A MULTI-ECHELON FACILITY LOCATION MODEL FOR A FOOD …

The Pennsylvania State University

The Graduate School

College of Engineering

A MULTI-ECHELON FACILITY LOCATION MODEL

FOR A FOOD SUPPLY CHAIN USING BENDERS DECOMPOSITION

A Thesis in

Industrial Engineering

by

Rajeev Murugaswamy

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

May 2011

ii

The thesis of Rajeev Murugaswamy was reviewed and approved* by the following:

Paul Griffin

Professor and Head of Industrial and Manufacturing Engineering

Thesis Advisor

A Ravi Ravindran

Professor of Industrial and Manufacturing Engineering

Vittal Prabhu

Professor of Industrial and Manufacturing Engineering

*Signatures are on file in the Graduate School

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ABSTRACT

For school children in Mexico, the cost of healthcare for obesity-related issues is increasing. The

Mexican government aims at incorporating healthy food habits among school children through a

National School Lunch Distribution Program, thereby reducing obesity-related health costs. For

this purpose, a food supply chain needs to be developed to transport different categories of food

from food growers to schools. One of the most important strategic problems to be addressed is to

design the location of distribution centers for the supply chain.

The objective of this thesis is to develop a facility location model to locate the distribution

centers for a food supply chain based on the idea of Mexican School lunch program. A multi-

commodity distribution network comprising farmers, distribution centers (DCs) and central

preparation kitchens (CPKs), is formulated using a mixed integer linear program for this

application. The CPKs are identified by grouping schools using clustering analysis based on their

proximity. The facility location model is then solved using Benders Decomposition algorithm for

different trial values with a maximum of 50 farmers, 50 DCs, 50 CPKs and 10 commodities. The

solution technique is implemented using MATLAB and the results are analyzed.

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TABLE OF CONTENTS

List of Figures .............................................................................................................................................. vii

List of Tables .............................................................................................................................................. viii

Acknowledgements ...................................................................................................................................... ix

CHAPTER 1: INTRODUCTION ......................................................................................................................... 1

1.1 Overview ....................................................................................................................................... 1

1.2 Problem Statement ....................................................................................................................... 2

CHAPTER 2: LITERATURE REVIEW ................................................................................................................. 3

2.1 Food Supply Chain ......................................................................................................................... 3

2.1.1 Conventional versus Food Supply Chain ............................................................................... 3

2.1.2 Characteristics of Food Supply Chain .................................................................................... 4

2.2 Global versus Local Food System .................................................................................................. 5

2.2.1 Global Food System .............................................................................................................. 5

2.2.2 Local Food System ................................................................................................................. 5

2.3 Environmental Intervention Strategies ......................................................................................... 6

2.3.1 Dietary Guidelines for Americans ......................................................................................... 7

2.3.2 Food Guide Pyramid .............................................................................................................. 7

2.3.3 National 5 A day for better health campaign ....................................................................... 8

2.4 National School Lunch Program (NSLP) ........................................................................................ 8

2.5 Research in Modeling Food Supply Chain ................................................................................... 10

2.6 Facility Location Problems .......................................................................................................... 12

2.6.1 Classification of Facility Location Problems ............................................................................... 13

2.6.2 Solution Approaches for Facility Location Problems .......................................................... 14

2.6.3 Research in Facility Location Problems ............................................................................... 16

CHAPTER 3: PROCESS METHODOLOGY ....................................................................................................... 18

CHAPTER 4: SUPPLY CHAIN DESIGN ............................................................................................................ 20

4.1 Supply Chain for Mexican School Lunch Program ...................................................................... 20

4.1.1 Farmers ............................................................................................................................... 20

4.1.2 Distribution Centers (DCs) .................................................................................................. 20

4.1.3 Central Preparation Kitchens (CPKs) ................................................................................... 21

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4.1.4 Schools ................................................................................................................................ 21

4.2 Supply and Distribution Flow Structure ...................................................................................... 22

4.3 Assumptions for the facility location model ............................................................................... 23

CHAPTER 5: CLUSTER ANALYSIS .................................................................................................................. 24

5.1 Need for Clustering ..................................................................................................................... 24

5.2 Clustering in Location Analysis .................................................................................................... 24

5.3 K-means Clustering Algorithm .................................................................................................... 25

5.4 Demonstration of Cluster Analysis in SAS ................................................................................... 26

5.4.1 Methodology ....................................................................................................................... 26

5.4.2 Location Coordinates for Schools ....................................................................................... 26

5.4.3 SAS Output: Clusters ........................................................................................................... 28

5.4.4 Locations of CPKs ................................................................................................................ 29

CHAPTER 6: FACILITY LOCATION MODEL FORMULATION .......................................................................... 31

6.1 Model Formulation: Multiple Sourcing DCs ................................................................................ 31

6.1.1 Notation .............................................................................................................................. 32

6.1.2 Decision Variables ............................................................................................................... 32

6.1.3 Objective Function .............................................................................................................. 32

6.1.4 Constraints .......................................................................................................................... 33

CHAPTER 7: BENDERS DECOMPOSITION HEURISTICS ................................................................................ 35

7.1 Benders Decomposition .............................................................................................................. 35

7.2 Benders Algorithm ...................................................................................................................... 36

7.3 Benders Algorithm in MATLAB .................................................................................................... 40

7.4 Analysis of Results ....................................................................................................................... 43

7.4.1 Benders Algorithm .............................................................................................................. 43

7.4.2 MILP .................................................................................................................................... 44

7.5 Sensitivity Analysis ...................................................................................................................... 46

7.5.1 Throughput Limit: 400 to 550 units .................................................................................... 46

7.5.2 Throughput limit: 350 to 500 units ..................................................................................... 47

CHAPTER 8: CONCLUSION AND FUTURE WORK ......................................................................................... 49

8.1 Limitations of the model ............................................................................................................. 49

8.2 Summary of Contributions .......................................................................................................... 50

8.3 Future scope ............................................................................................................................... 50

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REFERENCES ................................................................................................................................................ 51

APPENDIX A: ................................................................................................................................................ 55

I.SAS Code for Clustering Analysis........................................................................................................... 55

II.SAS Output: Clusters of Schools with Cluster membership ................................................................. 56

III.SAS Output: Cluster Means and Cluster Graph for Trial 2 values ....................................................... 60

IV.SAS Output: Cluster Means and Cluster Graph for Trial 3 values ....................................................... 61

V.SAS Output: Cluster Means and Cluster Graph for Trial 4 values ........................................................ 62

APPENDIX B: ................................................................................................................................................ 64

I.MATLAB Code for Benders Decomposition .......................................................................................... 64

II.MATLAB Output: DC Locations for Trial 1 values ................................................................................. 69

III.MATLAB Output: DC Locations for Trial 2 values ................................................................................ 72

APPENDIX C ................................................................................................................................................. 81

I.MATLAB Code for solving MILP ............................................................................................................. 81

II.MATLAB Output: .................................................................................................................................. 92

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LIST OF FIGURES

Figure 1: Conventional Supply Chain ........................................................................................................... 3

Figure 2: Food Supply Chain ........................................................................................................................ 3

Figure 3: Classification of Supply Chain Models [Min and Zhou 2002] ................................................... 10

Figure 4: Multi-facility Location Methods ................................................................................................. 14

Figure 5: Process Methodology .................................................................................................................. 18

Figure 6: Supply Chain Flow for School Lunch ......................................................................................... 20

Figure 7: Distribution Flow Structure for School Lunch ............................................................................ 22

Figure 8: Process Methodology for cluster analysis ................................................................................... 26

Figure 9: Map of Mexico showing the limits of coordinates ...................................................................... 27

Figure 10: SAS Output for Clustering Analysis ......................................................................................... 28

Figure 11: Map of Mexico showing locations of CPKs - potential sites for DCs....................................... 30

Figure 12: Model for facility location problem........................................................................................... 31

Figure 13: Process Methodology for Benders Decomposition ................................................................... 36

Figure 14: Convergence of Upper bound and Lower bound ....................................................................... 37

Figure 15: Qijkl Distribution graph for commodity 1 ................................................................................... 44

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LIST OF TABLES

Table 1: Cost of Healthcare in Mexico ......................................................................................................... 1

Table 2: Trial values for clustering analysis ............................................................................................... 28

Table 3: Cluster centers and their corresponding locations ........................................................................ 29

Table 4: Trial values of input parameters ................................................................................................... 40

Table 5: Processing times in seconds .......................................................................................................... 43

Table 6: Values of Qijkl for throughput limit 400 to 550 units .................................................................... 46

Table 7: Values of Qijkl for throughput limit 400 to 550 units .................................................................... 47

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ACKNOWLEDGEMENTS

I would like to take this opportunity to thank everyone, who has helped me in the successful

completion of this thesis.

First and foremost, I owe my deepest gratitude to my advisor Dr. Paul Griffin for his guidance,

encouragement and support throughout my thesis. He extended his support in a number of ways

and remained as a source of inspiration and motivation.

It‟s a pleasure to thank Dr.A.Ravi Ravindran, who accepted to be a reader for this work. I

definitely gained valuable knowledge from his courses, which helped me in this thesis.

I would like to express my sincere gratitude to Dr.Vittal Prabhu, who provided valuable

suggestions to select this field of research and accepted to be a reader.

I‟m indebted to many of my colleagues and friends, who helped me during the course of this

thesis, through their suggestions and pep talks.

Lastly, I would like to thank my mother, sister and brother-in-law for their ever-lasting love and

faith on me. That served as a constant motivation to complete my thesis.

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CHAPTER 1: INTRODUCTION

1.1 Overview

In Mexico, 26% of school children are obese and overweight, which is attributed to unhealthy

food habits. The cost of healthcare to treat obesity and its consequences is found to be an

increasing trend as shown below:

Year Cost of Healthcare for Obesity

2000 35 Billion pesos

2008 68 Billion pesos

2017 167 Billion pesos (estimated)

Table 1: Cost of Healthcare in Mexico

The Ministry of Public Education has initiated a National School Lunch Distribution Program

called PRONAES. The objective of PRONAES is to prevent the overweight and obesity among

school children 4 to 15 years old by incorporating healthy food habits into their school lunch

distribution program.

The School Lunch Program offers plenty of scope to design a food supply chain with problems

in the areas of supplier selection, distribution network, facility location, transportation, logistics,

etc. The focus of this work is to design the food supply chain and model the facility location

problem for locating the distribution centers. A secondary objective is to study the food supply

chain and its characteristics, environmental intervention strategies, National School Lunch

Program in USA and supply chain models in literature. This provides a good understanding of

the supply chain for the Mexican School Lunch Program.

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1.2 Problem Statement

The problem is to develop a multi-commodity, multi-echelon facility location model to identify

the locations of distribution centers for a food Supply Chain, as in Mexican School lunch

distribution program. This problem requires supply chain partners in different echelons to be

defined, with other assumptions for satisfying demand. A sub-problem is to identify the locations

of customer zones which have aggregated demand as the individual customers.

In the literature, single sourcing of DCs is generally assumed, i.e. a customer zone is served only

by one dedicated DC. In this problem, the DCs are considered as multiple-sourcing to the

customers i.e. multiple DCs can serve a single customer zone. This assumption is based on the

perishability and seasonality factors inherent in the food supply chain.

The remainder of this thesis is organized as follows: The literature is reviewed in Chapter 2

explaining the characteristics of food supply chain, National School Lunch program,

classification of supply chain models and facility location models. In Chapter 3, the process

methodology followed is presented. Chapter 4 explains the design of the food supply chain.

Clustering analysis is presented in Chapter 6 and facility location model formulation is

developed in Chapter 7. In Chapter 8, Benders Decomposition heuristics with equations are

specified and the results of the solution implemented in MATLAB are analyzed. Lastly, Chapter

9 concludes the thesis mentioning the limitations of the model and future scope.

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CHAPTER 2: LITERATURE REVIEW

2.1 Food Supply Chain

2.1.1 Conventional versus Food Supply Chain

A Supply Chain can be defined as “all the activities involved in delivering a product from raw

material through to the customer including sourcing raw materials and parts, manufacturing and

assembly, warehousing and inventory tracking, order entry and order management, distribution

across all channels, delivery to the customer, and the information systems necessary to monitor

all of these activities” [1]. Supply chain management is the process that integrates and

coordinates all the activities involved in the movement of raw material from suppliers until the

final delivery of the product to the customer.

Figure 1: Conventional Supply Chain

Figure 2: Food Supply Chain

The term „food chain‟ refers to the total supply process from agricultural production, harvest or

slaughter, through primary production and/or manufacturing, to storage and distribution, to retail

sale or use in catering and by consumers [2]. A food supply chain is a more dynamic and

complex system as it involves food safety, quality and temperature requirements, which directly

affect consumers. Although the overall supply chain structure is similar, supply chain partners

Farmer / Food

Grower Harvesting

Food Processor

Distributor Retailer /

Food Preparer

Food Consumer

Extraction of Raw Materials

Supplier Manufacturer Distributor Retailer Consumer

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for a food product differ from a conventional supply chain. The food supply chain starts from a

farmer or food grower, through to food harvest, food processing, distribution and finally to

consumer. In some cases, the supply chain involves a food preparer as in restaurants, worksite

canteens or other similar institutions.

