c 8 8888c8 888.8 8888: c8c 8 8 888.8 cuccc c8c 888.8 cc 8cc=8c c8c c88.8 . c 8c8zmz c8c 888.8 88:88c888z ccc 88 888.8c8.8 888.8c8.8 8.8 8888cc 888.8 8888: c8c c.8 888.. 83cc¥ c8c c88.cc 88cc 8cc88c c8c ccc c888cc c888cc cmmccocc +=8c88c88 8888888: 8888 c888 cocczcom 3 88888588 588888888 8.3888888 acqccccncco cccmccccachcm cc 88.”..ch 8888888c c8888< 888888 .c m 8 88 c 88cuccc888 m 8888 «.888c88c88 ocec 88ccccc888 888 :c88888 88c8888 88 8888888 .8.8 8c88c 161 0 Production Center Location © Production Center and Plant Location _.. Cashew and Products Routing Figure 5.9. Facility Location-Allocation Plan: Tanita II Forced into Solution (1982-83 season production figures). 162 0 Production Center Location © Production Center and Plant Location --*- Cashew and Products Routing Figure 5.10. Facility Location-Allocation Plan: Mtwara Forced into Solution (1982-83 season production figures). 163 0 Production Center Location © Production Center and Plant Locations __8- Cashew and Products Routing ./ /r’/. J . .... \1 ’ 4’. i l ,1“ .4-8 ‘3 b».#“ .fJ Figure 5.11. Facility Location-Allocation Plan: Tanita I and Il Forced into Solution (1982-83 season production figures). 164 5.5. Computational Experience User subroutines were written in FORTRAN V to implement the interfacing with computer code LINDO. Due to a finan- cial "squeeze," the code was only applied to solve fixed charge capacitated problems. Because integer problems tend to be difficult to solve, an addition of the Bender's cut and tightening of formulations by specifying only the .'s was found helpful in reducing the central J processing unit time (CPU). Table 5.8 shows a summary of allowed Ci computational requirements for solving the fixed charge capacitated facility location problems run on CDC CYBER 750. The computer time in this study compares favorably with that reported in other studies (cf. Soland, 1974). All problems were solved to optimum. One important aspect to note is that increases in the matrix size may lead to intolerably large amounts of com- puter time. With such large problems, use of data files (such as mathematical programming systems format files used in industry (MP5 files) for data input) need to be contem- plated. On LINDO, several observations may be made. A standard solution report by LINDO also includes items that may be used in sensitivity analysis, i.e., reduced costs and dual prices. Reduced cost is interpreted as the rate at which the objective function value will deteriorate if a variable currently at zero is arbitrarily forced to increase a small amount. If the units of the objective function are in 165 Table 5.8. Computational Requirements for Solving the Fixed Charge Problems. 1’" W Pr°blem m " Time“: s 11ml: 5 ”1216231 (hours) 1 3 4 .74 14.3 .14 2 4 7 .84 16.7 .16 3 13 19 2.31 28.9 .42 4 13 19 2.08 27.9 .39 5 13 19 1.94 28.6 .47 6* 4 6 0.4 _ _ 7* 6 8 4.5 _ _ *Compare Soland (1974) for similar fixed charge problems with zero fixed costs. n m supply center demand center lbte: Average central memory usage for the 13x19 problem was 2.43 H-H. Cost for obtaining one solution ranged from $3.80 (RG1) to $12.50 (R63). 166 dollars and the units of the variable are tonnes/year, then the units of reduced costs are dollars per tonne/year. Its value is the amount by which the variable profit contribu- tion of the variable must be improved before the variable in question would have a positive value in the optimal solu- tion. A variable in optimal solution, of course, will have a zero or negative reduced cost. A dual price associated with each constraint is the rate at which the objective function value will improve as the right-hand side (RHS) or constant term of the constraint is increased a small amount. The convention used in LINDO is that a positive dual price means that increasing the RHS in question will improve the objective function value while a negative dual price means increasing the RHS will cause the objective value to deteriorate. A zero dual price implies that small changes in the RHS will have no effect on the solution value. LINDO does not provide for senstivity (range) analysis for integer solutions due to the fact that the last integer solution found in the branch-and-bound search is not retained internally for further reference. This is cer- tainly a negative on LINDO. It should also be noted that LINDO does not do any scaling of the LP matrix. Therefore, the user needs to scale the rows and columns to avoid numer- ical problems. The rule of thumb is that nonzero coeffi- cients should not be less than 0.0001 or more than 105. LINDO assumes that the data are accurate. CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS The study reviewed handling and processing of cashew- nuts in Tanzania, and, where appropriate, comparison was made to other cashewnut-producing countries. The facility location-allocation model was developed and solved using a LINDO code. Bearing in mind the