1 KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY COLLEGE OF SCIENCE FACULTY OF PHYSICAL SCIENCE DEPARTMENT OF MATHEMATICS THE EFFECTIVE DISTRIBUTION OF MTN VOUCHERS IN THE KUMASI METROPOLIS (VEHICLE ROUTING PROBLEM) CASE STUDY: ASHCELL GHANA LIMITED. A DESSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN PARTIAL FULFILMENT FOR THE AWARD OF BACHELOR OF SCIENCE DEGREE IN MATHEMATICS BY ZAKARIA ABDUL RASHID ATTAH ZAKARIA SULEMAN ALHAJI ISSAH NAT MOHAMMED MAY, 2012
Transcript
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KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
COLLEGE OF SCIENCE
FACULTY OF PHYSICAL SCIENCE
DEPARTMENT OF MATHEMATICS
THE EFFECTIVE DISTRIBUTION OF MTN VOUCHERS IN THE KUMASI
METROPOLIS
(VEHICLE ROUTING PROBLEM)
CASE STUDY: ASHCELL GHANA LIMITED.
A DESSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN
PARTIAL FULFILMENT FOR THE AWARD OF BACHELOR OF SCIENCE DEGREE IN MATHEMATICS
BY
ZAKARIA ABDUL RASHID ATTAH
ZAKARIA SULEMAN ALHAJI
ISSAH NAT MOHAMMED
MAY, 2012
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DECLARATION
We declare that this work is an original piece of our own research under the supervision of
Mr. CHARLES SEBIL, Lecturer, Mathematics Department, KNUST, Kumasi. We further
declare that this work has never been produced partly or fully in any form except for those
sections that have been cited and duly acknowledged.
NAME OF STUDENT SIGNATURE DATE
ZAKARIA ABDUL RASHID ATTAH ………………… …………..
ZAKARIA SULEMAN ALHAJI ………………… …………..
ISSAH NAT MOHAMMED ………………… …………..
I declare that I have supervised the students in undertaking the study submitted herein and I
confirm that the students have my permission to present it for assessment.
SUPERVISOR SIGNATURE DATE
MR CHARLES ………………… …………
MAY, 2012
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DEDICATION
This work is humbly dedicated to our project supervisor Mr. Charles Sebil.
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ACKNOWLEDGEMENT
The first place of honour in the list of acknowledgement goes to GOD Almighty for His
protection and guidance throughout the writing of this project.
First of all we would like to thank our supervisor Mr. Charles Sebil for supporting us during the
project, and for his guidance and supervision. Even though he has little time at his disposal he
managed to fit us in his daily schedules.
Secondly, we would like to thank Mr. Mohammed Yakubu the territorial marketing controller of
MTN Ghana for his words of motivation and advice.
Last but not the least; we also thank Mr. Ali for giving us the opportunity to perform our thesis
research in his company. It was pleasant and informative working with him which we much
appreciate.
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ABSTRACT
The main problem confronting both large and small scale companies in any developing economy
is how to effectively distribute their goods in order to reduce their cost of distribution. In this
thesis we focus on a decision model for a real life problem. The problem reveals itself as a
vehicle routing problem; the effective distribution of MTN vouchers in the Kumasi Metropolis.
This study addresses the problem of finding the least cost routes from the depot to its various
branches in the metropolis. The thesis seeks to minimize the total distance each vehicle ply in a
day in order to serve its dedicated branches. The Clarke Wright Savings Algorithm was used in
the construction of a set of feasible routes in such a way that the total travelling distance was
minimized. Both the inter route and the intra route heuristic methods were used in minimizing
the total distance covered under the construction phase. After these heuristic methods were
applied it was found that, the total distance covered in a day was reduced by 10.9%. Thus we
were able to create a set of least cost routes such that the total travelling distance in a day was
reduced significantly from 97.9 kilometers to 87.2 kilometers.
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TABLE OF CONTENT
CONTENT PAGE
Declaration 2
Dedication 3
Acknowledgement 4
Abstract 5
CHAPTER ONE
1.0 Introduction 10
1.1 Background of study 10
1.2 Brief Background of case study 12
1.3 Problem statement 15
1.4 Objective of the study 16
1.5 Method of study 16
1.6 Significance 16
1.7 Scope of study 17
1.8 Limitations of study 17
1.9 Organization of the study 17
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CHAPTER TWO
2.0 Introduction 19
2.1 The Transportation Problem 19
2.2 The Vehicle Routing Problem 20
2.2.1 Classes of Vehicle Routing Problem 22
2.2.1.1 Vehicle Routing Problem with Time Windows (VRPTW) 22
2.2.1.2 The Travelling Salesman Problem. (TSP) 23
2.2.1.3 Capacitated Vehicle Routing Problem (CVRP) 25
2.2.1.4 Vehicle Routing Problem with Pick-Up and Delivery (VRPPD). 26
2.2.2 Real Life Application of the Vehicle Routing Problem 26
2.2.2.1 The Newspaper Delivery Problem 27
2.2.2.2 The Collection Problem 29
2.2.2.2.1 The Bin Packing Problem (BPP). 29
2.2.2.2.2 The Waste Collection Problem. 30
CHAPTER THREE
3.0 Introduction 35
3.1 Problem definition and formulation 35
3.2 Formulation of the General Vehicle Routing Problem 36
3.3 Notations 29 38
3.4 Methods of solution 39
3.4.1 The Exact Approach 39
3.4.2 The Approximation Approach 39
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3.4.2.1.0 Construction Heuristics. 40
3.4.2.1.1 Nearest Neighbour search 40
3.4.2.1.2 Clarke-Wright savings heuristic 40
3.4.2.1.3 The sweep algorithm 45
3.4.2.2 Improvement Heuristics 46
3.4.2.2.1 Intra-route exchanges 46
3.4.2.2.2 Inter-route Exchanges 47
3.4.3.0 Algorithms for solving λ − opt. 48
CHAPTER FOUR
4.1 Introduction 50
4.1 Model 50
4.2 Construction of the Basic Feasible Routes. 52
4.3 Improvement of the Basic Feasible Solution 55
4.3.1 Second Improvement Solution 57
CHAPTER FIVE
5.0 Introduction 58
5.1 Conclusion 58
5.1 Recommendation 59
REFERENCES 60
APPENDIX 61
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LIST OF FIGURES
Figure 3.1 Graphical representation of the Vehicle Routing Problem 37
Figure 4.1. The basic feasible route 55
Figure 4.2 First Improvement Solution 56
Figure 4.3 Second Improvement Solution 57
LIST OF TABLES
Table 3.1 Symmetric Distance table 42
Table 3.2 Demand vector table 42
Table 3.3 Symmetric Savings Matrix 43
Table 3.4 The Savings list 44
Table 4.1 Customer Demand 51
Table 4.2 The Symmetric Distance Matrix 51
Table 4.3 The Savings Matrix 52
Table 4.4 The Savings List. 53
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CHAPTER ONE
1.0 Introduction.
The chapter one consists of a brief description of the application of optimization in the
transportation problem, profile of the case study, the research question, the purpose, limitations,
significance, methodology and the scope of the study.
