Final Thesis(2011) Opoku Agyemang

8/10/2019 Final Thesis(2011) Opoku Agyemang

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MODELING TRANSSHIPMENT PROBLEM IN A MANUFACTURING

INDUSTRY: CASE STUDY: COCA-COLA BOTTLING COMPANY,

KUMASI

By

OPOKU AGYEMANG (B.Ed MATHEMATICS)

A Thesis Submitted to the Department of Mathematics,

Kwame Nkrumah University of Science and Technology, Kumasi,

In partial fulfillment of the requirement for the degree of

MASTER OF SCIENCE

Industrial Mathematics, Institute of Distance Learning

July, 2011


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DECLARATION

I hereby declare that this submission is my own work towards Master of Science degree

and that, to the best of my knowledge, it contains no material previously published by

another person nor material which has been accepted for the award of any other degree of

the University, except where due acknowledgement has been made in the text.

Opoku Agyemang(PG 3016609) . ..

Students Name and ID Signature Date

Certified by:

Mr. F.K. Darkwah .. ....

Supervisor Signature Date

Certified by:

Mr. F.K. Darkwah .

Head of Department Signature Date

Certified by:

Prof. I. K. Dontwi ..

Dean, IDL Signature Date


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ABSTRACT

In this thesis we focus on a decision model for a real world problem. The problem reveal

itself as a transshipment problem where the Coca-Cola Bottling Company, Kumasi,

transports its products from source to destinations through intermediate points as a

system consisting of multiple retail locations with transshipment operations among the

retailers.

The study address the problem as a transportation problem and seeks find an efficient way

to minimize the total transportation cost using the Dual Matrix Approach. Manual

computations were also used to obtain the optimal solution.


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TABLE OF CONTENT

CONTENTS PAGES

DECLARATION ................................................................................................................. ii

ABSTRACT ........................................................................................................................ iii

TABLE OF CONTENT ..................................................................................................... iv

LIST OF FIGURES........................................................................................................... vii

LIST OF TABLES ........................................................................................................... viii

DEDICATION .................................................................................................................... ix

ACKNOWLEDGEMENT................................................................................................... x

CHAPTER 1 ........................................................................................................................ 1

INTRODUCTION ............................................................................................................. 1

1.2 Backgroung of the Study .............................................................................................. 3

1.2.1 Mode of Transportation in Ghana .............................................................................. 4

1.2.2 Profile of the coca Cola Bottling Company Limited, (TCCBCGL)......................... 8

1.3 Statement of the Problem ........................................................................................... 11

1.4 Objectives .................................................................................................................. 12

1.5 Methodology .............................................................................................................. 12

1.6 Justification of the Study. ........................................................................................... 13

1.7 Organization of the Study........................................................................................... 14

CHAPTER 2 ...................................................................................................................... 16

LITERATURE REVIEW ................................................................................................. 16

2.0 Introduction ............................................................................................................... 16

2.1 Lateral Transshipment ................................................................................................ 18

2.2 Transshipment in a supply chain with decentralized retailers ...................................... 20

2.3 Centralized and decentralized systems........................................................................ 23


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CHAPETR 3 ...................................................................................................................... 30

METHODOLOGY ............................................................................................................ 30

3.0 Introdcution ............................................................................................................... 30

3.1 Characteristics of Transportation problem .................................................................. 30

3.1.2 The Transportation Problem: LP Formulations .................................................... 30

3.2 The Decision Variables .............................................................................................. 31

3.3. The Objective Function ............................................................................................. 31

3.4 Unbalanced Transportation Problem ......................................................................... 33

3.5 The Transportation Algorithm .................................................................................... 35

3.5.1 The Transportation Tableau..................................................................................... 35

3.6. Finding an Initial Solution ......................................................................................... 37

3.6.1 Northwest Corner Method ................................................................................... 37

3.6.2 Least Cost Method ............................................................................................... 37

3.6.3 Vogel Approximation Method (VAM) ................................................................. 38

3.6.4 Testing the Solution for Optimality ...................................................................... 38

3.7 Computing for Optimality ..................................................................................... 39

3.7.1 Optimality by MODI Method .................................................................................. 39

3.7.2 Stepping Stone Method ........................................................................................ 40

3.7.3 A Dual-Matrix Approach to the Transportation Problem ..................................... 41

3.8 Transshipment........................................................................................................... 49

3.9 The Transshipment Model .......................................................................................... 51

CHAPTER 4 ...................................................................................................................... 53

DATA COLLECTION, ANALYSIS AND MODELING ................................................ 53

4.1 Data Description ........................................................................................................ 53

4.2 The dual matrix solution method ................................................................................ 58

4.3 Interpretation of Results ............................................................................................. 66


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CHAPTER 5 ...................................................................................................................... 67

CONCLUSIONS AND RECOMMENDATION .............................................................. 67

5.1 Conclusion ................................................................................................................. 67

5.2 Recommendation ....................................................................................................... 67

REFERENCES .................................................................................................................. 68

APPENDIX ........................................................................................................................ 71


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LIST OF TABLES

CONTENTS PAGES

Table 4.1 Names of sources and destinations 54

Table 4.2 Distances (in kilometers) from Sources to Destinations 55

Table 4.3 Unit cost of Transporting a crate of Coca-Cola Product from sources to

destination 57

Table 4.4 Summary of the result of the data analysed 66

Table 4.5 Final Distribution Of Product 66


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DEDICATION

This work is dedicated to the Almighty God for His protection and divine favor towards

me. To all who contributed in one way or the other to make this dream a reality especially

my wife Joana and our children Phyllis, Janice and Pearl for their unfading love and

support.


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ACKNOWLEDGEMENT

I would like to express my gratitude to all those who helped me in diverse ways to

complete this thesis.

I thank my thesis advisor, Mr. Kwaku Darkwah for his invaluable suggestions,

encouragement and supervision throughout the study.

I am grateful to all my lecturers at the Mathematics Department, Kwame Nkrumah

University of Science and Technology, Kumasi for their immersed contribution in making

this programme successfully.

I also wish to commend and express my gratitude to the Managers of The Coca- Cola

Bottling Company, Kumasi, for providing the data for the study.

I would also like to thank my friends and course mates, Mr. Victor Atekpo, Miss Hilda

Baffour and Mr. Karikari Emmanuel for their invaluable suggestions and remark. Last

but not least, I would like to express my gratitude to my family for their support which

enabled me to accomplish this programme.


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CHAPTER 1

INTRODUCTION

The introduction of motor-vehicles, in the course of the twentieth century radically

transformed the economies of Africa. The increased mobility of people, products, raw

materials, information, goods and services led to the development of new economies.

There has been a tendency to see motor-vehicles as being attached solely to the state and

the political and economic elite, yet their impact stretches far beyond the elite and into the

everyday lives of people in the smallest villages at the furthest reaches of African states.

The bus, mammy truck, car, pick-up and so forth reach far beyond where railways, ferries

and boats cant reach. The introduction of railways had a tremendous impact on African

societies. However, from the 1940s onwards the train dwindled in importance, and has

come to be almost totally superseded by buses, trucks and Lorries. The extensive shanty

towns that have developed on the tracks of the shunting yards of Ghana railways in

downtown Accra is a typical example of this decline. In addition, in contrast to the motor-

vehicle, the train is bound to run on the tracks laid out for it. The train does not allow for

the initiative of a single individual or a small group of people. The capital input is such

that it requires state funding and is quite simply beyond the finances of small

entrepreneurs, whereas the purchase of a motor-cycle, taxi or truck is not. Jan-Bert,

(2005).