assumptions put forth and accepting the fact that it is insight rather than numbers that is important, one can say the following from the study: (1) (a) Cashewnut processing facility capacity changes showed a larger increase in total system costs between 50% to 75% compared with 75% to 100% capacity utilization. Above 75%, processing facility utilization seemed to be a utilization factor to aim toward under the circumstances. It should be pointed out, however, that 75% and 100% utilization solutions only provided five open plants as opposed to nine open plants with 50% utilization. Thus, a 50% utilization does answer some public-sector criteria, such as provision of a wider rural industrial base, income distribution to rural areas, and avoidance of the risks that pertain to centralization of facilities. Yet this 167 (2) (b) (a) (b) 168 does not rule out problems of decentralization, such as those of communication and facility dupli- cation (e.gc,special equipment and specially skilled labor). The robustness index (not necessarily as defined in statistics), based upon an optimum or near optimum solution set in static problems, can be used to sort out the robustness of a location in relation to other locations. In case of initial facility location problems, the so-called "green— fields" situation (the RI ranking for the poten- tial facilities based on projected demands) pro- vides insight to the facility location-allocation implementation strategy. In the case of a pres- ence of existing facilities, RI ranking can be used in the implementation of locational strategy, with present sites as fixed bases from which to move or with possibility of dropping old sites and' opening new ones over a longer planning horizon. The model effectively provides optimal or near optimal solutions to facility location-allocation problems, allowing for, where necessary, split shipments. The "allowable" processing plant-production center combination feature of the model for the input of assignment costs is very helpful in providing some answers to public sector issues, such as (A) (B) (C) (D) (E) 169 decentralization requirements and in stipulating effects of such issues as closures due to weather. (c) Although the interactive aspect of the model per- mits closer scrutiny of data inputs, with large problems, the process nevertheless tends to be tedious. Therefore, use of data files for data inputs may be necessary. Based on the foregoing discussion, it is recommended: Facility opening/closing strategies and shipment routes found in this study should be presented to the Cashew- nut Authority of Tanzania for consideration. There is a need to explore the computational experience for the factory operating costs with three piecewise linear segments instead of the reported fixed charge approach. The use of similar solution techniques for agricultural machinery management problems, such as location of farm machinery workshops and machinery depots in large mech- anized farms, should be explored. There is a need to explore a dynamic approach to the same cashew industry facility location-allocation prob- lem for comparison with static solutions reported here. The LINDO package needs to provide for a range test when solving integer problems in order to provide for sensitivity of solutions to small parameter changes. 170 (Jofieueu) IOIlVllSIIIHOV (Janillo) momw (1.1m mod-I91) SIIIIVTd/BIIIIVII IIIERIAL AUDI! 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Partitioning Production Year Sum of Squares by Use of T: 91200001100 YEAR and 10111.3 2 Effect 2 9 6 8 10 Q rzci SS F 10.2992 19.086911.693312.961519.9239 Linear -2 -1 0 +1 +2 11.2290 7(10) 1.80 105.88“ Quadratic +2 -1 -2 -1 +2 2.9112 7(19) 0.09 5.29: Cubic -1 +2 0 -2 +1 +1.57957(10) 0.09 2.35 Quartic +1 -9 +6 -9 +1 2.6393 70°) 0.01 1 Total 1.99 179 (F) Prediction Equation. The analysis shows significant (P50.10) linear and quadratic effects in the cashewnut production year. The prediction equation is given by: where: 1.6976 (from section (0)) <1 11 b. = 2(xi-7)(vi—T) = Q — 2 2 z(xi x) rzci i e b _ 11.2240, b _ 2.9112 0 O, 1- 7‘10), 2'. 7‘1!) 1, from tables of polynomials (Steel and )4 —I y N I Torrie, 1980) and: - k’(n’-k’) g9+1 * 515k ' 'ITIETTTT' Ek-1 Thus: x.-Y x.-Y z 1 and :2 = [( 1 1* ” " - I d d 12 E1: where d is spacing between consecutive x's, k is degree of polynomial, and n is the number of levels of a factor. 180 Therefore, the prediction equation for cashew production levels is: T = 1.6976 + (W1111<5§§)+1%111m5§§r - 3,51] 'T = 1.6976 + 0.0802x - 0.4810 + 0.0297[0.25x’ - 3x + 7] 'T = 1.6976 + 0.0802x - 0.4810 + 0.0074x= - 0.0891x + 0.2079 T = 1.4245 - 0.0089x +0.0074x2 Thus, the production center transformed production level (Y) prediction equation for the ten-year production period (x = production year) is: T = 1.4245 - 0.0089x + 0.0074xz APPENDIX G LAGRANGE MULTIPLIER TECHNIQUE 181 LAGRANGE MULTIPLIER TECHNIQUE General Description The Lagrange method of undetermined multipliers is designed to solve problems of the form: Optimize f(xi) i = 1,2,...,n subject to (x1,x2,...,xn) to satisfy the following equation: '1 _n N 3 gj(xi) = DJ 1 (_| TI _s N 3 where m < n. The general approach for the procedure begins by formu- lating the Lagrangean function: - m - ) - .