2.1.2 Characteristics of Food Supply Chain

Fisher [3] states that the nature of a product‟s demand needs to be considered before devising a

supply chain. Due to the predictable demand patterns, globalization of food markets and the

presence of competitive retailers, a food product can be considered as a functional product and

hence infer that an efficient supply chain is suitable for food products. However, a food supply

chain cannot be categorized as an efficient or responsive supply chain based on the demand

patterns. The nature of food consumer demand determines the characteristics of the food supply

chain. A food consumer demands chilled and fresh food, products with short shelf-life, own-label

food products, seasonal, regional and specialty foods at a low price [4]. Based on these demand

characteristics, a food supply chain should have the following characteristics: refrigerated

storage and transportation to supply safe and healthy food, time-critical environment, cross-

docking to reduce shelf-life, technology to measure and maintain food quality and information

technology to support the movement of products and smooth coordination in the supply chain.

Also, when designing the distribution, transportation and inventory models of the supply chain,

food characteristics such as perishability, fresh or processed food need to be considered.

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2.2 Global versus Local Food System

2.2.1 Global Food System

Global food can be understood as an integrated retail/wholesale buying system in which sources

within a country or across countries supply food products through retail supermarkets,

restaurants, and other institutional markets served by distributors. This usually involves longer

travel distances and associated higher fuel use and emissions [5]. Though local food system was

used in earlier days, global system gained popularity as a result of globalization, seasonality,

cheaper prices in certain parts of the world, etc.

2.2.2 Local Food System

Local food can be understood as food that is produced, processed, and distributed within a

particular geographic boundary that consumers associate with their own community [6].

According to the definition adopted by the U.S. Congress in the 2008 Food, Conservation, and

Energy Act (2008 Farm Act), the total distance that a product can be transported and still be

considered a “locally or regionally produced agricultural food product” is less than 400 miles

from its origin, or within the State in which it is produced”.

Based on the market arrangements that reach the final consumer, the local food can be classified

as:

1. Direct-to-consumer marketing

A local food marketing arrangement in which producers sell agricultural products directly

to the final consumers, such as sales-to-consumers, through farmers‟ markets, CSAs

(Community Supported Agriculture) or farm-stands.

2. Direct-to-retail / foodservice marketing

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A local food marketing arrangement in which producers sell agricultural products directly

to the final sellers, such as sales-to-restaurants, supermarkets, or institutions, like schools

and hospitals. Farm-to-school program connect schools with local farms to supply fresh,

local produce in order to improve student nutrition, educate students about food and

health, support local and regional farmers and the economy of the local community.

2.3 Environmental Intervention Strategies

Story et al [7] present an ecological framework to explain the significance of various food eating

environments and their interactions with people. A person‟s individual behavior to make healthy

food choices can occur only in an environment which has access to healthy food. The food

environment influences what people eat and hence environmental intervention strategies can help

improve the healthy eating behavior of people. Children consume at least two meals, plus snacks

in schools [8]; the school food environment affects what children eat and how much they eat.

Hence the school food environment provides many opportunities to deploy intervention

strategies to prevent obesity in children.

A study by European Commission [9] reviewed the effectiveness of school intervention schemes

in promoting the consumption of fruits and vegetables among school children. The results

showed a significant increase in the intake of fruits and vegetables. In addition, the schemes

influenced healthy eating behavior for a long time. The results suggest there is no overall

decrease in intake of fruits and vegetables after intervention programs in any of the studies.

Hence effective intervention strategies deployed in the school food environment can help reduce

the epidemic of obesity and overweight related health problems among children by increasing

healthy intake of food. Another study in Finland [10] observed that around 70-90% of school

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children eat school lunch. The results showed that children eating school lunch tend to make

food choices closer to nutritional recommendations when compared to those students not eating

school lunch in the same proportion.

The following are some of the nutritional intervention and guidelines that help to pass the

message to people and create awareness about health and nutrition:

2.3.1 Dietary Guidelines for Americans

The Dietary Guidelines for Americans [11] is published by U.S. Department of Health and

Human Services (HHS) and the U.S. Department of Agriculture (USDA) every five years to give

science-based advice on food and physical activity choices for health [DGA 2005

http://www.health.gov/dietaryguidelines/]. These guidelines are used by policy makers, nutrition

educators, nutritionists, healthcare providers and general public. The purpose of Dietary

Guidelines is to encourage people to eat fewer calories, remain more active and make healthier

and wiser food choices. It contains nutrition information, calorie needs, and recommendations

for nutrition intake, etc.

2.3.2 Food Guide Pyramid

The Food Guide Pyramid [12] is a pyramid-shaped visual guidance tool to help people consume

food from different groups of food. The pyramid also suggests different food from each of the

groups. It serves as a guide to eat healthy as per the recommendations of Dietary Guidelines for

Americans. Also, it helps to plan menu choices for different consumer preferences to align with

the goals of each group of food.

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2.3.3 National 5 A day for better health campaign

The National 5 A Day Program [13] gives a theory-based message to people to eat five or more

servings of fruits and vegetables daily for more than 10 years [14].

2.4 National School Lunch Program (NSLP)

The National School Lunch Program (NSLP) was established in the United States under the

National School Lunch Act in 1946 “to safeguard the health and well-being of nation‟s children,

encourage domestic consumption of nutritious agricultural commodities, by assisting states

through grants-in-aid and providing adequate supply of food and other facilities for the operation

and maintenance of non-profit school lunch programs” [15]. The main purpose behind the

development of this program was to alleviate the malnutrition and poverty among the nation‟s

children by providing school meals. Due to the excessive costs incurred by school districts for

equipment, infrastructure and labor to provide school meals, there was a need for permanent

federal funding to assure the continuity of the program over a period of years and NSLP served

this purpose.

Following the success of NSLP and the research contributions in nutrition, the Child Nutrition

act was passed in 1966. The purpose of this program was to strengthen NSLP under the authority

of the Secretary of Agriculture to meet more effectively the nutritional needs of the children. The

school food service funds were transferred from several federal agencies to USDA (United States

Department of Agriculture). This unifying of school food services under one agency helped to

achieve uniform standards of nutrition, management of funds and assurance of program

continuity. Also, another amendment of National School Lunch Act in 1970 set the uniform

national guidelines for determining eligibility criteria for free and reduced-price school lunches

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based on the income level of the families. More changes in 1990 required the NSLP to meet the

nutrition requirements of Dietary Guidelines for Americans [16].

Recently, the NSLP is more focused on addressing the problem of obesity and overweight

among children by providing nutritionally balanced, reduced-price or free lunches to school

children every day. Any child in a school participating in NSLP can purchase meals for free or at

reduced price through this program. Children from families whose income level is below 130

percent of the poverty level are eligible for free lunches. Children from families whose income

level is between 130 percent and 185 percent of the poverty level are eligible for reduced-price

lunches [17]. Children from families whose income level is over 185 percent of poverty level

need to pay full-price. In 2009, the NSLP was operating in over 101,000 public and non-profit

private schools providing lunches to more than 31 million children [18] each school day. The

USDA offers cash subsidies and donated commodities to participating schools in order to serve

school lunches as per the recommended nutritional guidelines.

As per the School Food Purchase study by USDA [19] in 1998, the food acquisition for NSLP

was classified as: commercially-purchased, USDA-donated commodities, processed food

products containing donated commodities among which 83% were from commercial purchases.

The food purchasing behavior was influenced by school participation size, access to markets,

availability of vendors, storage capacity, etc. Large school districts were involved in high volume

purchases compared to small school districts.

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2.5 Research in Modeling Food Supply Chain

Min and Zhou [20] discuss that a supply chain model should be reflective of the real-world

dimensions, yet not too complex to solve. Hence a model builder must compromise between

reality and model complexity. The key components of supply chain modeling are to identify the

supply chain drivers, supply chain constraints and supply chain decision variables. They classify

supply chain models as follows: i) deterministic models ii) stochastic models iii) hybrid models

iv) IT-driven models.

Figure 3: Classification of Supply Chain Models [Min and Zhou 2002]

While designing a supply chain, efficiency and responsiveness aspects of the logistics are built

into the model, whereas product quality is assumed not to influence the design. However, in the

case of a food supply chain, the product quality is intrinsic to the supply chain. Van der Vorst et

al [21] discuss about the specific requirements for modeling a food supply chain. A food supply

chain needs to preserve the food quality through the use of sophisticated environmental

conditioning techniques and the reduction of lead times. They developed an integrated

simulation modeling approach to design a food supply chain based on food quality and

sustainability in addition to logistics. A simulation environment is developed to implement this

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model and analyze the results. It is found that incorporating quality and sustainability has

improved the quality and speed of decision making.

Georgiadis et al [22] present a comprehensive model for addressing strategic issues in a food

supply chain, specifically capacity planning issues. A System Dynamics (SD) methodology is

employed for modeling and simulation of a dynamic system. SD approach incorporates casual

loop diagrams with stock and flow variables to model the system and provides useful qualitative

insights. Mathematically, the stock-flow diagram can be transformed into a system of differential

equations and numerically solved by simulation. A generic single-echelon model is presented

using this approach and is extended into a multi-echelon model. This model can be used to

identify effective policies and optimal strategies for strategic decision making problems.

The use of time-dependent quality information to design distribution systems for perishable food

is scarce in literature. Giannakourou et al [23] proposed a Time Temperature Integrators (TTI)

based distribution management system to monitor the quality status of the food products. The

least-shelf-life-first-out (LSFO) system based on time-dependent information is demonstrated

using Monte Carlo simulation and the results showed minimization of rejected products at the

consumer end.

The challenge in the food supply chain is to maintain the logistics performance when handling

temperature sensitive and perishable products (TSPP), which require shorter delivery times and

sophisticated storage and transportation systems. Cold Chain Management (CCM) for handling

the logistics of TSPP is discussed by Ju-Chia Kuo et al [24]. A Multi-temperature joint

distribution (MTJD) model is proposed to handle multiple products with different temperature

requirements with the objective of balancing between quick deliveries and diversifying goods. It

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uses cold chain equipment such as cold boxes, cold cabins, eutectic plates, etc. to enable storage

and distribution of products with different temperatures in same warehouses and trucks. This

integrates the processes and provides one-stop logistics for different categories of food.

Cross Docking Logistics (CDL) is an advanced logistics distribution strategy, which has the

advantages of Cross Docking and information based Systems. The application field of food cold

supply chain involves perishable products with strict requirements of time and low temperature.

Qiu Qijun et al [25] propose a new model of CDL consisting of information, transportation and

distribution module, to achieve the advantages in time, cost and space.

2.6 Facility Location Problems

One of the most important strategic problems when designing a supply chain is the design of its

physical distribution system, which involves the outbound logistics of products from

manufacturers to consumers. A location problem is a spatial resource allocation problem. In the

general location paradigm, one or more service facilities serve a spatially distributed set of

demands. The objective of a location problem is to identify the location of distribution facilities

such that it minimizes the costs or the distances involved or maximizes the customer

responsiveness. Following are some of the facility location decisions to be considered [26]:

Number of facilities

Location of facilities

Size of facilities

Vendor to warehouse mapping

Warehouse to customer mapping

Products stored in Warehouses (Small orders)

Products directly shipped by Vendors (Large orders)

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2.6.1 Classification of Facility Location Problems

Facility location problems for distribution planning can be classified as follows:

1) Continuous and Discrete: A location problem which explores every possible location

along a space continuum or plane is referred to as a continuous location problem; a

location problem which selects from a finite set of potential candidate facilities is referred

to as a discrete location problem.

2) Capacitated and Uncapacitated: A facility without capacity constraints is uncapacitated

and there is no restriction on demand allocation; a facility with a capacity limit is referred

to as capacitated. A capacitated facility location problem needs to be examined for single-

sourcing or multiple-sourcing.

3) Single-stage and Multi-stage: The distribution of products from only one stage of supply

chain to the customers is termed as single-stage location problems; the distribution of

products through more than one stage of supply chain, for instance, manufacturer to

customer via distributor, is termed as multi-stage location problems.

4) Single-commodity and Multi-commodity: In single-product models, the demand, cost

and capacity for several products can be aggregated into a single homogeneous product.

If the products are inhomogeneous, the demand and capacity for each product needs to be

considered separately in the model.

5) Deterministic and Stochastic: If the inputs to the model are known with certainty, they

are called deterministic models; if the model inputs are uncertain, they can be assumed to

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follow a probability distribution which is closest to their property and are called

stochastic models.

6) Static and Dynamic: The location problems modeled for one representative period are

called static; the dynamic models reflect cost, demand, capacities, etc. varying over time

within a given planning horizon.