Africa may possess but a minute proportion of the worlds motor-vehicles, precisely

because of the scarcity of transport there has been a tendency to see Africa as pre-

dominantly rural. Yet Africa is highly urbanized in sprawling cities that are often serviced


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solely by motor-vehicles. New companies have been created that transport people and

goods, these range from small single taxi companies to enormous freight enterprises. The

presence of motor-vehicles necessitated the development of roads, which in turn led to

further economic development. The increased accessibility stimulated and allowed for the

development and exploitation of resources which had been hitherto neglected; mining,

agriculture and industry all received a boost. In addition, the economic expansion and

increased mobility led to the development of, not only, the itinerant migrant laborers, but

also, the daily commuter , people essential to Africas formal economies, but heavily

dependent on the taxi and bus services of the informal economy.

The impact of the motor-vehicle in the informal economy has primarily been in the

service industry. African bus stations and transport depots are unthinkable without the

myriad of services provided by transport operators, food and drink sales, informal bars,

puncture repair men, welders, bush mechanics and many more. Drivers maintain their

concentration through the supply of stimulants, legal or otherwise, and passengers are

entertained and kept occupied by everything from acrobats to illegal copies of music

cassettes and book and pamphlet sellers.

Along the road villagers peddle handicrafts, agricultural produce, chickens, fish and as

well as bush meat and charcoal for city dwellers. New forms of corruption and taxation

have developed along African roads, and in many countries roadblocks have become an

important source of income for under paid civil servants. Associated with the informal

economy is the flourishing trade in second-hand cars, which has developed in the last

twenty years of the 20th century between Europe and West Africa, and Japan and Central


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season further impose a huge financial burden on organizations. It is therefore imperative

that managers of organizations make an improved management decisions to make better

utilization of resources at their disposal so as to minimize transportation costs.

The above, among other factors have contributed to the complexity of running a business

nowadays. The increasing rate of competition both domestically and abroad, high quality

requirement in the product and services, increasing awareness of environmental issues

have compelled organizations to improve their internal process rapidly in order to stay in

competitive. Many companies have come out with different marketing strategies focusing

on creating and capturing customer loyalty, translating customer needs into product and

service specification leading to high level quality products at reasonable cost.

The competitive nature of different brands satisfying the same customer needs in a

restricted market environment, organization utilize a large number of channels of

distribution in making their products available to the customers who may be spread in a

vast areas across the country though some may cater for foreign markets. Considered in

this perspective, modeling transshipment problem, simple as it seems, assumes a greater

significance. Transshipment is therefore the transfer of goods from one source to another

for further transportation to different destinations.

1.2.1 Mode of Transportation in Ghana

Road

Road transport is by far the dominant carrier of freight and passengers in Ghanas land

transport system. It carries over 95% of all passenger and freight traffic and reaches most


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communities, including the rural poor and is classified under three categories of trunk

roads, urban roads, and feeder roads. The Ghana Highway Authority, established in 1974

is tasked with developing and maintaining the country's trunk road network totaling

13,367km, which makes up 33% of Ghana's total road network of 40,186km.

The demand for urban passenger transport is mainly by residents commuting to work,

school, and other economic, social and leisure activities. Most urban transportation in

Ghana is by road and provided by private transport including taxis, mini-buses and

state/private-supported bus services. Buses are the main mode of transport accounting for

about 60% of passenger movement. Taxis account for only 14.5% with the remaining

accounted for by private cars.

One important trend in road transport (especially inter-city) is that there has been a shift

from mini-buses towards medium and large cars with capacities of 30-70 seats. There has

been a growing preference for good buses as the sector continues to offer more options to

passenger in terms of quality of vehicles used.

According to the Ministry of Roads and Transport, Ghanas road transport infrastructure

is made up of 50,620km of road network linking the entire country. These are under the

control of the Ghana Highways Authority (14,047 km), Department of Urban Roads

(4,063 km) and the Department of Feeder Roads (32,594 km). About 15.7% of the total

road network is paved.


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Railways

A triangular rail network of 950km link the three cities of Kumasi in the heart of the

country, Takoradi in the west and Accra-Tema in the east. The network connects the main

agricultural and mining regions to the ports of Tema and Takoradi. It has mainly served

the purpose of hauling minerals, cocoa and timber. Considerable passenger traffic is also

carried on the network.

There are firm plans by the Government to develop the rail network more extensively to

handle up to 60% of solid and liquid bulk cargo haulage between the ports and the interior

and /or the landlocked neighboring countries to the north of Ghana and elsewhere. The

government has set out seeking the necessary investment to restore the network, improve

speed and axle load capacity and replace worn-out rolling stock. Plans are far advanced to

privatize the State-owned Ghana Railways Corporation (GRC) through concession and to

provide much greater capacity for rail haulage of containers and petroleum products.

Air

The country is at the hub of an extensive international (and national) airline network that

connects Ghana to Africa and the rest of the world. Most major international carriers fly

regularly to Kotoka International Airport (KIA) in Accra, the main entry point to Ghana

by air. This is the result of Ghanas open skies policy, which frees an air space regulator

from the constraints on capacity, frequency, route, structure and other air operational

restrictions. In effect, the policy allows the Ghana Civil Aviation Authority (GCAA) to

operate with minimal restrictions from aviation authorities, except in cases of safety and

standards and/or dominant position to distort market conditions.

Ghana is working to position herself as the gateway to West Africa. KIA remains the

leading and preferred airport in the sub-region, having attained Category One status by the


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US Federal Aviation Administration (FAA) audit as part of their International Aviation

Safety Audit (IASA) programme. As at now, Ghana is one of the four countries in sub-

Saharan Africa in this category. The others are Egypt, South Africa and Morocco. It

handles the highest volume of cargo in the sub-region and has all the requisite safety

facilities, recommended practices and security standards.

A rehabilitation programme embarked upon since 1996 has brought about an expansion

and refurbishment and upbringing of facilities at the international terminal building, as

well as the domestic terminal. These terminals now have significantly increased traveler

and cargo capacity. The airports runway has been extended to cater for all types of

aircraft allowing direct flights from Ghana at maximum take-off weight without the need

for technical stops en-route.

Water

The Volta Lake was created in the early 1960s by building a dam at Akosombo and

flooding the long valley of the River Volta. It is the largest man-made lake in the world

stretching 415km form Akosombo 101km north of Accra, to Buipe in northern Ghana,

about 200km from Ghanas borderwith Burkina Faso.

As a waterway, the Volta Lake plays a key role in the Ghana Corridorprogramme by

providing a useful and low cost alternative to road and rail transport between the north

and the south. Ghana is in an advantageous position, by virtue of her seaports and inland

lake transport system, to service the maritime needs of land-locked countries to the north

of Ghana.


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Equatorial Coca-Cola Bottling Company - 68%

Government of Ghana - 32%

In year 2003, the Equatorial Coca-Cola Bottling Company of Barcelona, Spain bought

over the Ghana Government shares and assumed 100% ownership.

Mission Statement

The mission of TCCBCGL is to deliver high quality products and services that meet the

needs of our customers and consumers. To this end, we will manufacture and market

products which comply with the Coca-Cola Companys specifications and the

requirements of the consumers and endeavors to exceed.

Administrative Setup

Administratively, TCCBCGL is headed by a General Manager/CEO who is assisted by

eight Heads of Departments namely: Finance, Technical, Human Resource, Commercial

Manager, Supply Chain, Internal Control, and Administrative Plant Manager in Kumasi

and External Facilities Plant Manager in Accra. The company employs about 760 workers

and has about 31,000 customers, with over 8,000 Mini-Table operators and 77

independent Mini-Depot Operators, each of which employs at least 4 persons. Equally, the

Company outsources other non-core operators to outside bodies.

