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CHAPTER 1

INTRODUCTION

The Vehicle Routing Problem (VRP) is a generic name given to a

set of problems in which set of routes for a fleet of vehicles based at one or

several depots are to be formed for servicing the customers dispersed

geographically. The objective of the VRP is to form a route with lowest cost

to serve all customers. More than 50 years have elapsed since Dantzig and

Ramser introduced the VRP in 1959 (Dantzig and Ramser 1959). They

proposed the first mathematical programming formulation and algorithmic

approach. They also described VRP with a real-world application concerning

the delivery of gasoline to service stations. Clarke and Wright (1964)

proposed an effective greedy heuristic that improved on the Dantzig-Ramser

approach. After these two papers, many models and algorithms are proposed

for the optimal and approximate solution of the different versions of the VRP

(Toth and Vigo 2002).

1.1 VEHICLE ROUTING PROBLEM

Generally, distribution or collection of goods from customers to

depot is called as VRP or Vehicle Scheduling Problem. The distribution of

goods concerns the service, in a given time period, to a set of customers by a

fleet of vehicles, which are located in one or more depots. These vehicles are

operated by a set of crews (drivers), and perform their movements by using an

appropriate network. In particular, the solution of a VRP calls for the

determination of a set of routes, each performed by a single vehicle that starts

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and ends at its own depot, such that all the requirements of the customers are

fulfilled, with some operational constraints and the global transportation cost

is minimized. The operational constraints can be a vehicle capacity, route

length, time window, precedence relation between customers, etc. Figure 1.1

illustrates a VRP with 3 vehicles serving 10 customers forming 3 routes.

Figure 1.1 Vehicle Routing Problem

A single universally accepted definition of the VRP does not exist

because of the diversity of constraints encountered in practice. VRP is a well

known integer programming problem which falls into the category of NP-hard

problems, meaning that the computational effort required in solving this

problem increases exponentially with the problem size. For such problems, it

is often desirable to obtain approximate solutions, so they can be found fast

enough and are sufficiently accurate for the purpose. The Vehicle Routing

Problem (VRP) can be seen as a merge of two well-known problems:

Traveling Salesperson Problem (TSP) and Bin Packing Problem (BPP). TSP

as a special case when the number of vehicle is one and its capacity is infinity.

VRP is considerably more difficult to solve than a TSP of the same customer

size (Laporte 2007). VRP is one of the most important, and most studied

Combinatorial Optimization Problem (COP).

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1.2 VARIANTS OF VRP

Usually, in real world VRPs, many side constraints will appear.

Some of the most important types of VRPs are (Hrvoje Markovi et al 2004)

shown in Figure 1.2.

Figure 1.2 Basic Variants of VRP

Capacitated VRP (CVRP) - CVRP is a Vehicle Routing

Problem (VRP) in which a fixed fleet of delivery vehicles of

uniform capacity must provide service to known customer

demands for a single commodity from a common depot at

minimum transit cost. That is, CVRP is like VRP with the

additional constraint that every vehicle must have uniform

capacity of a single commodity (Ralphs, 1991).

VRP with Time Windows (VRPTW) - The VRPTW is the

VRP with the additional restriction that is, a time window is

associated with each customer, defining an interval [e0, l0]

wherein the customer has to be supplied where e0 and l0

represents the early and late time. The interval [e0, l0] at the

depot is called the scheduling horizon (Solomon, 1995).

VRP with Backhauls (VRPB) - The VRPB is the extension of

the VRP in which the customer set is partitioned into two

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subsets. One contains customers that require a given quantity

of product to be delivered and the second contains customers

where a given quantity of inbound products must be picked up

(Goetschalckx 1993).

Distance-Constrained VRP (DCVRP) – In DCVRP, each route

has a maximum length (or time) constraint instead of capacity

constraint.

Multi-Depot VRP (MDVRP) - A company may have several

depots from which it can serve its customers. If the customers

are clustered around depots, then the distribution problem

should be modeled as a set of independent VRPs. However, if

the customers and the depots are intermingled, then a MDVRP

should be solved. A MDVRP requires the assignment of

customers to depots. A fleet of vehicles is based at each depot.

Each vehicle originates from one depot, provide service to the

customers assigned to that depot, and return to the same depot.