=1Aj[gj(xi) bj] h(xi,xj) = f(xi J where Aj(j=1,2,...,m) are called Lagrange Multipliers. The key fact should be noted that for the permissible values of xi(i=1,2,...,n), gj(xi)-bj=0 for all i, (j=1,2,...,m). Thus h(xi,xj) = f(si). Therefore, it can be shown that if (xi,A.) = (x?,x§) is a local or global mini- J mum or maximum for the unconstrainted function h(xi,Aj), then (x?) is a corresponding critical point for the original problem. The method now reduces to analyzing h(xi,xj). This involves setting the (n+m) partial derivatives at zero; that is: 182 89. oh of m 1 _ _ F: 5?— - 2J=1Aj-a—x—_- 0 for k - 1,2,...,n k k fl==g(x)+b=0 fori=12 m axj J- i i D D..., and then the critical points would be obtained by solving these equations for (x1,x2,...,xn;x1,12,...,Am). Notice that the last m equations are equivalent to the constraints in the original problem, so only permissible solutions are considered. After further analysis to identify the global minimum or maximum of h(-), the resulting value of (x1,x2,...,xn) is the desired solution to the original problem. As an example, consider (Hellier and Lieberman, 1980): n = 2 and m = 1 The problem might be to: Optimize f(x1’x2) = x3 + XE = 1 subject to: g(x1,x2) = x1 + XE = 1 (i.e., in this case, (x1,x2) is restricted to be in circle of radius 1 whose center is at the origin). Therefore, in this case: h(x1,x2) = x1 + 2x2 - x[xfi + XE - 1] 183 so that: (1) 59%=2x1-2xx1=0 (2) a3—x"——=2-2ix2.0 (3) 9-,'%=[x=1+xf?-1]=o Hence: Equation (1) implies that A=1 or x1=0 If 1:1, Equation (2) implies that x2=1, x1=0 If x1=0, then Equation (3) implies that x2=¢1. Therefore, the two critical points (x1,x2) = (0,1) and (0,-1), providing the global maximum and minimum, respec- tively. APPENDIX H PREDICTION EQUATIONS: CASHEWNUT PRODUCTS FOR EXPORT 184 .x_ocmaa< m_;u mo Amy :_ cm>wm mg mfiasmxm co_um_:u~mo mwpms_pmm gmpmsmcma :oflmmmcmmm«4 .e.m mfinm» co ummmn wgm mmgsu_w wmmpcwocmax mm.o xno.o+wu.o um mm.o RN.— N©.o m.m Np+m Amzu mm.o xen.n+mm.m um mm.w w—.o ¢N.n m.m N®.mm m—mcgwx mm.o xmm.m1—o.mnuw Fo.mm mw.01 om.m1 w.m m—.vm mpaczmm x.xn.x.»nu~.. .3 + m up w >5xn X.>.n x > 0 “58:868. .5333 I I E: com. me 5328.39 c2883“. $232.23 3853mm c3333; .8. .u a Amzu oz< .mhzzz Y(I) DO 250 I - 1.N DP - CIJMIN - C(1.J) C 1F OPENING OF SITE 1 wILL HELP CUSTOMER J THEN ADD SAVINGS IF(DP.GT.O.) BEN(I) - BEN(1)+DP 250 CONTINUE 300 CONTINUE C GET ACTUAL TOTAL COST ACTUAL - GLOBAL+SBCOST HRITE(6.AOO)ACTUAL 900 FORMAT(/.' TRUE COST-'.F12.2./.' ADD THE CONSTRAINT:') C GENERATE THE BENDERS CONSTRAINT C DEFINE THE RON TYPE THE RHS CALL DEFROHI-I.SBCOST.IRN0.TRUBLE) C NON PUT IN THE COEFFICIENT FOR EACH VARIABLE DO 500 I - I.M IARG - I IF(BEN(I).NE.O.)CALL 1NSERT(IRNO.IARG.BEN(I).TRUBLE) IF(TRUBLE) GO TO 9000 500 CONTINUE C INSERT THE BOUND VARIABLE CALL INSERT(IRNO.H+I.1..TRUBLE) IF(TRUBLE) Go TO 9000 C SHON THE NEH CONSTRAINT CALL LOOK(IRNO.1RNO) RETURN 9000 HRITEI6.9001) . 9001 FORMAT(/.1x.' OUT OF SPACE IN NEHIP') C ABORT SOLUTION ACTUAL - GIANT RETURN END 190 AAAAAAAAAAAAAAAA‘AAAAAAAAAAAAAAAAAAAAAAAAAAAAA‘AAAAAAAAAAAA; CII'III -- "I" C. THE FOLLOWING SUBROUTINES ARE LINKED TO LINDO PACKAGE ** C* FOR SOLVING THE PUBLIC SECTOR ISSUES E.G. RELAXATION *3 C. OF COSTS AS SPECIFIED DY THE RESTRICTIONS IHPOSED ** C. BY THE HINIRUH AND HAXIHUH NUHBER OF FACILITIES ALLOHED *8 Ct TO IE OPEN. THE PROGRAH SOLVES THE SIMPLE FACILITY it C* LOCATIONAL PRODLEH USING A 'DENDERS‘ ALGORITHH. it C* THE SUBROUTINE IS LINKED TO LINDO IN PFN-DENHOPLINDO. ** C* REQUIRED INPUTS: NO. OF FACILITIES.FIXED AND ASSIGNHENT ** C* COSTS. HINIHUH AND HARIHUH OPEN FACILITIES ALLONED. 9* .AAAA‘AAAAAAAA‘AAAAAAAAAAAA‘AAA‘AAAA‘AL‘LAAAL‘AAAAAAAAAAAAAAA SUBROUTINE USER(INPROB) C NATRIX GENERATOR FOR FACILITY LOCATION NITN IENOERS CUT C NRITTEN BY H O DIHENGA (FROM SCHRAGE GKEPNER) JUNE.I989 COMMON/DESDATA/M.N.F(3O).C(3O.50) DIMENSION 1RO(5). VAL(s) LOGICAL TRUBLE INTEGER ALFANM(36).VNANE(8) INTEGER BLANK C 1/0 NUNBERS DATA 1NPUT/5/.LOUT/6/ C ALPHABET DATA ALFANM/IHO.IHI.IH2.IH3.IH9.IH5.IH6.1H7.IH8.IH9. + IHA.IHB.IHC.IHO.1HE.1HF.1HG.IHH.1HI.IHJ.1NK.1HL.IHM. + 1HN.1HO.IHP.IHQ.IHR.IHS.IHT.IHU.IHV.IHw.IHx.1HY.1H1/ BLANK - " " C MAXIMUH NUMBER OF PROCESSING PLANTS NDIM1 - 30 C HAXIHUH NUMBER OF PRODUCTION CENTERS (CUSTONERS) NDIM2 - so TRUBLE - .FALSE. 50 HRITE(6.IOOI) 1001 FORMATI/.' INPUT FOR PLANT AND TRANSPORTATION PROBLEH THE'./. + . NO. OF PROCESSING PLANTS AND PRODUCTION CENTERS: ') REAO(5.*)H.N IF(M.GT.NOIM1)wRITE(6.2OOIINDIMI IF(N.GT.NOIH2)NRITE(6.2002INDIM2 IF(M.LT.I)NRITE(6.2OO§)M IF(N.LT.I)NRITE(6.2006)N IF((M.GT.NDIM1).OR.(N.GT.NDIM2))GO TO 50 IF((M.LT.1).OR.