text: UFDC Home myUFDC Home | Help | RSS Group Title: dynamic, mixed integer location model applied to Florida citrus packinghouses Title: A dynamic, mixed integer location model applied to Florida citrus packinghouses CITATION THUMBNAILS PAGE IMAGE ZOOMABLE Full Citation STANDARD VIEW MARC VIEW Permanent Link: http://ufdc.ufl.edu/UF00102861/00001 Material Information Title: A dynamic, mixed integer location model applied to Florida citrus packinghouses Physical Description: Book Language: English Creator: Machado, Virgílio Azuíl Páscoa, 1950- Copyright Date: 1978 Record Information Bibliographic ID: UF00102861 Volume ID: VID00001 Source Institution: University of Florida Holding Location: University of Florida Rights Management: All rights reserved by the source institution and holding location. Resource Identifier: oclc - 04798904ltuf - AAB6225 Full Text A DYNAMIC, MIXED INTEGER LOCATION MODEL APPLIED TO FLORIDA CITRUS PACKINGHOUSES By VIRGILIO AZUIL PASCOA MACHADG A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1978 Ii~ Copyright by VIRGfLIO AZUfL PASCOA MACHADO 1978 TO MY BELOVED FATHER ACKNOWLEDGEMENTS The author wishes to thank the helpful assistance of Dr. D. S. Tilley, his supervisor and friend, for all the guidance, encouragement and support. Thanks also to Dr. L. Polopolus, whose critical revisions much improved this dissertation; to Dr. R. W. Ward and Dr. R. S. Leavenworth for their assistance. Special thanks to Dr. B. E. Melton, whose expertise with the computer packages opened doors to their effective utilization; to Nancy Melton who assisted in the execution of all the computer work; and to the Northeast Regional Data Center for providing the computational facilities. Nancy Waters was kind enough to dedicate herself to the manuscript typing. Molly Owen was superb in typing extensive early versions of this document and was helped here and there by Vera Anderson and Debbie Mixon. Susan Howard did the excellent art work. The author's friends were always prodigal in dispensing their love and attention and made "saudade" almost an unknown word. Special recognition is given herein to those men of great vision of the governments of the U.S.A. and Portugal who made the possibility of this educational experience a reality. There is a certain pride in having been found deserving of the financial support of the Florida Department of Citrus. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . . . . . . . . . . . iv LIST OF TABLES . . . . . . . . ... . . . .viii LIST OF FIGURES. . . . . . . . . ... . . . xii KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . ... .xiii ABSTRACT . . . . . . . . . . . . . . xv CHAPTER I PROBLEM DEFINITION . . . . . . . . ... .. 1 I-1. Problem Statement . . . . . . . . .. 1 I-2. Propositions and Objectives . . . . . . 4 II THE CHANGING FLORIDA CITRUS INDUSTRY . . . . . . 6 II-1. Florida Citrus Production . . . . . . . 6 11-2. Florida Grapefruit Production . . . . ... 16 11-3. Fresh Grapefruit Production . . . . . .. 22 11-4. Florida Citrus Packing. . . . . . . .. 28 11-5. Florida Citrus Processing . . . . . ... 34 11-6. Marketing Costs . . . . . . . . ... 34 11-7. Florida Citrus Markets. . . . . . . .. 39 III GENERAL PLANT LOCATION MODELS . . . . . . ... 40 IV SOLUTION PROCEDURES. . . . . . . . .... . 52 IV-1. Introduction. . . . . . . . .... . 52 IV-2. Linear Programming Procedures . . . . ... 53 IV-3. Transportation Model Procedures . . . . ... 55 IV-4. Combinatorial Procedures . . .. . . . 56 IV-5. Transshipment Procedures . . . . . . . 59 IV-6. Nonlinear Prograrning Procedures . . . ... 60 IV-7. Mixed-Integer Procedures. . . . . . . ... 61 TABLE OF CONTENTS--Continued CHAPTER V LOCATION MODEL FOR THE CITRUS INDUSTRY . . . . .. 65 V-I. Model Formulation . . . . . . . . . 65 V-2. Model Discussion . . . . .... . . . 70 V-2.1. Cost Minimization. . . . . .. . . 70 V-2.2. Size Limits and Economies of Scale. . . 71 V-2.3. Existing Packinghouses Utilization, Closing and Expansion . . . . . . .. 71 V-3. Dynamic Analysis . . . . .. . . . . 72 VI ESTIMATION OF MODEL PARAMETERS AND COEFFICIENTS. . . . 78 VI-1. Production and Demand Constraints . . . . .. 78 VI-1.1. Number of Products . . . . . .. 78 VI-1.2. Fruit Origins . . . . . . . 79 VI-1.3. Fruit Supply . . . . . . . .. 80 VI-1.4. Shipments to Markets . . . . . .. 85 VI-1.5. Processing Plant Capacity.. . . . ... 90 VI-2. Capacity Constraints and Cost Coefficients. . ... 97 VI-2.1. Description of Assembly Operations. . 97 VI-2.2. Assembly Cost . . . . . . .. 98 VT-2.3. Eliminations Transport Cost. . . . ... 99 VI-2.4. Distribution Costs . . . . . .. 100 VI-2.5. Packinghouse Locations . . . . .. 100 VI-2.6. Capacity Constraints . . . . . .. 104 VI-2.7. Packing Cost . . . . . . . .. 105 VI-2.8. Cost Coefficients . . . . . .. 117 VII EMPIRICAL RESULTS. . . . . . . . . . ... 123 VII-1. Probationary Exercises . . . . . . .. 123 VII-1.1. Model Simplification . . . . .. 123 VII-1.2. First Auxiliary Model . . . . .. 126 VII-1.3. Second Auxiliary Model .. . ..... 136 VII-1.4. Third Auxiliary Model . . . . .. 137 VII-1.5. Fourth Auxiliary Model . . . . .. 137 TABLE OF CONTENTS--Continued Page CHAPTER VII VII-2. VII-3. VII-4. VII-5. VIII SUMMARY VIII-1. VIII-2. VIII-3. Static Model Optimal Solution, 1981-82 Season . Dynamic Model Solution, 1975-76 through 1881-82 Duality Analysis and Secondary Computer Runs. . VII-4.1. Duality Analysis. . . . . . VII-4.2. "What If . .?" Analysis . . . VII-4.3. Sensitivity Analysis. . . . . VII-4.4. Trade-Off Analysis. . . . . . VII-4.5. Priority Analysis . . . . . Computational Experience. . . AND CONCLUSIONS . . . . . Summary . . . . . . Conclusions . . . . . Suggestions for Further Research APPENDICES APPENDIX A PRODUCTION AND SHIPMENTS DATA . . . APPENDIX B COST DATA . . . . . . . . APPENDIX C LOCATION MODEL REVIEW . . . . . APPENDIX D MIP CONTROL PROGRAM TO OBTAIN SECOND-BEST LIST OF REFERENCES . . . . . . . . . . SOLUTIONS BIOGRAPHICAL SKETCH . . . . . . . . . . . . vii S139 143 S 150 S 150 S 151 . 155 S 161 S164 168 171 171 173 178 182 201 209 231 232 249 LIST OF TABLES Table Page 1 Citrus Production for 1975-76 Crop Year . . . . . 7 2 Number of Carlots Received by 41 U.S. and Canadian Cities, 1975-76 Season. . . . . . . . . .. . 7 3 Nonbearing Citrus Acreage and Trees, and Trees Less than Twenty Years Old, County Percentage in 1965 and 1975. . 9 4 Counties with Gains in Citrus Acreage Between 1973 and 1975 12 5 County Production Estimated Rates of Growth, 1963-64 through 1975-76 Seasons . . . . . . . ... 14 6 County Grapefruit Acreage in 1975, Percentage of Nonbearing Grapefruit Acreage and Trees, and Percentage of Trees Less than Twenty Years Old in 1965 and 1975 ...... 17 7 Counties with Gains in Grapefruit Acreage Between 1973 and 1975. . . . . . . . . ... ..... 18 8 County Grapefruit Production Estimated Rate of Growth . 20 9 Number of Florida Grapefruit Trees by Type and Year Set, Percent of Trees Set by Year, January 1976 ...... 21 10 Average Box Yield Estimates and Percent Increase in Yields for Selected Tree Ages of Florida Grapefruit. . . ... 23 11 Fresh Florida Citrus Production, 1970-71 through 1975-76 Seasons . . . . . . . . ... ...... 25 12 Total Shipments, Number of Packinghouses and Average Ship- ments per Packinghouse, by County, 1955-56 and 1975-76 Seasons . . . . . . . . ... . . . 29 13 County Shipments Estimated Rate of Growth . . . ... 31 14 County Grapefruit Shipments Estimated Rate of Growth . 33 viii LIST OF TABLES--Continued Table Page 15 Agricultural Location Models . . . . . . ... 42 16 Main Features of Selected Location Models . . . ... 49 17 Box Yield Estimates by Age of Tree, Average of 1966-67 through 1975-76 Seasons. . . . . . . . .. 83 18 Percentage of Florida Citrus Used as Fresh Fruit, by Variety, 1973-74 through 1975-76 Seasons . . . ... 84 19 Fruit Supply, Indian River Origins (1=13), 1975-76 through 1981-82 Seasons. . . . . . . . . ... ... 86 20 Population of U.S. Market Regions and Canada, 1975-76 through 1981-82 Seasons . . . . . . . ... 92 21 Indian River Fruit Demand, U.S., Canadian Markets and Exports (K=8), 1975-76 through 1981-82 Seasons . . .. 93 22 Existing Citrus Cannery Locations in the East Coast Region of Florida, 1975-76 . . . . . . . . ... 96 23 Monthly Truck Rates for Oranges and Grapefruit, 1977 . .. 101 24 Existing Packinghouse Locations (J=10) and Volume Handled, 1975-76 through 1981-82 Seasons . . . . . ... 103 25 Average Packing Cost Function Variable Definitions and Hypothesized Relationships . . . . . . ... 106 26 Regression Results for Average Packing Cost Model . .. .108 27 Sample Average and Assumed Values of Selected Cost Function Variables. . . . . . . . . ... . . .109 28 Per Unit (v.) and "Fixed" Cost (f.) of Existing and New Packinghouses, and Operating Volume Range, per Packing- house . . . . . . . . .. . . . . . 116 29 Assembly Distances . . . . . . . . ... .. .119 30 Distribution Distances to Ports . . . . . ... 120 LIST OF TABLES--Continued Table Page 31 Distribution Distances to Processors . . . . . .. 121 32 Upper Bounds on xj. . . . ....... . ... . . 127 33 First Auxiliary Model Solution Costs, 1975076 Season . . 130 34 First Auxiliary Model Solution Flows, 1975-76 Season . . 131 35 First Auxiliary Model Transportation Costs, 1975-76 through 1981-82 Seasons . . . . . . .... 134 36 First Auxiliary Model, Routes Changed Between 1975-76 and 1981-82 Alternative Solutions . . . . . .... 135 37 Fourth Auxiliary Model Solution Summary, 1981-82 Season .140 38 Optimal Location, Number and Size of Citrus Packinghouses and Optimal Fruit Flows, Indian River Region, 1981-82 Season . . . .. . . . . . . 141 39 Total Cost and Packinghouse Configuration of Minimum Cost Solution to Dynamic Packinghouse Location Problem. . . 149 40 "Shadow" Prices of Fruit at the Origins, Indian River, 1981- 82 Season . . . . . . . .... ..... 152 41 "Shadow Prices of Fruit at the Markets and Ports, Indian River, 1981-82 Season. . . . . . . . . ... 153 42 Comparative Advantage of Packinghouse Locations and Sizes 154 43 Supply, Supply Range and "Shadow" Prices for the Short-Run Optimum, 1975-76 Season . . . . . . . .. 157 44 Demand, Demand Range and "Shadow" Prices for the Short-Run Optimum, 1975-76 Season . . . . . . . .. 159 45 Packinghouse Optimum Short-Run Operating Volume and Volume Range Over Which "Shadow" Prices Apply, 1975-76 Season 160 46 Total Cost and Integer Variables Values of Six Solutions, 1976-77 Season . . . . . . . . .. . . 162 LIST OF TABLES--Continued Table Page 47 Fourth Auxiliary Model Solution Summary, 1975-76 through 1977-78 Seasons . . . . . . . . ... .. . 165 48 Second Auxiliary Model Capacity Constraint Dual Activities 1975-76 Season . . . . . . . . ... .. . 167 49 Number of Variables, Constraints and Solution Time for Different Models . . . . . . . .... . 169 LIST OF FIGURES Figure Page 1 State of Florida, Interior and Indian River Citrus Marketing Districts . . . . . . . . .. 11 2 Grapefruit market shares, selected countries, 1975-76 26 3 World production of grapefruit . . . . . . 27 4 The ct best solutions to the static packinghouse location problem by season . . . . . . . 74 5 The orange production origins for fresh fruit assembly in Florida. . . . . . . . . ... . .81 6 Florida citrus U.S. market regions . . . . ... .91 7 Long-run packing cost functions for the Indian River . 111 8 Existing and optimum number, size and location of citrus packinghouses, Indian River, 1975-76 and 1981-82 Seasons . . . . . . . . ... .. .. 142 9 The 107 best and current solutions to the static packing- house location problem by season. . . . . ... 144 10 Minimum cost solution to the dynamic packinghouse location problem . . . . . . . . . . . .148 11 Total, variable and fixed costs of solutions 1 through 6 (Table 27), as a function of number of packinghouses of any size, 1976-77 season . . . . . .. .163 KEY TO SYMBOLS AND ABBREVIATIONS APC = Average packing cost E = Packout percentage F = Fixed cost H = Number of products I = Number of origins of products J = Number of plant locations K = Number of markets for final products L = Number of processing plants + K LTC = Long-run total cost M2 = 1 if technology level 2,0 otherwise M3 = 1 if technology level 3.0 otherwise N = Number of plants 0 = Natural logarithm of the proportion of oranges packed to grapefruit packed PL = Location, 1 for Indian River, 0 otherwise PO = Natural logarithum of percent of fruit delivered that is acceptable for fresh shipment xili PK = Percent of fruit that is not packed in standard 4/5 bushel cartons q = Quantity shipped and/or processed Q = capacity or total quantity Q = Large number S = Mean weekly shipments divided by the standard deviation of weekly shipments SL = Natural logarithm of number of weeks in which packinghouse packs fruit STC = Short-run total cost T = Time u = Average variable unit cost v = Average variable unit cost x = Integer number of plants open at a site y = Fraction of Q y. = Year, 1 for 1974, 0 otherwise 4 y. = Year, 1 for 1975, 0 otherwise Z = Objective function 8 = Constant, growth rate xiv Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A DYNAMIC, MIXED INTEGER LOCATION MODEL APPLIED TO FLORIDA CITRUS PACKINGHOUSES By VIRGILIO AZUIL PASCOA MACHADO December, 1978 Chairman: Dr. Leo Polopolus Major Department: Food and Resource Economics Southward movement in production location, the increasing importance of grapefruit as a fresh variety, increases in size of packinghouses and cost inflation have an effect on the efficiency of the Florida fresh citrus industry, and, therefore, on producer returns and consumer prices. An improvement in the industry's productive efficiency is achieved by optimizing the number, location, size, supply and market areas of the Indian River citrus packinghouses. Two models were developed and used. One is a location-allocation mixed-integer model with fixed costs, demand, supply and capacity con- straints. The integer variables are the number of packinghouses of the same size at each location. The other model used dynamic programming to find the minimum cost path through a matrix of static solutions, taking into consideration the cost of moving from one system configuration to another. The static solutions are obtained with the mixed-integer model for each season, and ranked by cost. Included in the model are oranges and grapefruit. Townships in the Indian River are aggregated into 13 