2.6.2 Solution Approaches for Facility Location Problems

Following are the methods to solve the multi-facility location problems [26]:

1. Optimization Methods

2. Simulation Methods

3. Heuristic Methods

Figure 4: Multi-facility Location Methods

1. Optimization Methods

Optimization methods result in a mathematically optimum solution or a solution of known

accuracy to a location problem. They can be considered as an ideal approach to a location

problem, but they prove to be useful only when the problem size is small. This approach

becomes computationally complex when applied to problems of large size, which are the case in

Multifacility Location Methods

Optimization Methods

Simulation Methods Heuristic Methods

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many real world problems. Hence it may result in lengthy processing times and a compromised

problem definition.

Mixed Integer Linear Programming (MILP) is another mathematical modeling approach which can

handle fixed costs with integer variables. The model is formulated with an objective function such as

minimizing the sum of transportation costs, fixed warehouse costs and handling costs and constraints such

as supplier capacity, customer demand and warehouse throughput. When the problem size is very large,

MILP models may require heuristics to obtain the solution quickly.

2. Simulation Methods

Simulation methods result in sub-optimum solutions to an accurately described problem. This is

particularly useful when handling some real world problems, which require accurate description. In this

case, a sub-optimum solution to an accurate problem description is preferred over an optimum solution to

an approximate problem description. A simulation facility location model refers to a mathematical

representation of a logistics system by algebraic and logic statements manipulated by computer. The

solution is found by repeatedly running the simulation model with different warehouse and allocation

options. However, this requires high degree of skill and intuition from the user.

3. Heuristic Methods

Heuristics are concepts based on experience and intuition that guide in problem solving and reduces the

average time to search for a solution. In location problems, heuristics provide insight into the solution

process, and allow good solutions to be obtained quickly from several alternatives. Some heuristics

include, drop heuristics, Lagrangian relaxation, Benders Decomposition, Dantzig-Wolfe decomposition,

etc.

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2.6.3 Research in Facility Location Problems

Location Theory was first introduced by Weber, who considered the problem of locating a single

warehouse which minimizes the travel distance between the warehouse and a set of spatially

distributed customers. He proposed a material index, to select the location, which if greater than

one, the warehouse should be in closer proximity with the source of raw material or it should be

closer to the market. Francis et al [27], in a survey of locational analysis, reviewed four different

classes of location problems: i) continuous planar ii) discrete planar iii) mixed planar iv) discrete

network.

Many research works have been carried out both in continuous and discrete optimization

problems. Aikens [28] discussed mathematical models for different classifications of location

problems. Brandeau et al [29] presented an overview of different types of location problems,

which have been formulated as optimization problems. A classification scheme is proposed to

categorize location problems along various criteria and finally solution approaches to location

problems are discussed. Hansen et al [30] presented the Big Square Small Square (BSSS)

technique to find the global optimum for continuous planar location problems. Drezner and

Suzuki [31] improved upon the BSSS by replacing the squares with triangles and proposed Big

Triangle Small Triangle (BTST) technique for global optimization of continuous location

problems. This technique requires the construction of bounds to the value of the objective

function to make it efficient.

Klose and Drexl [32] surveyed continuous location models (single-facility and multi-facility

weber problem), network location models (p-median and p-center problem) and mixed integer

programming models (capacitated, uncapacitated, multi-stage, multi-commodity, dynamic,

17

stochastic) with their mathematical formulations. Nozick et al [33] developed an integrated

discrete location model to locate distribution facilities for an automobile manufacturer in US.

The model considered all the demand points as candidates for facility sites and integrated facility

costs, inventory costs, transportation costs and customer responsiveness. The solution presented

multiple outputs with tradeoffs between cost and service to aid in decision-making.

Geoffrion and Graves [34] formulated a mixed integer linear program to design the distribution

system for a multi-commodity, multi-echelon problem. They were among the first to develop a

solution technique to solve this class of problems and implement it on a real world problem.

They used Benders Decomposition algorithm to solve by considering the linear programming

subproblem and decomposing it into independent transportation problems. Hindi and Basta [35]

discussed a two-stage multi-commodity location problem, in which the stages from supplier to

warehouse and warehouse to customer are addressed through separate variables and combined in

the formulation. They modeled the problem using mixed integer linear programming formulation

and solved it using branch and bound procedure to find the locations of warehouses and the

shipping schedule. Pirkul and Jayaraman [36] performed a computational study on a PLANWAR

model, a mixed integer programming model for multi-commodity, multi-echelon location

problem. They proposed an efficient heuristic based on Lagrangian relaxation of the problem,

which simplifies the problem by relaxing some of the constraints and introducing a penalty

function.

18

CHAPTER 3: PROCESS METHODOLOGY

The objective is to identify the location of DCs using facility location model, by considering the

Mexican School Lunch Program as the application. As a first step, the supply chain for this

problem is to be defined. Secondly, the demand points are aggregated using clustering analysis

based on their proximity. The facility location model is then formulated using mixed integer

linear programming (MILP) and the model is solved using Benders Decomposition algorithm.

The solution is obtained using MATLAB and finally the results are analyzed. The steps involved

in the process are mentioned in Figure 5.

Figure 5: Process Methodology

1. Define the supply chain partners, stages and

assumptions

2. Aggregate the demand points using clustering

analysis

3. Formulate the facility location model using MILP

4. Solve the model using Benders Decomposition

Algorithm

5. Analyze the results

19

The Process methodology is explained below:

Step 1:

The supply chain structure for the problem is defined which includes the terminologies,

suppliers, customers, stages in the distribution network and assumptions. This serves as the basis

in obtaining the variables and other parameters when formulating the model.

Step 2:

Clustering analysis is carried out in SAS to identify the location of central preparation kitchens

(CPKs) for clusters of schools. In this way, the demand is aggregated from schools to CPKs. The

CPKs thus identified serve as customers for the DCs. Also, the CPKs are considered as potential

candidates for the location of DCs. Hence there is a location problem to be solved to identify the

number and location of DCs.

Step 3:

The problem at hand is a discrete optimization problem, with a finite set of candidate DCs, which

are assumed to be all the demand points, i.e. CPKs. This location problem is formulated as a

mixed integer linear programming model, the objective of which is to minimize the

transportation costs and the fixed DC costs.

Step 4:

The model thus formulated is solved using Benders Decomposition algorithm, a heuristic to

solve large scale optimization problems of this class. The algorithm is implemented using

MATLAB.

Step 5:

The solution is analyzed and the results are discussed to gain useful insights.

20

CHAPTER 4: SUPPLY CHAIN DESIGN

4.1 Supply Chain for Mexican School Lunch Program

The general supply chain for Mexican School Lunch program is defined as shown in Figure 6.

The supply chain starts with farmers growing the food produce. The produce from farmers are

aggregated to supply the distribution centers (DCs). The DCs supply the Central Preparation

Kitchens (CPKs) based on their demand. The CPKs prepare and serve food to a group of schools

which are located in close proximity.

Figure 6: Supply Chain Flow for School Lunch

4.1.1 Farmers

Farmers grow organic produce and crops that are natural to the geographic location. Each farmer

grows several commodities with a specific agricultural production capacity. The DCs are

supplied from a set of local farmers that are dedicated to the DC served.

4.1.2 Distribution Centers (DCs)

Distribution centers act as intermediary between the suppliers and customers. Each DC has a

range of throughput that can be handled. The throughput is common across all commodities

Farmers DCs CPKs Schools

FLOW OF PRODUCT

FLOW OF DEMAND

21

based on an intensity factor. The produce received from farmers are sorted and distributed to

CPKs to satisfy the demand.

4.1.3 Central Preparation Kitchens (CPKs)

Central Preparation Kitchens are involved in the process of cutting, cleaning, cooking, portioning

and packing the fresh produce supplied by the distribution centers. After processing of food, the

cooked food is transported warm in special boxes daily to the schools. This necessitates the

location of central preparation kitchens within close proximity to the schools. Also, other ready-

to-eat food, snacks and beverages are transported corresponding to the demand from schools.

4.1.4 Schools

The schools are the end consumer of the food. Each school has a demand that is satisfied by

CPKs. The activities carried out in a school depend upon the infrastructure.

Schools with Kitchens

The processes carried out are cooking/heating of sandwiches, soup, meat, etc. before serving. All

other items such as beverages, snacks and fruits are served from storage.

Schools without Kitchens

The processes carried out are serving beverages, snacks and fruits from storage. The hot food is

received from CPKs daily for serving. It is assumed that hot meal is served at lunch time. Hence

supply reaches the school within a time window before the lunch time.

In the problem under consideration, all schools are assumed as without kitchens to allow for

uniformity in supply.

22

4.2 Supply and Distribution Flow Structure

Figure 7: Distribution Flow Structure for School Lunch

The structure of the supply and distribution is devised based on the problem under consideration.

The farmers are assumed local to DCs, since the Mexican School Lunch Program aims at

obtaining organic produce from local farmers. The DCs distribute the food products to the

Central Preparation Kitchens (CPK). It is assumed that a DC can serve many CPKs and a CPK

Distribution

Centers (DCs)

Central Preparation

Kitchens (CPKs) Schools Farmers

CPK 1

CPK 2

CPK 3

CPK 4

Farmer 1

DC 1

DC 2

Farmer 2

Farmer 3

Farmer 4

Farmer 1

Farmer 2

Farmer 3

Farmer 4

Cluster Analysis

Facility Location Model

23

can receive from many DCs in order to satisfy demand at CPKs. The multi-sourcing from DCs

assumed here is not so common but the application demands it due to food perishability and

seasonality. Finally, the CPKs cater to the demand of a fixed set of schools.

The CPKs are identified using cluster analysis and assigned to schools within close proximity.

Once the CPKs are thus established, the resulting distribution system is formulated as a facility

location model with the CPKs as customers.

4.3 Assumptions for the facility location model

The facility location model is a discrete optimization problem with the potential candidate sites

for DCs given. The assumptions are as follows:

1. A multi-facility location model for which number of DCs are specified by „p‟.

2. A multi-echelon model with farmers to DCs to CPKs.

3. A multi-commodity distribution model involving a maximum of 10 commodities.

4. A capacitated plant model with limited agricultural production capacities for farmers.

5. The time horizon is static.

6. A farmer supplies produce through 1 DC only i.e. dedicated farmers to DCs.

7. The DCs are assumed as multiple-sourcing, i.e. 1 CPK can be served by more than 1 DC.

24

CHAPTER 5: CLUSTER ANALYSIS

5.1 Need for Clustering

A location problem with candidate sites specified is a discrete location problem, whereas the

problem without any potential sites necessitates finding locations anywhere in a continuous

space [37]. The solution to a continuous problem is based on distance and as the solution can be

anywhere in the space, it‟s difficult to establish bounds for the search space and iterative

procedures may not converge. A discrete location problem is considered for this application,

which requires a set of potential candidate sites to choose the facilities which optimize the

objective. But, only customer locations i.e. locations of schools are given for this problem.

Hence cluster analysis is used to identify the potential candidate sites for DCs based on the

locations of schools.

5.2 Clustering in Location Analysis

Clustering is the process of grouping items by similarities among any of the attributes of the

items. The items within a cluster have a high degree of association between them whereas the

items in different clusters are distinct from one another [37]. Clustering is used in location

analysis to group the individual demand points into aggregated customer zones. In our problem,

the individual demands at schools can be transferred into aggregated demand at Central

Processing Kitchens (CPKs) by grouping schools based on their proximity. The objective of

clustering is to identify the locations of CPKs, identified by cluster centers and the schools

served by them, identified by cluster membership. The locations of CPKs, in turn, are used as the

potential candidate sites for the locations of DCs in the facility location problem.

25

Following assumptions are considered while performing clustering:

1) It is to be figured out whether the observations need to be standardized or not. This is

done to ensure a common unit of measure among the items being clustered. In our case,

the values of items are coordinates of locations of schools in latitudes and longitudes.

2) When the items are allowed to be members of more than one cluster, it is called

overlapping clusters. In our case, exclusive clustering is considered, with schools within a

cluster served exclusively be a single CPK, represented by that cluster.

3) Finally, the clustering method is chosen. There are two methods of clustering:

hierarchical and non-hierarchical. K-means clustering, a non-hierarchical method is

chosen to cluster the items. This is an iterative distance based clustering which is

explained below.

5.3 K-means Clustering Algorithm

Step 1: Specify the number of clusters based on logical intuition

Step 2: Specify the initial cluster centers arbitrarily

Step 3: Group all the items to their nearest cluster center using Euclidean distance

Step 4: Find out the centroids of each cluster (mean of distances of all the items to the

corresponding cluster center)

Step 5: Consider the centroids as new cluster centers

Step 6: The process is continued iteratively until two consecutive iterations result in the same

cluster centers

Note: This algorithm as presented only works if the number of clusters is specified in advance.

26

5.4 Demonstration of Cluster Analysis in SAS

5.4.1 Methodology

Figure 8: Process Methodology for cluster analysis

K-means Clustering is carried out in SAS by using the FASTCLUS procedure. First, the

coordinates (in latitudes and longitudes) of the locations of schools are fed as input for cluster

analysis. In this problem, the coordinates are generated randomly within the limits of Mexico.

The number of clusters „K‟ is specified and the FASTCLUS procedure forms the clusters using

the K-means algorithm. The cluster centers obtained in the output are the coordinates of the

locations of CPKs. The cluster membership of the schools represents the cluster to which they

belong to. In this section, cluster analysis is demonstrated by selecting the coordinate points

randomly.