Product Range:

TCCBCGL manufactures eight (8) brands of its products:

Coca-Cola Fanta Minute Maid

Sprite Krest Burn

Schweppes Bon-Aqua


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Seventeen (19) flavors are currently bottled under the above mentioned brands, namely:

Coca-Cola Fanta Orange Fanta Lemon

Fanta Fruit Cocktail Sprite Krest Bitter Lemon

Krest Ginger-Ale Krest Soda Water Krest Tonic Water

BonAqua drinking water Schweppes Tonic Water Fanta Pineapple

Schweppes Bitter Lemon Schweppes Soda Water Fanta blackurrant

Coke light Burn Energy drinks Schwepps Malt

Minute maid

Operations

The TCCBCGL operates two plants, Accra and Kumasi, made up of 5 production lines:

four in Accra plant and one in Kumasi plant. From a sixty percent (60%) market share in

1995, the company in 2005 controls eighty six percent (86%) and as at March 2007, the

company controls ninety five percent (95%) of the beverage industry in Ghana.

A market leader in its own right, TCCBCGL has established extensive marketing and

distribution networks since 1995 throughout the country. To date, the company has

created 31,000 new outlets; 8,000 Mini-Tables and 8,000 Electric Coolers.

Social Responsibilities & Community Relations Activities

TCCBCGL has made tremendous contributions in the following areas:

1. Education

Donation to the Otumfuo Education Trust Fund (US$10, 000. 00).


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US$50,000 - Graduate Fellowship at the Premier University - University of

Ghana, Legon.

Project Partner - Interest Initiative for Africa set up by the UNITED NATIONS.

US$10,000.00 support towards Mother & Child Development Foundation

(US$5000)Total Cost of organizing annual National Essay Competition

Child Educational support for staff

2. Health and Environment

Medical College, University of Ghana , Legon, Endowment Fund

Assistance to the Ghana AIDS Commission

Refreshment during vaccination exercise for children against childhood diseases

US$ 1m Waste Water Treatment Plant (Accra)

Awareness Seminars organized by EPA

Support for Ramsar Site

Sakumono Lagoon

(US$ 600,000) Waste Water Treatment Plant for Kumasi

Ambulance for 37 Military Hospital.

Sponsors of Top four premier leagues in Ghana in 2003

Co-Sponsors of Top four premier leagues in Ghana from 2004 to date.

Official Soft Drink Sponsorship package for Ghana at 50 Jubilee Celebration

1.3 Statement of the Problem

Transshipment problem emanating from transportation is one of the most significant areas

of logistic management because of its direct impact on customer service level and the

firms cost structure. Outbound transportation costs can account for as much as ten (10) to


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1.7 Organization of the Study

The study is made up of five chapters with the Introduction as the Chapter 1, chapter 2

consists of the literature review .The methodology used in the study is discussed in the

Chapter 3 while Chapter 4 is made up of the data collection, analysis of the data and

results. Finally, the Chapter 5 deals with conclusion and recommendations.


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Summary

The role of transportation in the market economy was discussed in this chapter. Types of

transportation system in the country, brief history of Coca-Cola Bottling Company,

justification and the objectives of the study were also discussed.


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CHAPTER 2

LITERATURE REVIEW

2.0 Introduction

There has been a sizable amount of work done on transshipment problems.

Transshipments have been studied for emergency replenishment and perishable goods

applications. Krishnan and Rao (1965) laid the groundwork for much of the transshipment

research by analyzing a two-location, single period inventory problem. In their model,

transshipments occurred after demand was known but before it had to be fulfilled.

Transshipments therefore served as an emergency way to fill demands that would have

otherwise gone unfilled. They assumed that all the retailers had identical cost parameters.

Tagaras (1989) did an extension of that model, examining a two-location problem where

cost parameters varied from facility to facility. He also established the conditions for

complete inventory pooling. Robinson (1990) discussed solution techniques for specific

cases of these types of problems over multiple periods.

All of these research work assumed that transshipments had a zero lead-time and occurred

instantaneously. When a product is transshipped, that product can be used to fulfill

demand in that period. These authors have shown and emphasized the risk pooling

benefits associated with these types of transshipment policies. Tagaras and Cohen (1992)

analyzed a problem where replenishment lead times from the supplier are non-negligible,

but they still assumed that transshipments had a zero lead-time


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Other research has shown that the transshipment center problem is important in supply

chain management. Most of the relevant existing research focuses on networking, and

does not consider the dynamics of the configurations in transshipment center units.

Hoppe and Tardos (2000) concluded that the transshipment problem is defined by a

dynamic network with several sources and sinks. There are no polynomial-time

algorithms known for most of these problems. In their paper, they gave the first

polynomial-time algorithm for the quickest transshipment problem. Their algorithm

provided an integral optimum flow. Qi (2006) presented a logistics scheduling model for

two processing centers that are located in different cities. Each processing center has its

own customers. When the demand in one processing center exceeds its processing

capacity, it is possible to use part of the capacity of the other processing center subject to

a transshipment delay.

Lee and Elsayed (2005) noted that space shortage occurred when the demand exceeds the

warehouse storage capacity. The additional space requirement is satisfied by considering

leasing storage space. The warehouse storage capacity problem is then formulated as a

non-linear programming model to minimize the total cost of owned and leased storage

space.

Aghezzaf (2005) proposed strategic capacity planning to solve warehouse location

problems in supply chains operating under uncertainty. He used a special Lagrangian


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relaxation method in which the multipliers are constructed from dual variables of a linear

program.

Heragu et al (2005) and Meng et al (2004) stated that the two primary functions of a

warehouse are temporary storage and protection of customer orders, packaging of goods,

after-sale services, repairs, testing, inspection of goods; and providing value added

services.. To perform the above functions, the warehouse is divided into several functional

areas such as reserve storage area, forward (order collation) area and cross-docking. They

used a mathematical model and a heuristic algorithm that jointly determined the product

allocation to the functional areas in the warehouse as well as the size of each area using

data readily available to a warehouse manager.

2.1 Lateral Transshipment

Lateral transshipment between stocking locations are used to enhance cost efficiency and

improve customer service in different ways.

In the first approach, transshipments are realized after the arrival of demand but before it

is satisfied. If there is inventory at some of the stocking locations while some have

backorder, lateral transshipments between stocking locations can work well. Moreover,

pooling the stocks can be viewed as a secondary source of supply for inventory shortages,

especially when transshipments between stocking locations are faster and cheaper than

emergency shipments from a central depot or backlogging of excess demand. A large

portion of the transshipment literature is dedicated to models of such emergency


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transshipment models. Krishnan and Rao (1965), Herer and Rashit (1999), Herer et al.

(2004), Robinson (1990).

In the second approach, transshipments between stocking locations is considered as a tool

to balance inventory levels of stocking locations during order cycles. In this approach, to

be able to guarantee a certain level of customer service in all stocking locations, an

amount of inventory is carried at each location in balance relative to each other. Inventory

levels can become unbalanced due to random variations in demand, where the term

imbalance refers to the deviation of the inventory position of stocking locations from the

average inventory position (Diks and de Kok, 1996). The system stock is redistributed

before demand is observed when the transshipments between locations during the system

order cycle is allowed. It is expected that such redistribution will decrease the total

shortages and will increase the service level.

In emergency models, transshipments respond to actual shortages. However, the purpose

of redistributing inventory before the realization of demand is to reduce the risk of

possible future shortages Hoadley and Heyman (1977). Tagaras (1999), therefore, refers

to these models as preventive models.

In Krishnan and Rao (1965), they permit transshipments between identical retailers after

demand is observed but before demand is satisfied. They consider a single-item inventory

distribution system where the item can be stored in each of the N stocking locations that


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are supplied by an upper echelon common source with infinite capacity. They model the

transshipment problem with independent stochastic demand for infinite horizon.

Hoadley and Heyman (1977) extend the identical cost model of Krishnan and Rao (1965)

to a two-echelon model. Their model assumes returns and transshipments, where

preventive transshipments are executed before the realization of demand.

Different from the Krishnan and Rao model, Robinson (1990) also assumes finite

horizons and proves the optimality of the base stock ordering policy. Herer and Rashit

(1999) solve the single-period model for two stocking locations with non-identical cost

structures taking into consideration fixed replenishment costs. Herer and Tzur (2001,

2003) develop optimal and heuristic algorithms for the dynamic transshipment problem

incorporating fixed replenishment and transshipment costs with a deterministic demand

structure for finite horizon.