origins of fruit that supply fruit to the packinghouses. Supply is determined by the number and age of trees and yield per tree, and estimated for the seasons of 1975-76 through 1981-82. Export quantities being shipped out of the major Florida ports are estimated together with quantities being shipped to domestic and Canadian markets. Assembly, packing and distribution costs are estimated in order to compute the model coefficients. Locations with existing packinghouses are Cocoa, Ft. Pierce, Titusville and Vero Beach. Economies of scale are included by considering packinghouses of different sizes on those locations. Potential packinghouse locations are Jupiter, Melbourne and Stuart. Capacity constraints are then defined as the capacity percentage that will insure packing all fruit during the peak weeks and a minimum degree of operating efficiency. A long-run packing cost function is statistically estimated by using a variance component model on cross sectional, time-series accounting data. Solutions to the model are obtained using IBM's MPSX (1972) and MIP (1973) computer programs. The results indicate that there should be fewer and larger packinghouses in the region. The solution Ls not sensitive to use of nonoptimal assembly and distribution routes, and to changes in transportation cost. To adjust to the proposed future efficient xvi marketing organization, the industry should phase out small plants operating at high unit costs. Large plants are desirable if they can be operated at low unit costs. xvii CHAPTER I PROBLEM DEFINITION I-1. Problem Statement Changes in the location of production, product mix, technology and costs have an adverse effect on the efficiency of the Florida citrus industry, and, consequently, on producer profits and consumer prices. Structural adjustments to these changes are needed. An improvement in productive efficiency can be achieved by optimizing the number, location, size, supply and market areas of citrus packinghouses in Florida. This will contribute to reduced distribution costs,accomnodating future out- put expansions without deterioration of market prices and grower returns. The adjustment of the present industry configuration and product flow to a lower cost level can be facilitated and encouraged by providing guidelines consistent with the minimization of industry costs over a planning horizon. The Florida citrus industry is quite large at the state, national and international level, in terms of production and market shares (Florida Crop and Livestock Reporting Service, 1976a). It employs con- siderable manpower and capital (Florida Department of Agriculture and Consumer Services, 1977). Successive commercial citrus tree inventories show that removal of citrus acreage and new plantings have been shifting production within the state (Florida Crop and Livestock Reporting Service, 1976a). Citrus production has been increasing and more than doubled - in the past twenty-five years, but production growth is substantially higher in south Florida counties (Florida Crop and Livestock Reporting Service, 1967 through 1976a). All counties with high growth rates have percentages of trees less than twenty years old. Counties with signif- icant citrus commercial acreage and high percentages of young trees are all south of Lakeland. This was not the case ten years earlier, in 1965, when more counties had high percentages of young trees. In the same ten years the state percentage of non-bearing trees had decreased. Counties with high percentages of non-bearing trees are also now limited to some south of Lakeland. Bearing acreage in Florida has been decreasing since 1970-71, although bearing acreage of seedless grapefruit continues to increase. Despite its small percentage of total crop utilization, the fresh citrus market is nevertheless an important source of potential profit. For grapefruit, tangelos and tangerines most of a grower's profit comes from fruit marketed as fresh (Growers Administrative Committee, 1976). Grapefruit is the most important of these. Expansion of foreign markets for fresh grapefruit has caused its fresh shipments to increase faster than production which is also expected to increase rapidly in the next five years. World production of grapefruit is dominated by the U.S. but other countries have been progressively increasing their shares in the international market (Myers, 1977). Southern counties in Florida have experienced high grapefruit produc- tion growth, increasing their share of the state output (Florida Crop and Livestock Reporting Service, 1967 through 1976a). Counties with high growth rates are expected to continue growing faster due to maturity of their large percentage of young grapefruit trees (Fairchild, 1977). Fresh shipments of grapefruit have been increasing faster in the East Coast and North Interior regions, presenting a marketing challenge which must be met (Florida Department of Agriculture and Consumer Services, 1964 through 1976). Large fresh fruit shipments were made in 1976 from packinghouses in the same counties where the average shipment per packinghouse is high and there is a large number of packers (Florida Department of Agricul- ture and Consumer Services, 1976). Not all these counties have been experiencing a comparable rate of growth in their shipments. Over the past twenty years the average size of packinghouses has increased. Volume of shipments over the season varies widely from as little as 30 boxes to more than 3 million 4/5 bushel boxes. Moreover, volume of shipments per packinghouse per county, varies from 8 to 800 thousand boxes packed. Capacity utilization of packinghouses has been reportedly low (Kilmer and Tilley, 1978). Costs have been increasing in all phases of the industry (Hooks and Spurlock, 1977 and Hooks and Kilmer, 1977 a and b) with reductions in returns to growers expected to continue under present marketing conditions (Myers, 1977). Increasing capacity utilization is an apparent short-run key to cost reductions. Furthermore, packinghouses that are capable of coordinating grove production and harvesting such that they receive a steady volume of high quality fruit are likely to have lower costs (Kilmer and Tilley, 1978). 1-2. Propositions and Objectives For the above problem the following propositions are formulated: a) An improved Florida citrus fruit assembly, packing and distribution system can be proposed which, if adopted, would minimize costs. b) The optimal locations and sizes of new packinghouses to accommo- date future output expansion, costs and other expected changes can be determined. c) If shipping routes and plants excluded from the efficient solution are not used, then substantial savings will be made by the industry. d) Knowledge of the most minimizing marketing network will help future industry decision making and provide a firm management tool. a) A least-cost plan for the adjustment of the industry to an optimal organization can be provided to management. The primary objective of this study is to determine a marketing network that will improve the productive efficiency of the Florida fresh citrus industry. Specifically the objectives are to: a) Determine the cost minimizing number, location, size, supply and market areas of citrus packinghouses in Florida, and assembly and distribution patterns. b) Determine the sensitivity of a minimum cost solution to changes in supply and other parameters and costs. 5 c) Present the citrus industry with a criterion and technique that can be used to provide guidelines for the industry or a plan of action for citrus firms to adjust towards a more efficient industry marketing organization. CHAPTER II THE CHANGING FLORIDA CITRUS INDUSTRY II-1. Florida Citrus Production World grapefruit production is dominated by the United States, which is also the largest producer of oranges and tangerines. Florida is a major factor in domestic citrus production and markets (Tables 1 and 2). In the 1975-76 crop year, 852,369 acres containing 70.547 million trees bore fruit in Florida (Florida Crop and Livestock Reporting Service, 1976b). The citrus industry accounted for 27 percent of Florida farm cash receipts in 1975. Approximately 54,000 persons are employed annually by the industry: 33,000 in field operations, 9,000 in processing plants and 12,000 in packing plants (Florida Department of Agriculture and Consumer Services, 1977). Fresh citrus was shipped by 170 packinghouses and processed in 52 registered citrus canning or concentrate plants during the 1975-76 season (Florida Department of Agriculture and Con- sumer Services, 1976). In 1975, commercial citrus acreage had decreased 0.7 percent since the first commercial citrus inventory was conducted in 1965. The new survey shows a removal of 40,518 acres between 1973 and 1975 surveys, more than offsetting the 28,789 acres of new trees set in 1974 and 1975, a net decline of 1.4 percent from the 864,098 acres in December, 1973 (Florida Crop and Livestock Reporting Service, 1976a). 6 Table 1 .--Citrus Production for 1975-76 Crop Year Florida United States World 1,000 Percent of 1,000 Percent of 1,000 Percent of Metric World Metric World Metric World Tons Production Tons Production Tons Production Grapefruit 1,803 50.0 2,574 71.0 3,624 100.0 Oranges and Tangerines 7,402 23.7 9,908 31.7 31,267 100.0 Source: Florida Crop and Livestock Reporting Service, 1976a. Table 2.--Number of Carlots Received by 41 U.S. and Canadian Cities, 1975-76 Season Florida Grapefruit: Oranges: Tangerines: U.S. Canada U.S. Canada U.S. Canada Cars 16,401 2,451 11,712 1,307 3,894 499 Percent 68.7 79.0 31.2 19.2 73.2 80.6 Cars 23,866 3,102 37,533 6,792 5,319 619 Total Percent 100.0 100.0 100.0 100.0 100.0 100.0 Source: Florida Crop and Livestock Reporting Service, 1976a. Half of the acreage that was planted in the calendar years 1974 and 1975 was reset planting. Land that had never before been used for citrus accounted for 25 percent of the new acreage. Of the acreage that was removed from commercial production, approximately 12.5 percent was taken out for commercial development. The percentage of nonbearing acres and trees, and the percentage of trees less than twenty years old, all decreased from 1965 to 1975 (Table 3). In 1975, Polk and Lake counties had the largest commercial citrus acreage (Figure 1). Acreage gains between the surveys of 1973 and 1975 were recorded in a few counties, i.e., Hendry, Indian River and St. Lucie (Table 4). All types of citrus showed net declines in commercial acreage be- tween 1973 and 1975, except seedless grapefruit. White seedless grape- fruit bearing acreage increased by 2,047 acres and pink seedless by 1,110 acres. Nonbearing acreage, however, increased by 1,641 and 2,718 acres bring the gains in total acreage to 3,688 and 3,828 acres for white and pink seedless grapefruit, respectively. Indian River and St. Luice are the counties with the largest acreage and number of seedless grapefruit trees (Florida Crop and Livestock Reporting Service, 1976b). St. Lucie and part of Indian River counties are in the Indian River District (Figure 1). This region is one of two Marketing Agreement Regulation Areas: Interior and Indian River (Code of Fegeral Regulations, 1975). The Marketing Agreements regulate size, grades, containers, shipping holidays, and prorated shipments. Under Marketing Agreement Table 3.--Nonbearing Citrus Acreage and Trees, and Trees Less than Twenty Years Old, County Percentage in 1965 and 1975 Nonbearing Nonbearing Trees Less than Acres Trees Twenty Years Old 1965 1975 1965 1975 1965 1975 - - - - - Percent - - - - - - Indian River:b Brevard 28.5 5.87 34.9 6.05 67.0 60.9 Indian River 45.5 14.9 48.4 16.7 74.9 82.2 Martin 70.6 10.8 73.1 10.2 97.6 97.4 Palm Beach 75.4 1.61 80.9 1.64 89.6 97.8 St. Lucie 44.4 8.79 49.7 9.40 80.7 78.4 Volusia 14.9 2.07 15.0 2.48 45.8 35.2 Interior: Alachua 21.1 0 25.0 0 66.7 45.6 Broward 16.7 0.482 21.9 0.105 60.6 33.7 Charlotte 68.5 7.32 71.1 7.62 93.6 94.9 Citrus 27.8 0.508 28.6 0.518 75.0 50.4 Collier 75.8 5.10 79.7 4.77 99.2 99.8 Dade 21.2 16.5 26.9 16.6 89.5 89.3 De Soto 40.4 25.2 43.8 31.1 81.2 89.9 Flagler 17.2 13.0 26.3 17.4 94.7 100.0 Glades 74.9 2.91 72.1 2.61 100.0 100.0 Hardee 31.5 6.55 34.4 7.28 85.6 76.3 Hendry 90.8 20.5 93.0 21.0 98.3 99.2 Hernando 31.6 1.89 32.0 1.92 83.2 68.0 Highlands 23.0 5.25 28.1 5.67 69.6 59.0 Hillsborough 31.4 4.73 34.2 5.15 79.7 63.8 Lake 23.8 2.52 26.7 2.75 75.5 50.7 Lee 20.8 8.81 21.1 10.1 63.3 92.3 Manatee 34.6 4.33 35.0 4.72 83.4 75.6 Marion 21.8 1.02 23.9 1.20 59.7 45.5 Okeechobee 72.3 11.8 70.1 12.3 95.7 97.0 Orange 14.3 2.61 15.6 2.62 51.1 35.3 Osceola 28.4 2.97 30.4 3.02 81.7 61.9 Pasco 28.2 2.59 29.2 2.52 76.8 60.3 Pinellas 11.8 0.849 12.4 0.814 29.2 24.6 Polk 14.5 3.20 16.8 3.51 48.1 42.6 Putnam 25.8 1.12 24.9 1.15 49.9 35.0 St. Johns 13.6 0 16.7 0 50.0 25.0 Sarasota 27.5 16.0 32.1 15.3 56.6 65.5 Table 3.--Continued Nonbearing Nonbearing Trees Less than Acres Trees Twenty Years Old 1965 1975 1965 1975 1965 1975 - - - - - -Percent - - - - - - Seminole 18.5 1.46 18.4 1.39 62.9 46.3 Sumter 21.9 6.99 22.3 6.14 81.2 75.0 State 29.2 6.59 33.1 7.80 71.0 66.0 aSet in 1945 and 1955 or later, respectively. with the exception of St. Lucie these countries have some townships in the Interior Region (Figure 1). Source: Florida Crop and Livestock Reporting Service, 1966 and 1976b. Indian River Marketing District 164,120 Acres All Citrus Interior Marketing District 688,249 All Citrus Figure 1. State of Florida, Interior and Indian Citrus Marketing Districts / ; r 'salt , Source: Florida Crop and Livestock Reporting Service, 1976b. Table 4 .