5.4.2 Location Coordinates for Schools

In the real scenario, the actual location coordinates of schools can be fed into SAS in a

spreadsheet or notepad file and cluster analysis is performed on this data. To generate the

coordinates with a close match to the real scenario, four rectangles are plotted in the map of

Mexico to obtain the limits for coordinates. Figure 10 shows the four regions in the map for

limits of coordinates.

Input coordinates of

schools

Specify number of clusters

Form clusters using K-Means

in SAS

Identify cluster center and cluster

membership

Obtain the locations of

cluster centers

27

Figure 9: Map of Mexico showing the limits of coordinates

The coordinate boundaries are:

1) Latitude: 26 to 31; Longitude: -106 to -110

2) Latitude: 20 to 29; Longitude: -101 to -105.5

3) Latitude: 19.5 to 26; Longitude: -97.5 to -100

4) Latitude: 17.5 to 19.5; Longitude: -96.5 to -102

For each region, the number of observations are selected in proportion to the size of the region.

The random numbers for coordinates of each region are generated in SAS with a uniform

distribution for the ranges obtained. The trial values assumed for clustering are mentioned in

Figure 11. For this demonstration, trial 1 is considered. In this instance, 250 schools are to be

served by the lunch program and it is assumed that 10 CPKs are allotted with about 25 schools

served by a single CPK. The locations of CPKs and thus the potential sites for DCs is found by

clustering using SAS with 10 clusters of CPKs and the schools served by them.

1

2

3

4

28

Trial Schools CPKs Potential DCs

1 250 10 10

2 500 20 20

3 750 30 30

4 1000 40 40

5 2500 100 100

Table 2: Trial values for clustering analysis

5.4.3 SAS Output: Clusters

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=10 Maxiter=10 Converge=0.02

Cluster Means

Cluster x y

ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

1 29.7281353 -108.2687411

2 25.2368048 -98.9485047

3 23.6449432 -102.9057181

4 21.2236815 -103.6562666

5 27.1764116 -107.6110550

6 27.9185067 -102.1327145

7 19.2491701 -98.0465284

8 22.5571665 -98.2537562

9 18.7900919 -100.4461423

10 26.8725068 -104.6037309

Figure 10: SAS Output for Clustering Analysis

29

The cluster means and the clustering graph for trial 1 obtained in the SAS output is shown in

Figure 12. The entire SAS code and the output for other trial values are attached in Appendix A.

The 250 observations are grouped in 10 clusters as shown in the output. The cluster means

obtained represent the coordinates for the locations of cluster centers, which are the locations of

CPKs. These need not be the exact coordinates, but a feasible location in close proximity to

them.

5.4.4 Locations of CPKs

The cluster means obtained in the SAS output are used to identify the locations of CPKs. The

locations are obtained using Google maps, which are given in Figure 13.

1 29.7281353 -108.2687411 Near Madera, Chihuahua

2 25.2368048 -98.9485047 Near Linares, Nuevo Leon

3 23.6449432 -102.9057181 Near Fresnillo, Zacatecas

4 21.2236815 -103.6562666 Near Tequila, Jalisco

5 27.1764116 -107.6110550 Near Guachochi, Chihuahua

6 27.9185067 -102.1327145 Near Sabinas, Coahuila

7 19.2491701 -98.0465284 Near Huamantla, Tlaxcala

8 22.5571665 -98.2537562 Near Cuauhtemoc, Tamaulipas

9 18.7900919 -100.4461423 Near Arcelia, Guerrero

10 26.8725068 -104.6037309 Near Hidalgo, Chihuahua

Table 3: Cluster centers and their corresponding locations

The locations of CPKs can be observed from the map in Figure 14. It can be inferred that the

locations obtained from cluster analysis is spread throughout the country. The CPKs obtained

therein are the potential locations for DCs, which can be used in the facility location model. The

coordinates for the southeast and northwest regions of the country are not specified when

generating random numbers and hence there are no clusters formed there. But in the real case,

the clusters will be located corresponding to the coordinates of schools fed as input.

30

Figure 11: Map of Mexico showing locations of CPKs - potential sites for DCs

31

CHAPTER 6: FACILITY LOCATION MODEL FORMULATION

6.1 Model Formulation: Multiple Sourcing DCs

Based on the supply chain design and the assumptions considered, the facility location model is

defined using mixed integer linear programming with an objective function and constraints. The

model is multi-echelon with supply from farmers to DCs to CPKs. The farmers have known

agricultural production capacities and each farmer supplies through 1 DC only i.e. dedicated

farmers to DCs. The DCs are assumed as multiple-sourcing, i.e. a CPK can be served by more

than 1 DC. The CPKs have demand for each commodity, which is known. With all these

considerations, the model for facility location problem is shown in Figure 12.

Figure 12: Model for facility location problem

CPK 1

CPK 2

CPK 3

CPK 4

Farmer 1

DC 1

DC 2

Farmer 2

Farmer 3

Farmer 4

Farmer 1

Farmer 2

Farmer 3

Farmer 4

32

6.1.1 Notation

i = farmers

j = DCs

k = CPKs

l = commodities

Cijkl = transportation costs from farmer „i‟ via DC „j‟ to CPK „k‟ for commodity „l‟

Fj = fixed facility costs at DC „j‟

Dkl = demand of commodity „l‟ at CPK „k‟

Pil = production capacity of farmer „i‟ for commodity „l‟

Tmin-j = minimum throughput at DC „j‟

Tmax-j = maximum throughput at DC „j‟

Il = intensity factor for commodity „l‟

6.1.2 Decision Variables

yj = binary variable to determine whether DC „j‟ is open or not

yij = binary variable to determine whether DC „j‟ is supplied by farmer „i‟ or not

Qijkl = quantity shipped from farmer „i‟ via DC „j‟ to CPK „k‟ for product „l‟

6.1.3 Objective Function

Minimize total supply chain costs = Transportation cost from farmer to DC to CPK for all

commodities + Fixed facility cost for DCs

Minimize Z = ∑i ∑j ∑k ∑l C ijkl Q ijkl + ∑j Fj yj

33

6.1.4 Constraints

1) yj = 1 DC „j‟ is open

0 Otherwise

2) yij = 1 DC „j‟ is supplied by farmer „i‟

0 Otherwise

3) ∑j yj = p

„p‟ is the number of open DCs, specified by the user.

4) ∑j yij = 1 ¥ i

Farmer „i‟ is located local to DC „j‟. Hence farmer „i‟ supplies via one DC only.

5) yij ≤ yj ¥ i,j

Farmer „i‟ supplies only to open DC „j‟.

6) Qijkl ≤ M yij ¥ I,j,k,l

∑i ∑j Qijkl = Dkl ¥ k,l

where „M‟ is assumed a large value of 10000.

Quantity shipped to any CPK „k‟ is equal to the demand at CPK „k‟.

Demand is met at all CPKs through multiple sourcing i.e. a CPK can be served by

multiple DCs.

7) ∑l Il ∑i ∑k Qijkl yij = Tj ¥ j

Quantity supplied through a DC „j‟, from a set of local farmers, is the total throughput of

DC „j‟.

34

An intensity factor Il is provided to measure the intensity of all the commodities from a

base unit for calculating throughput.

yij considers only the quantities supplied by those farmers who serve the DC ‟j‟

8) ∑i ∑k Qijkl ≤ ∑i Pil yij ¥ j,l

The quantity of a commodity „l‟ supplied by a set of farmers serving a DC „j‟ should be

less than the production capacities of the farmers for the commodity „l‟.

9) Tmin-j ≤ Tj yj ≤ Tmax-j ¥ j

The throughput of a DC is bounded by the maximum and minimum allowed throughput.

10) ∑i Pil ≥ ∑k Dkl ¥ l

The production capacities of all farmers for a commodity should be greater than the

demand at all CPKs.

11) Qijkl ≥ 0 ¥ i, j, k, l

The decision variables for quantity should be non-negative.

35

CHAPTER 7: BENDERS DECOMPOSITION HEURISTICS

7.1 Benders Decomposition

The facility location problem formulated belongs to the class of large scale optimization

problems, which in general cannot be solved by exact solution methods due to the combinatorial

complexity. Hence heuristic approaches are considered. After reviewing the literature, Benders

Decomposition is considered, because the problem structure in facility location problem suits

this approach well.

The Benders approach [38] partitions the facility location problem into two simpler problems: an

integer problem, known as Benders master problem, and a linear problem, known as Benders

subproblem. The subproblem is obtained by temporarily fixing the integer variables in the

original problem, hence relaxing the complicating integer variables. The dual of the subproblem

is used to obtain Benders cuts. The Master problem is one with a set of integer variables and

Benders cuts as constraints. The Benders algorithm solves these two simpler problems iteratively

and new Benders cuts are added as constraints to the master problem in each iteration. The

subproblem yields an upper bound and the master problem yields a lower bound. The algorithm

continues until the upper bound and the lower bound converge to an optimal solution for the

original mixed integer problem. Benders Decomposition can be used for large scale mixed

integer programming problems where the original problem is difficult, but the Benders

subproblem and the relaxed master problem are easier to solve.

36

Figure 13: Process Methodology for Benders Decomposition

The process methodology is outlined in Figure 13. The Benders subproblem and master problem

are first formulated for the problem. The Benders cut equations are generated from the dual of

the subproblem. The solution approach for Benders Decomposition using MATLAB is discussed

in the following section. The locations of DCs obtained from Benders algorithm is used to relax

the constraints in the remaining MILP problem which is then solved using MATLAB. Finally,

the results are analyzed.

7.2 Benders Algorithm

The Benders' Decomposition algorithm can be stated as follows:

Fix y‟ in the original problem; Benders subproblem

LB: - Inf

UB: + Inf

while e > ε (where „ε‟ is assumed a small value, say 5)

Solve subproblem dual

Obtain Benders cut By(y)

UB: By‟(y‟)

Add By(y) to master problem

Step 1

•Formulation of Benders Equations

•Benders subproblem and Master Problem

•Generation of Benders cuts

Step 2

•Solution to Benders Decomposition

•MATLAB Programming

•Convergence of Benders algorithm to obtain locations of DCs (yj)

Step 3

•Solution to MILP

•Input locations of DCs into the problem

•MATLAB Programming to obtain yij and Qijkl

Step 4

•Analysis

•Summarization of output

•Sensitivity Analysis

37

Solve master problem

if (master problem is infeasible) then

Stop. The original problem is infeasible.

else

Let (z‟,y‟) be the optimal value and solution to the master problem.

return (z‟,y‟)

LB: z‟

e: UB-LB

end

Upper bounds

ε = epsilon Lower bounds Iterations

Figure 14: Convergence of Upper bound and Lower bound

The equations for Benders decomposition are developed for the multi-commodity facility

location model as follows:

Step 1:

The objective function as specified in the model formulation,

Minimize Z = ∑i ∑j ∑k ∑l C ijkl Q ijkl + ∑j Fj yj

The binary variables yj and yij in the optimization problem are held fixed to satisfy the

constraints (1)-(5) involving those binary variables.

38

Step 2:

The Benders sub problem is obtained by fixing the integer variables,

Minimize Z = ∑i ∑j ∑k ∑l C ijkl Q ijkl + ∑j Fj yj‟

where yj‟ indicates the fixed values for DCs yj.

The sub problem is a transportation problem with DC costs known and can be written as,

Minimize Z = ∑i ∑j ∑k ∑l C ijkl Q ijkl

subject to constraints,

1) Qijkl ≤ M yij‟ ¥ i,j,k,l

∑i ∑j Qijkl = Dkl ¥ k,l

2) ∑l Il ∑i ∑k Qijkl yij‟ = Tj ¥ j

3) ∑i ∑k Qijkl ≤ ∑i Pil yij‟ ¥ j,l

4) Tmin-j ≤ Tj yj‟ ≤ Tmax-j ¥ j

5) ∑i Pil ≥ ∑k Dkl ¥ l

6) Qijkl ≥ 0 ¥ i, j, k, l

Step 3:

The dual of the subproblem is formulated to solve the problem. The dual variables Vk and Wijk

are introduced which represent demand constraints and setup constraints respectively.

Vk = min Cijk where j € O(y)

Vk represents the minimum cost to all CPKs through open DCs.

Wijk = 0 for j € O(y)

max { (Vk - Cijk),0 } for j € C(y)

Wijk represents the incremental cost when any of the closed DCs are open.

39

Using the dual variables, the Benders cut is obtained which eliminates all solutions that yield

more expensive solution to the original problem.

By‟(y‟) = ∑k Vk + ∑j ( Fj - ∑k Wijk ) yj

The Benders cut equation gives an upper bound to the solution.

Step 4:

The master problem is formulated as,

Min z

s.t. z ≥ By‟(y‟)

The optimal solution to the master problem yields a lower bound by solving for z. And the values

of y obtained for the optimal solution will be the new values of y for the next iteration.