(N.LT.1))GO TO 50 C ENTER THE OPENING RESTRICTIONS OF FACILITIES NRITE(6.IOOO) IOOO FORMAT(/.' INPUT NINIHUM AND HAXIHUH OPEN NO. OF PLANTS:') READ(S.*)FHNO.FHXO wRITE(6.1002) 1002 FORNATI' INPUT EACH FACILITY FIxED COSTS:') DO 10 I - 1.N NR1TE(6.90)I 9O FORMAT(Ix.19.': ., READ(5.*)F(1) IO CONTINUE C INITIALIZING ROHS CALL INIT C GENERATE RONS C THE OBJECTIVE FUNCTION C RIN SUMMATION C(I.J)*X(I.J)+SUMATION F(I)PY(I) CALL DEFRON(I.O..1RNO.TRUBLE) 191 C GENERATE THE FACILITY OPENING RESTRICTIONS CALL DEFRONI-I.FRNO.1RNO.TRUBLE) CALL DEFRON(I.FHxO.IRNO.TRUBLE) C GENERATE THE Y VARIABLES VNAHEII) - ALFANHI3S) VNAME(3) - BLANK VNAME(h) - BLANK VNARE(5) - BLANK VNAME(6) - BLANK VNAME(7) - BLANK VNAME(B) - BLANK NONz - 3 1RO(1) - I IROIZ) - 2 VAL(2) - I. 1R0(3) - 3 VAL (3) - I. DO 500 J - 1.M VNAME(Z) - ALFANM(J+I) VAL(I) - F(J) CALL APPCOL(VNAME.NONz.VAL.IRO.TRUBLE) 500 CONTINUE NR1TE16.IOOA) 1009 FORMAT(/.' INPUT ASSIGNMENTS COSTS') DO 20 K - I.H*N NR1TEI6.DO)K BO FORMAT(Ix.19.': 1) REAO(5.*)CIJ 20 CONTINUE C ADD THE 2 VARIABLE VNAME(I) - ALFANH(36) VNAME(Z) - NULL NONz - 1 VAL(I) - 1. CALL APPCOL(VNAME .NONz.VAL.1RO.TRUBLE) RETURN 2001 FORHAT(/.' NUHBER OF PLANTS EXCEEOS THE MAKIMUR ALLOHED- '.16) 2002 FORMAT(I.‘ NUHBER OF PROD.CENTERS EXCEEOS RAx ALLOHED- '.16) 2003 FORMAT(I.’ PLANT NO. '.16.' IS TOO LARGE.(IGNOREO)') 2009 FORMAT(I.’ CENTER NO.'. IB.‘ IS TOO LARGE.(IGNORED)') 2005 FORMAT(/.' NUHBER OF PLANTS 1.13.' IS LESS THAN I.') 2006 FORHAT(/.' NUHBER OF CENTERS '.13.' IS LESS THAN I.') 2007 FORHAT(/.' NUMBER OF PLANTS. '.13.' IS LESS THAN I.(IGNOREO)') 2008 FORHAT(/.' NUMBER OF CENTERS. '.13.' IS LESS THAN l.(IGNOREO)') END SUBROUTINE NEwIPIACTUAL.BOUND) C GENERATE NEXT BENDERS CONSTRAINT FOR THE FACILITY LOCATION C ALLOCATION PROBLEH . COHRON /DESDATA/M.N.F(30).C(30.SO) DIMENSION BEN(30).Y(3O) LOGICAL TRUBLE C OUTPUT UNIT DATA OUT/SI C LETS LOOK AT THE SOLUTION CALL OUTSOL GIANT - IO.E3O C GLOBAL HILL ACCUMULATE THE TRUE COST OF THIS SOLUTION 192 GLOBAL I 0. DO 100 I I 1.N IARG I I CALL REPVAR(1ARG.PRIMAL.DUAL) Y(I) - PRIMAL C ACCUMULATE THE FIXED COSTS OF OPEN PLANTS GLOBAL - GLOBAL+F(IIRPRIMAL C INITIALIzE THE BENDERS CONSTRAINT TO ZERO BENIII - O. 100 CONTINUE C SBCOST NILL ACCUMULATE THE SUBPROBLEH OR TRANSPORTATION COSTS SBCOST - 0. DO 300 J - 1.N C FIND THE CLOSEST OPEN PLANT CIJMIN - GIANT DO 200 I - 1.M IF(Y(I) .LT..9)GO TO 200 IF(CII.J) .LT. CIJMIN) CIJMIN - C(I.J) 200 CONTINUE C ACCUHuLATE THE SUBPROBLEM COST SBCOST - SBCOST+CIJMIN C NON COMPUTE AND SUM OVER 1 THE DUAL PRICES ON THE CONSTRAINTS CXII.J) >YII) . DO 250 I - 1.N DP - CIJHIN - C(I.J) C 1F OPENING OF SITE 1 NILL HELP CUSTOMER J THEN ADD SAVINGS IF(DP.GT.O.) BENII) - BEN(I)+DP 250 CONTINUE 300 CONTINUE C GET ACTUAL TOTAL COST ACTUAL - GLOBAL+SBCOST NRITE(6.AOO)ACTUAL 900 FORHAT(/.' TRUE COST-'.F12.2) C GENERATE THE BENDERS CONSTRAINT C DEFINE THE RON TYPE THE RHS CALL DEFRON(-I.SBCOST.IRNO.TRUBLE) C NON PUT IN THE COEFFICIENT FOR EACH VARIABLE DO 500 I - 1.N IARG - 1 IF(BEN(I).NE.O.)CALL INSERT(IRNO.IARG.BEN(1).TRUBLE) IF(TRUBLE) GO TO 9000 500 CONTINUE C INSERT THE BOUND VARIABLE CALL 1NSERT(IRNO.R+I.I..TRUBLE) IF(TRUBLE) GO TO 9000 C SHON THE NEN CONSTRAINT CALL LOOKIIRNO.IRNOI RETURN 9000 NR1TE(6.9OOI) 9001 FORMAT(/.Ix.' OUT OF SPACE IN NENIP') C ABORT SOLUTION ACTUAL - GIANT RETURN END APPENDIX J SAMPLE MODEL OUTPUTS HIN + 1? ++++++++++++++++++++ 193 SAMPLE MODEL OUTPUTS 650 YOOI + 680 Y002 + 930 YOO3 + 250 YOO9 + 280 Y005 312 YOO6 + 221 Y007 + 390 YOO8 + 390 YOO9 + 390 YOIO 390 YOII + 375 Y012 + 930 YOI3 + 59.9 20050001 + 66.9 20060001 25.2 20110001 + 25.2 20120001 + 21 20090002 + 21 20100002 36.1 20080002 + 92 20060002 + 29.8 20070003 + 27.8 20080003 36.1 20090003 + 36.1 20100003 + 92 20060003 + 28.9 20050009 37.8 20060009 + 59.3 20110009 + 59.3 20120009 + 37.8 20050005 32.1 20060005 + 33.7 20070005 + 91.8 20080005 + 91.8 20090005 91.8 20100005 + 33.7 20060006 + 18.1 20070006 + 31.9 20080006 60.5 20090006 + 60.5 20100006 + 10.2 20060007 + 66.9 20070007 91.2 20080007 + 127.1 20090007 + 127.1 20100007 51.3 20090008 + 100.2 20050008 + 137.7 20060008 129.7 20070009 + 100.1 20080009 + 26.2 20130009 299.2 20070010 + 229.9 20080010 + 125 20130010 + 20.5 20010011 27.2 20020011 + 95.5 20030011 + 25.1 20010012 + 31.9 20020012 22.1 20030012 + 90.2 20010013 + 97 20020013 + 65.2 20030013 100 20010019 + 93.8 20020019 + 125 20030019 + 61.2 20090019 66.7 20010015 + 600.5 20020015 + 91.7 20030015 115.3 20090015 + 12.7 20010016 + 12.7 20020016 + 25.2 20030016 216.5 20010017 + 223.3 20020017 + 191.6 20030017 377.9 20090017 + 932.8 20010018 + 939.6 20020018 907.9 20030018 + 959.9 20080018 + 100 20010019 + 16.8 20020019 75.1 20030019 + 259.1 20090019 SUBJECT TO 2) 3) I.) 5) 6) 7) 8) 9) IO) 11) 12) I3) 19) IS) 16) I7) 18) 19) 20) 21) 22) 23) 4. <- + <- + <- .5 + ... '9 4. 