--Counties with Gains in Citrus Acreage Between 1973 and 1975 County 1973 1975 Gain - -- ---Acres -- --- Indian River:a Indian River 52,261 56,206 3,945 St. Lucie 73,036 73,812 876 Interior: Dade 4,340 4,536 196 Flagler 182 192 10 Hendry 24,225 25,944 1,719 Marion 11,223 11,327 104 Okeechobee 4,087 4,162 75 St. Johns 125 127 2 Sarasota 1,449 1,661 212 Sumter 1,677 1,760 83 aIndian River county has some townships in the Interior Region (Figure 1). Source: Florida Crop and Livestock Reporting Service, 1974 and 1976b. regulations, grower committees make recommendations for marketing policy at the beginning of the season and for changes in regulations during the season. Florida citrus production has more than doubled in the past twenty- five years. Counties with the largest production are in the Interior and East Coast of Florida (Appendix Table A-i). Eight counties produced a total of almost 175 million boxes or 70.7 percent of the state total. The highest rates of production growth since the freeze of 1962 occurred in the southern Florida counties of Collier, Glades, Hendry, Martin and Palm Beach (Table 5). Data for the 1976-77 season are available from the Florida Crop and Livestock Reporting Service but are not used in this research because the freeze of January of 1977 rendered these data in- appropriate. Collier, Glades, Hendry, Martin and Palm Beach counties had an estimated production in the 1965-66 season of only 1,553 thousand boxes of fruit or 1.1 percent of the state total. Those same counties ten years later or in the 1975-76 season produced an estimated 21,009 thousand boxes of fruit or 8.5 percent of the state's estimated citrus production (Appendix Table A-l). Production growth is determined fitting an exponential model such as Bt Q = Qoe , to the data using ordinary least squares, and estimating 8. After differentation with respect to time and division by Q one obtains, dQ/dt =- e Q Q0eBt the percentage growth rate. Table 5.--County Production Estimated Rates of Growth, 1963-64 through 1975-76 seasons ^a b a r Percent Indian River: c Brevard 3.82 0.84 Indian River 8.52 0.95 Martin 25.4 0.98 Palm Beach 24.3 0.95 St. Lucie 8.13 0.95 Volusia 3.88 0.66 Interior: Broward 0.0821 0.03 Charlotte 18.0 0.97 Citrus 12.4 0.74 Collier 39.9 0.96 De Soto 10.3 0.91 Glades 25.1 0.96 Hardee 9.45 0.86 Hendry 33.9 0.96 Hernando 11.9 0.76 Highlands 4.98 0.88 Hillsborough 9.91 0.67 Lake 6.75 0.84 Lee 12.5 0.92 Manatee 11.7 0.78 Marion 5.44 0.72 Okeechobee 19.0 0.96 Orange 3.96 0.66 Osceola 7.98 0.86 Pasco 12.4 0.78 Pinellas 2.63 0.24 Polk 3.60 0.68 Putnam 1.64 0.29 Sarasota 7.87 0.67 Seminole 2.85 0.49 Sumter 6.89 0.73 State 6.83 0.90 aEstimate of 8, defined in footnote, page 13. correlation coefficient = T Et InQ Et InQg T Et2 (Zt)2 / T E(InQ 2 (ZnQt CWith the exception of St. Lucie these counties have some townships in the Interior Region (Figure 1). Other counties have seen their share of the state citrus output increased. Counties with an estimated citrus production growing at a higher rate than the state growth rate had their share of the estimated production increased from 29.4 percent to 44.6 percent between the 1965-66 and 1975-76 seasons (Appendix Table A-i). Brevard and Volusia counties in the northern half of the Indian River have much lower growth rates than the counties located in the southern half of the Indian River. There are indications that the shift of citrus production cowards southern and coastal counties is going to continue. All the counties with the highest rate of growth had more than 90 percent of the trees twenty years old or younger, as of January, 1976 (Table 3). Counties with a growth rate greater than average had more than 70 percent of the immature trees, with somewhat lower figures for Citrus, Pasco, and Sarasota counties, all of them, however, with more than 50 percent of immature trees (Table 3). Dade county supplies only limes and lemons, which are not considered in this study although some of this fruit is packed in the summer off-season in packinghouses that pack oranges and grapefruit. A higher percentage of nonbearing trees also predominates in the southern half of Florida. Counties with significant citrus commercial acreage and more than the statewide percentage of nonbearing trees as of January, 1976,were DeSoto, Hendry, Indian River, Lee, Martin, Okeechobee, St. Lucie and Sarasota, all south of Lakeland (Table 3). Those counties had 3,751.7 thousand nonbearing trees or 68 percent of the state total. Counties in 1975 that had a higher percentage of young trees than in 1965 are also all south of Lakeland (Table 3). One finds that in 1965 high percentages of young were not as prevalent in southern Florida, but extended throughout the West Coast and Interior regions. Note that all citrus producing counties in 1965 had a percentage of nonbearing acres and trees above the 1975 state average (Table 3). 11-2. Florida Grapefruit Production Grapefruit represents the most important fresh fruit category and the one whose production growth has been very high in the Indian River region. Grapefruit acreage in Florida amounted to 137,909 acres or 16 percent of total citrus acreage in 1975-76. There was a slight decrease in nonbearing acreage from 16.9 to 14.5 percent and in non- bearing trees from 19.5 to 17.3 percent between 1965 and 1975. The percentage of trees less than twenty years old, however, increased from 46.8 to 60.5 percent in the same period (Table 6). This is in sharp contrast with the situation for all citrus acreage and trees (Table 3). In 1965, Indian River, Lake, Polk and St. Lucie counties had the largest grapefruit acreage. Acreage gains between the surveys of 1973 and 1975 were recorded for most counties, particular Indian River and St. Lucie (Table 7). Grapefruit production has increased only 50 percent in Florida in the past twenty-five years. Production for the five year period Table 6.--County Grapefruit Acreage in 1975, Percentage of Nonbearing Grapefruit Acreage and Trees, and Percentage of Trees Less than Twenty Years Old in 1965 and 1975 Grapefruit Nonbearing Nonbearing Trees Less than Acreage Acres Trees Twenty Years Old 1975 1965 1975 1965 1975 1965 1975 Acres - - - Percent - - - - - - Indian River:a Brevard 3,442 9.13 12.3 8.10 12.6 50.0 34.8 Indian River 30,477 29.7 23.4 30.4 27.4 64.8 79.0 Martin 5,682 57.8 10.1 62.9 9.20 94.3 98.2 Palm Beach 3,405 65.9 0.294 75.9 0.318 83.9 98.8 St. Lucie 20,050 35.8 16.5 39.7 19.0 76.0 75.4 Volusia 681 13.8 9.69 13.2 11.2 26.4 32.0 Interior: Broward 389 3.15 0 2.70 0 81.1 21.8 Charlotte 250 29.5 8.40 38.5 9.52 84.6 74.4 Citrus 38 5.88 5.26 0 8.70 100.0 26.1 Collier 529 76.0 36.7 100.0 42.2 100.0 99.6 De Soto 1,400 9.00 26.2 9.52 26.4 41.3 64.8 Glades 75 0 0 0 0 0 100.0 Hardee 960 6.35 26.4 4.88 30.9 48.8 55.6 Hendry 4,404 95.9 38.8 96.9 40.3 98.5 99.4 Hernando 152 0 4.60 0 5.66 16.7 23.6 Highlands 4,771 4.49 11.5 4.51 13.7 16.2 37.4 Hillsborough 2,593 6.40 15.7 7.11 18.8 38.4 38.9 Lake 12,564 5.39 5.83 5.69 7.70 42.8 25.0 Lee 430 0.373 1.16 0 0.771 31.6 77.9 Manatee 2,149 10.8 16.4 10.8 17.6 41.6 50.3 Marion 407 3.85 2.21 3.13 3.34 18.8 12.7 Okeechobee 978 62.0 29.0 61.9 32.3 95.2 98.3 Orange 2,445 4.42 5.32 4.15 5.16 33.7 22.2 Osceola 1,078 6.67 22.4 7.14 21.4 53.6 45.6 Pasco 1,515 7.86 7.59 8.08 8.32 39.4 32.8 Pinellas 1,176 4.04 1.70 4.42 1.51 16.6 11.7 Polk 25,282 2,28 5.58 2.48 6.41 13.9 17.5 Putnam 39 2.15 0 0 0 0 6.45 Sarasota 323 16.9 17.3 16.0 17.9 40.0 52.0 Seminole 216 0.850 4.17 0 4.64 24.0 9.27 Sumter 9 0 0 0 0 0 0 State 137,909 16.9 14.5 19.5 17.3 46.8 60.5 With the exception of St. Lucie these counties have some townships in the Interior Region (Figure 1). Source: Florida Crop and Livestock Reporting Service, 1966 and 1976b. Table 7 .--Counties with Gains in Grapefruit Acreage Between 1973 and 1975 County 1973 1975 Gain - - - Acres - - - - Indian River:a Brevard 3,207 3,442 235 Indian River 26,536 30,477 3,941 Martin 5,441 5,682 241 St. Lucie 27,652 30,050 2,398 Interior: Citrus 32 38 6 Collier 528 529 1 Hardee 858 960 102 Hendry 3,607 4,404 797 Hillsborough 2,587 2,593 6 Lake 12,486 12,564 78 Manatee 1,910 2,149 239 Marion 398 407 9 Okeechobee 954 978 24 Osceola 895 1,078 183 Pasco 1,443 1,515 72 Sarasota 227 323 46 Sumter 7 9 2 aWith the exception of St. Lucie these counties have some townships in the Interior Region (Figure 1). Source: Florida Crop and Livestock Reporting Service, 1974 and 1976b. 19 1971-72 through 1975-76 oscillated around an average of 46.8 million boxes per season. Counties with the largest production are in the Interior and East Coast of Florida. Five counties in these two regions produced a total of almost 37 million boxes or 74.7 percent of the state total (Appendix Table A-2). Grapefruit production is expected to increase rapidly between the 1976-77 and 1981-82 seasons or at a rate of 2 million boxes per year. Collier, Glades, Hendry, Martin and Palm Beach counties in South Florida also have high rates of grapefruit production growth (Table 8). These counties had an estimated grapefruit production in the 1965-66 season of only 224 thousand boxes or 0.64 percent of the state total. Ten years later or in the 1975-76 season these South Florida counties produced an estimated 4,659 thousand boxes or 9.5 percent of the state estimated grapefruit production (Appendix Table A-2). A high percentage of nonbearing grapefruit trees predominates in the southern half of Florida. Counties with more than the state- wide percentage of nonbearing grapefruit trees as of January, 1976 were in three separate areas in Florida (Table 6), Indian River, Okeechobee, Osceola and St. Lucie counties in the East Coast, Collier and Hendry counties in the South, and De Soto, Hardee, Hills- borough, Manatee and Sarasota counties in the West Coast. These countries had 1,477.2 thousand nonbearing grapefruit trees or 82 per- cent of the state total. The age distribution of trees has an important impact on long- range production expectations. As can be seen in Table 9, 37 percent Table 8.--County Grapefruit Production Estimated Rate of Growth 8a rb 83 r Percent Indian River:c Brevard 0.232 0.08 Indian River 7.26 0.91 Martin 32.7 0.95 Palm Beach 24.9 0.93 St. Lucie 7.78 0.92 Volusia 3.43 0.51 Interior: Broward 1.72 0.42 Charlotte 7.02 0.83 Citrus 45.9 0.92 Collier 45.9 0.92 De Soto 3.48 0.58 Glades 159.0 0.77 Hardee 1.89 0.36 Hendry 38.3 0.94 Hernando 3.78 0.36 Highlands 1.40 0.34 Hillsborough 1.68 0.18 Lake 3.61 0.57 Lee 3.02 0.64 Manatee 5.73 0.48 Marion 2.59 0.40 Okeechobee 16.7 0.90 Orange 2.10 0.34 Osceola 4.33 0.58 Pasco 4.10 0.44 Pinellas -1.54 -0.14 Polk 0.580 0.14 Putnam -4.88 -0.52 Sarasota 4.43 0.47 Seminole -0.726 -0.10 State 4.18 0.86 a Estimate of S, defined in footnote, p. 13. bT Zt InQt Zt *.1nQt bCorrelation coefficient = ____Et _nQt -t Z-lnQt / T Et2 (Zt) / T E(nQ,) (ZlnQt)2 With the exception of St. Lucie these counties have some townships in the Interior Region (Figure 1). Table 9.--Number of Florida Grapefruit Trees by Type and Year Set, Percent of Trees Set by Year, January 1976 Year Set Seedy White Pink Total Percent Seedless Seedless of Total - - - 1,000 trees 1951 or earlier 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 Total 1233.1 2.5 0.7 1.7 0.7 1.0 1.2 3.3 2.0 9.5 6.1 10.6 8.0 7.9 43.4 36.5 66.6 20.3 15.6 14.7 10.2 31.1 10.1 23.4 9.7 1569.9 1553.7 13.6 11.6 17.7 14.2 7.9 6.5 69.9 49.8 96.2 99.2 63.3 128.5 232.9 650.0 609.9 528.9 220.7 131.4 98.2 174.7 151.3 216.7 261.9 198.0 5607.6 1086.5 68.2 73.9 45.6 37.1 6.7 5.8 24.9 7.4 16.5 4.8 12.6 12.6 32.0 155.5 156.1 170.0 123.5 109.1 68.5 107.4 169.9 298.8 286.6 141.3 3221.3 3873.3 84.3 86.2 65.0 52.0 15.6 13.5 98.1 59.2 122.2 110.1 86.5 149.1 272.8 849.8 802.5 765.5 364.5 256.1 181.4 292.3 352.3 525.6 571.9 349.0 10,398.8 Percent 37.2 0.8 0.8 0.6 0.5 0.2 0.1 0.9 0.6 1.2 1.1 0.8 1.4 2.6 8.2 7.7 7.4 3.5 2.5 1.7 2.8 3.4 5.1 5.5 3.4 100.0 Source: Fairchild, 1977. 22 of the existing grapefruit trees have attained full maturity. Only 9 percent of the grapefruit trees are in the 13-24 year ranges and thus will reach full maturity during the next 12 years. Furthermore, over one-half (53.8 percent) of all existing grapefruit trees are less than 13 years of age, of which 20 percent were less than five years old in 1976 (Fairchild, 1977). As emphasized by Fairchild (1977), the major implication of age distribution for future grapefruit production is that the full growth effect of the heavy plantings of the mid-sixties will be felt over the next 12-18 years. The effect of tree age distribution on citrus production is ex- plained by Fairchild (1977) using the data provided in Table 10. The average box yields per tree for five, ten, fifteen, twenty, and twenty- five year old trees illustrates the influence of the age distribution on total production. The bottom part of Table 10 compares the yields for trees of various ages. For example, an average 25 year old Pink Seedless tree yields 282 percent more fruit than an average 5 year old Pink Seedless tree. The effect of age distribution is even more dramatic when plantings vary considerably from year to year, as is the case with Florida citrus. 11-3. Fresh Grapefruit Competition Florida grapefruit is expected to face increasing competition from both domestic and foreign sources. This competition will increase the need to work toward a more efficient fresh fruit packing industry. 23 Table 10.--Average Box Yield Estimates and Percent Increase in Yields for Selected Tree Ages of Florida Grapefruit Grapefruit Tree Age White Pink Seedy Seedless Seedless - - - Boxes/tree - - - - 5 yr. 2.00 1.81 1.52 10 yr. 3.71 3.61 2.85 15 yr. 5.14 5.29 3.97 20 yr. 6.53 6.72 5.07 25 yr. 8.21 6.92 6.78 ---------------------------------------------------------------------- - Percent increase in yield - - 25 yr. versus 5 yr. 310 282 346 25 yr. versus 10 yr. 121 92 138 25 yr. versus 15 yr. 60 31 71 25 yr. versus 20 yr. 26 3 34 20 yr. versus 15 yr. 27 27 28 15 yr. versus 10 yr. 39 47 39 10 yr. versus 5 yr. 86 99 88 aCalculated as Older Age Yield Younger Age Yield Younger Age Yield Source: Fairchild, 1977. 24 As noted earlier, Florida accounts for a large share of domestic, Canadian and world markets, and seedless grapefruit was the only type of citrus that showed gains in acreage between 1973 and 1975, accounting for 50 percent of Florida citrus production for fresh use (Table 11). Fresh shipments of grapefruit have been increasing at a rate faster than the general growth rate in production. According to Myers (1977), this trend results mainly from the expansion of foreign markets for fresh fruit. Fresh sales of Florida grapefruit increased 23 percent during 1971-72 through 1975-76 while the total Florida crop increased by only 4 percent. While increased sales occurred in the domestic and Canadian markets, the major factor affecting the fresh growth rate was the development of European and Japanese markets. Japan has played the primary role since 1972, accounting for 73 percent of all offshore Florida grapefruit exports in 1975-76. Florida grapefruit market shares were 32 percent in France and 12 percent in the Netherlands in 1975-76 (Figure 2). In the domestic market, Florida's share of total U.S. grapefruit production has declined due to a stronger growth rate in Texas and California-Arizona. By 1984-85, it is projected that Florida will pro- duce 70 percent of the grapefruit produced in the U.S., down from 76 percent in 1971-72 and 73 percent in 1975-76 (Myers, 1977). Israel, Argentina and South Africa are the only three other coun- tries currently producing significant quantities of grapefruit (Figure 3). Israel, the largest producing country outside the U.S., increased production 36 percent between 1971-72 and 1975-76. Argentina increased 0 0 .Q m m co r, 0 0 0 I-i E4J 0 Co rJ 0 ia 01 c, .