Ozdemir et al. (2006) incorporate transportation capacity such that transshipment

quantities between stocking locations are bounded due to transportation media or the

location transshipment policy. They develop a solution procedure based on infinitesimal

perturbation analysis to solve the stochastic optimization problem, where the objective is

to find the policy that minimizes the expected total cost of inventory and shortage.

2.2 Transshipment in a supply chain with decentralized retailers

Traditional work on transshipment focuses on the optimal inventory and transshipment

policies for a vertically integrated supply chain (Krishnan and Rao, (1965); Tagaras,


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(1989); Robinson, (1990); Wee and Dada, (2005); Herer et al., (2006), There are two

streams of recent research that study transshipment in decentralized supply chains.

One stream examines a horizontally decentralized supply chain; that is, transshipment

occurs between locations that are not owned by one firm. The upstream supplier of the

locations is not explicitly modeled in this stream of research. In particular, Rudi et al.

(2001) compare the equilibrium inventory levels under transshipment and under no

transshipment.

The second stream studies a vertically decentralized supply chain with a single

manufacturer and a chain store retailer. Assuming a normal demand distribution, Dong

and Rudi (2004) show that, under mild assumptions, the manufacturer is better off from

transshipment. Zhang (2005) generalizes the results of Dong and Rudi (2004) to an

arbitrary demand distribution. This work differs from the existing literature as it examines

transshipment in a completely decentralized supply chain taking into consideration both

the downstream retail competition (in inventory) and the upstream manufacturers

decisions.

Lee and Whang (2002) consider the transfer price in a secondary market, which is similar

to transshipment, though in their model there are an infinite number of retailers and the

retailers are price-takers. Rudi et al. (2001) analyze several cases where two retailers with

asymmetric bargaining powers set the transshipment price. However, both papers assume

that the manufacturers wholesale price is fixed.


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When manufacturers decisions are fixed, it is well recognized that the (inventory)

centralization can be beneficial to the retailers due to the pooling effect Chen and Zhang,

(2006)). Anupindi et al. (2001), Granot and Sosic (2003) and Sosic (2006) consider the

scenario of retailers competition, that is, the retailers unilaterally determine the

inventory they stock, but cooperatively determine how much inventory they want to share

through transshipment.

Three papers take into account the vertical interaction between the manufacturer and

retailers in the process of retail centralization. Under an endogenous wholesale price,

Netessine and Zhang (2005) compare the inventory levels of the decentralized retailers

and a chain store in the cases where the retailers products are complementary and

substitutable.

Anupindi and Bassok (1999) analyze an alternative to transshipment, i.e., customer

search. The manufacturer may prefer retail decentralization or centralization, depending

on the rate of customer search. While the two papers consider retail centralization in

contexts other than transshipment, both papers do not discuss the impact of centralization

on the retailers. Ozen et al. (2008) show that if retailers reallocate inventories after

observing demand signals, the retailers are better off but the manufacturers profit may

either increase or decrease.


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2.3 Centralized and decentralized systems

This subsection is structured by first considering centralized systems with a single echelon

followed by centralized systems with two echelons and finally decentralized systems

where each stock point (retailer) aims to maximize its own profit.

One echelon centralized systems.

The first research to consider a reactive mode of transshipment was that by Krishnan and

Rao (1965). They consider a model that is similar to the periodic lateral transshipment

model of Gross (1963) which has negligible transshipment times, but they aim at

minimizing cost through transshipments once all demand is known. This type of model is

continued through the work of Robinson (1990), who provides an optimal solution for a

multi-location, multi period model. However, this solution can only be determined for

networks with either two non identical locations or any number of identical locations. For

more than two non-identical locations, a LP based heuristic solution procedure is

proposed and shown to perform well for a number of scenarios.

Returning to the multi location problem of Robinson, a similar model is that of Herer et

al. (2006). They consider a more general cost structure and use LP and a network of

framework to produce a method which is shown to be more robust than that of Robinson.

Further developments are done by Ozdemir et al. (2006) who look at putting capacity

constraints on the transport network and observe that these restrictions change the

system's inventory distribution and increase the total cost.

An alternative approach is taken by Hu et al. (2005) to develop the multi location

problem. They calculated a simplified model which can be used to approximate ordering

policies under certain conditions. These conditions are that the system contains a small


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supply. For a two location system an optimal policy is computed and further analysis

examines when this policy finds most benefit in transshipping.

Whilst results for systems which transship after all demand in the period has been

observed give useful insights into the transshipment problem, in practice it is often more

likely that continuous demand will be observed and each instance may trigger a reactive

transshipment. For such a problem Archibald et al. (1997) consider a two location system

where both experience Poisson demand. Demand that is not met from stock on hand at a

location or by lateral transshipments is lost to the system (and can be emergency ordered).

Archibald et al, (1997) show that an order-up-to policy is optimal. Moreover, they prove

that there exist threshold times dependent on the inventory level, such that a location

should only fulfill a lateral transshipment request if the time until the next ordering

opportunity is less than the threshold time. Archibald (1997) continues this line of

research for a multi- location setting. He examines the performance of three proposed

heuristics. The results show that all three partial pooling heuristics outperform both no

pooling and complete pooling. Out of the three, the least conservative appears to work

best over the range of test settings. This heuristic determines for each location whether

there is at least one other location that benefits from transshipping to it and, if so, fulfills

transshipment requests from not only that location but any location. A useful extension of

this work is by Archibald et al. (2007), who look at the multi- product case where each

location only has a fixed capacity.

Further work on multi -location systems is undertaken in (Archibald et al. 2008; 2009).

These papers consider the real world situation of a tyre retailer with a large network of

locations. Archibald et al. (2009) look to mitigate the problem of dimensionality with this


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type of system by approximating the dynamic programming value function. This is done

by a pair wise decomposition, which considers two locations at a time and has been

shown to improve upon the previous heuristics proposed in Archibald (2007) and also

upon complete pooling. One restriction on this model is that all locations must have the

same review period. Archibald et al. (2008) relaxes this restriction by using a two step

heuristic that first calculates a static policy for determining which location meets a

demand, and then applies dynamic programming policy improvement.

A separate direction for research is the study of dynamic deterministic demand systems.

Herer and Tzur (2001) develop a solution for a two location problem. Looking at

determining optimal ordering and transshipment decisions over a finite horizon, they

examine the key properties of the system. These properties form a framework that allows

this type of model to be solved in polynomial time. This problem is later extended by

Herer and Tzur (2003) to a multi-location system.

Finally, Herer et al. (2002) look more generally at the usefulness of transshipments under

the term `legility' which looks to provide a lean and agile inventory system. By looking at

some of the previously discussed models they show that transshipments help to improve

system performance under these two criteria and produce a way of analyzing this

information.

Two echelon centralized systems

In a system with two echelons there are several ways in which stock outs can be satisfied

through emergency stock movements. Lateral transshipments are one possibility but there


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could be situations where it is beneficial also to perform emergency shipments from the

central warehouse. Wee and Dada (2005) consider this problem with five different

combinations of transshipments, emergency shipments and no movements at all and

devises a method for deciding which setup is optimal under a given model description.

This research allows the structure of the emergency stock movements to be established.

Dong and Rudi (2004) examine a different aspect by looking at the benefits of lateral

transshipments for a manufacturer that supplies a number of retailers. They compare the

case where the manufacturer is the price leader to the case of exogenous prices. For

exogenous prices, it is found that retailers benefit more when demand across the network

is uncorrelated.