4. + - 12 YOOI + 10010011 + 10010012 + 10010013 + 10010019 10010015 + 10010016 + 10010017 + 10010018 + 10010019 0 - 12 YOOZ 9» 10020011 + 10020012 «9 10020013 «9 10020019 10020015 + 10020016 + 10020017 + 10020018 + 10020019 0 - 10 YOO3 + 10030011 + 10030012 + 10030013 + 10030019 10030015 + 10030016 + 10030017 + 10030018 + 10030019 0 - 5 Y009 + 10090008 + 10090019 + 10090015 + 10090017 10090019 <- O - 5 YOOS + 10050001 + 10050009 I 10050005 + 10050008 <- - 5 1006 + 10060001 + 10060002 + 10060003 + 10060009 10060005 + 10060006 + 10060007 + 10060008 mu 197 LP OPTIHUH FOUND AT STEP NEH INTEGER SOLUTION AT BRANCH OBJECTIVE FUNCTION VALUE 1) VARIABLE Y001 Y002 YOO3 Y009 Y005 Y006 Y007 Y008 Y009 YOIO YOII Y012 Y013 10050001 10060001 10110001 10120001 10090002 10100002 10080002 10060002 10070003 10080003 10090003 10100003 10060003 10050009 10060009 10110009 10120009 80050005 3329-39930 VALUE .000000 .000000 1.000000 .000000 1.000000 .000000 1.000000 .000000 1.000000 1.000000 .000000 .000000 .000000 1.767000 .000000 .000000 .000000 9.506000 .000000 .000000 .000000 1.106000 .000000 .000000 3.217000 .000000 1.179000 .000000 .000000 .000000 1.992000 59 11 PIVOT REDUCED COST .000000 -19.600000 930.000000 ‘69.000000 260.000000 “28.000000 76.500000 .000000 13.675000 .000000 .000000 51.000000 89.000000 .000000 27.600000 .000000 .000000 73.900000 .000000 72.900000 59.600000 68.275000 .000000 122 10060005 10070005 10080005 10090005 10100005 10060006 10070006 10080006 10090006 10100006 10060007 10070007 10080007 10090007 10100007 10010008 10050008 10060008 10070009 10080009 10130009 10070010 10080010 10130010 10010011 10020011 10030011 10010012 10020012 10030012 10010013 10020013 10030013 10010011 10020011. 10030011 10010011. 10010015 10020015 10030015 10010015 10010016 10020016 10030016 10010017 10020017 10030017 10010017 10010018 10020018 10030018 10080018 10010019 10020019 10030019 10010019 RON 2) 3) d O O \J \D O O O N o O U N O o o ’- . 0‘ N O O O o .- o O \J N O O O ISBEB 58.300000 3.200000 35.900000 .000000 .000000 72.300000 .000000 37.900000 31.100000 31.100000 .000000 .000000 18.900000 18.900000 98.900000 10.900000 .000000 101.500000 .000000 .000000 25.700000 .000000 .300000 .000000 29.166667 90.000000 .000000 57.166667 68.100000 .000000 29.166667 90.100000 .000000 29.166667 27.100000 .000000 .000000 29.166667 567.100000 .000000 87.900000 81.666667 55.800000 .000000 79.066667 90.000000 .000000 299.600000 79.066667 90.000000 .000000 87.900000 79.066667 .000000 .000000 292.800000 DUAL PRICES 59.166667 58.300000 199 .000000 63.800000 9.000000 68.000000 11.300000 35.900000 .000000 .000000 33.200000 96.875000 135.500000 -58.900000 '21.000000 -36.100000 '32.900000 -91.800000 “29.900000 -78.200000 -109.200000 -136.000000 “260.500000 -95.500000 '22.100000 '65.200000 -125.000000 -91.700000 -25.200000 '191.600000 -907.900000 '75.100000 1) 1.526000 5) .000000 6) .000000 7) .000000 8) .000000 9) .000000 10) .191000 11) 1.901000 12) .000000 13) .000000 11) .000000 15) .000000 16) .000000 17) .000000 18) .000000 19) .000000 20) .000000 21) .000000 22) .000000 23) .000000 21) .000000 25) .000000 26) .000000 27) .000000 28) .000000 29) .000000 30) .000000 31) .000000 32) .000000 33) .000000 10. ITERATIONS- 122 Inaucuzs- 11 DETERH.- 96.000E 0 39) YOOI >- 0 £10 INTEGER-VARIABLES- 13 BOUND on OPTIHUN: 2151.109 ENUHERATION COMPLETE. IRANCNESI 11 LAST INTEGER SOLUTION IS THE DEST FOUND PIVOTs- 122 .2()0 650 1001 + 680 1002 + 930 1003 + 250 1009 + 280 1005 + 312 1006 + 221 1007 + 390 1008 + 390 1009 + 390 1010 1 + + + + + 4 + 4 + + + + + + + + + + + + + NUIOV'UII'N-‘dod rUNU‘U‘tN—MUC‘PUW SUBJECT TO 2) 3) 9) 5) 6) 7) 8) 9) 10) 11) 12) 13) 11) 15) 16) I7) 18) 19) - 12 1001 + 10010011 + + 10010015 + 10010016 + (I O - 12 1002 + 10020001 + + 10020005 + 10020006 + + 10020010 + 10020011 + + 10020015 + 10020016 + (a 0 10010012 + 10010017 + 10020002 + 10020007 + 10020012 + 10020017 + - 10 1003 + 10030011 + 10030012 + + 10030015 + 10030016 + (a 0 10030017 + 10 10010018 + 010013 + 10020003 + 10020008 + 10 020013 + 10020018 + 10030013 + 10030018 + 390 1011 + 375 1012 + 930 1013 + 10020001 + 10020002 10020003 + 10020009 + 10020005 + 10020006 + 10020007 10020008 + 10020009 + 10020010 + 10020011 + 10020012 10020013 + 10020019 + 10020015 + 10020016 + 10020017 10020018 + 10020019 + 59.9 10050001 + 66.9 10060001 25.2 10110001 + 25.2 10120001 + 21 10090002 + 21 10100002 36.1 10080002 + 92 10060002 + 29.8 10070003 + 27.8 10080003 36.1 10090003 + 36.1 10100003 + 92 10060003 + 28.9 10050009 7.8 10060009 + 59.3 10110009 + 59.3 10120009 + 37.8 10050005 2.1 10060005 + 33.7 10070005 + 91.8 10080005 + 91.8 10090005 .8 10100005 + 33.7 10060006 + 18.1 10070006 + 31.9 10080006 .5 10090006 + 60.5 10100006 + 100.2 10060007 + 66.9 10070007 .2 10080007 + 127.1 10090007 + 127.1 10100007 .3 10090008 + 100.2 10050008 + 137.7 10060008 9.7 10070009 + 100.1 10080009 + 26.2 10130009 9.2 10070010 + 229.9 10080010 + 125 10130010 + 20.5 10010011 .5 10030011 + 25.1 10010012 + 22.1 10030012 + 90.2 10010013 .2 10030013 + 100 10010019 + 125 10030019 + 61.2 10090019 .5 10010015 + 91.7 10030015 + 115.3 