-I ie Ul 03 64 C w 0 Cp m 0 a 0 r. (e a) u (n (U c 0 0 EI aN Ln m 0n *4 O -1 Ln r- Ln CN F- CM (n in 0>o q qa7 O 0 Cl- C 0 rl rO Ln 1. O -Cl 4- 0 Oa r" eN I *4 1.1 --4 I ,-. '0 Ll \o a (M ,-l O m CO N N I I I r" i'- I"- I"- I, I- 0 ~4 (N l. N N N N N 26 LA. o 3 c - in IIn E-4 * I. .I -.. .. .. . .... .. . 0 Z N v~ on -a 0 U (U $4 ( 4-4 r 94 -4 0 U crr ..*.. ... ................ ;;=- ............. ... "eon:. .. .......-=. ....................... : .. ........... 0''' ............J .... .... .... ... .....~ ..... .~U ....... .......... .................6 ~U .. .. .. . .. ...... .... ...............( .......... ( 6*C- r-d ..~... .~..~= ~ =i=i-....-...1 : : ............'' ''' ............... ........ cn:::::: .... .. .. ........ .... " '''.I- -'-' X .......... ........:::::::..-.... .........~ ~~~ 0' 00 0 -4--M --4 Ci cc I -4 -0 CL U -'' ::::::::::....0 I ''-' : ::' ~~`'` ~ ~;~ ''' ' :-'0 0. ..... '''''' ''0' .:..::;:5':.::..:. ,, ~,:.:,,~~~::~:~:: Xi..... '''=-:'::'':''--4 0 . ='':='::.:.;:::::::=C.. (fl::== ::= production by 78 percent over the same 5 year period. Total world sup- plies increased 12 percent from 1971-72 through 1975-76. Myers (1977) also reports that grapefruit production in countries other than the U.S. is projected by the FAO to expand rapidly by 1981-82. Israel and Cuba are expected to be then the major world suppliers of grape- fruit other than the U.S. Other important countries will be Argentina, South Africa and Central America. Overall, world supplies of grape- fruit are expected to increase approximately 34 percent between 1975-76 and 1981-82. Given the increasing levels of domestic and foreign competition, the Florida citrus industry will perhaps face increased pressure to improve packing industry efficiency. 11-4. Florida Citrus Packing Size and concentration of packinghouses throughout the state vary widely. Shipments from packinghouses ranged from 30 boxes to more than three million 4/5 bushel boxes in the 1975-76 season (Appendix Table A-3). Packinghouses in six counties in the Interior and East Coast of the state--Highlands, Indian River, Lake, Orange, Polk and St. Lucie-- shipped large amounts of fresh fruit, exceeding one million 4/5 bushel boxes each. Average quantity of fruit shipped per packinghouse per county ranged between 6 and 800 thousand boxes (Table 12). All the above counties, plus De Soto, Duval and Hillsborough had an acreage of fruit shipped per packinghouse above the state average of 446 thousand boxes per packinghouse in 1975-76 (Table 12). Table 12.--Total Shipments, Number of Packinghouses per Packinghouse, by County, 1955-56 and and Average Shipments 1975-76 Seasons Total Fresh Number of Average Shipments Citrus Shipped Packinghouses per Packinghouses 55-56 75-76 55-56 75-76 55-56 75-76 - /5 hb boxes -- - A/I h"i bxvc- Indian River:a Brevard Indian River Martin Palm Beach St. Lucie Volusia Interior: Alachua Broward Citrus Dade De Soto Duval Hardee Hernando Highlands Hillsborough Lake Lee Manatee Marion Orange Osceola Pasco Pinellas Polk Putnam Sarasota Seminole Sumter Other 2,303,048 5,028,016 m-- 60,748 4,610,648 990,258 266 48,886 48,918 5,966 1,221,310 1,002,298 30,776 562,746 3,459,962 3,603,150 7,615,730 1,210 63,618 1,939,910 15,848,806 434,648 730,214 3,484,288 26,067,280 449,006 80,784 3,403,310 82,698 2,330,044 16,102,331 16,147 90,484 9,882,085 669,307 50,117 28 447,499 585,610 6,984 432,770 2,102,618 1,912,549 9,196,442 28,432 505,451 8,124,932 177,153 470,195 21,761,642 77,088 745,736 42,492 83,239,498 75,758,196 241 170 345,392 445,636 aWith the exception of St. Lucie these counties have the Interior Region (Figure 1). some townships in Source: Florida Department of Agriculture and Consumer Service, 1956 and 1976. 255,894 419,001 60,748 461,065 76,174 266 12,472 24,459 5,966 610,655 501,149 10,259 187,582 494,280 360,315 217,592 1,210 63,618 149,224 480,267 144,883 243,405 316,753 637,251 44,901 80,784 378,146 338,341 596,383 16,147 18,097 449,136 95,615 --m 7,168 28 447,449 585,610 6,984 432,770 700,978 478,137 540,967 -- 7,108 63,181 507,808 88,576 156,732 805,987 38,544 372,868 21,246 State .I - - .rl u urs 30 The variability in fruit shipped per packinghouse also shows itself in the capacity utilization as measured by Kilmer and Tilley (1978). From a sample of 23 Florida citrus packinghouses for three seasons (1973-74 to 1975-76), a maximum annual capacity was defined as the maximum monthly output through a packinghouse during three seasons multi- plied by 11. Capacity utilization is then equal to the volume packed in a season as a percentage of maximum annual capacity. The minimum value found for the firms in the sample was 19 percent, and the maximum 90 percent. The average was 50 with a standard deviation of 14 percent. Using the same growth model as before, and data shown in Appendix Table A-4, the shipment growth rates vary from negative values to almost 20 percent (Table 13). Some counties, e.g., Charlotte, Collier, Dade, Osceola and Sarasota, are not now shipping citrus although they did in the past (Appendix Table A-4). High growth rates can be found mostly in the southern East Coast in Broward, Indian River, Martin, Palm Beach and St. Lucie counties. Other counties with high growth rates include Hardee, Hernando, Hillsborough, Lake and Sumter (Table 13). These counties shipped approximately 37 million boxes of fruit in 1975-76 or 50 percent of the state's shipments. The four fastest growing counties in terms of fresh citrus shipments had a very small share of the state total in 1975-76 with only 546 thousand boxes or 0.7 percent. Over time there has been a clear trend towards fewer and larger packinghouses, using quantity shipped as a measure of packinghouse size. While in 1955-56 there were 241 facilities shipping an average of about 345 thousand boxes, in 1975-76 only 170 plants were operating. Table 13.--County Shipments Estimated Rate of Growth a rb Percent Indian River:c Brevard -0.019 -0.34 Indian River 7.00 0.95 Martin 19.5 0.35 Palm Beach 19.4 0.76 St. Lucie 5.25 0.83 Volusia 3.70 0.77 Interior: Broward 9.75 0.77 Citrus -30.8 -0.62 De Soto 1.06 -0.18 Duval 0.212 -0.036 Hardee 16.4 0.17 Hernando 17.2 0.64 Highlands 2.40 0.57 Hillsborough 3.75 0.62 Lake 4.95 0.71 Manatee 07.48 -0.75 Marion 1.49 0.16 Orange -1.46 -0.36 Osceola -19.4 -0.87d Pasco -8.08 -0.62 Pinellas -2.77 -0.37 Polk 1.06 0.37 Putnam -3.09 0.33 Sarasota -60.0 -0.77d Seminole 2.20 0.26 Sumter 8.12 0.23 State 2.46 0.73 aEstimate of 8, defined in footnote, page 13. Correlation coefficient = T Et InQt t lnQt ST -Zt2 (Zt)2 / T E(lnQt)2 (lnQt)2 CWith the exception of St. Lucie these counties have some townships in the Interior Region (Figure 1). dNo shipments in 1975-76. This is a reduction of 29 percent in the number of packinghouses. In 1975-76, however, total shipments had decreased by 9 percent below the 1955-56 level, but the average shipments per packinghouse were at more than 445 thousand boxes, an increase of 29 percent since 1955-56 (Table 12). Furthermore, packinghouses with an output exceeding 100 thousand boxes handled 95 percent of the fruit packed in the 1975-76 season, an increase from the 90 percent packed in 1955-56. Despite the state trend of declining number of packinghouses the following counties have more packinghouses in 1975-76 than before: Broward, Indian River, Manatee, Martin, Palm Beach and Sumter. With the exception of Manatee and Sumter, all counties are in the Southern East Coast region. Four counties (Hardee, Hernando, Hillsborough and Lake) have experienced reductions in the number of packinghouses although shipments have been increasing over the past ten years (Table 13). The grapefruit shipment growth rates vary from negative values to 30 percent per year (Appendix Table A-5 and 14). Charlotte, Collier, Hardee, Osceola and Sarasota counties no longer ship grapefruit (Appen- dix Table A-5). Two regions in Florida have high grapefruit shipments growth rates: The East Coast with Broward, Indian River, Martin, Palm Beach and St. Lucie counties; Central Florida with Lake, Seminole and Sumter counties (Table 14). These counties shipped almost 26 million boxes of grapefruit in 1975-76 or 67 percent of the state grapefruit shipments. The three fastest growing counties in terms of fresh grape- fruit shipments had a very small share of the state total in 1975-76, with slightly over 50 thousand boxes or 0.13 percent. Table 14.--County Grapefruit Shipments Estimated Rate of Growth ea r b Percent Indian River:c Brevard 0.108 0.02 Indian River 6.93 0.94 Martin 30.7 0.64 Palm Beach 18.6 0.67 St. Lucie 6.78 0.86 Volusia -0.181 -0.03 Interior: Broward 9.32 0.40 Citrus -16.3 -0.26 Duval -13.7 -0.82 Hernando -5.61 -0.25 Highlands -8.17 -0.75 Hillsborough 0.809 0.18 Lake 4.96 0.55 Manatee -11.6 0.69 Marion -7.16 -0.48 Orange -0.848 -0.22 Osceola -12.4 -0.63 Pasco -14.5 -0.68 Pinellas -14.6 -0.89 Polk -0.608 -0.28 Putnam -16.7 -0.80 Sarasota -66.9 -0.77 Seminole 3.82 0.17 Sumter 22.7 0.38 State 3.05 0.79 aEstimate of defined in footnote, page 13. bCorrelation coefficient = T .* t InQt Et J nQt T t2 (Et)2 / T Z(lnQt)2 (ZlnQt)2 cWith the exception of St. Lucie these counties have some townships in the Interior Region (Figure 1). 34 II-5. Florida Citrus Processing About 84 percent of Florida citrus production was processed in 1975-76, including eliminations from packinghouses. The percentage of citrus processed has been practically at the same level over the past decade, after increasing from between 60 and 70 percent twenty years ago (Florida Crop and Livestock Reporting Service, 1976a). Approximately 200 million boxes were processed in 1975-76 by 52 plants. The number of plants has been quite stable over the past twenty years. In 1955-56, 57 plants processed about 84 million boxes. Thus, the increase in output of canned and concentrated products has been accompanied by a substantial increase in average number of boxes pro- cessed per plant or from 1.5 million in 1955-56 to 4 million in 1975-76 (Florida Department of Agriculture and Consumer Services, 1956 through 1976). Since 1955-56 several processing plants have closed while new plants opened in the same or different counties. While there are no packing- houses in Dade, Hendry and Highlands counties, there are processing plants. The opposite is true in Broward, Citrus, Duval, Marion, Putnam and Sumter counties. II-6. Marketing Costs Costs have been rising in all stages of the citrus industry, as reported by Hooks and Spurlock (1977) and Hooks and Kilmer (1977 a and b). 35 Marketing cost increases imply a need to improve citrus industry efficiency in order to remain competitive in world and domestic markets. Orange picking and roadsiding costs in the 1975-76 season were 85 cents per box, an increase of about 1 percent from 1974-75. Grape- fruit and tangerine picking and roadsiding were estimated to be 69 and 158 cents per box, respectively. Hauling costs for all types of fruit were estimated to be 15.4 cents per box, an increase of about 1 percent from 1974-75 levels. Total citrus picking and hauling costs increased slightly in 1975-76 and were higher than in any preceding season except 1973-74. Total picking and hauling cost estimates for oranges and grapefruit were higher by 1.2 and 1.4 percent, respectively, from 1974-75 season estimates. Tangerines remained virtually unchanged. Some of the season-to-season variation in cost for each service is due to changes in the firms in the sample. The long-term trend shows that costs have increased almost steadily over the past 19 seasons before 1975-76 (Appendix Table B-1). Orange picking and hauling costs decreased relative to the delivered- in value of oranges for 1975-76. Appendix Table B-2 shows the relation- ship between picking and hauling cost and the per-box delivered-in price of oranges reported by the Florida Canners Association. During the first five seasons as reported in Appendix Table B-2, picking and hauling costs averaged 16.4 percent of the per-box delivered-in price of oranges. During the last five year period, picking and hauling costs averaged 36.8 percent of the delivered-in price for a box of oranges used in con- centrate. In the 1975-76 season, picking and hauling costs represented 32 percent of the delivered-in value of fruit. 36 Appendix Table B-3 shows how total picking and hauling costs have changed compared with the average reported figure for the 1960-61 through 1964-65 seasons--the base period. With few exceptions, the increases for all types of fruit have been very steady. For the 1975-76 season, orange and grapefruit picking and hauling costs are about twice the levels during the base period, while tangerine costs are estimated to be 1.7 times the average level reported during the base period. Hooks and Spurlock (1977, p. 10) show how the components of total orange picking and hauling costs have changed in absolute magnitude and relative to their base period levels. Picking labor, the largest component, has shown the greatest increase relative to the base period. While Appendix Table B-3 shows that total picking and hauling costs for oranges in 1975-76 were 198 percent of the base period level, picking labor has increased to 227 percent of the average in the base period. From the base period, (1960-61 through 1964-65) total picking and hauling costs for oranges are estimated to have increased 49.77 cents and picking labor costs have increased 28.28 cents; that is, 57 percent of the in- crease is accounted for by picking labor cost increases. Labor costs have increased relatively more than any other component and also account for a major share of the total absolute increase. Packing costs for two 4/5 bushel cartons of Florida oranges and grapefruit are estimated at $2.22 and $2.27, respectively, for the 1975-76 season. The estimates are 1.8 and 2.9 percent above 1974-75 season estimates. Tangerine packing costs for two 4/5 bushel cartons are estimated at $2.59, up 1.6 percent from previous season estimates. The cost to pack two 4/5 bushel cartons of tangelos was $2.35. 