1.1 Background of study.
Optimization is the method that seeks to minimize or maximize a real function by systematically
choosing the values of real variables from a feasible region. Optimization problem can be linear
or non linear depending on the type of model used in solving the problem.
Linear programming refers to planning that allocates resources in the optimal way so as to
minimize cost and maximize profit. In linear programming the resources are known as the
decision variables. The linear function that is to be either minimized or maximized is called the
objective function. The linear equations and inequalities in the linear program which define the
feasible set of the problem are called the constraints.
The word linear indicates that, the criterion for selecting the best values of the decision variables
can be described by a linear function of the variables. Optimization with its numerous
applications provides a lot of opportunities for companies to make decisions especially when it
comes to the area of transportation.
To be successful in today's highly competitive marketplaces, companies must strive for greatest
efficiency in all of their activities and completely utilize any possible opportunity to gain a
competitive advantage over other firms. Among many possible activities is cost minimization in
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the transportation of product and logistics which is regarded as one of the core areas presenting
enormous opportunities.
In the retail industry, the primary goal for all organizations is to satisfy their customers. The
factors affecting the success of this objective depend primarily on the decisions that the
organization makes. A principal factor includes the ability of the company to ensure that the
supply of their products equals the expected demand of the product from their customers while
ensuring that costs are minimized. In order to accomplish these goals, an important area of focus
is effective supply chain management. It also includes strategic network optimization including
the number, location, and size of warehouses, distribution centers, and facilities.
Based upon the initial decisions made by the company as to where to locate their assets, the next
decision is how to allocate the product in the most efficient manner to meet the demand of their
customers.
Most of the manufacturing companies in Ghana utilize vehicles in transporting their products to
their customers as that is the easiest and most affordable in the country.
The use of vehicles in transporting goods also falls under another application of optimization
known as the Vehicle Routing Problem (VRP).
In the literature, the basic VRP comprises of a set of vehicles, customers and a depot.
The Vehicle Routing Problem (VRP) can be defined as a problem of finding the optimal routes
of delivery or collection from one or several depots to a number of cities or customers, while
satisfying some constraints. Collection of household waste, gasoline delivery trucks, delivering
of newspapers are few of the real life applications.
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It can also be defined as the problem of designing least cost routes for identical vehicles of
known capacities, which run from a central depot to a set of geographically dispersed customers
with non-negative demand. The total demand and the length of a route must not exceed the total
capacity and the total distance travelled allowed for a vehicle. The vehicles will return to the
depot after servicing customers who have been assigned to them.
Optimization with its numerous application also helps solves the vehicle routing problem which
is one of the commonest transportation problems. The applications of optimization are
tremendous and cut across all fields of life.
1.2 Brief background of case study
Mobile Telecommunication Network (MTN) is a South Africa-based multinational mobile
telecommunications company with its branches spread across as many as 21 African and Middle
East countries with its head office in Johannesburg.
MTN, the leading provider of telecommunications services in emerging markets within Africa
and the Middle East, entered the Ghanaian market following the acquisition of Investcom who
owned the then Areeba in 2006. [www.mtn.com].
Equipped with a proven record of technological innovation and a corporate culture that thrives
on understanding telecommunications in emerging markets, MTN continues to consolidate its
leadership position in the country. MTN has a total of 9,894,074 out of the 20,419,635 total
numbers of mobile subscribers in the country as of September 2011.This gives them 48.45% of
the total number of mobile phone subscribers in Ghana. [www.nca.org.gh]
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MTN understanding that the best way for subscribers to gain a competitive edge in a local
market offers different segments which suits people’s life styles and economic situations thereby
creating different types of tariff plans for their subscribers and they include:
Pay as you go thus prepaid customers
Pay monthly
The prepaid customer as the name implies must recharge before accessing the services of the
network. These are MTN subscribers who pay for their call credits before utilizing the airtime.
They are commonly referred to as Pre-paid customers.
The pay monthly customer is also referred to as the MTN VIP. This customer uses the services
of the network and pays at the end of every month. [www.mtn.com].
MTN as any other service provider wants to meet the demand of its precious customers and this
influences them to produce different kinds of credit vouchers for its customers. The MTN credit
vouchers ranges from the two, five, seven cedis and ten cedis. MTN in its own assessment of the
economy of Ghana brings in the one cedis vouchers occasionally to meet the standard of the
average subscriber.
There exist two types of recharge procedures that can be used by the prepaid customer and this
includes the airborne recharge and the physical recharge.
The physical recharge consists of the numerous credit vouchers whilst the airborne consist of the
transfer system.
MTN in their policies to minimize the risk of distribution cost officially assign the task of
distribution of their products to some registered private companies and this type of distribution
can be classified as; the Producer to the Wholesaler to the Retailer and to the Consumer type of
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distribution. In this the wholesaler buys in bulk from the producer and stores the goods for later
resale to retailers.
In all, MTN have fourteen dealers of which are grouped into two:
Territorial dealers
Free dealer
The territorial dealers are strictly assigned a particular area to operate which is prohibited from
any other dealer to distribute MTN products in that particular area.
The free dealer can operate in all over the country with the exception of specific areas dedicated
to the services of the territorial dealer.
In view of this, Ashcell Company Limited is the official territorial dealer in the Ashanti Region.
Ashcell’s main objective is to distribute all MTN products (vouchers, logistic, souvenirs etc) in
the Ashanti region such that:
There is constant supply of the products.
The prices for the products across the whole region are stabilized.
Distribution of the products is done in four main ways and they are through:
The branches
Wholesalers
Sub dealers
Local agency scheme (LAS).
Products from the MTN headquarters in Accra are deposited in the head office of Ashcell which
serves as the main source of MTN products in the Ashanti region. The products are then
distributed to the branches across the region.
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From the branches the sub dealers and the wholesalers pick the products up before it get to the
distributor in the local area scheme.