Step 5:

The difference between the upper bound and the lower bound is calculated and is represented by

„e‟.

e=UB-LB

The iteration is continued till the upper bound and the lower bound converge to an optimum

solution i.e. a smaller value of „e‟. Once the convergence is reached, the optimal solution is the

minimum transportation cost and the locations of DCs are obtained as „y‟ values.

The locations of DCs obtained using Benders algorithm relaxes the constraints related to yj in the

MILP problem, which is easier to solve. This problem is solved to obtain the allocation of

farmers to DCs, throughput of DCs and Qijkl.

40

7.3 Benders Algorithm in MATLAB

The Benders decomposition equations are developed and the algorithm is implemented using

MATLAB. The fixed DC costs and transportation costs are assigned reasonably based on

assumptions.

The trial values used in the facility location model are given in Table 4. It can be seen that each

trial set is used in MATLAB with different commodity levels such as 2, 4, 6, 8, and 10.

Trial 1 2 3 4 5

Schools 250 500 750 1000 2500

Farmers (i0) 10 20 20 25 40

DCs (j0) 10 20 30 40 100

CPKs (k0) 10 20 30 40 100

Commodities (l0) 2,4,6,8,10 2,4,6,8,10 2,4,6,8,10 2,4,6,8,10 2,4,6,8,10

No. of DCs (p) 2 4 6 8 20

Table 4: Trial values of input parameters

Following are the steps performed in MATLAB for implementing the Benders algorithm:

1) The values for number of farmers (i0), DCs (j0), CPKs (k0), commodities (l0) and

number of open DCs (p) are defined based on trial values. It is to be noted that j0 denotes

the number of potential DCs and p denotes the number of open DCs.

2) The transportation costs tcijkl (4-D matrix of i0 x j0 x k0 x l0) and fixed costs fcj for DCs

(1 x j0) are specified.

3) The open DCs denoted by integer variable „y‟ is randomly assigned based on p value, i.e.

if p=2, then any two y‟s are randomly given values of 1. And their location indices are

41

found using find(), indicating which DCs are open. Two sets are formed for open and

closed DCs.

O(y): open DCs with a y value of 1 and

C(y): closed DCs with a y value of 0.

4) Once the DCs are set, all the farmers are assigned to open DCs to initiate the iteration for

Benders algorithm. One farmer is assigned to one DC only as farmers are assumed local

to DCs in the formulation. Also, it‟s assumed that one DC should be assigned atleast

three farmers. This is considered as a set covering problem and solved using binary

programming to get the allocations of farmers to DCs. The number of assignments is

equal to the number of farmers, since each farmer has one assignment. This is done to

initiate the loop.

5) By fixing the integer variables yj and yij, the problem becomes Benders subproblem. The

next step is to obtain the dual of this problem, in which the concept of shadow price is

used. For this purpose, the dual variables Vk and Wijk are defined.

6) The minimum cost for each of the CPKs is calculated by considering only open DCs in

„j‟ and the farmers „i‟ allotted to them. The minimum cost for each of the commodities

for a CPK is summed to obtain Vk.

Vk = min Cijk for i,j,k j € O(y)

7) The dual variable Wijk is then calculated using Shadow price concept.

Wijk = 0 for j € O(y)

Max {(Vk - Cijk),0} for j € C(y)

42

8) The Benders cut are obtained using the following equation:

b(y) = ∑ + ∑ - ∑ yj

The solution to benders cut yields an upper bound which implies that the sum of local

optimum obtained is an upper limit but higher than the global optimum.

9) The master problem is then formulated by using the benders cuts as constraints and

solved using Integer programming solver.

Min z

s.t. z ≥ b(y)

The solution to the master problem yields lower bound. The optimal solution to this

problem gives the new y-values for the next iteration of the algorithm. For every

iteration, new benders cuts are generated and are added as constraints to the master

problem.

10) The value of „e‟, the difference between the upper bound and the lower bound, is

calculated at the end of every iteration. A smaller value of „e‟ signifies upper bound and

lower bound converging to an optimal solution and this acts as the terminating criteria for

the algorithm.

e = UB - LB

The algorithm is run for different trial values and the elapsed processing time is

measured.

43

7.4 Analysis of Results

7.4.1 Benders Algorithm

The Benders algorithm for the multi-echelon, multi-commodity problem is solved using

MATLAB and the results are analyzed. The processing times for different trial values are

summarized in Table 5.

p0=2 p0=4 p0=6 p0=8 p0=10

i0=10 i0=20 i0=30 i0=40 i0=50

j0=10 j0=20 j0=30 j0=40 j0=50

k0=10 k0=20 k0=30 k0=40 k0=50

l0=2 0.092 0.597 1.930 4.577 10.090

l0=3 0.148 0.678 2.191 5.694 12.770

l0=4 0.167 0.781 2.625 6.930 16.050

l0=5 0.177 0.846 2.980 8.196 18.890

Table 5: Processing times in seconds

From this table, it can be inferred that the processing times increase with increase in the number

of commodities. Hence the complexity of a multi-commodity model is more than a single

commodity model.

The input values assumed for the problem are given in Table 4. For input values of i0=10; j0=10;

k0=10; l0=2 and p=2, the DC locations obtained in the output are [3,9] as given in the following

portion of MATLAB output:

New y values

0 0 1 0 0 0 0 0 1 0

New DC Locations

3 9

Elapsed time is 0.080416 seconds.

44

7.4.2 MILP

When the integer variables (yj) for DC locations are fixed using the output from Benders

algorithm, it relaxes the constraints involving yj in the resulting MILP, which is much easier to

solve. This is solved using mixed integer optimization solver in MATLAB to yield the allocation

of farmers to DCs (yij) and the quantity shipped from farmer „i‟ through DC „j‟ to CPK „k‟ for

commodity „l‟ specified by Qijkl. Also, the throughput of open DCs are obtained based on the

throughput constraints.

Figure 15: Qijkl Distribution graph for commodity 1

DC 3

Farmer 5

Farmer 7

Farmer 8

Farmer 9

Farmer 1

Farmer 2

Farmer 3

Farmer 4

Farmer 6

Farmer 10

DC 9

CPK 1

CPK 2

CPK 3

CPK 4

CPK 5

CPK 6

CPK 7

CPK 8

CPK 9

CPK 10

97

17

81

43

25

77

53

69

38

63

27

68

10

35

67

76

21

99

39

63

27

10

68

35

67

63

99

21

76

39

563 505

Throughput: 320

Throughput: 185

45

The input values for this MILP are yj = [3,9] from Bender‟s algorithm and other relaxed

constraints with respect to yj from the model formulation. The capacities of farmers and demands

of CPKs for commodities 1 and 2 are assumed such that the sum of the capacities at farmers is

greater than the sum of the demand at CPKs as shown in Figure 14.

The following are the assumed values of capacities of farmers and demand at CPKs:

pc=[[25;77;53;69;97;38;17;81;43;63;],[27;17;24;84;37;45;69;84;79;44;]];

dc=[[27;68;10;35;67;63;39;99;21;76;],[24;49;85;42;17;60;35;22;73;54;]];

The output to this MILP gives the allocation of farmers to DCs as follows:

Farmers allotted to DC 3: 5, 7, 8, 9

Farmers allotted to DC 9: 1, 2, 3, 4, 6, 10

The distribution graph in Figure 14 gives the Qijkl.

However, it is observed from Figure 14 that the Qijkl is allotted only through single sourcing of

DCs. Further analysis and rerun of the program indicated that single sourcing observed is by

chance in this case. To further validate this, the capacities at farmers are changed as follows:

pc=[[10;20;30;40;50;60;70;80;90;100;],[27;17;24;84;37;45;69;84;79;44;]]; dc=[[27;68;10;35;67;63;39;99;21;76;],[24;49;85;42;17;60;35;22;73;54;]];

This resulted in a distribution graph with multiple sourcing. Hence it can be concluded that the

model and program developed works well for multiple sourcing and the values of capacities and

demands play a role in determining how the quantities are allotted. Hence a seasonal commodity

from a few farmers may be allotted through multiple DCs to CPKs.

46

7.5 Sensitivity Analysis

Based on the results obtained for the MILP problem, sensitivity analysis is carried out to analyze

the changes in Qijkl when throughput constraint is modified.

7.5.1 Throughput Limit: 400 to 550 units

MATLAB Output:

The farmers allotted to DC 3 are:

3

4

5

8

10


1

2

6

7

9

Throughput of DC 3 is:

504.3600


410.9300


The output of non-zero Qijkl values obtained in MATLAB is summarized in Table 6.

Q4311 1 Q2912 2

Q6911 26 Q4312 22

Q8321 68 Q7922 49

Q5331 10 Q10,3,3,2 85

Q9941 35 Q7942 42

Q3351 67 Q1952 17

Q2961 63 Q10,3,6,2 60

Q5371 39 Q5372 35

Q4381 99 Q3382 22

Q4391 21 Q1992 73

Q6,9,10,1 76 Q7,9,10,2 54

Table 6: Values of Qijkl for throughput limit 400 to 550 units

47

7.5.2 Throughput limit: 350 to 500 units

MATLAB Output:


1

2

4

5

7


3

6

8

9

10


435.2600


480.0300


The output of non-zero Qijkl values obtained in MATLAB is summarized in Table 7.

Q5311 27 Q8912 24

Q2321 68 Q1322 49

Q3931 10 Q5332 36

Q8941 35 Q3932 49

Q4351 67 Q6942 42

Q4361 63 Q8952 17

Q8971 39 Q3962 60

Q9981 99 Q6972 35

Q10,9,9,1 21 Q2382 22

Q1,3,10,1 2 Q2392 73

Q9,9,10,1 74 Q2,3,10,2 54

Table 7: Values of Qijkl for throughput limit 400 to 550 units

Processing time range: 23 to 32 seconds

48

Based on the analyses with different values for the limits of throughput, it is inferred that

throughput should be set reasonably i.e. it should not be drastically different from the demand at

CPKs. For instance, if the combined demand at CPKs is 1000 units, then the throughput limit for

the DCs should be set in such a way that they can meet the demand. Otherwise, the throughput is

bound to go beyond its limit to satisfy the demand, which is a hard constraint given as equality.

49

CHAPTER 8: CONCLUSION AND FUTURE WORK

The supply chain is designed with Mexican School lunch program as the application. A multi-

echelon, multi-commodity facility location model is developed using mixed integer linear

programming (MILP). For the discrete location problem considered, the potential candidate

locations are identified from the end customer locations using clustering analysis in SAS. The

facility location problem is solved using Benders Decomposition approach. The Benders

algorithm is used in MATLAB to identify the locations of DCs. The DCs obtained are used to

relax the constraints involving the integer variable for DCs yj in the MILP. The resulting

problem is solved in MATLAB to obtain the allocation of farmers to DCs yij, the quantity

shipped from farmer „i‟ through DC „j‟ to CPK „k‟ for commodity „l‟ specified by Qijkl and the

throughput of open DCs.

8.1 Limitations of the model

The model developed has some limitations based on the assumptions made. Following are some

of the limitations observed:

1) The initial „y‟ values fixed can have a bearing on the number of iterations and there are

techniques to fix „y‟ values to reduce the number of iterations. But in our case, the initial

„y‟ values for DC locations are assigned randomly to obtain the Benders subproblem.

2) The demand at all CPKs for all commodities is assumed to be satisfied. Hence demand

constraints are given as hard equality constraints in the model formulation.

50

3) The variable operating costs of DCs are not considered in the model formulation. The

throughput constraints can influence the operating costs in the objective function and a

more advanced variation of Benders heuristics is required to solve the problem.

4) In the benders decomposition algorithm, the assignment of farmers to DCs is made

randomly using set covering problem to start the iteration.

8.2 Summary of Contributions

1) Development of a methodology to identify locations of customer zones (CPKs) from

individual customers (schools) using clustering analysis in SAS.

2) Implementation of Benders Decomposition in a multi-commodity multi-echelon food

supply chain location problem.

3) Implementation of the solution procedure in MATLAB to identify the locations of DCs

using Benders Decomposition algorithm.

4) Implementation of the solution procedure in MATLAB to solve the resulting MILP to

obtain the allocation of farmers to DCs and the amount Qijkl.

8.3 Future scope

A program can be developed for a more advanced variation of Benders heuristics with operating

costs involved. Also, the Benders Decomposition algorithm can be applied to both single

sourcing and multiple sourcing DCs and the results obtained can be compared to identify the

efficiency parameters. A real world problem can be tested with these approaches to identify the

effect and relevancy of the solution.