37 Packing costs have increased steadily over the past 16 seasons with the greatest increases in direct and indirect operating expenses, as reported in Hooks and Kilmer (1977a, p. 15). Specifically, material costs increased 5 and 9 percent for oranges and grapefruit while labor costs increased one percent for oranges and decreased three percent for grapefruit. Other direct operating costs were unchanged for oranges but decreased 10 percent for grapefruit. Finally, administrative costs decreased 10 percent for oranges but increased 21 percent for grape- fruit. Trends in orange and grapefruit packing costs for the past 17 seasons are reported in Appendix Table B-4. In general, packing costs have increased over time; however, the rate of cost increase slowed in 1975-76 when compared with the change between the 1973-74 and 1974-75 seasons. The relationship between average packing costs and the average FOB value of packed fruit for 4/5 bushel cartons of oranges and grape- fruit in the Interior and Indian River regions for the past nine seasons is shown in Appendix Table B-5. The changes in the ratio have been cyclical. The value of fruit has shown greater variability than costs which have tended to increase over time (Hooks and Kilmer, 1977). Processing, warehousing and selling costs remained reasonably stable until the 1971-72 season. As shown in Hooks and Kilmer (1977b, p. 16), costs for processing, warehousing and selling 48 six ounce cans of con- centrate in cases were at only 107 percent of the 1965-68 average unit the 1971-72 season. Since the 1971-72 season the index has risen to 153 percent in 1974-75 and declined slightly to 145 for 1975-76 season. 38 While 1975-76 processing costs were estimated at 145 percent of 1965- 1968 levels, 1975-76 prices were at only 131 percent of 1965-1968 levels. Changes in the cost categories that make up the total processing, warehousing and selling expenses for 48 six-ounce cans of concentrate are shown in Hooks and Kilmer (1977, p. 17). Materials, other pro- essing expenses and warehousing expenses are the items that increased the most relative to 1965-1968 levels while materials, selling and other expenses have shown relatively small increases. Hooks and Kilmer (1977, pp. 18-19) also show how costs of producing other products have changed since the 1961-62 seasons. Kilmer and Tilley (1978) estimated that approximately 10 cents in the cost of packing a box could be saved with a volume increase relative to capacity such that capacity utilization will increase by 10 percent. Also found was that firms with a large capacity and the ability to procure sufficient volume at a steady rate throughout the season in which they are open would tend to have lower costs than firms that have greater difficulty achieving favorable levels of the supply related variables. Soule (1974) discusses how careful handling can increase the per- centage of fruit harvested that reaches the consumer's table. Kilmer and Tilley (1978) report that packout percentage among a sample of pack- inghouses ranges from 30 percent to 88 percent. The cost differential between these two extremes is 28 cents per box or 11.4 percent of an average firm with a packout percentage of 88, which is a substantial potential cost reduction. Packout percentage is the proportion of fruit delivered to the packinghouse that meets quality standards for fresh fruit shipments. 39 11-7. Florida Citrus Markets The major markets for Florida citrus are the large metropolitan areas in the Northeast of the U.S.: Baltimore, Boston, Chicago, New York and Philadelphia. All these markets received more than two thousand truck carlots in 1975-76. In Canada the major market is Toronto. Other major export markets are Japan, France and Germany, each with more than 1 million 4/5 bushel boxes. This fruit is shipped mainly through the Tampa, Fort Everglades and Jacksonville ports (Florida Crop and Livestock Reporting Service, 1976a). CHAPTER III GENERAL PLANT LOCATION MODELS Selection of the optimal location, number and size of facilities in a region has been the focal point for a variety of economic studies. These include the location of educational and health service facilities as well as agricultural and other types of processing plants. In this chapter several classes of mathematical models which represent alterna- tive approaches to making such selections are cited. The earliest developments in location theory are those by Thunen (1826) and Laundardt (1872) and continued into this century by Weber (1909) and Losch (1940) forming what is known as classical location theory. The assumptions of those models differ to a certain extent from those which can be made with respect to the problem of improving the efficiency of the citrus industry marketing (Appendix C). Particularly, the set of possible locations is the plane, that is, the geographical area where the production and markets are localized. These models provide benchmark or ideal solutions which do not account for limita- tions on the solution such as a finite and discrete set of potential and existing locations. Thus one has to go beyond the models of classical location theory. 41 The need to introduce a discrete space as the set of location possibilities led first to the use of the linear programming techniques developed by Dantzig in 1947 (Schrader, 1961), and the transportation models (Judge, Havlicek and Rizek, 1964), discussed in Appendix C. Simultaneously, in order to include economies of scale in plant operation, Stollsteimer (1961) developed a model which evaluates all possible combinations of facilities and locations to determine the optimal minimum cost solution, and King and Logan (1964) adapted the transshipment model developed by Orden (1956) to plant location analyses (Appendix C). Several nonlinear programming models have also been used as is pointed out in Appendix C. Separable programming introduced by Miller in 1963 is used in a location-allocation context by Crowder (1967). Candler, Snyder and Faught (1972) formulate the problem as one of con- cave programming. Graves (1972) uses quadratic programming for which solutions had been provided earlier by Beale (1955) and Wolfe (1959). Beale also studied mixed-integer problems in 1958 and such a formula- tion is used by Ballinski (1964) for a location-allocation model. Mixed- integer models are reviewed in Appendix C. Many other contributions have been made to each one of these types of models as shown in Table 15. The model classifications used in Table 15 and throughout this chapter are not exclusive. All models that follow a classical location theory approach are grouped under that designation. Linear programming models which do not have or do not exploit the special characteristics of a transportation problem are 42 Table 15.--Agricultural Location Models Model Author(s) Application Classical Location Theory Olson (1959) Williamson (1962) Miller and Henning (1966) Araji and Walsh (1969) Oppen and Hill (1970) Ballestero Pareja (1971) Oppen (1972 and 1974) and Oppen and Scott (1976) Lee (1975 and 1977) Linear Programming Schrader (1961), Schrader and King (1962) and King and Schrader (1963) Egbert and Heady (1963) Martin (1964) Langemeier (1968) and Langemeier and Finley (1971) Leunis (1968) Lee (1968) and Lee and Snyder (1970) Freeman (1969) Shumway (1969) and Shumway et al. (1970) Free (1970 and 1971) Milk Processing Meat Processing Livestock Markets Grain Distribution Grain Distribution Sugar Beet Scales Soybean Processing Wastewater Treatment Cattle Feeding Field Crops Poultry Industry Cattle Feeding Soybean Processing Feed Industry Snap Beans Processing Field Crops and Vegetables Soybean Processing Table 15.--Continued Model Author(s) Application Alexander and Ashley (1970) Dietrich (1971) Goodwin and Crow (1973) Byrkett (1974) and Byrkett et al. (1976) Thomas (1975) Fedeler and Heady (1976) Milk Processing Cattle Feeding Meat Processing Cattle Feeding Milk Processing Grain Distribution Transportation Pherson and Firch (1960) Judge et al. (1964 and 1965), Havlicek et al. (1964) and Rizek et al. (1965) Carley (1966) Fraase and Anderson (1970) Weindlmaier (1971) Hopkins et al. (1971) Hardy (1972) and Hardy et al. (1973) Rao (1973) Fraase et al. (1974) Doeksen and Oehrtman (1976) Meat Processing Meat Processing Milk Processing Malt Plants Meat Processing Feed Industry Medical Clinics Grain Distribution Pasta Processing Fire Fighting Combinatorial Stollsteimer (1961 and 1963), Sammet and Courtney (1975) and Stollsteimer et al. (1975) Pear Packing Table 15.--Continued Model Author(s) Application Mathia (1962 and 1963) and Mathia and King (1962) Peeler (1963) and Peeler and King (1964) Siebert (1964) Polopolus (1965 and 1968), Popolus and Strebeck (1965) and Lopez (1966) Sanders and Fletcher (1966) Warrack (1967) and Warrack and Fletcher (1970 a and b) Courtney (1968) O'Dwyer (1968 a and b) Halvorson (1968 and Ladd and Halvorson (1970) Chern (1969) and Chern and Polopolus (1970) Hicks (1970) and Hicks and Badenhop (1971) Huie (1970) Dawson and Warrack (1971) Eddleman (1972) Moore (1972) and Moore and Courtney (1973) Kedar (1972) Tilley (1973) Kuehn (1973) Sweet Potato Processing Egg Packing Citrus Industry Vegetable Processing Egg Packing Feed Industry Feed Industry Milk Processing Turkey Processing Citrus Industry Livestock Markets Meat Processing Hog Markets Hospitals Cotton Ginning Citrus Industry Hog Markets Livestock Markets Table 15. --Continued Model Author(s) Application Tosterud (1973) and Tyrchniewicz and Tosterud (1973) Baumel et al. (1973), Lifferth (1974) and Ladd and Lifferth (1975) Fuller and Washburn (1974) Fuller (1975a) King and Logan (1964) and Logan and King (1964) Hinton (1964) Hurt and Tramel (1965) Leath and Martin (1966 and 1967) and Leath (1970) Bobst (1966) and Bobst and Waananen (1968) Cassidy (1968), Toft et al. (1970) and Cassidy et al. (1970) Strawn (1969) C Ferguson and McCarthy (1970) Ferguson and Pemberton (1970), McCarthy et al. (1971) and Ferguson et al. (1972) Blair et al. (1970) E Stennis (1970) and Stennis et al. (1971) Jesse (1970) and Hudson and Jesse (1972) C Grain Distribution Grain Distribution Cotton Ginning :otton Ginning leat Processing 4ilk Processing leat Processing ;rain Distribution ilk Processing teat Processing orn Distribition lool Processing gg Packing lilk Processing :otton Ginning Table 15 .--Continued Model Author(s) Stammer (1971) Holroyd (1972) and Holroyd and Lessley (1973) Haas (1972 and 1974) and Via and Haas (1976) King et al. and King (1973) King and Seaver (1974) Stennis and Hurt (1974 and 1975) Boehm and Conner (1976 a and b and 1977) Nonlinear Programming Crowder (1967) Kloth (1970) and Kloth and Blakley (1971) Stennis (1970) and Stennis et al. (1971) Holder et al. (1971) and Candler et al. (1972) Graves (1972) Howard (1975) Kilmer (1975) and Kilmer and Hahn (1978) Baritelle and Holland (1975) Holland and Baritelle (1975) Schools Application Feed Industry Honey Packing Hog Markets Potato Packing Apple Packing Meat Processing Milk Processing Milk Processing Milk Processing Milk Processing Rice Milling Farm Inputs Farm Inputs Milk Processing Table 15.--Continued Model Author(s) Application Mixed-Integer Barton (1974) Meat Processing Fuller (1975b) and Fuller et al. (1976) Cotton Ginning Cleveland (1976) and Cleveland and Blakley (1976) Cotton Ginning Daberkow (1976) and Daberkow and King (1977) Emergency Medical Facilities Hilger et al. (1977) Grain Distribution 48 listed separately. Models formulated on the basis of an optimization of an objective function over all possible combinations of plant numbers, sizes and locational patterns are classified as combinatorial, although for each combination they might be a transportation or trans- shipment model. Transshipment implies shipment between an origin and a destination through an intermediary point where some sort of product transformation takes place. Nonlinear programming models are all those whose objective function is nonlinear, mostly due to a nonlinear pro- cessing cost function. Mixed-integer models are those which include continuous and integer decision variables. These integer variables are associated with the operation or not of a specific plant. The major features of the combinatorial, transshipment, nonlinear and mixed integer models are listed and summarized in Table 16. Among these developments, the works of Siebert (1964), Chern and Polopolus (1970) and Kedar (1972) have special interest for not only extending the existing models but also applying them directly to the citrus industry. These models are discussed in Appendix C. The only model and solution procedure with all of the desirable characteristics shown in Table 16 is the mixed-integer model developed by Barton (1974) for a meat processing firm. He could not solve and analyze the firm's actual problem because all the essential data were not provided, so an illustrative example was used. The model has characteristics specific to meat marketing which do not apply to the citrus industry. None of the agricultural location models addresses the issue of the optimal dynamic adjustment of the existing structure of the optimal 49 Table 16.--Main Features of Selected Location Models Type of Number of Stages Multiple Model Author(s) Transportation Products (assembly and Processing distribution) Combinatorial Stollsteimer (1961, 1963) 1 1 No Siebert (1964) 1 1 No Polopolus (1965 1 1 Yes Chern and Polopolus (1970) 1 1 No Kedar (1972) 2 1 Yes Tyrchniewicz and Tosterud (1973) 2 1 No Fuller (1975a) 1 1 No Ladd and Lifferth (1975) 3 2 Yes Transshipment King and Logan (1964) 2 1 No Hurt and Tramel (1965) n-i n Yes Leath and Martin (1966( n-i n Yes Bobst and Waananen (1968) 2 1 No Stennis and Hurt (1974, 1975) 1 1 No Separable Programming Kloth and Blakley (1971) 2 1 No Baritelle and Holland (1975) 2 1 Yes Concave Programming Candler et al. (1972) 2 1 Yes Quadratic Programming Graves (1972) 2 1 Yes Mixed Integer Balinski (1964) 1 1 No Efroymson and Ray (1966) 1 1 No Barton (1974) 8 3 Yes Fuller et al. (1976) 1 2 No Hilger et al. (1977) 2 1 No Table 16.