For the endogenous price case, modeled as a Stackelberg game, the manufacturer exploits

his leadership to increase his benefits, leaving retailers worse of if they use

transshipments. These results are restricted to demand that follows a normal distribution,

but Zhang (2005) extends them to general demand distributions

Decentralized systems

A decentralized system is one in which each stocking point operates to meet its own

goals. Chang and Lin (1991) consider when it is beneficial for such a system to actually

operate as a more centralized system by using transshipments. They compare a

decentralized model with a centralized model and deduce some properties that, if met,

show that costs will be reduced if the operation shares resources.

A related model to this is that of Zhang (2005) who studies whether independent vendors

benefit by co-operating as a grand coalition. The problem is modeled as a general


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newsvendor situation with N retailers. Using a game theoretic approach, it is shown that if

retailers cooperate then they can always achieve a higher profit. This shows that no

retailer has an incentive to leave the grand coalition and illustrates that centralized

ordering and transshipments can also be beneficial for decentralized systems. A limitation

of this study is that transshipment costs are not included.

If it is beneficial for a system to include more co-operations, the next stage is to establish

how co-operation can be established. Two papers which consider this are Rudi et al.

(2001) and Hu et al. (2007). Both consider newsvendor type models with a manufacturer

and two retailers. In Rudi et al. (2001) transshipment prices are determined in advance by

an accepted authority, for example by the manufacturer who would like to stimulate stock

sharing and is also willing to invest in an information system to provide accurate stock

level data.

In Hu et al. (2007), necessary and sufficient conditions are derived for the existence of

transshipment prices that induce retailers to make jointly optimal decisions. The research

focuses on finding linear transshipment costs which will induce co-operation but it is

shown that these do not always exist. This highlights an area of future research which

could consider more complex pricing structures.

An extension of this type of model to N retailers is discussed by Anupindi et al. (2001).

They drop the assumption of predetermined transshipment prices, and instead apply a rule

for allocating the additional profits from transshipments. This rule uses a price that is

based on the dual of the transshipment problem. It is shown that this rule is always in the


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core of the corresponding transshipment game. However, Granot and Sosic (2003) show

that if retailers can decide how much to share (i.e. partial pooling), and then it may happen

that no residual inventory is distributed and hence no additional profit is gained.

Granot and Sosic (2003) also identify a class of allocation rules that results in complete

pooling, but that is not in the core of the corresponding transshipment game. It can be

shown that this allocation rule leads to a farsighted stable grand coalition for symmetric

retailers Sosic (2006)).

The ideas of several authors with different views were discussed in this section. This

paper seeks to expand the ideas of the authors whose names have been mentioned above

to minimize cost using transshipment concept.


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CHAPETR 3

METHODOLOGY

3.0 Introdcution

This chapter of the research work discusses the transportation and transshipment

problems.

3.1 Characteristics of Transportation problem

The transportation deals with the distribution of goods from several points of supply, such

as factories, often known as sources, say m sources to a number of points of demands,

such as warehouses often known as destination, say ndestinations.

Each source is able to supply a fixed number of units of products, usually called capacity

or availability and each destination has a fixed demand, usually known as requirement.

Movement of goods or products are usually across a network of routes that connect each

point serving as a source and another point acting as a destination thus supply routes and

demand routes respectively. Each source has a given supply while each sink has a given

demand and the routes connecting the two has a given transportation cost per unit of

shipment. The objective is schedule shipment from source to destination so that the total

transportation cost is minimized so as to maximize profit.

3.1.2 The Transportation Problem: LP Formulations

Suppose a company has m warehouses and n retail outlets. A single product is to be

shipped from the warehouses to the outlets. Each warehouse has a given level of supply,

and each outlet has a given level of demand. We are also given the transportation costs


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between every pair of warehouse and outlet, and these costs are assumed to be linear.

More explicitly, the assumptions are:

The total supply of the product from warehouse iis aiwherei= 1, 2, m.

The total demand for the product at outlet j is bj,wherej= 1, 2,n.

The cost of sending one unit of the product from warehouse i to outlet j is

equal to Cij, where i= 1, 2, m and j= 1, 2, n. The problem of interest is to

determine an optimal transportation scheme between the warehouses and the

outlets, subject to the specified supply and demand constraints.

3.2 The Decision Variables

A transportation scheme is a complete specification of how many units of the product

should

be shipped from each warehouse to each outlet. Therefore, the decision variables are:

Xij= the size of the shipment from warehouse ito outletj,where i= 1, 2, mand

j = 1, 2. . .n. This is a set of m nvariables.

3.3. The Objective Function

Consider the shipment from warehouse ito outletj. For any iand anyj,the transportation

cost per unit is Cij; and the size of the shipment is Xij. Since we assume that the cost

function is linear, the total cost of this shipment is given by CijXij. Summing over all i and

alljnow yields the overall transportation cost for all warehouse-outlet combinations. That

is, our objective


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function is:

1 1

m n

ij

i j

C X i j= =

3.3.1 The Constraints

Consider warehouse i.The total outgoing shipment from this warehouse is the sum

xi1+xi2+ + xin.

In summation notation, this is written as

1

n

ijj

X=

Since the total supply from warehouse i is ai,the total outgoing shipment cannot exceed

ai. That is, we must require

1

, ( 1, 2... )

m

ij i

i

X a j n=

= (1)

Consider outletj. The total incoming shipment at this outlet is the sum

x1j+x2j + +xmj.

In summation notation, this is written as.

1

m

i

X ij=

Since the demand at outlet j is bj, the total incoming shipment should not be less than bj .

That is, we must require:

1

, ( 1, 2... )

m

ij j

i

X b j n=

= (2)

This results in a set of m + nfunctional constraints. Of course, as physical shipments, the

Xijs should be non negative. This is a linear program with m ndecision variables, m +

nfunctional constraints, and m nnon negativity constraints. With the above discussion,

we can now assume without loss of generality that every transportation problem comes

with identical total supply and total demand. This gives rise to what is called the balanced

transportation problem. Finally, under the assumption that


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1 1

m n

i ji j

a b= =

= (3)

holds for a balanced transportation problem and we have

Minimize

1 1ij

m n

iji j

C X

Subject to

1

, ( 1, 2... )

m

ij i

i

X a j n=

= = (4)

1

, ( 1, 2... )

m

ij j

i

X b j n=

= = (5)

With1 1

m n

i ji j

a b= =

=

and Xij 0 for i=1,2,mand j=1, 2,n

3.4 Unbalanced Transportation Problem

In many real situations demand exceeds supply and vice versa. This leads to what is

termed an unbalanced transportation problem. An unbalanced transportation problem can

be changed to a balanced one by adding a dummy row (source) with cost zero and the

excess demand is entered as a requirement if total supply is less than the total demand. On

the other hand if the total supply is greater than the total demand, then introduce a dummy

column (destination) with unit cost being zero and the excess supply is entered as the

requirement for dummy destination. For an unbalanced transportation problem we have

1 1

m n

i ji j

a b= =

(6)


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Two cases can be considered hee and they are

Case (1).

Introduce a dummy destination in the transportation table. The cost of transporting to this

destination is all set equal to zero. The requirement at this destination is assumed to be

equal to

Case (2) .

Introduce a dummy origin in the transportation table; the costs associated with are set to

be equal to zero and the availability is

7

1

.

1

n

j

j

m

i

i ba

81 1

m

i

n

j

ji ba

911

m

i

i

n

j

j ab


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3.3.2 Converting unbalanced problem to a Balanced Transportation Problem

An unbalanced transportation problem can be converted to a balanced one by adding a

dummy row (source) with cost zero and the excess demand is entered as a requirement if

total supply is less than the total demand. On the other hand if the total supply is greater

than the total demand, then introduce a dummy column (destination) with cost zero and

the excess supply is entered as the requirement for dummy destination.

3.5 The Transportation Algorithm

The transportation algorithm consists of three stages

1. Find a transportation pattern that uses all the products available and satisfies all the

requirement. This is called developing an initial solution.

2. Test the solution for optimality. If the solution is optimal stop but if not move to

stage three.