10090015 + 12.7 10010016 .2 10030016 + 216.5 10010017 + 191.6 10030017 7.9 10090017 + 932.8 10010018 + 907.9 10030018 59.9 10080018 + 100 10010019 + 75.1 10030019 + 259.1 10090019 10010019 10010019 10020009 10020009 10020019 10020019 10030019 10030019 - 5 1009 + 10090008 + 10090019 + 10090015 + 10090017 + 10090019 <- 0 - 5 1005 + 10050001 + 10050009 + 10050005 + 10050008 <- - 5 1006 + 10060001 + 10060002 + 10060003 + 10060009 * 10060005 + 10060006 + 10060007 + 10060008 <- - 5 1007 + 10070003 + 10070005 + 10070006 + 10070007 + 10070009 + 10070010 <- - 10 1008 + 10080002 + + 10080007 + 10080009 4 - 10 1009 + 10090002 + + 10090007 <- 0 ° 10 1010 + 10100002 + + 10100007 <- O - 10 1011 + 10110001 + 0 10080003 + 10080010 + 10090003 + 10100003 + 10110009 <- ° 8 1012 + 10120001 $ 10120009 <- - 10 1013 + 10130009 1 10020001 + 10050001 + . 1.767 10020002 + 10090002 + ' 9-506 10020003 + 10070003 1 + 10060003 - 1.323 10020001 1 10050001 + - 1.171 1mmw5+1wwm5+ + 10090005 +. 10100005 - 10130010 <- 10060001 + 10100002 + 10080003 + 10060009 + 10060005 + 3-371 0 10080005 + 10080006 10080018 <- 10090005 + 10090006 10100005 + 10100006 0 0 0 10110001 + 10080002 + 10090003 + 10110001 + 10070005 + 0 10120001 10060002 10100003 10120009 10080005 0 m) 2H SUB SUB SUB soo< uoo< d-n-o-o—o—o—n—n NfimtuudOWONGNBI-WN-o 18: 20: d 1? 10020006 + + 10100006 I 10020007 + + 10100007 I 10020008 + 10020009 + 10020010 + 10020011 + 10020012 + 10020013 + 10020019 + 10020015 + 10020016 + 10020017 + 10020018 + 10020019 + 1001 1002 1003 1009 1005 1006 1007 1008 1009 1010 1011 1012 1013 U004 0.004 1 0" n n moo< 000-¢ N00< fl 0 8 8 \fl . .. \H 000-( 201 10060006 + 0.812 10060007 + 0-935 10090008 + 10070009 + 10070010 + 10010011 + 10010012 + 10010013 + 10010019 + 10010015 + 10010016 + 10010017 «1 10010018 + 10010019 + 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9000< n n 1 D c>-¢:-< --¢:«< ns-c>-¢ \»-—¢:-< I ’ ----- cn-n, . ------- on H n 8 ’ g - O #0001000)! ”0000000)! “00009800,! 10070006 + 10070007 + 10050008 + 10080009 + 10080010 + 10030011 - 10030012 - 10030013 - 10030019 + 10030015 + 10030016 - 10030017 + 10030018 + 10030019 + d o d 0'000NO0X ---d----------d 0‘ 080008800” - - ,- '1 " o~c>c>c>nac>c311 d o d d N000N00x ---------------‘-d 10080006 + 10090006 10080007 + 10090007 10060008 I 10130009 I 10130010 I 9.067 0£% 0.969 10090019 I 10090015 I 5 10090017 I 10080018 I 10090019 I 0000.000” 00000000)! . d d - d 0-‘00N00’I d-f00NO0X N-I00N00X “#0010003! P-‘00NO0X Ui-IOONOOX ...-......u....-.‘-‘ d d d 0 d ----u-u----------—-d 0.567 2.092 0.055 1.077 0.293 0.515 0.139 0.125 O-‘OONOOX ‘ d d - - d - ------o-u-o-c---od-d d d c 202 10080006 50070006 10060006 10100005 10090005 10080005 50070005 50060005 50050005 10120009. 10110001“ 5006000.“ 50050001“ I50060003 50100003 «50090003 50080003 500110003 500-60002 50080002 10100002 10090002 50120001 50110001 50060001 10050001 50020019 50020018 50020017 0 - - - 1- - - - - 1 ......................... 1 - 1- - -1 - - - - - ........ 1--------1-------------- 1 1 1- - - -1 - - - - - ........... 1------------- 1 - -1 - v - - - - - .......... 1-----1---------- 1 - - - v - - - - - 1 -1 - - - - - - - - ............................... - - -1 - - - - - - .............. 1------------ 1 1 -1 - -1 - - - - - - ........... 1-1--------------- 1 1 - - v - - - - - - ........ 1---------------- 1 - - - - - - - - 1- - - - - - - -1 - . - ..-: -----.---.1--. mm wwmnunuuunnnwnnn I50 50 50 50 I50 50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 I50 50 I50 I50 I50 I50 I50 50 203 010016 090015 .. 1 1 - 1 - .- 1 1 1 1 030015 010015 090019 1 1 1 .- 1 1 1 - - 1 1 030019 010019 030013 .. 1 o 1 1 1 1 1 1 - 1 010013 030012 010012 a 1 1 1 1 1 1 a .. o 1 030011 010011 130010 080010 070010 130009 080009 070009 060008 050008 090008 100007 090007 080007 070007 060007 100006 090006 8 8 N 1 1 M<<<<<<<<< 88888 888888888888888 8 88888888888 C888‘ 88888 .- .. 1 1 1 1 1 1 .- 10090019 1 .. .- 10030019 10010019 1 .- 1 1 1 .- u. 1 - 1 1 10080018 1 1 1 1 10030018 10010018 1 1 1 .- 1 1 1 1 .- .. 1 10090017 1 1 1 10030017 10010017 I O 0 0. .0 O. C. O. I. .0 C. .0 .0 O. “u.” 50030080 123956709w 11: 12: 19: 15: 16: 17: 18: 19: 20: 21: 22: 23: 29: 25: 26: 27: 28: 29: )0: 31: 33: LP OPTIHUH rouuu 'AT STEP 1 - Cd 0 o a - o o o o u c n o c o - o c o o o -d - - - o O - - o - - - o a - - - - o - - - 2(14 d - ------ c -- - - -- - ------- IIIIIIIIIIIIIIIIIIIAAAA -1-1-1-1-1:-1-I)1c:n-o-1-1)n>»>1>u> NEH INTEGER SOLUTION AT BRANCH 1) VARIABLE Y001 Y002 Y003 7009 Y005 Y006 Y007 1008 Y009 Y010 Y011 Y012 Y013 A0020001 20020002 X0020003 10020009 90020005 20020006 90020007 10020008 90020009 X0020010 10020011 20020012 .000000 1.000000 .000000 1.000000 .000000 1.000000 1.000000 .000000 1.000000 .000000 .000000 .000000 .000000 1.767000 .000000 .000000 .000000 .000000 .000000 .935000 000000 2.092000 .055000 9.067000 .686000 OBJECTIVE FUNCTION VALUE 2913.79650 VALUE 53 2 2 3 1 12 PIVOT REDUCED COST 68.000000 92.900000 39.000000 50.000000 