--Extended Capacity Considering Exact Efficient Constraints Existing -Solution Search Other Facility Procedure No Yes No Yes Yes No Yes Yes No No No No No Yes No Yes No No Yes Yes Yes Yes Yes Yes No NA No No No Yes No Yes Yes No Yes Yes No No Yes Yes No No Yes No Yes Yes Yes NA Yes Yes Yes Yes Yes No Piecewise linear cost function NA Discounted costs No Institutional constraints No Piecewise linear cost function No Multiperiod and piecewise linear cost function No No No No Institutional constraints No No No Inventory costs No NA Yes Yes Yes Yes Multiperiod and piecewise linear cost function Yes Multiperiod - - proposed for the end of the planning horizon. Sweeney and Tatham (1976) present a synthesis of a mixed-integer location model with a dynamic programming procedure for finding the optimal sequence of configurations over multiple periods. A mixed-integer programming formulation is selected in this study because it is the only one that is truly optimizing and that allows multiple secondary optimization runs at relatively low computer cost. These secondary runs allow the use of the procedure developed by Sweeney and Tatham (1976) to solve the dynamic model. A solution procedure is classified as exact when it provides the user with the global optimum solution. An efficient procedure avoids evaluating every possible combination of flows and locations. The solution procedures associated with the models listed in Table 16 are classified according to these criteria. They are analyzed in detail in Chapter IV. CHAPTER IV SOLUTION PROCEDURES IV-1. Introduction This chapter contains a presentation and discussion of the pro- cedures used to obtain solutions to the location models presented in Chapter III. A solution procedure to the problem of improving the efficiency of an assembly and distribution system has the following major steps: a) determine the origins and supply of initial product and market demand of final products; b) select potential plant locations; c) estimate transfer and plant cost; d) determine the optimum plant number, size and location; e) study the solution behavior under different industry situations. The main approaches used to determine the optimum number, size and location of plants can be classified as follows: a) iterative actualization of plant cost; b) evaluation of all possible combinations; c) analysis of cost effects of successively opening or closing plants; d) approximation of concave costs by a polygonal cost curve; e) analysis of local optima; f) mixed-integer algorithms. 53 Linear programming, transportation, transshipment or network algorithms have been used to obtain solutions to the continuous distri- bution subproblems. The combinatorial procedure gives an optimal solution at the expense of lengthy computations for reasonably complex cases. Concave programming allows the determination of the global optimum if all local optima are considered. Most mixed-integer algorithms converge to the optimal solution. All other procedures do not assure that an optimum is obtained. Most procedures can be stopped when a solution is obtained that is close enough to a lower bound. Both concave programming and mixed- integer algorithms update the current lower bounds during the computa- tional procedure. The analysis of solution behavior under different situations is generally done using sensitivity analysis and/or multiple runs of the model with the data changed to correspond to the new condi- tions. IV-2. Linear Programming Procedures When using linear programming as a location model (King and Schrader, 1963) the primal provides the following information: a) the location of plants and input usage; b) shipment patterns of products and inputs; c) equilibrium prices and consumption of final product. 54 The dual solution provides the following information: a) the input prices of factors consistent with equilibrium flow; and b) the cost associated with introducing activities not in the optimum solution. To analyze an industry, sets of prices, consumption and transfer cost information are divided into specific sets of "situations" and the cost for the industry is minimized for each of these situations (Martin, 1964). To minimize total assembly and processing costs, an iterative pro- cedure is followed. First, each plant is assigned the minimum value of processing costs, usually associated with a large plant operating at capacity. A solution is then obtained to determine the volume of processing under these conditions in each plant. Second, processing costs used in the first step are compared for consistency with the actual cost of processing the volume of product indicated by the first solution. Third, processing costs are revised and another solution obtained as in step one. The steps are repeated until no further adjustments in processing costs need to be made. It is recognized that this procedure may not yield an optimum answer. The final step in this analysis consists of varying the situations assumed in the analysis, and determining the effect on the optimum number, size and location of plants. IV-3. Transportation Model Procedures The transportation model is a special class of linear programming models for which computing routines have been developed to obtain solu- tions more efficiently. The transportation model was initially conceived to give a minimum transport cost in satisfying a given set of needs from a given set of sources. The need of each location and the capacity of each source are predetermined. An analytical procedure, similar to the one described for the linear programming model, can be used, incorporating the transportation model in the total assembly and processing cost minimization step. The same iterative method is used, with a solution for each iteration being obtained faster and more efficiently, but still with no guarantee that an optimum answer will be obtained. Another heuristic procedure devised by Rao (1973) consists of solving the transportation model to get a set of potential locations. Then, the location pattern obtained is modified in such a way that all the facilities in the new location pattern have full capacity utilization to the extent possible. The criterion which justifies the transfer of volume is that the profit by this transfer should more thar. offset the loss. In this process some locations will get eliminated and the facil- ities in the remaining locations will have full capacity utilization. The process is repeated for each facility size. That particular size for which the total cost is a minimum is then selected. IV-4. Combinatorial Procedures The procedure for cost minimization varies with the presence or absence of economies of scale in plant operations and the way in which processing costs are influenced by plant location. For a model where it is assumed that there are economies of scale in plant operations with plant costs independent of plant locations, the problem of mini- mizing the sum of processing and transfer costs is accomplished in two steps (Stollsteimer, 1963). The first step is to obtain a transfer cost function that has been minimized with respect to plant locations with varying number of plants. N There are E ( ) possible combinations of locations. The second step n=l is to add to the minimized total transfer costs, the processing costs with varying numbers of plants, yielding a total transfer and processing cost function minimized with respect to plant locations for varying numbers of plants. With constant marginal processing costs in any given plant, and a positive intercept in the plant cost function, the total cost of processing a fixed quantity of material, Q, will increase by an amount equal to the intercept value of the plant cost function with each increase in plant number (Chern and Polopolus, 1970). Stollsteimer (1963) noted that the first difference of total trans- fer costs with respect to plant numbers is negative or zero, and the second difference is positive or zero with empirical applications yielding positive second differences. This gives a total transfer cost function of the number of plants that is an envelope to a set of total transfer 57 cost points. This envelope is decreasing at a decreasing rate. When added to a constant increasing total plant cost function, a minimum or extreme point is obtained, whenever the decrease in transfer costs equals the increase in total processing costs. The minimization procedure proposed by Stollsteimer (1963) has been classified as a set of linear programming problems, automated to select the least cost among repeated solutions with both plant numbers and locations permitted to vary (Stollsteimer et al., 1975). Further work by Hoch (1965), however, shows that it is possible to obtain a negative second difference, even though the empirical evidence indicates that the probability of this occurring may be low. As Hoch (1965) puts it, transfer costs as a function of number of plants is a decreasing monotonic concave function only for some special cases. In other cases (apparently rare in occurrence), the function may not be concave. This implies that local minima are possible for total costs, where total costs equal transfer costs plus processing costs. This has the practical implication that, to find the minimum minimorum (or minimum of all possible cases), one cannot terminate computations when a local minimum has been obtained but must work through all possible cases. This, in fact, has already been indicated by Heady and Candler (1958, p. 371) when a transportation problem is solved taking fixed costs into account. A numerical example exhibiting negative second differences of transfer costs is also given by Hoch (1965). Although the correction involved in Hoch's note (1965) does not affect Stollsteimer's major conclusions (1963), it contains a major rule for any combinatorial model: all possible cases must be evaluated. The complete enumeration rule for a combinatorial model is, of course, applicable to the generalizations of the Stollsteimer (1963) model which consider multiple products (Polopolus, 1965), discontinuous plant cost (Chern and Polopolus, 1970), collection and distribution costs (Tyrchniewicz and Tosterud, 1973), multiperiods, capacity expan- sions and economies of scale in transportation (Ladd and Lifferth, 1975). Failure to recognize all possible combinations has led to sub- optimal solutions as in Chern and Polopolus (1970), (Fuller, 1975), and Chern, (1969, p. 35), (Machado, 1975). Thus, the combinatorial procedure yields an optimal result at the expense of lengthy computations (Warrack and Fletcher, 1970b). All citrus industry location models (Siebert, 1964) Chern (1969), Kedar (1972) developed until the present are extensions of the Stollsteimer model (1961). As it stands, the problem of optimizing the distribution system of citrus has not been solved. None of the procedures proposed before dealt effectively with it. The major drawback encountered is that previous studies relied upon complete enumeration which when not pursued to its fullest extent, may lead to suboptimal results. Several sub- optimization procedures have been suggested and used by Warrack and Fletcher (1976), Kuehn and Hamburger (196), Shannon and Iqnizio (1970), Hardy (1972), Hardy et al. (1973), Fuller (1975a), Sielken (1973) and Fuller and Sielken (1978). None of the procedures necessarily obtain the optimal solution. 59 IV-5. Transshipment Procedures Basically, the cost minimization procedure for the transshipment model modifies the transportation problem by specifying that the product is shipped twice. Each production and processing area is designated as a possible shipping point. Then, the cost of shipping directly from a production area to a market is given a very high cost thus forcing the product to be shipped into a processing plant (or warehouse) before it is shipped to the distribution center. The computational advantage of the transshipment formulation over the alternative linear program is pointed out by King and Logan (1964). For a single product, 30 origins, 20 potential locations and 30 markets case, a 60 x 60 matrix is required. This compares with an alternative formulation requiring 90 equations and 1,800 activities. Market demands are either known or estimated in the first steps of the procedure. To consider economies of scale in plant operations, an iterative procedure identical to the one described above for the linear programming procedure is proposed by King and Logan (1964). This procedure is used in the cost minimization step and may not provide an optimum answer, when economies of scale exist in processing. An investigation of the transshipment model solution sensitivity to the processing cost function is performed by Toft et al. (1970). Using the sensitivity analysis common in the transportation model it is possible to determine: a) the change in processing costs necessary to alter the actual location of processing operations; 60 b) the effects on the solution caused by localized variations in transportation costs; c) a mapping of optimum solutions for continuous variations in either the position or shape of the processing cost curves. The iterative procedure used by Blair et al. (1970) to arrive at the minimum cost solution is similar to the linear programming procedure suggested by Martin (1964) and the iterative eliminations approach sug- gested by Warrack and Fletcher (1970). For the King and Logan (1964) iterative technique to approximate an optimum plant location, Stennis and Hurt (1975) listed three techni- ques which can be utilized to save computer time and expense. These techniques involve eliminating in an iterative fashion some potential plant locations from the admissable set in subsequent solutions. IV-6. Nonlinear Programming Procedures Kloth and Blakely (1971) used separable programming in the cost minimization step. The implementation of separable programming is based on the fact that each nonlinear, separable function can be approximated by a piecewise linear function, known as polygonal approximation. The solution is obtained for the problem defined in terms of a polygonal approximation of each separable function; thus the solution reached is an approximation of the true solution. Each polygonal approximation is represented by linear equations together with certain logical restric- tions on the variables