51

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55

APPENDIX A:

I.SAS Code for Clustering Analysis

data coord;

/*Creating random numbers for the coordinates of 4 regions of Mexico*/

/*Latitudes and longitudes are represented by the variables lat and lon*/

/*250 locations are generated to represent 250 schools*/

do i = 1 to 62;

lat=5*ranuni(5)+26;

lon=-4*ranuni(5)-106;

output;

end;

do i = 63 to 175;

lat=9*ranuni(5)+20;

lon=-4.5*ranuni(5)-101;

output;

end;

do i = 176 to 220;

lat=6.5*ranuni(5)+19.5;

lon=-2.5*ranuni(5)-97.5;

output;

end;

do i = 221 to 250;

lat=2*ranuni(5)+17.5;

lon=-5.5*ranuni(5)-96.5;

output;

end;

run;

/*K-means Clustering on random data set*/

proc fastclus data=coord maxc=10 maxiter=10 out=clus1;

title 'K-means clustering using coordinates';

/* K is the number of clusters defined by maxc */

var lat lon;

run;

proc print data=coord;

title 'Observations: Location of Schools';

run;

Proc sort data=clus1 out=sor;

by cluster;

run;

proc print data=sor;

title 'Cluster of CPKs with the schools served';

run;

proc freq;

tables cluster*i;

run;

56

proc candisc anova data=clus1 out=can;

class cluster;

var lat lon;

/*Canonical Discriminant Analysis of Iris Clusters*/

run;

legend1 frame cframe=ligr label=none cborder=black

position=center value=(justify=center);

axis1 label=(angle=90 rotate=0) minor=none;

axis2 minor=none;

proc gplot data=Can;

plot Can2*Can1=Cluster/frame cframe=ligr

legend=legend1 vaxis=axis1 haxis=axis2;

/*Plot of Canonical Variables Identified by Cluster*/

title 'Cluster Analysis - CPKs and Schools';

run;

II.SAS Output: Clusters of Schools with Cluster membership

Cluster of CPKs with the schools served

Obs i lat lon CLUSTER DISTANCE

1 1 30.6241 -109.402 1 1.44446

2 2 30.9956 -107.188 1 1.66570

3 3 29.0401 -109.386 1 1.31175

4 4 29.5745 -108.634 1 0.39611

5 8 29.9256 -107.944 1 0.38017

6 10 30.4191 -106.530 1 1.87087

7 12 29.9639 -107.745 1 0.57479

8 15 30.5645 -109.540 1 1.52196

9 16 29.2090 -108.132 1 0.53679

10 17 29.2803 -108.582 1 0.54672

11 18 30.7415 -107.944 1 1.06399

12 19 30.7761 -107.449 1 1.33072

13 21 30.1632 -107.422 1 0.95186

14 22 28.4697 -108.693 1 1.32794

15 23 28.5118 -109.376 1 1.64464

16 25 28.7829 -106.664 1 1.86233

17 26 30.4026 -107.301 1 1.17984

18 29 30.3709 -108.769 1 0.81481

19 31 29.0596 -108.867 1 0.89737

20 33 29.4266 -106.920 1 1.38217

21 34 29.6734 -108.258 1 0.05575

22 35 28.5818 -109.503 1 1.68479

23 38 29.9623 -108.229 1 0.23761

24 39 30.7462 -108.959 1 1.23023

25 40 30.0459 -107.402 1 0.92299

26 41 29.1300 -109.384 1 1.26516

27 43 28.9359 -108.517 1 0.83019

28 44 29.2954 -108.104 1 0.46287

29 45 30.4001 -109.759 1 1.63462

30 46 29.2831 -107.977 1 0.53184

31 47 29.0391 -109.613 1 1.51085

32 53 28.6506 -109.941 1 1.98932

33 54 29.7263 -109.319 1 1.04996

34 55 29.5850 -106.251 1 2.02234

35 56 29.8228 -108.718 1 0.45929

36 57 30.2964 -106.913 1 1.47010

37 58 29.9294 -107.501 1 0.79383

38 59 30.2810 -109.135 1 1.02799

57

39 60 29.7107 -106.509 1 1.76009

40 132 25.0398 -101.016 2 2.07659

41 176 24.9574 -99.690 2 0.79197

42 178 24.8172 -97.784 2 1.23816

43 181 25.2895 -99.768 2 0.82148

44 183 24.4718 -98.500 2 0.88700

45 185 25.2602 -97.998 2 0.95037

46 186 25.6075 -98.665 2 0.46647

47 188 25.8052 -98.710 2 0.61652

48 189 25.3198 -99.493 2 0.55053

49 190 25.2774 -99.547 2 0.59994

50 195 25.7013 -97.838 2 1.20414

51 199 24.8651 -98.277 2 0.76717

52 203 25.6857 -99.452 2 0.67428

53 207 25.1316 -98.542 2 0.41975

54 209 24.8889 -98.449 2 0.60899

55 217 25.6705 -99.448 2 0.66186

56 64 24.9744 -103.519 3 1.46392

57 66 23.8796 -104.430 3 1.54197

58 68 22.0743 -101.985 3 1.82037

59 71 24.0166 -105.140 3 2.26543

60 73 24.2895 -101.698 3 1.36935

61 75 24.1384 -101.675 3 1.32620

62 76 21.9976 -101.078 3 2.46057

63 77 23.5548 -104.265 3 1.36241

64 79 24.5957 -103.058 3 0.96283

65 81 24.6831 -103.832 3 1.39115

66 82 21.8292 -101.189 3 2.49903

67 86 23.8464 -105.135 3 2.23801

68 95 22.4948 -101.001 3 2.22548

69 98 23.9980 -102.200 3 0.78918

70 103 24.9751 -102.552 3 1.37641

71 105 23.2357 -104.531 3 1.67563

72 106 22.6212 -102.905 3 1.02378

73 113 22.9588 -102.402 3 0.85094

74 116 23.8467 -104.124 3 1.23451

75 123 23.1846 -101.051 3 1.91092

76 127 24.7672 -103.413 3 1.23146

77 129 24.6376 -101.645 3 1.60479

78 134 24.0874 -103.500 3 0.74069

79 137 23.3799 -104.346 3 1.46432

80 141 24.1462 -101.607 3 1.39238

81 145 22.4015 -102.846 3 1.24490

82 146 23.1262 -104.155 3 1.35308

83 148 23.8396 -103.775 3 0.89040

84 151 23.1501 -102.698 3 0.53678

85 152 23.0620 -101.271 3 1.73580

86 158 22.7238 -102.207 3 1.15632

87 161 23.9311 -102.700 3 0.35260

88 165 23.9378 -104.979 3 2.09355

89 166 25.5432 -101.888 3 2.15373

90 69 22.0452 -103.096 4 0.99420

91 70 20.9035 -102.765 4 0.94686

92 72 20.9211 -105.353 4 1.72350

93 78 21.0996 -104.320 4 0.67509

94 80 21.4107 -103.393 4 0.32282

95 89 21.0819 -104.312 4 0.67065

96 91 20.8370 -102.267 4 1.44212

97 92 21.3703 -102.808 4 0.86112

98 93 21.6017 -102.579 4 1.14128

99 94 21.0226 -103.211 4 0.48842

100 97 21.9815 -103.632 4 0.75823

101 99 21.6044 -103.193 4 0.60003

102 100 20.1461 -104.529 4 1.38651

103 101 20.9872 -105.068 4 1.43126

104 107 20.6457 -104.773 4 1.25752

105 108 20.3728 -104.099 4 0.95913

106 109 20.9064 -102.005 4 1.68149

107 111 20.0478 -104.887 4 1.70209

108 115 21.1145 -102.186 4 1.47459

109 117 21.9667 -104.053 4 0.84212

58

110 118 21.5391 -103.409 4 0.40052

111 119 20.9568 -104.541 4 0.92405

112 121 21.6159 -105.462 4 1.84806

113 128 20.9092 -104.478 4 0.87940

114 138 21.7984 -102.870 4 0.97424

115 139 22.2759 -103.683 4 1.05251

116 140 21.5253 -103.236 4 0.51710

117 149 21.7033 -101.886 4 1.83363

118 150 20.3690 -102.280 4 1.61995

119 155 22.0068 -104.409 4 1.08654

120 156 21.2644 -104.273 4 0.61850

121 157 21.9291 -102.184 4 1.63229

122 168 20.4063 -104.416 4 1.11600

123 173 21.2391 -104.657 4 1.00075

124 5 27.2449 -107.331 5 0.28797

125 6 26.4830 -108.477 5 1.10970

126 7 27.3349 -107.344 5 0.31079

127 9 27.8153 -107.799 5 0.66584

128 11 28.2059 -106.255 5 1.70235

129 13 27.8947 -106.900 5 1.01092

130 14 26.8631 -109.807 5 2.21847

131 20 27.9881 -109.362 5 1.92974

132 24 27.2386 -108.712 5 1.10247

133 27 26.1205 -108.491 5 1.37451

134 28 27.3372 -109.991 5 2.38543

135 30 26.9953 -106.987 5 0.64948

136 32 26.5678 -108.258 5 0.88804

137 36 27.7226 -106.297 5 1.42294

138 37 26.5177 -107.435 5 0.68191

139 42 27.0649 -106.784 5 0.83433

140 48 27.7660 -107.064 5 0.80412

141 49 26.5343 -106.760 5 1.06646

142 50 27.7702 -106.555 5 1.21180

143 51 26.5026 -106.869 5 1.00230

144 52 27.9228 -107.344 5 0.79255

145 61 26.4304 -107.164 5 0.86949

146 62 26.7368 -107.068 5 0.69879

147 65 27.8899 -101.665 6 0.46901

148 74 28.3167 -102.945 6 0.90445

149 83 28.1865 -101.592 6 0.60318

150 85 28.7661 -101.376 6 1.13605

151 90 28.5788 -102.423 6 0.72145

152 114 28.1858 -103.390 6 1.28562

153 124 26.9659 -102.161 6 0.95297

154 125 27.7489 -101.104 6 1.04271

155 126 28.0282 -103.340 6 1.21216

156 131 27.7292 -101.001 6 1.14717

157 133 26.5860 -101.240 6 1.60400

158 142 26.5719 -101.565 6 1.46116

159 147 28.9220 -101.918 6 1.02612

160 153 28.4534 -103.520 6 1.48677

161 159 26.6129 -101.375 6 1.50981

162 160 28.3094 -103.288 6 1.21987

163 163 28.4022 -102.339 6 0.52566

164 172 28.9159 -102.948 6 1.28808

165 174 27.2818 -101.332 6 1.02321

166 179 20.4363 -99.136 7 1.61122

167 184 19.6651 -97.620 7 0.59545

168 191 20.1939 -98.379 7 1.00135

169 196 19.7353 -98.081 7 0.48736

170 202 19.7846 -97.872 7 0.56324

171 211 19.8702 -98.562 7 0.80692

172 215 20.1058 -97.554 7 0.98803

173 218 19.8606 -98.705 7 0.89836

174 219 19.5754 -98.248 7 0.38320

175 221 19.1538 -96.825 7 1.22549

176 223 19.4367 -99.111 7 1.08081

177 224 19.1830 -98.387 7 0.34713

178 227 18.1960 -98.806 7 1.29820

179 228 18.0804 -96.753 7 1.74344

180 229 18.6737 -98.514 7 0.74156

59

181 234 18.9468 -97.152 7 0.94392

182 235 17.5549 -98.277 7 1.70989

183 237 18.7746 -97.679 7 0.60014

184 238 19.4319 -97.981 7 0.19428

185 242 18.5117 -96.742 7 1.49897

186 244 19.3998 -99.122 7 1.08641

187 245 18.6339 -96.702 7 1.47901

188 248 19.4870 -98.486 7 0.49956

189 249 19.2886 -98.424 7 0.37910

190 177 24.0768 -97.666 8 1.62942

191 180 21.2166 -99.221 8 1.65307

192 182 22.5210 -97.548 8 0.70712

193 187 21.2676 -99.928 8 2.11318

194 192 23.2101 -98.003 8 0.69933

195 193 23.9732 -98.023 8 1.43477

196 194 23.0344 -97.628 8 0.78723

197 197 20.9095 -98.200 8 1.64853

198 198 21.3255 -98.063 8 1.24639

199 201 23.5882 -98.046 8 1.05167

200 204 22.3241 -98.735 8 0.53464

201 205 21.2474 -97.831 8 1.37646

202 206 23.5114 -97.542 8 1.19040

203 208 22.3957 -97.617 8 0.65692

204 210 22.5156 -99.396 8 1.14302

205 212 23.4651 -97.750 8 1.03823

206 213 21.9204 -98.138 8 0.64716

207 214 23.0747 -99.274 8 1.14388

208 216 22.5703 -98.285 8 0.03377

209 220 22.9958 -98.182 8 0.44448

210 120 20.2863 -101.025 9 1.60434

211 164 20.3859 -101.314 9 1.81666

212 175 20.9391 -101.219 9 2.28392

213 200 20.6262 -99.665 9 1.99553

214 222 17.6378 -99.738 9 1.35271

215 225 18.9415 -99.287 9 1.16861

216 226 18.1058 -101.396 9 1.17031

217 230 18.4421 -101.802 9 1.39981

218 231 19.2245 -99.303 9 1.22298

219 232 17.6228 -101.402 9 1.50854

220 233 18.2975 -101.835 9 1.47341

221 236 17.7983 -99.508 9 1.36507

222 239 18.1644 -100.345 9 0.63389

223 240 18.1555 -101.033 9 0.86423

224 241 18.2106 -99.231 9 1.34612

225 243 18.6702 -99.369 9 1.08373

226 246 17.7877 -101.621 9 1.54453

227 247 18.9127 -99.683 9 0.77282

228 250 18.8031 -99.701 9 0.74529

229 63 27.3284 -104.990 10 0.59730

230 67 28.1818 -104.931 10 1.34966

231 84 25.6381 -104.160 10 1.31174

232 87 25.9686 -103.376 10 1.52492

233 88 25.7425 -103.169 10 1.82597

234 96 25.0519 -105.408 10 1.99020

235 102 25.8977 -104.872 10 1.01114

236 104 27.0208 -104.087 10 0.53761

237 110 27.3828 -105.063 10 0.68636

238 112 26.8028 -104.437 10 0.18110

239 122 28.3600 -105.079 10 1.56159

240 130 27.4821 -105.279 10 0.90935

241 135 25.3467 -105.497 10 1.76793

242 136 25.2124 -104.554 10 1.66083

243 143 28.9405 -104.877 10 2.08603

244 144 28.5867 -104.392 10 1.72719

245 154 26.5277 -104.523 10 0.35408

246 162 26.6357 -103.281 10 1.34380

247 167 26.9356 -104.565 10 0.07414

248 169 27.7372 -104.776 10 0.88172

249 170 28.1874 -105.457 10 1.56763

250 171 26.2277 -104.511 10 0.65139

60

III.SAS Output: Cluster Means and Cluster Graph for Trial 2 values


2 500 20 20

Cluster Analysis - CPKs and Schools

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=20 Maxiter=10 Converge=0.02

