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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
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