The objective of the problem is to service all customers while

minimizing the number of vehicles and travel distance

(Hjorring, 1995). Giosa et al (2002) compared assigning of

customers to depot using six heuristic for multi-depot

VRPTW.

VRP with Pick-Up and Delivering (VRPPD) - The Vehicle

Routing Problem with Pick-up and Delivering (VRPPD) is a

VRP in which the possibility that customers return some

commodities is contemplated (Righini, 2000). So in VRPPD,

it should be taken into account that the goods returned by

customers to the delivery vehicle must fit into it. These

restrictions make the planning problem more difficult and can

lead to bad utilization of the vehicles capacities, increased

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travel distances or a need for more vehicles. Hence, it is usual

to consider restricted situations where all delivery demands

start from the depot and all pick-up demands shall be brought

back to the depot, so there are no interchanges of goods

between the customers. Another alternative is relaxing the

restriction that all customers have to be visited exactly once.

Another usual simplification is to consider that every vehicle

must deliver all the commodities before picking up any goods

(Savelsbergh 1995).

Split Delivery VRP (SDVRP) - SDVRP is a relaxation of the

VRP wherein it is allowed that the same customer can be

served by different vehicles if it reduces overall costs. This

relaxation is very important if the sizes of the customer orders

are as big as the capacity of a vehicle (Dror and Trudeau

1990; Archetti et al 2006)

Stochastic VRP (SVRP) - Some values (like number of

customers, their demands, serve time or travel time) are

random. Stochastic VRP (SVRP) is a VRP where one or

several components of the problem are random (Laporte,

1998). Three different kinds of SVRP are:

o Stochastic customers: Each customer vi is present with

probability pi and absent with probability 1-pi.

o Stochastic demands: The demand di of each customer is a

random variable.

o Stochastic times: Service times and travel times tij are

random variables.

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In SVRP, two stages are made for getting a solution. A first

solution is determined before knowing the realizations of the random

variables. In the second stage, a recourse or corrective action can be taken

when the values of the random variables are known (Steward and Golden

1980; Jaillet 1987).

Periodic VRP (PVRP) - In classical VRPs, typically the

planning period is a single day. In the case of the Period

Vehicle Routing Problem (PVRP), the classical VRP is

generalized by extending the planning period to a specified

number of days (Angelelli and Speranza 2002).

1.3 APPLICATIONS OF VRP

There are several important problems that must be solved in real-

time. The main applications that motivate the research in the field of the real-

time VRPs are listed below (Ghiani et al 2003).

i. Dynamic fleet management: Several large scale trucking

operations require real-time dispatching of vehicles for the

purpose of collecting or delivering shipments. Important

savings can be achieved by optimizing these operations.

ii. Vendor-managed distribution systems: In vendor-managed

distribution systems, distribution companies estimate

customer inventory level in such a way to replenish them

before they run out of stock. Hence, demands are known

beforehand in principle and all customers are static. However,

because demand is uncertain at some time, some customers

(usually a small percentage) may run out of stock and have to

be serviced urgently.

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iii. Couriers: Long-distance courier need to collect locally

outbound parcels before sending them to a remote terminal to

consolidate loads. Also, loads coming from remote terminals

have to be distributed locally. Most pick-up requests are

dynamic and have to be serviced the same day if possible.

iv. Rescue and repair service companies: There are several

companies providing rescue or repair services (broken car

rescue, appliance repair, etc.).

v. Dial-a-ride systems: Dial-a-ride systems provide

transportation services to people between given origin–

destination pairs. Customers can book a trip one day in

advance (static customers) or make a request at short notice

(dynamic customers).

vi. Emergency services: Emergency services comprise of police,

fire fighting and ambulance services. By definition, all

customers are dynamic. Moreover, the demand rate is usually

low so that vehicles become idle from time to time. In this

context, relocating idle vehicles in order to anticipate future

demands or to escape from downtown rush hour traffic jam is

a major issue.

vii. Taxi cab services: In taxi cab services, almost every customer

is dynamic. As in emergency services, relocating temporary

idle vehicles is an issue.

viii. Refuse collection: It is an activity rendering used products or

wastes and moving them to some points where further

treatment is taken care. In real life waste collection vehicle

routing problem with time windows (VRPTW), with

consideration of multiple disposal trips and drivers’, with

workload balancing and compact route of each vehicle need to

be found.