Cluster Means

Cluster x y


1 27.9377995 -111.6902172

2 28.5049801 -93.5480162

3 22.2566838 -113.5467155

4 24.4081798 -91.6889736

5 16.6370756 -95.3406202

6 17.0175055 -101.3837301

7 25.8862028 -97.0124251

8 16.8809396 -106.9053747

9 20.0673995 -97.7296398

10 22.6578881 -87.4370703

11 17.9993189 -86.6321226

12 16.8076689 -91.1370202

13 20.9084630 -93.1052651

14 27.5524345 -106.9076527

15 27.5693643 -87.8287063

16 24.5410752 -101.4603616

17 22.2047588 -105.4788969

18 28.8849074 -100.2694870

19 17.0998444 -112.6263467

20 22.2798363 -109.6957704

61

IV.SAS Output: Cluster Means and Cluster Graph for Trial 3 values


3 750 30 30

Cluster Means

Cluster x y


1 28.6841574 -109.0362649

2 16.2197196 -94.6856579

3 16.8256566 -86.2843865

4 27.4709328 -105.3964950

5 28.7057151 -91.8129595

6 25.4640948 -100.1101309

7 22.6937119 -95.5438608

8 19.1030049 -97.3108174

9 22.2370523 -100.0952215

10 25.5260981 -88.4090443

11 16.3693621 -99.1767180

12 19.4210815 -92.4862182

13 23.6657815 -85.9371160

14 16.3473157 -113.4690238

15 16.5129529 -105.5732150

16 20.8582790 -87.9983431

17 17.3524922 -102.2088967

18 28.6411269 -98.3943066

19 25.0740240 -109.0089477

20 20.5241168 -108.2053554

21 27.5254384 -95.5245032

22 23.3072822 -104.6725502

23 27.8264370 -113.0092655

24 28.5165020 -86.9124967

25 24.1600364 -92.0090488

26 16.5327653 -109.7874611

27 28.4171016 -101.3970133

28 22.6253969 -112.7987165

29 19.2788172 -113.4171711

30 16.4862357 -90.4894498

62

V.SAS Output: Cluster Means and Cluster Graph for Trial 4 values


4 1000 40 40

Cluster Means

Cluster x y


1 28.8137680 -110.9027693

2 28.0261664 -94.8495655

3 23.3289502 -107.1532065

4 26.1418100 -104.5833899

5 20.0025391 -109.4791347

6 15.8614824 -94.0167697

7 16.4402390 -109.6571266

8 25.4954236 -88.4066110

9 21.5782759 -94.7166690

10 28.8598667 -88.6446806

11 23.2486450 -86.0308901

12 25.6520463 -98.0672405

13 16.3895213 -90.4566197

14 16.3855280 -113.2208851

15 19.2670915 -92.3667265

63

16 16.0686666 -86.1995973

17 21.0919889 -88.7944746

18 28.7870669 -98.2044549

19 22.1601706 -98.7922154

20 27.9061076 -85.9812709

21 18.9166365 -86.4572681

22 22.0457448 -103.2142683

23 19.4081645 -98.7143125

24 19.4563180 -106.2508383

25 28.7750751 -101.4003161

26 24.6323164 -113.5833947

27 16.2382358 -105.4301357

28 19.2801868 -113.6503446

29 29.0231238 -106.3543369

30 28.1581936 -91.6882626

31 26.3383876 -108.1283209

32 17.2109194 -101.8770824

33 23.6380446 -91.6125805

34 28.1070758 -113.7147022

35 21.9435821 -112.1284683

36 24.7873946 -110.1785123

37 17.6612377 -96.0378442

38 25.6776926 -101.4979535

39 16.1594838 -98.9434466

40 24.4669865 -94.5827785

64

APPENDIX B:

I.MATLAB Code for Benders Decomposition

tic;

% Given Values i0=10; % Farmers j0=10; % DCs k0=10; % CPKs l0=2; % Commodities p0=2; % Number of open DCs

% Transportation cost, Cijkl; Range 1-10 tc=rand(i0,j0,k0,l0)*9+1;

%Fixed cost for DCs is defined; Range 1-10 fc=rand(1,j0)*9+1;

% 'p' Open DCs initial random assumption n1=p0; n2=j0-p0; y=zeros(1,n1+n2); d=randperm(n1+n2); y(d(1:n1))=1; ind0=find(y); % Open DC Locations cind=find(y==0); % Closed DC Locations

disp('Given value of p:'); disp(p0); disp('Initial y values:'); disp(y); disp('DC Locations:'); disp(ind0);

% Yij Farmers to DCs initial assumption % Set covering problem %Objective: To minimize the sum of (fixed cost * farmer integer var) f0=zeros(i0,j0); f=rand(p0*i0,1)*10+10; % Fixed cost of farmers Aeq=zeros(i0,p0*i0); beq=ones(i0,1); % One farmer should serve 1 DC only A=zeros(p0,p0*i0); b=zeros(p0,1); b(:,:)=-3; % One DC should be served by atleast 3 farmers

% Defining the LHS of inequality constraint cc=1; while(cc<=p0*i0) for i=1:p0 for j=1:i0 A(i,cc)=-1;

65

cc=cc+1; end; end; end;

% Defining the LHS of equality cc=1; while(cc<=i0*p0) for i=1:i0 Aeq(i,cc)=1; cc=cc+1; end; end;

% Solving the set covering problem to allot the farmers [x,fval] = bintprog(f,A,b,Aeq,beq);

% Defining the matrix for allotment of farmers to open DCs count=1; for i=ind0 for j=1:i0 f0(j,i)=x(count); count=count+1; end; end;

% Finding the index - which farmers are allotted to which DCs [ii iind]=find(f0);

cat=[ii,iind]; disp('Farmers-->DCs'); disp(cat);

% Printing the DCs allotted to each farmer for i=1:p0 str=sprintf('The farmers alloted to DC %d are :',ind0(i)); disp(str); count=1; for j=1:i0 if(cat(j,2)==ind0(i)) disp(cat(j,1)); count=count+1; end; end; end;

tempx=zeros(i0,l0,k0); v=zeros(k0,1); % Defining the dual variable v qq=zeros(l0,1);

w0=zeros(i0,j0,k0); % Defining the dual variable w w=zeros(i0,j0,k0); tempw=ones(i0,j0,k0,l0);

A1=zeros(1,j0+1);

66

b1=zeros(1,1); Aeq1=zeros(1,j0); Beq1=zeros(1,1);

% e is defined as the difference between upper bound and lower bound. % The iteration continues till the difference is very small meaning % benders algorithm converged to an optimal solution.

e=200;%This starts the iteration iter=1;

while(e>10)

% Solving for the dual variable v ind=find(y); % finding the open DC locations

for k=1:k0 for l=1:l0 for i=1:i0 % Finding the min cost among open DCs to all CPKs tempx(i,l,k)=tc(cat(i),cat(i,2),k,l); end; qq(l)=min(tempx(:,l,k)); end; v(k)=sum(qq); % Minimum cost for each CPK routed through % open farmers and DCs - sum of two commodities end;

% Solving for the dual variable w for j=1:j0 for k=1:k0 for i=1:i0 for l=1:l0 % Finding the incremental cost among closed DCs % w.r.t v(k) for all CPKs tempw(i,j,k,l)=tc(i,j,k,l); end; tempw(i,ind,:,:)=0; w0(i,j,k)=sum(tempw(i,j,k,:)); end; % w(j,k)=sum(tempw(i,j,k,l)); end; end;

for k=1:k0 for i=1:i0 for j=1:length(cind) w(i,j,k)=max((v(k)-w0(i,cind(j),k)),0); end; end; end; % Obtaining the Benders cut sv=sum(v); BD=zeros(j0,1);

67

for j=1:j0 BD(j)=fc(j)-sum(sum(w(:,j,:))); end;

UB=sv+sum(BD(ind)); % Upper bound for Benders Decomposition

disp('Benders Cut'); disp(sv);

for i=1:j0 disp(BD(i)); disp('y'); disp(i); end;

% Master Problem and Inserting Benders cut into the Master problem

% Cost coefficents in Objective Function f1=zeros(j0+1,1); f1(1)=1; f1(2:j0+1)=0;

% Lower bounds lb1=zeros(j0+1,1);

% Upper bounds ub1=Inf(j0+1,1); ub1(2:j0+1)=1;

% Integer variables M1=(2:j0+1); e1=2^-24;

A_temp=zeros(1,j0+1); b_temp=zeros(1,1); A_temp(1,1)=-1;

Aeq1(1,2:j0+1)=1; Beq1=p0;

% Defining the new Benders cut as an Inequality constraint for j=2:j0+1 A_temp(1,j)=BD(j-1); b_temp(1)=-sv; end;

% Adding the new Benders cut to the previous constraints for j=1:j0+1 A1(iter,j)=A_temp(1,j); b1(iter,1)=b_temp(1,1); end;

% Solving the Master Problem using Integer Programming [x1,f1val,status]=IP1(f1,A1,b1,Aeq1,Beq1,lb1,ub1,M1,e1);

68

%x1=round(x1); %f1val=round(f1val);

LB=f1val; % Lower bound for Benders Decomposition e=UB-LB;

for i=1:j0 y(i)=x1(i+1); end;

iter=iter+1;

disp('New y values'); disp(y); disp('New DC Locations'); ind=find(y); disp(ind);

end;

toc;

69

II.MATLAB Output: DC Locations for Trial 1 values

Trial Farmers i0 DCs j0 CPKs k0 Commodities l0 No. of DCs p

1 10 10 10 2 2

Given value of p:

2

Initial y values:

0 0 0 0 0 1 0 1 0 0

DC Locations:

6 8

Optimization terminated.

Farmers-->DCs

3 6

6 6

10 6

1 8

2 8

4 8

5 8

7 8

8 8

9 8


3

6

10


1

2

4

5

7

8

9

Benders Cut

34.1073

7.2015

70

y

1

4.2810

y

2

-2.5857

y

3

9.0449

y

4

4.5849

y

5

7.4159

y

6

7.0217

y

7

4.5729

y

8

2.2371

y

9

2.9039

y

10

New y values

0 0 1 0 0 0 0 0 1 0

New DC Locations

3 9

Benders Cut

34.1073

71

7.2015

y

1

4.2810

y

2

-339.5608

y

3

9.0449

y

4

4.5849

y

5

7.4159

y

6

-333.1326

y

7

4.5729

y

8

2.2371

y

9

2.9039

y

10

New y values

0 0 1 0 0 0 0 0 1 0

New DC Locations

3 9


72

III.MATLAB Output: DC Locations for Trial 2 values

Trial Farmers i0 DCs j0 CPKs k0 Commodities l0 No. of DCs p

2 30 30 30 3 6

Given value of p:

6

Initial y values:

Columns 1 through 14

0 0 0 1 1 0 0 0 0 0 0 0 1 1


0 0 0 0 0 0 0 0 0 0 0 1 0 0


0 1

DC Locations:

4 5 13 14 26 30

Optimization terminated.