For the proper distribution of the products Ashcell has zoned the whole region into two and
within these two zones are thirty one branches of Ashcell.
The Kumasi zone contains fourteen branches of which the factors considered before sitting them
includes traffic, demand, population etc. [Mr. Ali General Manager Ashcell Company Limited]
1.3 Problem statement
Failure in making optimal location and inventory decisions will ultimately result in loss of profit.
There are many contributing factors towards optimal decision-making, but a key issue is the
method of transportation uncertainty.
Currently the main problem is how to assign a particular vehicle to a route to minimize the total
transportation cost whilst satisfying route and the available constraints to serve their customers
precisely at the right time. As stated previously, ensuring customer satisfaction, and as result,
ensuring company success depends largely on the fact that supply of a product is equal to the
expected demand. If an organization fails to accomplish this, they are at the risk of losing
customers and profit.
1.4 Objective of the study.
The effective distribution of goods always raises a transportation problem in which most
businesses find it difficult to solve. The main objective of this project is to:
Minimize the cost of transportation of MTN vouchers in the Kumasi Metropolis.
Selecting the best transportation path with the least distance for each vehicle to use.
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1.5 Method of study
Vehicle routing is a member a category of linear programming known as the network flow
problem which deals with the distribution of goods from sources to a number of points of
destination. It involves finding an initial solution and developing an improved solution. This
process continues until an optimal solution is reached.
Search on the internet will be used to obtain the related books, articles and journals. Books and
previous works from the KNUST Library and the Mathematics Department’s library will be very
useful in the course of the project.
1.6 Significance of study
In today’s world where telephone communication has become inevitable there is the need to
address the factors that affect the cost of communication. Minimizing transportation cost will
further reduce the cost of the vouchers considerably and this in one way or the other helps boost
the revenue of the local retailers, as subscribers buy frequently when the prices are relatively
low.
The cost of bulk transportation of credit vouchers in Ghana is part of price build-up to price of
the vouchers therefore its reduction will result in reduction of prices of general goods and
services. Higher prices of credit vouchers will result in low patronage in buying the vouchers and
this can collapse the small scale retailer’s business thereby increasing the unemployment
population in Ghana. Therefore a significant reduction in cost of bulk transportation of MTN
products will impact positively on the economy of Ghana.
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1.7 Scope of study
Kumasi metropolis is a very large area constituting ten sub metros. Various forms of economic
activities like manufacturing, petty trading, and hawking among others, take place in the
metropolis.
This study is only limited to Kumasi metropolis. Data for the study was collected from MTN
Ghana ltd, Ashcell Ghana ltd, and individuals in this area. Ash Cell Ghana Ltd which gave us
most of the data is the sole distribution company that distributes MTN products in the Ashanti
region of Ghana.
1.8 Limitations of study
This study as every human product has its own limitations it came along with. One of these
limitations is how to access the transportation cost of the airborne recharge system since it does
not necessarily needs a vehicle to transport it to the destination.
As urbanization is on the rise in every regional capital MTN keeps expanding its geographical
area it considers part of the metropolis. This makes it very difficult to get a clear cut boundary
enclosing Kumasi Metropolis.
1.9 Organization of study.
Chapter one consists of a brief description of the application of optimization in the transportation
problem, profile of the case study, the research question, the purpose, limitations, significance,
methodology and the scope of the study.
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Chapter two contains the Literature Review. Chapter three covers the methods to be used in this
particular study. Chapter four covers data collection, analysis and discussion. The last chapter
covers conclusion and recommendations.
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CHAPTER TWO
LITERATURE REVIEW
2.0 Introduction
This chapter presents a global overview of the Vehicle Routing Problem (VRP). First, the
transportation problem is explained before stream lining to the vehicle routing itself. Most of the
written works about the vehicle routing problem and some variants of the vehicle routing
problem have been explained in this chapter. The real life applications of the routing problem are
also included in this chapter.
2.1 The Transportation Problem.
One of the most important and successful applications of optimization to solving business
problems has been in the physical distribution of products usually referred to as transportation
problem.
The purpose of transportation problem is basically to minimize the cost of shipping goods from
one location to another so the demands of each destination area are satisfied and every shipping
location operates within its capacity.
The basic transportation problem was originally developed by Hitchcock (1941).
Efficient methods of solution derived from the simplex algorithm were developed in 1947.
This involves Koopmans (1947) research on the potentialities of linear programs for the study of
problems in economics which was basically an update on Hitchcock proposals hence the name
Hitchcock-Koopmans’s transportation problem.
The transportation problem can be modeled as a standard linear programming problem, which
can then be solved by the simplex method. However, because of its very special mathematical
structure, it was recognized early that the simplex method applied to the transportation problem
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can be made quite efficient in terms of how to evaluate the necessary simplex-method
information (variable to enter the basis, variable to leave the basis and optimality conditions).
Under the transportation problem emerges the vehicle routing problem in short (VRP) which has
a great effect on the cost of transporting goods from a source to a destination.
2.2.0 The Vehicle Routing Problem
The Vehicle Routing Problem (VRP) is one of the most important and challenging optimization
problems in the field of Operations Research. It consists of designing the optimal set of routes
for a fleet of vehicles parked at a central depot in order to service a given set of customers with a
fixed demand. The VRP originated from the Travelling Salesman Problem (TSP), a special case
of the VRP in which only one vehicle with ‘sufficient’ capacity is available.
It was introduced by Dantzig and Ramser (1959) and was developed by Clarke and Wright
(1964). Their paper appeared in the journal of Management Science concerning a fleet of
gasoline delivery trucks between terminal and a truck number of service stations supplied by the
terminal. The problem formulated in the Dantzig and Ramser’s paper given the name
“Dispatching Problem” and many years later was coined the name “Dantzig and Ramser’s
Problem” and “Vehicle Routing Problem” respectively.
In the literature, the basic VRP is comprised of a set of vehicles, customers and a depot.
It can also be defined as the problem of designing least cost routes for identical vehicles of
known capacities, which run from a central depot to a set of geographically dispersed customers
with non-negative demand. Each customer is to be fully serviced exactly once (typically by one
vehicle). The total demand and the length of a route must not exceed the total capacity and the
total distance travelled allowed for a vehicle. The vehicles will return to the depot after servicing
customers who have been assigned to them.
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Due to the significant economic benefit that can be achieved by optimizing the routing problems
in practice, more and more attention has been given to various extensions of the VRP that arise
in real life.