35.500000 12.000000 69.500000 307.000000 3 3 90.000000 90.000000 30.000000 87.000000 80.000000 .000000 50.200000 25.100000 23.900000 29.100000 31.300000 .000000 9.900000 .000000 .000000 .000000 .000000 119 20020013 20020019 20020015 20020016 20020017 20020018 20020019 20050001 20060001 20110001 20120001 20090002 20100002 20080002 20060002 20070003 20080003 20090003 20100003 20060003 20050009 20060009 20110009 20120009 20050005 20060005 20070005 20080005 20090005 20100005 20060006 20070006 20080006 20090006 20100006 20060007 20070007 20080007 20090007 20100007 20090008 20050008 20060008 20070009 20080009 20130009 20070010 20080010 20130010 20010011 20030011 20010012 20030012 20010013 20030013 20010019 20030019 20090019 20010015 20030015 20090015 \0 O O 19 0‘ O O O ‘1 J- O O O 2()5 .000000 .000000 .000000 .000000 .000000 .000000 .000000 2.100000 5.700000 .000000 .000000 .000000 .000000 23.900000 21.000000 .000000 .000000 .000000 .000000 5.900000 .000000 .000000 52.500000 52.500000 19.600000 .000000 12.900000 18 .000000 9 . 700000 9.700000 9.300000 .000000 10.300000 31.100000 31.100000 39.000000 17 .000000 38.300000 65.900000 65.900000 .000000 5mew 86.900000 79.800000 97.200000 .000000 199.300000 172.000000 98 . 800000 7 . 800000 23.900000 12.900000 .000000 27.500000 93.100000 87.300000 102.900000 .000000 97 . 000000 69 .600000 59.100000 APPENDIX K SAMPLE SHIPMENT PLANS 20010016 20030016 20010017 20030017 20090017 20010018 20030018 20080018 20010019 20030019 20090019 Row 2) 3) 9) S) 6) 7) 8) 9) 10) 11) 12) I3) 19) 15) 16) I7) 18) 19) 20) 21) 22) 23) 29) 25) 26) 27) 28) 29) 30) 31) 32) 33) NO. ITERAT BRANCHES- 39) sun INTEGER-VARIABLESI BOUNO ON OPTIHUH: ENUHERATION COMPLETE. BRANCHESI IONS. YOOI >- 2()€5 .000000 .000000 .000000 3.100000 .000000 203.800000 .000000 169.500000 .000000 316.200000 .000000 920.100000 .000000 385.800000 .000000 907.000000 .oooooo 87.300000 .000000 53.000000 .000000 192.900000 SLACK DUAL PRICES .000000 98.500000 .000000 60.200000 .000000 39.100000 3.713000 .000000 .000000 8.900000 .955000 .000000 .000000 11.300000 .oooooo 8.300000 .359000 .000000 .000000 .000000 .000000 36.000000 .000000 36.000000 .000000 35.000000 .000000 -61.200000 .oooooo -21.000000 .000000 ~36.100000 .000000 -37.800000 .0ooooo -32.100000 .000000 -29.900000 .000000 -61.200000 .000000 —51.300000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 .000000 -61.200000 199 12 0ETERH.- 1.000: 0 0 13 1998.502 LAST INTEGER SOLUTION IS THE BEST FOUND 12 PIVOTSO 199 207 NIN 65 11 + 6800 12 + 93 13 + 25 19 + 280 15 + 312 16 + 221 17 + 390 18 + 390 19 + 390 1A + 390 18 + 375 1C + 930 10 + z SUBJECT TO 2) 11 + 12 + 13 + 19 + 15 + 16 + 17 + 18 + 19 + 19 + 18 + 1C + YD >- 6 3) 11 + 12 + 13 + 19 + 15 + 16 + 17 + 18 + 19 + 1A + 18 + 1C + 10 <- 13 END 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 9 5 6 7 8 9 A 8 c O z I: 8 D 8 8 C C C C C C C C C I HIN 2: 1 1 1 I 1 1 1 1 1 1 1 1 1 > 6 3: 1 1'1 1 1'1 1 1'1 1 1'1 1 '< 8 LP OPTINUN EOUNO AT STEP 11 LP OPTIHUH IS IP OPTINUN NEH INTEGER SOLUTION AT BRANCH O PIVOT 11 OBJECTIVE FUNCTION VALUE 1) 996.000000 VARIABLE VALUE REOUCEO COST 11 1.000000 -297.000000 12 .000000 6988.000000 13 1.000000 -269.000000 19 1.000000 ~287.000000 15 1.000000 -32.000000 16 1.000000 .000000 17 1.000000 -91.000000 18 .000000 78.000000 19 .000000 78.000000 19 .000000 78.000000 18 .000000 78.000000 1: .000000 63.000000 10 .000000 118.000000 2 .000000 1.000000 RON SLACK DUAL PRICES 2) .000000 -312.000000 3) 7.000000 .000000 NO. ITERATIONS- 11 BRANCHES- 0 DETERH.- 1.000E 0 9) z >- 0 END . INTEGER-VARIAOLES- 13 BOUND ON OPTIHUH: 996.0000 ENUNERATIDN COMPLETE. BRANCHES- 0 PIVOTSO 19 208 LAST INTEGER SOLUTION IS THE BEST FOUND 209 22 3 2.2.30 :3 2 35.3 02.; 9.22.8 52266:. a: :3... ~23 .359 73.8. ...—32.8 ...... Jen—a 9.3383 ~ 22.8. of 3 3.23 :N.m E.- Couu: 9:333; .3328: ... m m. m m m ...... m m. m m m m m m m .. . m u w 1 n u m n u a m u n 1 u p 1 u u 0 s . B . J . cul- . 8. n“- vl u m s “u a n .0 up a N l ‘- a.=a==. mp ...-32 m M 2 38.: ...... 1. = 2 22.8 93$ N w 2 u 2...! at... 53% J N a .88: W... .10... a .8226... R ~22 m? N; 8... u. a h ...e. ”H mm a .6... “on ~om.. .65., 595.. r. 1. a ..__. mu Mm . .2.... mNP Sn. «Pm com n8~ 259.. ,2. 662 soo.. 9. mm m : 5.... m u ~ _ .«.c._ P Acocv Accev o. a. A. o. m. e. a. N. P. o. a a h o m o n ~ P 9.. 2.8 . .3 .3 as: .2325: A.V ...d_u..1_..d. .ucoewo :— mmacozu Co muumuum ”mm\~ma_ - .... cam LoC ..mucco» c. m0==°s__._u«.-_..d‘ .Aummv :o_uc~___u= xu_unqcu - ~.~§ cam ”Amocco» c. mucaos<. =o_m ucmsn.zm .m 217 OH H I. 0 H n ' I. I. S I. In 1 u 1. J n W I" W! 19 . 9| “- 0 a "U .l. m II m u u so. .... w a u m .m m u m m u m m m u u u a m S I I I J I U I. n... .II N 0 ... M I I 5 up n . . .- a.=u==— mp 22:: 8 NF 3.9.: m L 2 2 2...... H W or u :2... m? 034. 0 0 o 7.3.: H J 0 339.28: ~90 09.: m m s :3: :2 a... n n o :3: ... 3 m o ' ..... o~h New ”u “w . .g.... o mm m __ ...... mNP an, arm can mo~ an» ac. goo soo.. mm ~o°.