3. Use the stepping stone method or other method to obtain an improved solution and

return to stage two

3.5.1 The Transportation Tableau

The Simplex tableau serves as a very compact format for representing and manipulating

linear programs. The transportation tableau represents for transportation problems that are

in the standard form. For a problem with msources and nsinks, the tableau will be a table

with mrows and ncolumns. Specifically, each source will have a corresponding row; and

each sink, a corresponding column. For ease of reference, we shall refer to the cell that is

located at the intersection of theithrow and thejthcolumn as cell (i, j). Parameters of

the problem will be entered into various parts of the table in the format below.


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Figure 1. Transportation Problems Matrix

That is, each row is labeled with its corresponding source name at the left margin; each

column is labeled with its corresponding sink name at the top margin; the supply from

source i is listed at the right margin of the ith row; the demand at sink j is listed at the

bottom margin of thejthcolumn; the transportation cost Cijis listed in a sub cell located at

the upper-left corner of cell (i,j); and finally, the value ofXijis to be entered at the lower-

right corner of cell (i, j).

1.

The sum of product of theXijand Cijis the cells is the objective function

ij ijC X

2.

Sum ofXij across row give source supply constraint ij ij

X a

3. Sum ofij

X across column give destination constraintij ji

X b with 0ij

X


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3.6. Finding an Initial Solution

In finding an initial solution the following methods are used:

3.6.1 Northwest Corner Method

An initial solution can be found by the North-West Corner Method. It will be recalled

that prior to applying the Simplex Algorithm an initial solution had to be established in

the simplex tableau. This method requires that we start from the upper left-hand cell or the

NorthWest Corner of the table and allocate units to shipping routes as follows;

1. Allocate as much goods as possible to the selected cell and adjust the associated

amounts of supply and demand by subtracting the allocated amount.

2. Cross out the row or column with zero supply or demand to indicate that no

further assignments can be made in that row or column. If a row and a column add

up to zero simultaneously, cross out one only.

3. If exactly one row or column is left out uncrossed, stop. Otherwise, from the

current cell move to the right cell if a column has been crossed out or below if a

row has been crossed out. Go to (1)

3.6.2 Least Cost Method

The Least Cost Method uses the following algorithm step follows;

Step 1: Assign as much goods as possible to the cell with minimum unit transportation

cost

Step 2: Cross out the satisfied row or column and adjust supply and demand.

Step 3: If both a column and arrow are satisfied simultaneously, cross out only one.

Step 4: Stop when only one row or column is left uncrossed, otherwise, continue


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Step 5:Locate the next cell having the least cost and go to step one.

The NorthWest Corner Method does not utilize shipping cost. It can only yield an initial

basic feasible solution easily but the total shipping cost may be very high. The Least Cost

Method uses shipping cost in order to come up with a basic feasible solution that has a

least cost

3.6.3 Vogel Approximation Method (VAM)

The algorithm for VAM is as follows;

Step 1:Compute column penalties for each column by identifying the least unit cost and

the next least unit cost in that column and taking the positive difference. This is the

column penalty for the column. In a similar way we may compute the row penalty for

each row as the positive difference between the least unit cost and next least unit cost in

the row. Column penalties are below the columns after the demand values and row

penalties are shown to the right of each row after the supply values

Step 2: Find the cell for which the value of the row and the column penalties is greatest.

Allocate as much goods to this cell as the row supply or column demand will allow. This

implies that either a supply is exhausted or a demand is satisfied. In either case delete the

row of the exhausted supply or the column of the satisfied demand.

Step 3:Calculate row and column penalties for the remaining rows and columns and go to

step 2, repeat the process until a basic feasible solution is found.

3.6.4 Testing the Solution for Optimality

The method for testing a solution for optimality can only be applied if one essential

condition is satisfied. Thus, the number of cells (routes) used must be equal to one less


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than the sum of the number of rows and the number of columns. In the general case, when

we have m sources andn destinations the number of occupied cells must be (m+n-1).

When the number of occupied (allocated) cells is less than this, the solution is said to be

degenerate. This can be resolved by creating an artificially occupied cell, that is we place

a zero in one of the unused cells and then treat that cell as if it were occupied.

3.7Computing for Optimality

For computation of optimality, the following methods are used:

3.7.1 Optimality by MODI Method

To test a solution for optimality there is the need to calculate an improved index for each

cell. As part of the steps in this process we must compute;

1. LetRibe the cost variable for each row and

2. Rjbe the cost variable for each column

If Cij is the unit cost in the cell in the ithrow andjthcolumn of transportation tableau then

we can obtain the above values by setting

Ri +Kj =Cij

for the occupied (used) cells.

After all equations have been written, setR1 =0

Step 1:Solve for the system of equations for allRandKvalues

Step 2;having calculated for the RandKvalues, we now calculate for each unused cell

an improvement indexIijusing the formula

Iij=CijRiKj

Step 3: Select the largest negative index.


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iii. Since the cell which carried the allocation m now has a zero allocation, it is deleted

from the solution and is replaced by the cell in the circuit which was originally

unoccupied and now has an allocation m.

iv. The result of the re- allocation is a new basic feasible solution. The cost of this new

basic feasible solution is m less than the cost of the previous BFS.

3.7.3 A Dual-Matrix Approach to the Transportation Problem

Similar to the stepping-stone method, the occupied cells are called basic cells, and all

other empty cells are called non-basic cells in the dual-matrix approach. The main idea of

the dual-matrix approach is

1. To obtain a feasible solution to the dual problem and its corresponding matrix.

2. Then the duality theory is used to check the optimal condition and to get the

leaving cell.

3.

All non-basic cells are evaluated in order to get the entering cell.

4. Finally, the entering cell replaces the leaving cell and the matrix is updated.

Advantages of the Dual Matrix Approach

A new approach, the dual-matrix approach, to the transportation problem approach

considers the dual of the transportation model, starts from a good feasible solution, and

uses a matrix to get next better solution until an optimal solution is obtained The approach

adopts the linear algebra to solve .the transportation problem. A new concept, virtual cells,

is introduced in this approach.

The dual-matrix approach can be applied to both balanced and unbalanced transportation

problems. An unbalanced transportation problem is not required to be converted into a


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balanced; problem, unlike the stepping- stone method. Another advantage over the

stepping-stone method is that the dual-matrix method does not have the degeneracy

problem. The third feature of the approach is no path tracing. The disadvantage of the dual

matrix approach is t.hat the approach needs an (m + n) x (m + n) matrix.

The dual-matrix approach is presented as follows:

Step 0 Initialization:

Step 0.1: Set A = (b1, b2,bn, -a1, -a2,-am).

where aiand bjrepresent supply and demand respectively

Step 0.2: Set ui= 0; (i = 1, 2, ... , m) and let

vj= cij = min {Cij, i= 1,2, ..., m}; j = 1,2, ..., n.

Ties can be broken arbitrarily. The corresponding cells to Cij are (ij, j) (j= 1, 2 ,n),

respectively.

Step 0.3: Let the basic cell set T = {(i1, 1), (i2, 2), .... , (in,n), (1, 0), (2, 0), ... , (m, 0)}

The cells (1,0), (2, 0), ... , (m, 0) are called virtual cells because they do not exist in the

original transportation problem matrix.

Step 0.4:Let the matrix D= [dij]; i, j= 1, 2, m+ n;

where dij1 2

; , 1,2,..........

1 , 1,2........

1 1, 2.......... , , ,..........where

1 , 1, 2,..................................

0 .

ij

n

ij

D d i j m n

i j n

i n j n i n i n id

i j n n n m

otherwise

=

and compute the objective: w=

1 1

j j i ib v a u

m n

i j

(10)


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Step 1 Determination of the leaving cell:

Step 1.1 Compute vector Y = AD

Step 1.2 Find the smallest value ykin the elements of Y, is the smallest. Ties can be

broken arbitrarily.

Step 1.3 If yk 0, the solution is optimal (both the dual and primal), stop.