The main objective of the VRP is to minimize the distribution costs for the individual carriers,
and can be described as the problem of assigning a collection of routes from a depot to a number
of geographically distributed customers, subject to certain constraints. The most basic version of
the VRP has also been called vehicle scheduling, truck dispatching or simply the delivery
problem [Joubert, 2007]. It has a large number of real life applications and comes in many forms,
depending on the type of operation, the time frame for decision making, the objective and the
type of constraint that must be adhered to.
The basic VRP consists of designing a set of delivery or collection routes, such that:
Each route starts and ends at the depot.
Each customer is called at exactly once and by only one vehicle.
The total demand on each route does not exceed the capacity of a single vehicle, and
The total routing distance is minimized.
Due to some constraints such as load, distance and time, a single vehicle may not be able to serve
all the customers. The problem then is to determine the number of vehicles needed to serve the
customers as well as the routes that will minimize the total distance travelled by the vehicles.
Many methods have been proposed in the last 50 years and these include the exact, heuristics and
metaheuristics. Heuristics proposed up until around 1980 are surveyed in Christofides et al.
[1979], while the most successful heuristics until the new millennium are surveyed in Laporte
and Semet [2002] and Gendreau et al. [2002]. The most recent advances in metaheuristics have
been surveyed in Cordeau et al. [2004]. The best heuristic for the problem at the moment is the
metaheuristic proposed by Mester and Bräysy [2005]. Quite a lot of attention has been given to
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exact methods for the VRP in the recent years and substantial advances in the size of problems
that can be solved to optimality have been achieved.
Most research has gone into developing branch and cut methods and valid inequalities for the
problem. The two most successful branch and cut algorithms are the one proposed by Lysgaard
et al. [2004] and Blasum and Hochstättler [2000]. Recently it has been shown that the
combination of column generation and cutting planes is a powerful approach for the VRP and the
branch-and- cut-and-price algorithm proposed by Fukasawa et al. [2005] must be considered as
the best.
2.2.1.0 Classes of Vehicle Routing Problem
2.2.1.1 Vehicle Routing Problem with Time Windows (VRPTW)
One of the most important extensions of the basic VRP is the Vehicle Routing Problem with
Time Windows (VRPTW). This variant introduces the additional restriction that a time window
is associated with each customer, defining an interval in which arriving at the customer is
allowed.
Moreover there is a time window at the depot, which guarantees that each route must start and
end within the time window associated with the depot.
The time windows are either soft or hard.
A hard time window has a strict lower bound and upper bound, i.e., if a vehicle arrives before the
lower bound of the customer time window an additional waiting time on the route is taken into
account and if a vehicle arrives after the upper bound of the customer time window a solution
becomes infeasible. A soft time window also adds a waiting time when a vehicle arrives too
early at a customer, but when a vehicle arrives too late the solution remains feasible, however the
total travel cost will be penalised by some amount.
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Because the VRPTW contains waiting time, an extension of the objective of the basic VRP is
needed. The objective of the VRPTW is not only to minimize the number of vehicles required
(vehicle cost) and the total travel distance (travel cost), but also the total travel time incurred by
the fleet of vehicles (travel time cost).
It can be reviewed as a combined vehicle routing and scheduling problem which often arises in
many real-world applications. It is to optimize the use of a fleet of vehicles that must make a
number of stops to serve a set of customers, and to specify which customers should be served by
each vehicle and in what order to minimize the cost, subject to vehicle capacity and service time
restrictions. The problem involves assignment of vehicles to trips such that the assignment cost
and the corresponding routing cost are minimal. The VRPTW can be defined as follows:
Let G = (V, E) be a connected digraph consisting of a set of n + 1 nodes, each of which can be
reached only within a specified time interval or time window, and a set E of arcs with non-
negative weights representing travel distances and associated travel times. Let one of the nodes
be designated as the depot. Each node i, except the depot, requests a service (demand) of size d .
2.2.1.2 The Travelling Salesman Problem. (TSP)
One of the problems that can be considered as an illustration of the delivery problem is the
travelling salesman problem.
During the past decades, considerable research on vehicle routing and scheduling problems has
been carried out. One of the earliest and also the simplest routing problem is the Traveling
Salesman Problem (TSP), in which the shortest tour to visit a number of cities must be
determined for a salesman who starts from and terminates at the same city.
This problem was later extended to the Multiple Traveling Salesman Problem (m-TSP), in
which there are multiple salesmen and they all start at and return to the same city, which is
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referred to as the depot. In the late fifties, Dantzig and Ramser (1959) introduced the VRP,
which can be viewed as an m-TSP with customer demands and vehicle capacity.
To cast this as a real-world problem, the synonymous salesman has decided to visit some cities
on a map to ply his wares, and wishes to spend a minimum on travel. On a complete graph with
one vertex for each city, whose edges are weighted with the financial cost of each intercity
journey, the solution to the TSP is the route of least expense.
TSP and VRP are the two most widely studied combinatorial optimization problems. There are
numbers of extensions of TSP but the main constraint of the algorithm is to visit customers from
a depot and customer has to be serviced to meet a particular demand.
However, in some problems customers are selected according to the profit gained by choosing
them and generally when a single vehicle is involved, the problem is the TSP with profits, (TSP-
P). There are many applications of TSP-P. TSP-P has two opposite objectives, one is supporting
to collect profits and the other is limiting the travel costs. Two of the objectives can be combined
in the objective function or one of the objectives can be constrained with a specified bound
value. Both objectives are addressed in the objective function; finding a circuit minimizing travel
costs minus collected profit. This problem is defined as the profitable tour problem by
Dell’Amico et al. (1995).
As described the TSP has feasible solutions on graphs which contain a Hamiltonian circle.
Schabauer, Schikuta and Weishaup have worked on to solve traveling salesman problem
heuristically by parallelizing of self-organizing maps on cluster architectures. Allan Larsen
investigated the dynamics of the vehicle routing problem in order to improve the performances
the existing algorithms and as well develop new algorithms. Jorg and Hermann Gehring worked
on vehicle routing problems on time windows in which they designed an optimal set of routes
that will service the entire customers with constrains being taken care of properly. Their
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objective function minimizes both the total distance travelled and the number of salesmen being
used.
2.2.1.3 Capacitated Vehicle Routing Problem (CVRP)
In the TSP one is given a set of cities and a way of measuring the distance between each city.
One has to find the shortest tour that visits all cities exactly once and returns back to the starting
node.