Farmers-->DCs

3 4

11 4

12 4

20 4

23 4

26 4

2 5

4 5

10 5

15 5

19 5

30 5

9 13

18 13

24 13

7 14

13 14

25 14

1 26

5 26

8 26

14 26

17 26

6 30

16 30

21 30

22 30

73

27 30

28 30

29 30


3

11

12

20

23

26


2

4

10

15

19

30


9

18

24


7

13

25


1

5

8

14

17


6

74

16

21

22

27

28

29

Benders Cut

114.7719

9.8443

y

1

6.1026

y

2

1.1750

y

3

9.2530

y

4

8.0471

y

5

9.9692

y

6

1.5998

y

7

7.7365

y

8

9.4030

75

y

9

3.8232

y

10

5.0134

y

11

9.4110

y

12

1.8494

y

13

3.7367

y

14

1.3406

y

15

8.1866

y

16

6.6269

y

17

5.9978

y

18

1.9657

y

19

1.3634

y

76

20

7.5343

y

21

4.5932

y

22

4.4876

y

23

8.3341

y

24

7.8334

y

25

5.7379

y

26

5.8644

y

27

4.6520

y

28

4.9384

y

29

6.9913

y

30

New y values


0 0 1 0 0 0 1 0 0 0 0 0 1 0

77


1 0 0 0 1 1 0 0 0 0 0 0 0 0


0 0

New DC Locations

3 7 13 15 19 20

Benders Cut

114.7719

9.8443

y

1

6.1026

y

2

-3.4420e+003

y

3

9.2530

y

4

-3.4351e+003

y

5

9.9692

y

6

1.5998

y

7

7.7365

y

8

9.4030

y

78

9

3.8232

y

10

-3.4381e+003

y

11

9.4110

y

12

1.8494

y

13

3.7367

y

14

-3.4418e+003

y

15

-3.4350e+003

y

16

6.6269

y

17

5.9978

y

18

1.9657

y

19

1.3634

y

20

79

7.5343

y

21

4.5932

y

22

4.4876

y

23

8.3341

y

24

7.8334

y

25

5.7379

y

26

5.8644

y

27

4.6520

y

28

4.9384

y

29

6.9913

y

30

New y values


0 0 1 0 0 0 1 0 0 0 0 0 1 0


80

1 0 0 0 1 1 0 0 0 0 0 0 0 0


0 0

New DC Locations

3 7 13 15 19 20


81

APPENDIX C

I.MATLAB Code for solving MILP

tic;

odc=[3,9]; % Open DCs cdc=[1,2,4,5,6,7,8,10]; % Closed DCs

%Objective function cost co-efficient f=rand(2020,1)*20+20; % Trial values: i=10;j=10;k=10;l=2 % 1000 continuous variables for Qijkl and % 20 binary variables for Yij

% Integer variables M=(2001:2020); e=2^-24;

%Lower Bounds lb=zeros(2020,1);

%Upper Bounds ub=inf(2020,1); ub(2001:2020)=1;

%Linear equalities Aeq=zeros(1630,2020);

% Quantities belonging to closed DCs are assigned zero % Commodity 1 for i=1:20 Aeq(i,i)=1; end;

for i=21:70 Aeq(i,i+10)=1; end;

for i=71:100 Aeq(i,i+20)=1; end;

for i=101:150 Aeq(i,i+30)=1; end;

for i=151:180 Aeq(i,i+40)=1; end;

for i=181:230 Aeq(i,i+50)=1; end;

82

for i=231:260 Aeq(i,i+60)=1; end;

for i=261:310 Aeq(i,i+70)=1; end;

for i=311:340 Aeq(i,i+80)=1; end;

for i=341:390 Aeq(i,i+90)=1; end;

for i=391:420 Aeq(i,i+100)=1; end;

for i=421:470 Aeq(i,i+110)=1; end;

for i=471:500 Aeq(i,i+120)=1; end;

for i=501:550 Aeq(i,i+130)=1; end;

for i=551:580 Aeq(i,i+140)=1; end;

for i=581:630 Aeq(i,i+150)=1; end;

for i=631:660 Aeq(i,i+160)=1; end;

for i=661:710 Aeq(i,i+170)=1; end;

for i=711:740 Aeq(i,i+180)=1; end;

for i=741:790

83

Aeq(i,i+190)=1; end;

for i=791:820 Aeq(i,i+200)=1; end;

% Commodity 2 for i=821:870 Aeq(i,i+210)=1; end;

for i=871:900 Aeq(i,i+220)=1; end;

for i=901:950 Aeq(i,i+230)=1; end;

for i=951:980 Aeq(i,i+240)=1; end;

for i=981:1030 Aeq(i,i+250)=1; end;

for i=1031:1060 Aeq(i,i+260)=1; end;

for i=1061:1110 Aeq(i,i+270)=1; end;

for i=1111:1140 Aeq(i,i+280)=1; end;

for i=1141:1190 Aeq(i,i+290)=1; end;

for i=1191:1220 Aeq(i,i+300)=1; end;

for i=1221:1270 Aeq(i,i+310)=1; end;

for i=1271:1300 Aeq(i,i+320)=1;

84

end;

for i=1301:1350 Aeq(i,i+330)=1; end;

for i=1351:1380 Aeq(i,i+340)=1; end;

for i=1381:1430 Aeq(i,i+350)=1; end;

for i=1431:1460 Aeq(i,i+360)=1; end;

for i=1461:1510 Aeq(i,i+370)=1; end;

for i=1511:1540 Aeq(i,i+380)=1; end;

for i=1541:1590 Aeq(i,i+390)=1; end;

for i=1591:1600 Aeq(i,i+400)=1; end;

% Single sourcing farmers: Ej Yij = 1 For all i Aeq(1601,[2001 2011])=[1 1]; Aeq(1602,[2002 2012])=[1 1]; Aeq(1603,[2003 2013])=[1 1]; Aeq(1604,[2004 2014])=[1 1]; Aeq(1605,[2005 2015])=[1 1]; Aeq(1606,[2006 2016])=[1 1]; Aeq(1607,[2007 2017])=[1 1]; Aeq(1608,[2008 2018])=[1 1]; Aeq(1609,[2009 2019])=[1 1]; Aeq(1610,[2010 2020])=[1 1];

% Demand equalities: EiEj Qijkl = Dkl Aeq(1611,(1:10:991))=1; Aeq(1612,(2:10:992))=1; Aeq(1613,(3:10:993))=1; Aeq(1614,(4:10:994))=1; Aeq(1615,(5:10:995))=1; Aeq(1616,(6:10:996))=1; Aeq(1617,(7:10:997))=1; Aeq(1618,(8:10:998))=1;

85

Aeq(1619,(9:10:999))=1; Aeq(1620,(10:10:1000))=1; Aeq(1621,(1001:10:1991))=1; Aeq(1622,(1002:10:1992))=1; Aeq(1623,(1003:10:1993))=1; Aeq(1624,(1004:10:1994))=1; Aeq(1625,(1005:10:1995))=1; Aeq(1626,(1006:10:1996))=1; Aeq(1627,(1007:10:1997))=1; Aeq(1628,(1008:10:1998))=1; Aeq(1629,(1009:10:1999))=1; Aeq(1630,(1010:10:2000))=1;

beq=zeros(1630,1); beq(1601:1610)=1;

% capacities at all farmers > demand at all CPKs %for i=1:2 % while(sum(pc(:,i))<sum(dc(:,i))) % pc=round(rand(10,2)*90+10);% Defining capacities at farmers Pil % dc=round(rand(10,2)*90+10);% Defining Demand at CPKs Dkl % end; %end;

pc=[[25;77;53;69;97;38;17;81;43;63;],[27;17;24;84;37;45;69;84;79;44;]]; dc=[[27;68;10;35;67;63;39;99;21;76;],[24;49;85;42;17;60;35;22;73;54;]];

count=1611; for l=1:2 for i=1:10 beq(count)=dc(i,l); count=count+1; end; end;

%Linear Inequalities A=zeros(408,2020);

co=1; for i=1:100 if(ceil(i/10)==3) A(co,[i 2001])=[1 -10000]; co=co+1; end; if(ceil(i/10)==9) A(co,[i 2011])=[1 -10000]; co=co+1; end; end;

co=21; for i=101:200 if(ceil(i/10)==13) A(co,[i 2002])=[1 -10000]; co=co+1;

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end; if(ceil(i/10)==19) A(co,[i 2012])=[1 -10000]; co=co+1; end; end;





co=121;

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for i=601:700 if(ceil(i/10)==63) A(co,[i 2007])=[1 -10000]; co=co+1; end; if(ceil(i/10)==69) A(co,[i 2017])=[1 -10000]; co=co+1; end; end;




co=201; for i=1001:1100 if(ceil(i/10)==103) A(co,[i 2001])=[1 -10000]; co=co+1; end; if(ceil(i/10)==109) A(co,[i 2011])=[1 -10000]; co=co+1;

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end; end;





co=301; for i=1501:1600 if(ceil(i/10)==153) A(co,[i 2006])=[1 -10000]; co=co+1;

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end; if(ceil(i/10)==159) A(co,[i 2016])=[1 -10000]; co=co+1; end; end;





90

A(401,[21:30,121:130,221:230,321:330,421:430,521:530,621:630,721:730,821:830,

921:930])=1; A(401,(2001:2010))=[-25 -77 -53 -69 -97 -38 -17 -81 -43 -63]; A(402,[1021:1030,1121:1130,1221:1230,1321:1330,1421:1430,1521:1530,1621:1630,

1721:1730,1821:1830,1921:1930])=1; A(402,(2001:2010))=[-27 -17 -24 -84 -37 -45 -69 -84 -79 -44]; A(403,[81:90,181:190,281:290,381:390,481:490,581:590,681:690,781:790,881:890,

981:990])=1; A(403,(2011:2020))=[-25 -77 -53 -69 -97 -38 -17 -81 -43 -63]; A(404,[1081:1090,1181:1190,1281:1290,1381:1390,1481:1490,1581:1590,1681:1690,

1781:1790,1881:1890,1981:1990])=1; A(404,(2011:2020))=[-27 -17 -24 -84 -37 -45 -69 -84 -79 -44];

% Throughput constraints A(405,[21:30,121:130,221:230,321:330,421:430,521:530,621:630,721:730,821:830,

921:930])=1; A(405,[1021:1030,1121:1130,1221:1230,1321:1330,1421:1430,1521:1530,1621:1630,

1721:1730,1821:1830,1921:1930])=0.89; A(406,[81:90,181:190,281:290,381:390,481:490,581:590,681:690,781:790,881:890,

981:990])=1; A(406,[1081:1090,1181:1190,1281:1290,1381:1390,1481:1490,1581:1590,1681:1690,

1781:1790,1881:1890,1981:1990])=0.89; A(407,[21:30,121:130,221:230,321:330,421:430,521:530,621:630,721:730,821:830,

921:930])=-1; A(407,[1021:1030,1121:1130,1221:1230,1321:1330,1421:1430,1521:1530,1621:1630,

1721:1730,1821:1830,1921:1930])=-0.89; A(408,[81:90,181:190,281:290,381:390,481:490,581:590,681:690,781:790,881:890,

981:990])=-1; A(408,[1081:1090,1181:1190,1281:1290,1381:1390,1481:1490,1581:1590,1681:1690,

1781:1790,1881:1890,1981:1990])=-0.89;

b=zeros(408,1); b(405:406)=500; %Maximum Throughput b(407:408)=350; %Minimum Throughput

[x,f1val,status]=IP1(f,A,b,Aeq,beq,lb,ub,M,e);

xf3=x(2001:2010); xf9=x(2011:2020);

f3=find(xf3);% Farmers alloted to DC 3 f9=find(xf9);% Farmers alloted to DC 9

disp('The farmers allotted to DC 3 are:'); disp(f3); disp('The farmers allotted to DC 9 are:'); disp(f9);

Q=zeros(10,10,10,2);

co=1; for l=1:2 for i=1:10 for j=1:10 for k=1:10

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Q(i,j,k,l)=x(co); co=co+1; end; end; end; end;

% Throughput of DCs b1=sum(x([21:30,121:130,221:230,321:330,421:430,521:530,621:630,721:730,821:8

30,921:930])); b2=sum(x([1021:1030,1121:1130,1221:1230,1321:1330,1421:1430,1521:1530,1621:16

30,1721:1730,1821:1830,1921:1930])); b3=sum(x([81:90,181:190,281:290,381:390,481:490,581:590,681:690,781:790,881:8

90,981:990])); b4=sum(x([1081:1090,1181:1190,1281:1290,1381:1390,1481:1490,1581:1590,1681:16

90,1781:1790,1881:1890,1981:1990]));

% Intensity factor

I1=1.0; % Commodity 1 I2=0.89; % Commodity 2

t1=b1*I1+b2*I2; t2=b3*I1+b4*I2;

disp('Throughput of DC 3 is :'); disp(t1); disp('Throughput of DC 9 is :'); disp(t2);

toc;

92

II.MATLAB Output:


1

2

4

5

7


3

6

8

9

10


435.2600


480.0300


Qijkl Values:

Commodity 1 (l=1):

val(:,:,1,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 27.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,2,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 68.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

93

0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,3,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 10.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,4,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 35.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,5,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 67.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,6,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 63.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

94

0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,7,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 39.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,8,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 99.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,9,1) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 21.0000 0 val(:,:,10,1) = 0 0 2.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

95

0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 74.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

Commodity 2 (l=2):

val(:,:,1,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 24.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,2,2) = 0 0 49.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,3,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 49.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 36.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

96

val(:,:,4,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 42.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,5,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 17.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,6,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 60.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,7,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 35.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

97

0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,8,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 22.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,9,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 73.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 val(:,:,10,2) = 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 54.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0 0 0 0.0000 0 0 0 0 0 0.0000 0

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