Otherwise, the leaving cell is the kthcell (ik, jk) T

Step 2 Determination of the entering cell

Step 2.1:Let

Q = = and P = =

Step 2.2: If for all non-basic cells, ifpi - qj0, then the dual problem is not bounded,

and the original primal problem has no feasible solution, and stop.

Otherwise, compute

ij=Cij+ uivj if piqj> 0

Step 2.3: Find the smallest value stin all ij,. The cell (s,t) is the entering cell. Ties can

be broken arbitrarily.

Step 3: Updating


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Step 3.1 Update the matrix D

Step 3.1.1: For the elements of column kin D:

d = - dik l= 1, 2 ,m+n;

Step 3.1.2:For the elements of other columns inD:

dlr= dlr+ (ds+ n rdtr) dik

1,2,... 1.....r k m n= - +

1, 2,...l m n= +

Step 3.2: Update the basic cell set T:

Replace the kth cell (ik, jk) in Twith the entering cell (s, t)

Step 3.3: Update the objective value:

Compute

i= ui- stpi i= 1, 2, m:

V 1 = vj- stqj j= 1, 2, n:

and the objective:

=1 1

j j i ib v a u

m n

i j

Go to step 1

The initialization procedure (Step 0) is; to obtain an initial feasible solution, by setting ui

= 0 and vjbeing the smallest cost in the columnj of Cij; obviously, they meet the

constraint set (10) in the dual problem.1 1

n m

j j i ij i

b v a u= =

-=


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Subject to j i ijV U C- (i=1,m; j=1,n) (10)

j i ijV U C- (i = 1m; j=1n)

Here ,all uiandvjare dual variables

The matrixD is an (m+ n) x (m +n) matrix, which can be divided into four sub -matrices

as follows:

1. The upper left sub-matrix is an n x n identity matrix.

2.

The upper right sub-matrix is an n x mmatrix: If the cell (i,j) is a basic

cell (corresponding to cij), then the element (j, i) in this sub-matrix is (-1). All

other elements in this sub-matrix are zero (0).

3. The lower left sub-matrix is an mx n zero matrix.

4. The lower right sub-matrix is an mx mnegative identity matrix.

During the main procedure of the dual-matrix approach: Step 1 is to get the leaving cell:

similar to getting a leaving variable in the simplex method. As a matter of fact, the initial

feasible solution in the dual-matrix approach is a very good starting point. From the

objective function in the dual, it is obvious that uishould be the smallest. The smallest is

(0) for all ui

On the other hand vjshould be the larger, the better. However, due to the constraint set

(2), a vj can only be the minimum value of cijin the columnj.

Step 2 is to obtain the entering cell by evaluating all non-basic cells: which is similar to

the stepping-stone method. Finally: the matrix D and other relevant data are updated


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1 1 1

2 22

3 411

03 3 st

v v q

v qvf

- = - = - =

And

=

1 1j j i ib v a u

m n

i j

= 2450

Now Y = AD = (400,350, 50, 250, 50). Since all yk > 0 Stop.

So the optimal solution is obtained with the objective = 2450, with x11= 400, x32= 350,

x21= 50, x20= 250, and x30= 50

If a dummy destination is introduced to make the problem balanced with the cost 0s for

those dummy cells, the objective will be the same as above, with x11= 400, x32= 350, x21

= 50, x23= 250, x33= 50, and x30= 0. Here, there are 6 basic cells since now it is a 33

transportation problem, and x23 and x33 are the dummy cells in this newly created

balanced problem. Two virtual cells x20= 250, and x30=: 50 in, the original problem can

be explained as the dummy cells in the balanced problem. However, the virtual cell (3, 0)

in the solution of the balanced problem is really a virtual cell because it really does not

exist

3.8 Transshipment

In a transportation problem, shipments are allowed only between source-sink pairs. In

many applications, this assumption is too strong. For example, it is often the case that

shipments may be allowed between sources and between sinks. Moreover, there may also

exist points through which units of a product can be transshipped from a source to a sink.

Models with these additional features are called transshipment problems. Interestingly, it


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turns out that any given transshipment problem can be converted easily into an equivalent

transportation problem.

As each transshipment point can both receive and send out products, it plays the dual roles

of being a sink and a source. This naturally suggests that we could attempt a reformulation

in which each transshipment point is split into a corresponding sink and a

corresponding source. A little bit of reflection, however, leads us to the realization that

while the demand and the supply at such a pair of sink and source should be set at the

same level (since there is no gain or loss in units), it is not clear what that level should be.

This is a consequence of the fact that we do not know how many units will be sent into

and hence shipped out of a transshipment point. Fortunately, upon further reflection, it

turns out that this difficulty can actually be circumvented by assigning a sufficiently-

high value as the demand and the supply for such a sink-source pair and by allowing

fictitious shipments from a given transshipment point back to itself at zero cost.

More specifically, suppose the common value of the demand and the supply at the

corresponding sink and source of a given transshipment point is set to h;and suppose x

units of real shipments are sent into and shipped out of tha t transshipment point. Then,

under the assumption that his no less thanx, we can interpret this by saying: (i) a total of

hunits of the product are being sent into the corresponding sink, of which xunits are sent

from other points (or cities) and h xunits are sent (fictitiously) from the transshipment

point to itself; and (ii) a total of h units of the product are being shipped out of the

corresponding source, of which x units are shipped to other points (or cities) and h x

units are shipped (fictitiously) back to the transshipment point itself.

Notice that since a shipment from a transshipment point back to itself is assumed to incur

no cost, the proposed reformulation preserves the original objective function. The only


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remaining question now is: What specific value should be assigned to h? The default

answer to this question is to let h equal to the total supply in the original problem. Such a

choice is clearly sufficient because no shipment can exceed the total available supply. It

follows that we have indeed resolved the difficulty alluded to earlier.

3.9 The Transshipment Model

A transshipment model is a transportation model with intermediate destinations between

source and destination (sink).

Constraints

Constraints involving source and destination are similar. That is, everything leaving

source must not exceed supplies and everything entering destination must not exceed

demand. Furthermore everything entering intermediate point must equal everything

leaving it.

Given m pure supply nodes with supply ai, n pure demand nodes with demand bj, and

transshipment nodes .Suppose the unit transportation cost from supply node i to

transshipment node k is Cikand the unit transportation cost for transshipment node k to

demand node jisCjkThe transshipment problem can be formulated as

Subject to


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CHAPTER 4

DATA COLLECTION, ANALYSIS AND MODELING

4.1 Data Description

The data was obtained from Coca Cola Bottling Company. The company has two

production points or source in Ghana, one in Accra and the other in Kumasi. The data was

obtained from Kumasi; hence the Kumasi Plant source would be used as the main source.

Their products are shipped by road from this plant (source) to their Mini Depot Offices

(MDOs) before they are transported to final destination hereby referred to as Manual

Distribution centers.

The data is a quantitative data which is made up of the distances from sources to the

destinations. The table 4.1 is a display of names of towns acting as sources and

destinations. Columns 1and 2 are the various sources and the names of the towns in which

these sources are located respectively. Column 3 is a list of codes representing the towns

serving as sources. Columns 4 and 5 show the destinations and the towns representing

these destinations respectively. Column 6 indicates the codes of the destinations.


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Table 4.1 Names of sources and destinations

The following distance data in table 4.2 which indicates the distances from sources to

destination were also obtained from the Coca- Cola Company.

The codes of the various sources are listed in columns one and two. Columns three to

column fourteen are the distances from the various sources to destinations.