The capacitated vehicle routing problem (CVRP) considers the movement of a set of vehicles to
a set of dispersed customers. In the CVRP we are given a depot, a set of n customers, a set of m
vehicles and a measured distance. Every vehicle has a capacity Q and every customer
i ∈ {1, . . . , n} has a demand d . The task in the CVRP is to construct vehicle routes such that all
customers are served exactly once and such that the capacities of the vehicles are obeyed. This
should be done while minimizing the total distance traveled.
The Capacitated Vehicle Routing Problem (CVRP) can be described as follows:
Let G = (V’, E) an undirected graph is given where V’ = {0, 1 . . . n} is the set of n+1 vertices
and E is the set of edges. Vertex 0 represents the depot and the vertex set V = {1… n}
corresponds to n customers. A non-negative cost C is associated with each edge {i, j} ∈ E.
The di units are supplied of from depot 0 (we assume d = 0). A set of m identical vehicles of
capacity K is stationed at depot 0 and must be used to supply the customers. A route is defined as
a least cost simple cycle of graph G passing through depot 0 and such that the total demand of
the vertices visited does not exceed the vehicle capacity.
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2.2.1.4 Vehicle Routing Problem with Pick-Up and Delivery (VRPPD).
A subclass of vehicle routing problems is pickup (collection) and delivery problems. In this class
of problems we are given a number of requests and a fleet of vehicles to serve the request. Each
request consists of a pickup at some location and a delivery at another location. The cost of
travelling between each pair of locations is given. The problem is to find routes for each vehicle
such that all pickups and deliveries are served and such that the pickup and delivery
corresponding to one request is served by the same vehicle and the pickup is served before the
delivery. Again a number of additional constraints are often enforced, the most typical being
capacity and time window constraints.
The general pick up and delivery problem (GPDP) is introduced in order to be able to deal with
various complicating characteristics found in many practical pickup and delivery problems, such
as transportation requests specifying a set of origins associated with a single destination or a
single origin associated with a set of destinations, vehicles with different start and end locations,
and transportation requests evolving in real time.
Many practical pickup and delivery situations are demand responsive, thus, new transportation
requests become available in real-time and are immediately eligible for consideration. As a
consequence, the set of routes has to be optimized at some point to include the new
transportation requests. Observe that at the time of the optimization, vehicles are on the road and
the notion of depots becomes void.
2.2.2.0 Real Life Application of the Vehicle Routing Problem
The VRP is of great practical significance in real life. It appears in a large number of practical
situations, such as transportation of people and products, delivery service and garbage collection.
One can therefore easily imagine that all the problems, which can be considered as VRP, are of
great economic importance, particularly to the national development. The economic importance
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has been a great motivation for both companies and researches to try to find better methods to
solve VRP and improve the efficiency of transportation.
2.2.2.1 The Newspaper Delivery Problem.
Many researches on this area indicated that VRP was and is still a great tool for minimizing the
total cost of delivery or total travel time in the newspaper industry.
In real world, fleet of transportation is very complicated. Number of trips, links (path) and cost
are to be considered. Transportation often involves routing vehicles according to customer given
time allowance that determines the customer’s satisfaction level. Therefore, all publishers
intensively improve and adjust company’ strategies by pertaining their internal resources with
external resource (the market).
The competitive advantage can be achieved by concentrating all the available resources on one
basic strategy which is to shorten delivery time. The short delivery time if administered
efficiently and effectively could also result in less distribution cost. This may be the ultimate
choice since a declining enterprise had difficulty to increase sales. [A. Harrison, and R. Van
Hoek (2008), P. Toth, D. Vigo (2002).]
The Newspaper Distribution Problem (NDP) involves the downstream movement of newspaper
from the printing process to the hand of readers. The NDP can be viewed as a hierarchical
distribution problem. That means the newspaper delivery involves at least two distinct stages.
The first stage is from the production facility to the transfer points and the second stage is from
the transfer points to customers Rochat and Taillard (1995). NDP is an example of a perishable-
good production and distribution problem. People who are working in publishing companies
classify physical newspaper as perishable goods because they could be lost in significant value if
delivered late or over printed Bramel and Simchi-Levi (1995).
28
A newspaper distribution problem for a metropolitan daily Korean newspaper was studied and
then developed a delivery plan using a branch-and-bound heuristic with simulated annealing
S. Ree, B.S Yun 1996. Before that Hurter, M. Van Buer (1996) develop a deterministic approach
to a medium sized newspaper production/distribution problem in which they employ a greedy
heuristic followed by an Or-Opt route improvement heuristic. The problem was smaller and
involved only one printing press and more importantly considered only a single product delivery
to each zone.
Thus, each zone contained its own routing problem. Also, Regret Distance Calculation algorithm
was selected for agent allocation, a Modified Urgent Route First algorithm for vehicles
scheduling, and a Weighted Savings algorithm for routing in addressing the optimal agent
allocation, vehicle scheduling and routing for a major newspaper in Korea, the experiment
showed that the formulation could significantly reduce delivery costs and delay. [Song, Lee &.
Kim (2002)].
In Daganzo1981 a newspaper delivery problem for the city of San Francisco was considered as
an application of a formulation developed for predicting the distance traveled by fleets of
vehicles in distribution problems. The formulation was a variant of the “cluster-first, route-
second” approach to solve vehicle routing problems.
In a follow up to Daganzo1981 work, Van Buer, Woodruff, and Olson (1999) extended the
solution method to include metaheuristics, simulated annealing and tabu search. Its approach was
deterministic and one of the main findings was that recycling trucks to create more routes while
using fewer vehicles can lead to significant cost reductions.
NDP is also vital in the newspaper industry provided that it is directly tied to customer service
level. Late delivery of a newspaper may result in the loss of customers or may result in the
29
shutting down of a production line if numbers of customers are rapidly reduced [A. Boonkleaw,
S. Suthikannarunai, R. Srinon, Engineering Letters, 18:2, EL_18_2_09 Advance].
2.2.2.2 The Collection Problem.
Essentially, the VRP for collection is dealing with the same type of constraints as in a delivery
problem when constructing vehicle routes. Thus, this problem also attempts to determine the
number of vehicles needed to serve the customers as well as the routes that will minimize the
total distance travelled by the vehicles. However, the vehicle for the collection problem is empty
when it starts from the depot, whereas the vehicle for the delivery problem begins its route
loaded with customers’ goods that need to be delivered. In the collection problem vehicles will
collect goods from a set of customers and return to the depot at the end of the working day.