in the equations. The plant location problem with fixed supplies and demands can be regarded as an example of concave programming. Candler et al. (1972) 61 discussed an algorithm capable of eventually reaching a global optimum for concave programming problems. The most significant consequence of a concave objective function is that there may be and usually will be, many local optima. That is to say, there will be many solutions each of which is better than any other solution in its immediate neighborhood. The algorithm, developed utilizing Tui's ideas (1964), does an orderly examination of the local optima. Given that an exhaustive search of the solution space is possible, determining how many local optima should be examined is an economic problem of balancing potential gains against known additional computing expenditures. IV-7. Mixed-Integer Procedures To solve the mixed-integer plant location model, Balinski (1954) used a partitioning theorem of Benders (1962) to reduce the problem to one of solving a sequence of increasingly more complex integer programs whose only variables are x's. At each step, the integer x's produced by the integer program are used to determine an optimal solution to a continuous dual problem, which is in turn used to produce a new integer program. It can be shown that this method will converge to the optimal solution of the problem in a finite number of steps. The technique of Benders decomposition is also used by Hilger et al. (1977). A branch and bound algorithm is used by Efroymson and Ray (1966). Branch and bound is a finite general purpose mixed-integer programming technique developed by Land and Doig (1960). The basic idea is to solve a sequence of linear programming problems (not necessarily meeting the integer restrictions) that give progressively improved lower bounds1 on the value of the solution to the mixed integer problem. To illustrate the technique one assumes a mixed integer formulation with objective function Z, and zero-one integer variables x. The problem is first solved as a linear program (without the integer requirements on the x's) giving a value of Z0. If all the x's are integer, then the problem is solved. If some x. is fractional,then it is first fixed at zero and the linear program again solved producing Z1, and 2 then fixed at one and the linear program solved producing Z2. It is clear that: (4.1) Z = man (Zl, Z2), is a new lower bound on the value of the solution. What is happening is that a tree is being constructed whose nodes are represented by the Z's and the corresponding values of the fixed x's. So far two nodes have been constructed (Z1, xj, = 1) and (Z2, xj = 0). One now "branches" on the node determined by Z by fixing some other fractional x., first at zero, then at one to determine two nodes Z3, Z4. One continues by branching on (4.2) Z = min [(Z3', Z4' max (Z, Z2)], 'Here one assumes that the problem is a minimization one. The technique produces upper bounds for maximization. 2The general technique can handle cases where the integer variables can assume more than two values. 63 which is a new, improved lower bound. Of course, one needs only keep track of "terminal" nodes, and if any node is infeasible, no branches can emanate from it. The process terminates when a node is reached where all the x's are iteger and its value is less than or equal to that of any other terminal node. A variety of node selection rules and branching rules can be used to achieve computational efficiencies in solving the plant location prob- lem. Khumawala (1972) discussed several rules which perform well in solving this problem. Francis and White (1974) illustrated the procedure with a plant location problem involving five customers and three plant sites. As Efroymson and Ray (1966) mention, the chief difficulty with branch and bound is computational. If a large number of linear programs have to be solved and the computing time for each linear program is high, the method could become prohibitively expensive. Thus, to use branch and bound on plant location, the problem must be formulated in such a way that the linear programming problem without the integer restrictions can be efficiently solved. In reporting their computational experience, Efroymson and Ray (1966) indicate that a number of 50-plant, 200-customer problems were solved. The average solution time is reported to be about 10 minutes on a IBM 7094 computer. Khumawala (1972) solved 16 test problems of size 25 by 50 and reports average solutions times of approximately 10 seconds on a CDC 6500 computer. A zero-one mixed integer programming computer code is available from McCarl et al. (1973) and an adaptation from Tyrrell and Barton 64 (1976), which can be used to solve small problems, up to 150 rows and 450 columns. Among the packages commercially available, IBM (1973) has an optional mixed-integer programming feature for the Mathematical Pro- gramming System-Extended (MPSX), for problems with up to 4095 integer variables. Fuller et al. (1976) recognized that the transportation cost minimization of the location problem could be formulated as a network problem. The efficiency of this solution approach depends on the solu- tion speed of the network code and the number of subproblems that actually have to be solved. The total number of subproblems is 2 -1. In order to avoid explicitly solving all subproblems, a branch and bound implicit enumeration procedure is employed (Fuller et al., 1976). CHAPTER V LOCATION MODEL FOR THE CITRUS INDUSTRY V-1. Model Formulation A general location model of the citrus industry is presented in this chapter. The static model formulation for a single season is given. A discussion of the flexibility of the static model to accomo- date various assumptions is made in Section V-2. The model in Section V-3 provides a plan for adjustment toward a more efficient marketing organization over time. There are H citrus fruits produced at several groves, I, with given variable yields. The quantity, Qhi, of fruit,h, supplied at each origin, i, is sent to a packinghouse at one of the possible locations, J, where quantity qhijk is packed and shipped to one Df the markets, K, and quantity qhijl is eliminated and shipped to one of the processors, L. There is an estimated consumer demand, Qhk' for each fruit, h, at each market, k. This demand is satisfied by shipping qhijk from groves, I, to markets, K, via potential packinghouse loca- tions, J. The quantity, Qhl. of each fruit, h, that each processor, 1, is capable or willing to receive from packers cannot be exceeded by the sum of the quantities, qhijl' shipped from all origins, I, through all possible packinghouse locations, J, and sent to processing. There are 66 lower, Qmin' as well as upper bounds, max' on the allowable total annual throughput of each packinghouse site. If a packinghouse is open at site j, variable x. takes the value 1, and is 0 otherwise. The products Qminxj and Qmaxxj either equal zero when x. equals zero or Qminand Qma, when xj equals one. The sum of the quantities, qhijk' going through location j is either zero when x. equals zero or is between Qmin and Qmax The percentage of each fruit, h, from each origin, i, going through a packinghouse location, j, that is actually packed is the packout, ahi, and the percentage eliminated is 100 = ehi. The proportional quantities packed and eliminated are qhijk and qhijl' which can be sent to any of the,-K, markets, and, L, processing centers. Supply of each fruit, h, from all origins, I, that is actually packed, ehi/100Qhi' will satisfy demand, Qhk, at all markets, K. Quantities shipped, qhijk and qhijl, are always nonnegative. Possible locations for packinghouses are selected beforehand, but the particular sites to be used are to be selected so as to result in the least total assembly and distribution costs. The objective function has three components. The first is total variable cost, for all types of fruit H, origins I and location J, of assembling qhijk units of fruit h from origin i, packing them at location j and then shipping to market k at a cost per unit of vhijk, for all markets K. The second is the total variable cost of assembling a certain quantity qhijl of fruit h, from origin i, which is eliminated at j, and shipped to a processing plant at 1, at a cost per unit of u The third is the hijl sum of all fixed costs f. incurred when fruit is packed at location j, J in which case x. is 1. Packing costs are expressed as fixed charges (imposed for the sites actually used) plus a linear per unit charge constant between the bounds Qmin and Qmax. Transportation costs are taken to be linear with respect to distance. There are two types of decision variables in the model: continuous and discrete. These last ones are the integer variables x that can only take the values 0 and 1. There are HIJL continuous variables q and J integer variables x. The model contains HI supply constraints, HL demand and processor constraints, 2J capacity constraints and HIJ packout constraints, and H supply-demand balance equations, bring the total to H(I (1 + J) + L + 1) + 2J. The number of possible solution vectors x is 2 -1. The model will determine which packinghouse sites to use, what size and how many packinghouses to have at each selected site, which markets and processing plants should be served by each packinghouse, and what the pattern of transportation flows should be for all fruit. This is to be done so as to meet the estimated demand at minimum total assembling costs subject to the fruit supply and packinghouse throughput constraints. The model can be written as the following mixed integer linear program: H I J (5.1) minimize Z = Z Z E h=l i=l j=l h=l i=l j=l subject to, supply constraints, J K (5.2) Z Z qhijk j=l k=l J + j j=! K Vhijkhijk + k=l L J S hijk hiji+ J' I=KI j- L S 1 < Q=K= hijl -- qhi' I=K=I h = . H, k = K, demand constraints, J K (5.3) hijk q hk j=1 k=l h - processing constraints, I J (5.4) Z q < 0 i=l j=l hij-- hl capacity constraints, H I K (5.5) E Z k hijk > Qmnx , h=l i=l k=l h = 1, . H, 1 = K+l, .., L, j = 1, J, H (5.6) Z h=l I K i l kq < hij axx i=l k=l- a= h = l, ., H, k = l, ., K j = 1- .. J, packout constraints, K (5.7) 7qhijk k= l e hi 1 eh i hij 1=K+I 100-ehi .hij, i h = ., H, L = 1, , j = 1, . ., J, supply-demand balance equations, I K (5.8) Z e hi > - i=tivi00t k= con nonnegativity constraints, h = 1, . ., H, for all h, i, j, k, 1, 0, 1 constraints, 1, if plant is open at site j, 0, if otherwise, j = 1, . ., J. J The objective function (5.1) gives the cost when xj packinghouses j=l are to be located at those sites corresponding to positive-valued xj. Constraints (5.2) indicate that the quantities shipped to market k and to processor 1 cannot exceed the quantities supplied. By (5.3) and (5.4) all demands for market k must be met by some combination of plants, and cannot exceed processors' capacity. Constraints (5.5) and (5.6) keep the total throughput at site j either equal to zero when x. equals zero or between Qmin and Qmax. Constraints (5.7) relate the capacity of each fruit packed at location j with the volume of eliminations going to processing. Constraint (5.8) is included to make sure that (5.9) qhijk' qhijl > O' 70 total demand does not exceed total supply. Nonnegativity and integer restrictions on the decision variables qhijk' qhijl and xj are given by (5.9) and (5.10), respectively. V-2. Model Discussion V-2.1. Cost Minimization Cost minimization is the most widely used objective in the design of physical distribution systems. The obvious reason is the relative ease with which costs can be calculated. Revenue response is not explicitly considered. Given the conditions of perfect competition, the equilibrium solution for cost minimization is equivalent to the solution for profit maximization, that is, cost minimization is equiva- lent to profit maximization only if revenues are independent of the location-allocation design. Such independence would mean revenues are not influenced by the level of customer service measured by the speed and reliability with which customer's orders are filled. Since this is very unlikely, cost minimization and profit maximization are seldom equivalent. Sales (and therefore revenue) may be considered in the form of minimum sales constraints. Market service need not be totally ignored. Network configuration restrictions and assumed transportation, order processing, and inventory policies can be imposed which provide satis- factory market service levels. By use of "what if" sensitivity analysis, 1 For an extensive discussion of location model features, see Geofrion (1975). 