SOURCES TOWNS CODE DESTINATIONS TOWNS CODE

1 Ahinsan AHI 1 Agona AGO

2 Techiman TEC 2 Konongo KON3 Meduma MED 3 Dormaa DOR

4 Feyiase FEY 4 Daban DAB

5 Kejetia KEJ 5 Kintampo KIN

6 Obuasi OBU

7 Sunyani SUN

8 Ejisu EJI

9 Dompoase DOM

10 Santasi SAN


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Table 4.2 Distances (in kilometers) from Sources to Destinations

FEY MED KEJ TEC SUN EJI SAN DOR KON KIN DAB AGO DOM OBU

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14

AHI S1 5 11 5 121 130 22 7 209 54 176 4 40 5 64

FEY S2 - 16 10 126 135 27 12 214 59 181 9 45 3 69

MED S3 16 - 6 120 131 28 13 210 55 177 11 29 16 65

KEJ S4 10 6 - 116 125 22 6 204 53 171 5 35 10 59

TEC S5 126 120 116 - 40 138 122 96 170 36 126 156 126 175


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Source one (S1) is seen as a pure source. Sources S2, S3, S4 and S5, are the junctions,

that is, they are serving as both sources and destinations. Destinations D8, D9, D10, D11,

D12, D13 and D14, are pure destination nodes.

An average fuel cost of 2.10cedis is incurred in transporting products peer kilometer. The

ratio of this amount to the truck load of 400crates was found to be 5.25 10-3 . This

amount was used to multiply all the distances in table 4.2 to obtain the unit cost in

transporting products from sources to the various destinations. This is summarized in

table 4.3.

Column one and row one are the list of the various sources and destinations respectively.

Columns two to column fourteen display the unit cost of transporting products from

sources to the destinations. The last column indicates the supply quantities and the last

row shows the demand quantities (in thousands) from January 2009 to December 2009.


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Table 4.3 Unit cost of Transporting a crate of Coca-Cola Product from sources to destination

DESTINATION

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 supply

SOURCE

S1 0.026 0.058 0.026 0.635 0.683 0.120 0.037 1.097 0.284 0.924 0.021 0.819 0.026 0.336 5001

S2 - 0.084 0.053 0.662 0.709 0.140 0.063 1.124 0.278 0.95 0.047 0.236 0.016 0.362 2107

S3 0.084 - 0.032 0.630 0.688 0.150 0.068 1.103 0.289 0.929 0.058 0.152 0.084 0.341 1143

S4 0.053 0.032 - 0.609 0.656 0.120 0.032 1.071 0.31 0.898 0.026 0.184 0.053 0.31 3612

S5 0.662 0.630 0.609 - 0.210 0.720 0.641 0.504 0.893 0.189 0.662 0.21 0.662 0.919 3187

Demand 822 952 928 914 847 998 800 1599 1050 714 1084 2312 1057 946


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4.2 The dual matrix solution method

Step 0: Initialization

Step 0.1: Set

822000, 952000, 928000, 914000, 847000, 998000, 800000,

1599000, 1050000,741000,1084000, 2312000, 1057000, 946000,

5001000, 2107000, 1143000, 3612000, 3187000

Step 0.2

1 2 3 4 5

1 2 3 4 5 6 7

8 9 10 11

Set 0

and

v 0.026, v 0.032, v 0.026, v 0.609, v 0.21, v 0.12, v 0.032,

v 0.504, v 0.31, v 0.189, v 0.021

u u u u u

12 13 14, v 0.152, v 0.016, v 0.31

Where 1 2 14v , v ,....................v are the minimum costs in each column of the cost matrix.

Step 0.3: Set1,1 , 4,2 , 1,3 , 4,4 , 5,5 , 1,6 , 4,7 , 5,8 , 2,9 , 5,10 ,

1,11 , 3,12 , 2,13 , 4,14 , 1,0 , 2,0 , 3,0 , 4,0 , 5,0T

Where T is the basic cell. The cells 1,0 , 2,0 , 3,0 , 4,0 and 5,0 are

called virtual cells because they do not exist in the original transportation

problem matrix.

Step 0.4: forming the matrix D which is m n m n matrix.

Let the matrix

1 2

; , 1,2,..........

1 , 1,2........

1 1, 2.......... , , ,..........where

1 , 1, 2,..................................

0 .

ij

n

ij

D d i j m n

i j n

i n j n i n i n id

i j n n n m

otherwise


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D=

1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1

0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1

0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1

Step 0.5:Computing initial feasible solution by setting the objective:


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1 1

2911625n m

j j i i

j i

b v a u

Step 1: Determination of the leaving cell

Step 1.1: Compute Y AD

822000,952000,928000,914000,847000,998000,

800000,1599000,741000,1084000,1156000,1057000,

946000,1169000,1050000, 1169000, 104000,0

Y

Step 1.2:The smallest value kY in the elements ofY , that is the value of theth

k element

in Y is the smallest..

FromY , the thk value is 1169000 , meaning 17k and the leaving cell in the

set T , that is, ,k ki j is 3,0 .

Step 1.3:The least value kY

in Y is 1169000 0 , hence the solution is not optimal.

Step 2: Determination of leaving cell

Step 2.1:

1,11 1

2,2 2 2

,

. .. .and

. .. .

. .. .

n kk

n kk

n mnk n m k

ddq p

dq d p

Q P

q pd d


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Hence,

1 2 19

1,17 2,17 19,17

, ,.......................................................

, ,............................................

0,0,0,0,0,0,0,0,0,0,0, 1,0,0

TQ q q q

d d d

and

1 2 5

15,17 16,17 19,17

, ,...............................

, ,.............................................

0,0, 1,0,0

TP p p p

d d d

Step 2.2: For all non basic cells if 0i jp q , then the dual problem is not bounded and

the original primal problem has no feasible solution.

Among all the non basic cells, the following have positive (piqj)

1,12 2,12 4,12 5,12

Hence we set ij ij i jC u vq = + - for all 0i jp q

min 0.058,0.084,0.032,0.667

min 0.032

st

Therefore the entering cell ,s t is 4,12

Step 3: Updating

Step 3.1: Updating the matrix D

Step 3.1.1:For the elements of column k in D


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We set , 1, 2, ...lk lk d d l m n= - = +

Let

1,171,17

2,172,17

19,1719,17

..

..

..

..

.

dd

dd

B

d d

0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0TB

Step 3.1.2: For the elements of other columns in D

, ( )lk lk s n r tr d d d d B+= + -

1,2,... 1.....r k m n= - +

1,2,...l m n= +

11 11

1212

181 121

11

..

..

..

..

.

nn

dd

dd

d d B C

d d

1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0T

C


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The new D becomes:

D=

1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1

0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1

0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1

Step 3.2: Update the basic cell set T. Replace the thk cell (3,0) inTwith the entering

cell ,s t .

1,1 , 4,2 , 1,3 , 4,4 5,5 , 1,6 , 4,7 , 5,8 , 2,9 , 5,10 , 1,11 ,

1,12 , 2,13 , 4,14 , 1,0 , 2,0 , 3,12 , 4,0 , 5,0

Step 3.3: Update the objective values by computing:


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The least value in Y is greater than zero (0), hence the solution is optimal.

The optimal solution obtained with:

Objective 2952009

with:

11 42 13 44

55 16 47 58

29 510 111 112

213 414 412

822000, 952000, 928000, 914000

847000, 998000, 800000, 1599000

1050000, 741000, 1084000, 1169000

1057000, 946000, 1143000

X X X X

X X X X

X X X X

X X X

= = = =

= = = =

= = = =

= = =

Table 4.4 is a summary of the results on the data analyzed. Column one shows the

sources from which the supplies are being made and column two is list of destinations

receiving the supplies from the sources. The number of shipment made from each source

to its destination is displayed in column 3. Column 4 displays the unit cost of shipment

from each source to a destination. The last column talks about the total shipment cost

from each source to its destination.


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Table 4.4 Summary of the result of the data analysed

FROM TO SHIPMENTCOST PER UNIT

(IN CEDIS)SHIPMENT COST

Source 1 Destination 1 822000 0.026 21372

Source 1 Destination 3 928000 0.026 24128

Sour

Documents

Final Thesis(2011) Opoku Agyemang