Some applications of collection problems that can be found in the literature are cash collection
(e.g. Lambert, Laporte and Louveaux, 1993), collection of raw materials for multi-product
dehydration plants (e.g. Tarantilis and Kiranoudis, 2001a; Tarantilis and Kiranoudis, 2001b), and
milk collection (e.g. Caramia and Guerriero, 2010).
2.2.2.2.1 The Bin Packing Problem (BPP).
The BPP can be classified under the collection problem since it involves the collection of waste
from residential and commercial places. This is normally formulated by considering a given
finite set of numbers (the item sizes) and a constant K, specifying the capacity of the bin, what is
the minimum number of bins needed? Naturally, all items have to be inside exactly one bin and
the total capacity of items in each bin has to be within the capacity limits of the bin. This is
known as the best packing version of BPP. The TSP is about a travelling salesman who wants to
visit a number of cities. He has to visit each city exactly once, starting and ending in his home
town. The problem is to find the shortest tour through all cities. Relating this to the VRP,
30
customers can be assigned to vehicles by solving BPP and the order in which they are visited can
be found by solving TSP.
2.2.2.2.2 The Waste Collection Problem.
In general a waste collection system involves the collection and transportation of solid waste to
disposal facilities. This essential service is receiving increasing attention from many researchers
due to its impact on the public concern for the environment and population growth, especially in
urban areas. Because this service involves a very high operational cost, researchers are trying to
reduce the cost by improving the routing of waste collection vehicles, finding the most suitable
location of disposal facilities and the location of collection waste bins as well as minimizing the
number of vehicles used. There is an additional constraint that needs to be considered in solving
this problem. Instead of returning to the depot to unload the collected goods, in a waste
collection problem vehicles need to be emptied at a disposal facility before continuing collecting
waste from other customers. Thus, multiple trips to the disposal facility occur in this problem
before the vehicles return to the depot empty, with zero waste. A complication in the problem
arises when more than one disposal facilities are involved. Here one needs to determine the right
time to empty the vehicles as well as to choose the best disposal facility they should go to so that
the total distance can be minimized. For example it may not be optimal to allow the collection
vehicle to become full before visiting a disposal facility.
A study by Simonetto and Borenstein (2007) tested a decision support system called SCOLDSS
on a real life waste collection problem in Porto Alegre, Brazil. By using SCOLDSS, they stated
that it is possible to obtain a mean reduction of 8.82% in the distance to be covered and a
reduction of 17.89% in the weekly number of trips by the collection vehicles. This result is very
significant to Municipal Department of Urban Cleaning (DMLU) because it can represent
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savings of around 10% of the DMLU annual budget for solid waste collection per year,
considering the operational and maintenance costs.
Increasing quantities of solid waste due to population growth, especially in urban areas, and the
high cost of its collection are the main reasons why this problem has become an important
research area in the field of vehicle routing.
Chang, Lu and Wei (1997) applied a revised multi objective mixed-integer programming model
(MIP) for analyzing the optimal path in a waste collection network within a geographic
information system (GIS) environment. They demonstrated the integration of the MIP and the
GIS for the management of solid waste in Kaohsiung, Taiwan.
Computational results of three cases, particularly the current scenario; proposed management
scenario (without resource equity consideration) and modified management scenario (with
resource equity requirement) are reported. Both the proposed and the modified management
scenarios show solutions of similar quality. On average both scenarios show a reduction of
around 36.46% in distance travelled and 6.03% in collection time compared to the current
scenario.
Mourao and Almeida (2000) solved a capacitated arc routing problem (CARP) with side
constraints for a refuse collection VRP using two lower-bounding methods to incorporate the
side constraints and a three-phase heuristic to generate a near optimal solution from the solution
obtained with the first lower-bounding method. Then, the feasible solution from the heuristic
represents an upper bound to the problem. The heuristic they developed is a route-first, cluster-
second method.
Bautista and Pereira (2004) presented an ant algorithm for designing collection routes for urban
waste. To ascertain the quality of the algorithm, they tested it on three instances from the
capacitated arc routing problem literature (i.e. Golden, DeArmon and Baker, 1983; Benavent et
32
al, 1992; and Li and Eglese, 1996) and also on a set of real life instances from the municipality
of Sant Boi del Llobregat, Barcelona. Computational results for Golden, DeArmon and Baker
(1983) and Benavent et al (1992) were within less than 4% of the best known solution, and for Li
and Eglese (1996) dataset up to 5.08%.
Mourao and Amado (2005) presented a heuristic method for a mixed CARP, inspired by the
refuse collection problem in Lisbon. The proposed heuristic can be used for directed and mixed
cases. Mixed cases indicate that waste may be collected on both sides of the road at the same
time (i.e. narrow street), whereas waste for the directed cases only can be collected on one side
of the road. They reported computational results for the directed case on randomly generated
data and for the mixed case on the extended CARP benchmark problems of Lacomme et al.
(2002). Computational results for the directed problem, involving up to 400 nodes show the gap
values (between their lower bound and upper bound values computed from their heuristic
method) varying between 0.8% and 3%. For the mixed problem, comparison results with four
other heuristics namely, extended Path-Scanning, extended Ulusoys, extended Augment-Merge
and extended Merge are reported. They stated that they were able to get good feasible solutions
with gap values (between the lower bound values obtained from Belenguer et al (2003) and their
upper bound values) between 0.28% and 5.47%.
Li, Borenstein and Mirchandani (2008) solved a solid waste collection in Porto Alegre, Brazil
which involves 150 neighbourhoods, with a population of more than 1.3 million. They design a
truck schedule operation plan with the purpose of minimizing the operating and fixed truck costs.
In this problem the collected waste is discarded at recycling facilities, instead of disposal
facilities. Furthermore, the heuristic approach used in this problem also attempts to balance the
number of trips between eight recycling facilities to guarantee the jobs of poor people in the
different areas of the city who work at the recycling facilities. Computational results indicate that
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they reduce the average number of vehicles used and the average distance travelled, resulting in a
saving of around 25.24% and 27.21% respectively.
Mourao, Nunes and Prins (2009) proposed two two-phase heuristics and one best insertion
method for solving a sectoring arc routing problem (SARC) in a municipal waste collection
problem. In SARC, the street network is partitioned into a number of sectors, and then a set of
vehicle trips is built in each sector that aims to minimize the total duration of the trips. Moreover,
workload balance, route compactness and contiguity are also taken into consideration in the
proposed heuristics.