71 the effects of various sales and customer service level policies can be explored. Measuring the revenue response of various design alternatives is extremely difficult and so is the application of sales, customer service and profit maximization objectives to physical distribution system design (Barton, 1974). V-2.2. Size Limits and Economies of Scale If the maximum capacity of a packinghouse is never exceeded at any location, there is no need to consider opening more than one plant at each possible site. Stipulation of lower and upper bounds on capacity permits a piecewise-linear representation of economies of scale and other non- linearities (or even discontinuities) in packinghouse costs as a func- tion of throughput. Alternative packinghouses are simply introduced at a given site with different size ranges controlled by in and Q , -mm max with fj and v i specialized accordingly. For instance, a piecewise 3 hijk linear packinghouse cost function with three segments would require three alternative packinghouses (small, medium and large) each with f. and vhijk dictated by the corresponding segment of the packinghouse cost function. More than one packinghouse of the same size may be open at each site. V-2.3. Existing Packinghouse Utilization, Closing and Expansion To consider expansion or closing of existing packinghouses existing facilities are included among the J locations in the model. Expansion of existing packinghouses can be considered using the technique described in the previous section to permit a piecewise linear representation of economies of scale. Expanded packinghouses at a given site are introduced in the model with size controlled by Qmin and Qmax' with f. and vhijk specialized accordingly. V-3. Dynamic Analysis It is understood that once an optimal solution to (5.1 5.10) is provided for some time in the future, one of the first questions asked is: How should the industry move from its present structure to the optimum in the future? If the optimal solution does not differ much from the present industry configuration, the analysis described in Section V-2 above provides helpful suggestions. Consider for instance the case where only some locations need to be abandoned. Then by closing those plants over the period considered, the final optimum stage is attained. Pri- ority analysis (Section VII-6.5) establishes those plants that should preferably be closed first, producing the largest savings. If some existing facilities are to be expanded, then again priority analysis shows those that will bring the highest reduction in costs. A combina- tion of both strategies can be outlined to open and close packinghouses simultaneously. The results of the priority analysis relative to new locations are a guide to further industry relocation. Another possible approach is the use of a multiperiod model. If the relocation of the industry from period to period can only be made at a certain cost, a new situation might arise where some configurations might result for certain years whose costs are only compensated by savings realized over the entire period. For example, a configuration which yielded the second best static solution in each planning season could quite possibly yield the long-run optimal location, since no relocation costs would be necessary over the planning horizon. Consider now a matrix (Figure 4) of static solutions, Zt for each season, t, ranked by cost. The optimal solution is found by using dynamic programming to find the minimum cost path through the matrix, taking into consideration, at each stage, the cost of moving from one system configuration to any other (Sweeney and Tatham, 1976). The computational feasibility of this procedure is dependent on the number of stages and the number of alternative solutions that must be considered at each stage. The computational time required by any dynamic programming procedure increases linearly with the number of stages and exponentially with the number of states. Therefore, the number of configurations to be considered at each stage is the most critical factor from a computational point of view. Let Ztl denote the cost of the best static configuration in period t and Ztc any other solution. Then Z1 = ZZt is the sum of the minimum t cost configurations for the entire planning horizon. Since no reloca- tion expenses are considered, Z1 is a lower bound on the value of the optimal multiperiod solution. Let Zu be an upper bound corresponding to any feasible solution to the multiperiod problem. Then it can be shown (Sweeney and Tatham, 1976) Rank Ordered Solutions (Low to High cost) Season t Z1l Z12 13 Z21 Z22 z23 Z2c* Z31 "32 z33 ZT2 ZT3 ZTc* Z3c* z2c Z3c3 ZTc z Icl Figure 4. The c best solutions to the static packinghouse location problem by season. ZIc* 75 A u 1 that if Z = Z Z is the maximum possible improvement that can be u made over the solution corresponding to Z then in period t, no static solution with value Ztc such that c>c*, with Z t Z < Z and t tc*t tl - Ztc*t+ -Ztl > Z may become part of an optimal multiperiod solu- tion. Thus it is only necessary to consider the c* best static solutions in each period for possible inclusion in an optimal multi- period solution. Since the number of static solutions ranked in each period depends upon Z it is desirable to have a good upper bound, Zu available. Any feasible solution, such as maintaining the current configuration over the entire planning period, can be used to determine an initial value for Zu. Better upper bounds can be generated as the solution procedure progresses. The following approach is recommended by Sweeney and Tatham (1976). Using the initial value for Z rank order the ct best solutions in each period where c < c*. This may be some pre- t t determined number of solutions, or one may rank order solutions until Z Z > constant. In the former case ct will be fixed across tct tl t periods and in the latter c will vary. A new (and hopefully better) upper bound can then be generated by using dynamic programming to find the multiperiod solution considering as alternatives in each period the c best static solutions. This new upper bound is clearly a feasible solution,and if it consists of a sequence of optimal static u solutions, Ztl, it is the optimum solution. Using the new value for Z one can recompute Z = Zu Z If now Zt- Ztl > Z for all periods 76 one is finished, and Zu is the value of the optimal multiperiod solution. If not, more solutions must be rank ordered in those periods where Ztt tct Z < Z Since the maximum possible improvement for further ranking tl is given by (5.11) 2i = maxA I tct+ Ztl, 0 1 t one might chose to terminate if Z1 is sufficient small. The value of Z is a measure of the maximum possible opportunity loss associated with implementing the solution associated with Zu. Computation cost also limit the number of solutions to be ranked. To obtain the next ct best static solution for the short-run problems a MIP search-limiting process is used (IBM, 1973). This is preferred to the procedure of adding additional constraints, and followed by Sweeney and Tatham (1976), because of its computational simplicity. The dynamic programming model that is used to find the multiperiod optimum defines Z (s) = minimum policy cost when in configuration s with T T more stages to go to final stage, r (s) = a decision yielding Z (s), and computes (5.12) ZT(s) = minimum [ Zs + Z (r) ] for T = 1, 2, ., (sr T- (sr) where Zsr is the cost of configuration s plus the cost of changing from configuration s to r. The recursion (5.12) is equivalent to the method of finding a shortest route in a network. The reason why this dynamic model is preferred over a multiperiod model derives mostly from the fact that solutions for each season are obtained independently from each other. Data inaccuracies for future periods do not affect results obtained for other seasons. Thus for each season one is able to rank order a reasonable number of alternatives which is often of interest. This procedure of obtaining solutions for each season also follows Geoffrion's recommendation (1975) for solving the same model with different requirement scenarios. The computational difficulty of a multiperiod model when compared to this dynamic model should also be considered. The number of integer variables and row constraints, both limiting factors in computational feasibility, are multiplied by the number of seasons considered and the number of constraints is increased by the between periods constraints that have to be added to the model. The dynamic programming model that gives the minimum cost solution over the seasons considered can be solved by hand for a problem of reasonable size. Most important of all, however, is the fact that the dynamic model presented and used in this study includes the costs of moving from one plant configuration to another. Obviously, these costs influence the sequence of configurations that the dynamic programming procedure will select as optimal for the multiperiod problem. CHAPTER VI ESTIMATION OF MODEL PARAMETERS AND COEFFICIENTS To determine the cost minimizing number, location, size, supply and market areas of citrus packinghouse, using the model of Section V-2, parameters and coefficients of the model need to be computed. In this chapter the number of products (Section VI-1.1), origins (Section VI-1.2), supply (Section VI-1.3), market demand (Section VI- 1.4), and processing capacity (Section VI-1.5) are established. Following a description of the assembly operations (Section VI- 2.1), transfer costs are estimated (Sections VI-2.2 through VI-2.4). Present and potential packinghouse locations are defined in Section VI-2.5 and packing costs are discussed in Section VI-2.6. Capacity constraints for existing and new packinghouses are defined in Section VI-2.7. The computation of the model's cost coefficients is discussed in Section VI-2.8. VI-1. Production and Demand Constraints VI-1.1. Number of Products The major Florida citrus fruits are oranges, grapefruit, tangerines, Temples and tangelos. Orange and grapefruit represent 79 percent of Florida fresh citrus shipments (Table 11). Fresh grapefruit offer a remarkable marketing challenge as detailed in Section 11-3. The 79 inclusion of tangerines, Temples and tangelos in the number of products in the model would bring to 96 the percentage of fresh citrus production under analysis. Thus, the model could include any number of products between two and five depending on level of aggregation. Since model size (Section V-2) is proportional to the number of products considered the study is made for oranges (h=l) and grapefruit (h=2). This is justified by the fact that other varieties mature between the peak orange and grapefruit packing seasons and can be packed without adding capacity. Furthermore, production of specialty fruit is not concen- trated in any one set of origins and thus its inclusion in the model is not likely to affect the locations of packinghouses. VI-1.2. Fruit Origins Practically all Indian River fruit is packed in the area. Some packers outside the region pack Indian River fruit for marketing reasons. While the Indian River and Interior production regions are technically geographically adjacent to one another, there is little fruit production in the area near the southern two-thirds of the boundary. Furthermore, given the characteristics of the Indian River region, as discussed in Chapter II, it is selected for application of the model. Data on citrus production are available by section and land grant in the State of Florida. This is the lowest level of aggregation by which fruit origins can be represented. Sections can be grouped by township and range, and these by county. There are 34 orange and/or grapefruit producing counties in Florida with 23 counties accounting for 97.4 percent of the total production (Appendix Table A-I). 80 Since Chern (1969) considered the county as too aggregative for the purpose of his study, he aggregated townships into larger units of less than county size. For convenience and accuracy in measuring the distance from production origins to plant sites, an attempt was made to define a production origin as consisting of approximately nine town- ships. However, due to the geographic restrictions and the uneven dispersion of producing townships, some production origins were specified to have more or less than nine townships. In this study the Indian River region is divided in 13 fruit origins as in Chern (1969), and indexed, i, according to Figure 5. Although, in reality, shipments flow from groves to packinghouses, to markets, it is assumed that those flows take place between central reference points. This assumption implies that costs of shipments be- tween all sets of groves in region i, packinghouse in j, and markets in k, or processing plants in 1, do not differe from the cost of shipment between the centers of those regions. A town or city, as large and as close as possible to the center of the origin, is selected to represent the assembly point in each production origin. VI-1.3. Fruit Supply For each origin, i, and each type of fruit, h, the supply to the packinghouses is calculated as: Qhi = Number of crees (age, variety, location) x yield (age, variety) 9 average fraction packed fresh (variety) x (100/packout percentage (type of fruit, region). Figure 5. The orange produc- tion origins for fresh fruit assembly in Florida, in Chern (1969). Source: Chern, 1969, p. 47. 82 The number, age, variety and location of trees as of January of 1976 were obtained from the Florida Crop and Livestock Reporting Service (1976b). Data were accumulated for oranges and grapefruit at each origin. Trees designated as "unidentified" by the Florida Crop and Livestock Reporting Service in the round orange and grapefruit sections of the report are prorated to the identified varieties according to the dis- tribution of total identified trees. Trees set in 1975 are considered one year old in the 1975-76 season. Yield data are published by Fairchild (1977) for early and mid- season oranges, Valencia oranges, seedy grapefruit, white seedless grape- fruit, and pink seedless grapefruit (Table 17). For the 1975-76 season, the number of trees of each age and variety, at each origin, multiplied by the appropriate yield gives total produc- tion. Only a percentage of this total production is packed as fresh fruit, and certain varieties are more predominantly packed fresh (Table 18). Thus, total production of each variety is adjusted for each origin by the average of the percentage used fresh in the past three years (Table 18). These more recent data are assumed to better reflect the present consumer preferences for the different types of fresh fruit. Furthermore, fruit shipped from origins to packinghouses exceeds fruit used fresh and shipped from the packinghouses. Packout, or the percentage of fruit going through the house that is actually packed, is 64.8 percent for oranges and 71.9 percent for grapefruit in the East Coast (Hooks and Kilmer, 1977a). Dividing total quantity used fresh by the packout percentage of each kind of fruit one obtains an estimate of the total supply of fruit from each origin, Qhi' Table 17.--Box Yield Estimates by Age of Tree, Average of 1966-67 through 1975-76 Seasons Age Oranges Grapefruit of Tree Early and White Pink Mid Season Valencias Seedy Seedless Seedless - - - - - - 1 3/5 bu. boxes/tree - - - - - 1-3 0. 0. 0. 0. 0. 4 0.81 0.66 1.22 1.59 1.48 5 1.03 0.85 1.52 2.00 1.81 6 1.25 1.05 1.82 2.41 2.14 7 1.48 1.26 2.09 2.78 2.49 8 1.72 1.47 2.34 3.09 2.86 9 1.95 1.69 2.60 3.39 3.24 10 2.19 1.90 2.85 3.71 3.61 11 2.43 2.12 3.11 4.01 3.99 12 2.66 2.33 3.36 4.33 4.36 13 2.86 2.48 3.57 4.59 4.67 14 3.01 2.63 3.77 4.87 4.98 15 3.28 2.78 3.97 5.14 5.29 16 3.48 2.93 4.18 5.42 5.60 17 3.69 3.08 4.38 5.69 5.92 18 3.90 3.23 4.58 5.96 6.23 19 4.10 3.38 4.79 6.23 6.54 20 4.32 3.57 5.07 6.53 6.72 21 4.56 3.78 5.41 6.86 6.76 22 4.79 3.99 5.76 7.21 6.79 23 5.02 4.21 6.09 7.54 6.84 24 5.25 4.42 6.44 7.88 6.88 25 & over 5.48 4.63 6.78 8.21 6.92 Source: Fairchild, 1977. 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