~ who 595.9 mm .; ~ _ 3:... 0 p 2 2 2 p m. p p p P . See :58 e a n ~ 9 o o o s o m . n ~ p an» ‘au § Aav Aqv .~_.~u .o__usao.. A_. .__.~u._-_.... .co.u:_om ouc— __ ou_cop mc_ugou n—.mu cam Lo» Ammcco» :. mucaoe\m2=oncmz ooov m o N N _ m _ m _ o _ w ucmsmo co co ovN omN mNm ONe ome m.N w .e ONN cm? on mm mmN om_ om. o.< o .m ONS omm oFN om. mo_ om om o.m m .N ome com oeN om, mm on m o.m < .P u a m. a o m < 882 N A. m m n m N F mxu «pmou wcmscm_mm< awe: - mcmpcwu _mc=p_=uwcm< mcmpcmu cwmama .mpmoo “coacmwmm< new cox“; .— w~amh 225 Agricultural Center (Potential Repair Center) Road Network Mileage (50 km) Regional Boundary lllv‘ll ' I C) Map 1. Machinery Repair Center Hypothetical Problem: Routes and Mileage. 226 MIN 5 Y001 + 3 Y002 + i Y003 + 2.5 YOGA + 5 10010001 + 30 10020001 + 180 10030001 + #50 10060001 + 30 10010002 + 30 10020002 + 15 10030002 + £20 100k0002 + 75 10010003 + 105 10020003 + 255 10030003 + 525 100h0003 + 180 10010006 + 150 10020005 + 75 1003000“ + 270 1006000h + 2&0 10010005 + 210 10020005 + 60 10030005 + 260 XOONOOOS + 360 10010006 + 330 10020006 + 180 10030006 + 90 100h0006 + 550 10010007 + 620 10020007 + 270 10030007 + 60 100k0007 SUBJECT TO 2) ' 8 Y001 + 10010001 + 10010002 + 10010003 + 1001000h + 10010005 + 10010006 + 10010007 (I 0 3) - 8 Y002 + 10020001 + 10020002 + 10020003 + 1002000b + 10020005 + 10020006 + 10020007 (I 0 h) - 8 Y003 + 10030001 + 10030002 + 10030003 + 1003000‘ + 10030005 + 10030006 + 10030007 <- 0 5) - 8 YOOL + 100N0001 + 10080002 + 100h0003 + 1006000h + 100h0005 + 100h0006 + 10050007 (I O 6) 10010001 + 10020001 + 10030001 + xoonoom - 0.8 7) 10010002 + 10020002 + 10030002 + 100b0002 - 1 8) 10010003 + 10020003 + 10030003 + 100h0003 I 1.8 9) 10010001 + 10020001 + 1003000h + 100b000h - 1.8 10) 10010005 + 10020005 + 10030005 + 100h0005 I 1.2 11) 10010006 + 10020006 + 10030006 + 100h0006 - 2 12) 10010007 + 10020007 + 10030007 + 100h0007 - 0.9 END SUB Y001 1.00 SUB Y002 1.00 SUB Y003 1.00 SUB YOOL 1.00 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 h 1 2‘3 1 1 2 3 h 1 2 3 h 1 2 3 k 1 2 3 h 1 Y Y Y Y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 h 1 1 l 1 2 2 2 2 3 3 3 3 h 8 h k 5 5 5 5 6 6 6 6 7 1: 5 3 B A 5 I C C B B B C I C C C C C I C C C B C C C C B C 2:-8 ' I ' I ' I ' I ' I ' I ' I 3: l-8l I I] I I 1 I I ‘l I I] I I 1 I I ‘0 I I ‘0: -8 ' I ' I ' I ' I ' I ' I ' 5: -8 "I ' I ' I ' I ' I ' I 6: I I I ’0] 1 ‘1 I I I I I I I I I I I I I I 7: I I 1 ' 1 ‘ I I I I I I 8: I I I 1 1 ' 1 I I I I 9: I I I I I I I I I I I ‘0' ' ‘0 I I I I I I '0: I I I I I I 1 1 1 ‘ I I 11: ' ' ' ' . . ' I I I I ‘2: I I I I I I I I I ‘I NOODNOO)‘ NOOOWOO)‘ NOOO?OO¥ WONG‘U‘IPWN—o O —o —c N -I a. on C C B HIN 1 d ‘ IIIIIIIAAA 4ND>>-I 1'1 < 227 LP OPTINUH FOUND AT STEP NEH INTEGER SOLUTION AT BRANCH 1) VARIABLE Y001 Y002 YOO3 YOOb 10010001 10020001 10030001 100h0001 10010002 10020002 10030002 100h0002 10010003 10020003 10030003 100b0003 10010001 1002000h 1003000B 100b000k 10010005 10020005 10030005 100b0005 10010006 10020006 10030006 100AOOO6 10010007 10020007 10030007 100N0007 RON OBJECTIVE FUNCTION VALUE 606.500000 VALUE 1.000000 .000000 1.000000 1.000000 .800000 .000000 .000000 .000000 .000000 .000000 1.000000 .000000 1.800000 .000000 .000000 .000000 .000000 .000000 1.800000 .000000 .000000 .000000 1.200000 .000000 .000000 .000000 .000000 2.000000 10 3 PIVOT REDUCED COST 5.000000 3.000000 h.OOOOOO 2.500000 .OOOOOO 25.000000 175.000000 Ah5.000000 15.000000 15.000000 .OOOOOO 305.000000 .000000 30.000000 180.000000 650.000000 105.000000 75.000000 .OOOOOO 195.000000 180.000000 150.000000 .000000 180.000000 270.000000 250.000000 90.000000 .000000 390.000000 360.000000 210.000000 0 ”0°00 DUAL PRICES 13 228 2) 5.Iooooo .000000 3) .000000 .000000 A) 5.000000 .000000 5) 5.100000 .000000 6) .000000 -5.000000 7) .000000 -15.000000 8) .000000 -75.000000 9) .000000 -75.oooooo I0) .oooooo -60.oooooo II) .oooooo -9o.oooooo 12) .000000 -60.oooooo N0. ITERATIONS- I3 BRANCHES- 3 DETERH.- 1.000E o 13) Y001 >- 0 END INTEGER-VARIABLEs- h BOUND 0N OPTIMUM: 602.9063 ENUMERITION COMPLETE. BRANCHES- 3 PIVOTs- I3 LAST INTEGER SOLUTION IS THE BEST FOUND 229 KEY C) Agricultural Center C) Designated Workshop Facility Location __ILRouting of Repair Jobs (000) Map 2. Machinery Unit Location Plan Showing Repair Volume Routings REFERENCES REFERENCES Abernathy, N. J., and J. C. Hershey. 1972. A spatial- allocation model for regional health-services planning. Opns. Res. 22:400-410. Ademosun, O. C., and D. H. Noble. 1982. A location- allocation model for the Nigerian oil palm industry assuming an infinite location space. Agricultural Sys. 9:15-27. Ahmed, S. 1959. Cashew nut, a dollar earning crop. Punjab Fruit J. 22(79):17-18. Akinc, U., and B. M. Khumawala. 1977. An efficient branch and bound algorithm for the capacitated warehouse loca- tion problem. Management Sci. 23:585-594. Albuquerque, S. D.; M. V. Hassan, and K. R. Shetty. 1958. 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