Ogwueleka (2009) proposed a heuristic procedure which consists of a route first, cluster second
method for solving a solid waste collection problem in Onitsha, Nigeria. Comparison results with
the existing situation show that they use one less collection vehicle, a reduction of 16.31% in
route length, a saving of around 25.24% in collection cost and a reduction of 23.51% in
collection time.
Gottinger (1988) proposed a network flow model for regional solid waste management that
minimizes a single objective function of the total costs of transportation, processing, and
construction. Some models aim to maximize the average separation distance; some maximize the
minimum separation distance, and others minimize the number of people within some critical
distance or impact radius.
Archetti and Speranza (2004) developed a heuristic algorithm called SMART-COLL for a
problem motivated by waste collection in Brescia, Italy. In their problem skips are collected from
customers and the vehicle can carry only one skip at a time. They call the problem the 1-skip
collection problem. They considered skips of different types and time windows are imposed on
both the customers and the disposal facilities. Computational experience was reported for real
world data involving 51 customers and 13 disposal facilities.
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Bodin et al (2000) considered a sanitation routing problem they called the rollon-rolloff vehicle
routing problem. In this problem trailers, in which waste is collected, are positioned at
customers. A tractor (vehicle) can move only a single trailer at a time.
Tractor trips involve, for example, moving an empty trailer from the disposal facility to a
customer and collecting the full trailer from the customer. A key aspect of their work is that they
assume that the set of trips to be operated is known in advance (so the problem reduces to
deciding for these trips how they will be serviced by the tractors).
They presented four heuristic algorithms and gave computational results for problems involving
up to 199 trips and a single disposal facility
35
CHAPTER THREE
METHODOLOGY
3.0 Introduction
A lot of methods for solving the vehicle routing problem have been proposed in the past years.
The exact methods like the branch and bound, branch and cut and approximations methods like
the heuristics, the metaheuristics and the genetic algorithms have been proved efficient for
solving the vehicle routing problem (VRP).
In this chapter, the general problem definition is given, a mathematical formulation of the vehicle
routing as well as some solution methods for solving the basic VRP have been elaborated
comprehensively.
3.1 Problem definition and formulation
A number of identical vehicles with a given capacity are located at a central depot. They are
available for servicing a set of customer orders. Each customer order has a specific location and
size. Travel costs between all locations are given. The goal is to design a least cost set of routes
for the vehicles in such a way that all customers are visited once and vehicle capacities are
adhered to.
Let G = (V, A) be a directed graph with vertex (node) set V = {v , v ,…v } and route set A.
Each customer i order a non-negative demand d . The edges in A = {(i, j): i, j ∈ N, i < j}
represent the connections between nodes.
The cost associated to each edge (i, j) is given by C .
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3.2 Formulation of the General Vehicle Routing Problem
To obtain a mathematical formulation of the basic VRP, first some notation and definitions are
needed.
Let G = (V, A) be a directed graph with vertex (node) set V = {v , v ,…v } and route set A.
Then the input data becomes:
Vertex v corresponds to the depot;
Vertices {v , v ,… v } correspond to the customers (nodes), take 푉 = {v , v ,… v }
A set of M = {1, . . . ,m} identical vehicles (a homogeneous fleet), each vehicle with
capacity K, is available at the depot;
The vehicles must return to the depot they originated from;
Each customer i ∈ V is associated with a known demand d ≥ 0 to be delivered
(assume d = 0 and d ≤ K for all i ∈ V );
A non-negative cost C is associated with each route (i, j) ∈ A representing the travel
cost between vertices i and j (assume C = 0).
Q specify the quantity of goods that a vehicle carries when it leaves customer i to
service customer j.
The binary variables X are used as vehicle flow variables that take value 1 if a vehicle
travels directly from customer i to customer j, and 0 otherwise;
Thus X = 1 if vehicle v drives from node i to node j 0 otherwise
2.1
The objective function of the model becomes:
Minimize C( , ) ∈ ∈
X 2.2
Subject to the constraints:
37
X ∈
= 1 ∀ j ∈ V 2.3
X ∈
− X ∈
= 0 ∀ p ∈ V 2.4
d X ≤ Q ≤ (K − d )X ∀ i, j ∈ V, i ≠ j 2.5
Q ≥ 0 ∀ i, j ∈ V, i ≠ j 2.6
Equation 2.2 is the objective function which is to minimise the transportation cost by generating
the shortest feasible set of routes.
Constraint 2.3 suggests that, each customer is visited once by exactly one vehicle.
Constraint 2.4 guarantees that, if a vehicle visits a customer, it must also depart from it.
Constraint 2.5 the sum of the demands of the customers visited by each vehicle does not exceed
the given vehicle capacity K
Graphical Representation of the Vehicle Routing Problem
Figure 3.1
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3.3 Notations
Weighted graph: A weighted graph is a graph with labels (weight) at every edge in a graph.
Weights are usually numbers and in most cases are positive, but this may vary due to the nature
of the graph that is being evaluated.
Directed graph: Directed graphs are graphs with edges directed to specific vertices. Directed
graph can be defined as an ordered pair G: = (V, A) with V is a set, whose elements are called
vertices or nodes and A is a set of ordered pairs of vertices, called directed edges, arcs, or arrows.
Travelling salesman problem
The Traveling Salesman Problem (TSP) is the method used to find the cheapest way of visiting
all of a given set of locations and returning to the starting point as quickly as possible.
Vehicle route: is defined to be a path that starts from and ends at the depot, and is denoted as
r = (v , v , … v , v ) where v = v = 0 represent the depot, and v ∈ {1, . . . , n} for
i ∈ {1, . . ., h} are customers.
Feasible route: is a route that covers each customer at exactly once and for which the total load
does not exceed the vehicle capacity, i. e. , vi ≠ vj ( ∀i, j ∈ {1, . . . , h}and i ≠ j)and
d
≤ 푞. The cost of the route is calculated by C = C
C
Feasible solution: is composed of m feasible routes, denoted by x = {r , . . . , r }.
The cost of a feasible solution is the sum of cost of all the routes i. e C = C
.
The optimal solution 퐗∗ is the solution that has the minimum cost. i.e
퐗∗ = min C ∀i ∈ (1, … , m).
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3.4.0 Methods of solution
Solving the vehicle routing problem can be done in many ways but are classified into two main
categories; the exact solution methods and the approximation methods.
3.4.1 The Exact Approach
For small problems, exact approaches are proposed that evaluate implicitly, every possible
solution to obtain the best solution. A well-known exact method is the branch and bound method,
which consists of a systematic implicit enumeration of all feasible solutions. The branch and
bound algorithm searches the complete space of solutions for a given problem for the best
solution. However, explicit enumeration is normally impossible due to the exponentially
increasing number of potential solutions. The use of bounds for the function to be optimized
combined with the value of the current best solution enables the algorithm to search parts of the
solution space only implicitly.
Using lower and upper bounds on the optimal objective value, more and more subsets of the
feasible solutions will be rejected, such that the optimal solution appears.
Another exact approach is the branch and cut method, a hybrid of the branch and bound method
and the cutting plane method. The cutting plane method adds linear inequalities, called cuts, to
the problem in order to define as small as possible feasible set of the objective values. To prevent
a slow convergence to the optimal value, the structure of the problem can be used to generate
very good cuts.
3.4.2.0 Approximation Approach.
Approximation algorithms are special classes of heuristic that provide a solution that is near to
optimal. Heuristics are approximation algorithms that aim at finding good feasible solutions
quickly. They can be roughly divided into two main classes; the construction heuristics followed
by the improvement heuristics.
40
3.4.2.1.0 Construction Heuristics.
Construction methods gradually build a feasible solution by selecting arcs based on minimising
the total cost of transportation which can be the travel cost, time or distance.
A route construction heuristic quickly builds a feasible solution, but usually not the optimal one.
The most well-known route construction heuristic algorithms are the nearest neighbour search,
savings algorithm, sweep algorithm and the cluster first route second method.
3.4.2.1.1 Nearest Neighbour search
This heuristic starts at an arbitrary customer most especially the nearest to the depot,
subsequently it chooses the nearest customer as the next one to visit and so on, until a feasible
solution is obtained.
Starting with a vehicle, until this current vehicle is full, we keep inserting the nearest unvisited
customer as long adding this customer does not exceed the capacity of this vehicle. Then we
select the next vehicle, and repeat the above, until either all the vehicles are full or until all
customers have been served.
3.4.2.1.2 Clarke-Wright savings heuristic
Another well-known route construction heuristic is the Clarke-Wright savings heuristic. This
savings heuristic starts with an initial allocation of each customer to a separate route. That is the
method initially assumes that each customer is served by its own vehicle. Next, two customers
are to be served by the same vehicle as long as their capacity constraints are not violated Then
for each pair of customers the cost savings of joining those customers on one route are
calculated. Based on the values of these savings, the customers are joined into routes starting
with the customer combination yielding the largest cost savings until no further savings can be
achieved.
41
Determining the order in which customers are combined into a certain vehicle route is done by
calculating the savings for a pair of customers:
The savings S for a pair of customers V and V is defined as the savings in terms of distance that
would be realized if these two customers would be served right after each other by the same
vehicle instead of each by their own vehicle.
S = C + C − C
The algorithm has a parallel and a sequential variant. The difference between the two is that the
parallel version builds multiple routes at a time, whereas the sequential version builds one route
at a time. In the parallel version it can happen that, when the savings list has been processed,
unassigned customers are assigned to their own vehicle, exceeding the total amount of available
vehicles m.
The savings algorithm is used to construct feasible routes after the following procedures are
followed.
1. Calculate the savings for every pair of customers using S = C + C − C
2. List the calculated savings in descending order of magnitude, creating the “Savings list.”
3. Then for each savings pair S on the savings list, starting from the pair with the highest
savings include path (i, j) in a route if no capacity constraints will be violated.
Note that if:
Neither i nor j have already been assigned to a route, in which case a new route is
initiated including both i and j.
Exactly one of the two points (i or j) has already been included in an existing route and
that point is not interior to that route (a point is interior to a route if it is not adjacent to the depot
in the order of traversal of points), in which case the link (i, j) is added to that same route.
42
Both i and j have already been included in two different existing routes and neither point
is interior to its route, in which case the two routes are merged.
4. If the savings list has not been exhausted, or reached a negative saving return to step 3.
Otherwise the algorithm terminates and the solution to the VRP consists of the routes created so
far. If any unassigned customers remain, they must be served by their own vehicle.
For example, consider the symmetric distance matrix in Table 3.1 for 5 customers (n = 5) and
demand vector given in Table 3.2. Assume that we have 2 vehicles available (m = 2) and the
capacity K is equal to 100. We will outline how both the sequential and the parallel version
Appendix 1. Codes for the generation of the distance saved matrix. (Microsoft Excel).
Private Sub cmdmatrixentry_Click() t = TextBox1.Text ' u = t Range(Cells(t + 1, 1), Cells(t + 1, 10)).Select 'select the row below your matrix Selection.Font.Bold = True ' ActiveCell.FormulaR1C1 = " ‘YOUR SYSTEM IS DISPLAYED ABOVE" ' For i = 1 To u For j = 1 To u ActiveSheet.Cells(i, j) = InputBox("Enter your element a" & " " & i & "," & j) ' If i = u And j = u Then ' when entries are all entered CheckInputs 'start the error checking procedure End If Next j Next i cmdmatrixentry.Enabled = False End Sub Sub CheckInputs() 'error checking procedure For i = 1 To u ' For j = 1 To u ' ' If ActiveSheet.Cells(i, j).Value <> "" And ActiveSheet.Cells(i, i).Value <> 0 And
msg = "You either typed a non-numeric value,entered zero for a diagonal element or left a space empty" & vbCrLf '
msg = msg & "Click ENTER MATRIX to input matrix again" ' MsgBox msg, vbInformation cmdmatrixentry.Enabled = True ' cmdmatrixentry.SetFocus ' End Sub Private Sub cmdsolve_Click() cmdsolve.Enabled = False Dim p As Integer ' Dim n As Double ' Dim m As Double '
62
Dim z As Integer ' p = TextBox1.Text ' Range(Cells(p + 17, 1), Cells(p + 17, 3)).Select Selection.Font.Bold = True ActiveCell.FormulaR1 YOUR SOLUTION IS DISPLAYED BELOW" z = TextBox1.Text Range(Cells(1, 1), Cells(z, z)).Select 'reading matrix' p = TextBox1.Text n = p Dim a() Dim s() ReDim a(n, n) ReDim s(n, n) For i = 1 To n For j = 1 To n a(i, j) = ActiveCell.Cells(i, j) Next j Next i For i = 1 To n For j = 1 To n s(i, j) = a(1, i) + a(1, j) - a(i, j) 'MsgBox (s(i, j)) If i = j Then ActiveCell.Cells(i + 18, j).Value = 0 Else ActiveCell.Cells(i + 18, j).Value = s(i, j) End If Next j Next i End Sub