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ix. Newspaper distribution: Distribution of newspaper is similar

to VRP, where it is to improve the distribution activity.

Distribution activity is a part of the strategic planning of the

company which aims to reduce number of staff members and

to reduce the cost occurred in the distribution process.

1.4 SOLUTION TECHNIQUES

Both exact and approximation algorithms are applied for solving

VRP and its variants. Many algorithms are developed for VRP, mostly

heuristics and metaheuristics are applied because the exact algorithms are not

guaranteed to find the optimal tours within reasonable computing time when

the numbers of cities are large.

1.4.1 Exact Algorithms for VRP

Exact methods guarantee that the optimal solution is found if the

method is given sufficient time and space. As stated initially, a simple

enumeration is out of question, so exact methods must use more clever

techniques. The worst case running time for NP-Hard problems are still going

to be high. Construction of exact algorithms that solve NP-hard problems in

polynomial time is possible, unless NP = P. For some classes of problems,

there are hopes of finding algorithms that solve problem instances occurring

in practice in reasonable time.

One of the most successful exact algorithm for solving CVRP is the

K-tree method of (Fisher 1994) that succeeded in solving a problem with 71

customers. However, there are smaller instances that have not been exactly

solved yet. To treat larger instances, or to compute solutions faster, heuristic

methods must be used. Ralphs et al (1995), implemented a parallel branch-

and-cut algorithm, and found an optimal solution for two instances and

provided a proof of optimality.

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Exact methods such as branch-and-cut, branch-and-price and their

combination branch-cut-and-price are used to solve VRP (Wolsey 1998).

Blasum and Hochstättler (2000) used branch-and-cut for VRP. The best

algorithm based on two-index vehicle flow formulations is due to Naddef and

Rinaldi (2002). Toth and Vigo (2002) applied branch-and-bound for solving

VRP.

Laporte and Louveaux (1993) proposed an exact algorithm based

on the integer L-shaped method for solving stochastic VRP. Gendreau et al

(1995) have considered the SVRP in which both customers and their demand

are random and formulated it as a nonlinear stochastic integer program and

proposed a method based on L-shaped method.

1.4.2 Heuristic Algorithm for VRP

Heuristics are solution methods that can quickly find a feasible

solution with reasonable quality. There are no guarantees about the solution

quality though, it can be arbitrarily bad. The heuristics are based on these

experiments and tested empirically; comments about the quality of the

heuristic can be made. Heuristics are typically used for solving real life

problems because of their speed and their ability to handle large instances.

Several heuristic methods are proposed to search in a limited space

to produce good quality of result in a modest computational time for VRP,

which are broadly classified as:

Classical heuristics

Metaheuristics

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1.4.2.1 Classical heuristics

Classical heuristics developed between 1960 and 1990. The class of

classical heuristics is divided into three groups: constructive heuristics, two-

phase heuristics and improvement methods.

Constructive heuristics

Laporte and Semet (2002) defined construction heuristics as

follows: “Constructive heuristics gradually build a feasible solution while

keeping an eye on solution cost; they do not contain an improvement phase

per se”. Many construction heuristics for vehicle routing problems have been

proposed during the last 40 years. Many construction heuristics for vehicle

routing problems are categorized into three classes: insertion heuristics,

savings heuristics, and clustering heuristics.

Insertion heuristics build a solution by inserting one customer at a

time. Insertion heuristics can build one route at a time (sequential insertion

heuristics) or build many or all routes parallelly (parallel insertion heuristics).

The choice of inserting a customer and where to insert a customer is what

differentiates the insertion heuristics. A very simple insertion heuristic could

choose to insert the customer that reduces the overall cost at least. The Mole

and Jameson (1976), expands one route at a time and Christofides et al (1979)

applies in turn sequential and parallel route construction procedures.

Savings heuristics initially build a solution where each customer is

served by a vehicle on its own route. Routes are then merged one by one

according to some criteria. Savings algorithms vary by the criterion used for

merging routes (that is, saving obtained by merging two routes) and also by

how routes are merged. For the CVRP, the most simple merge operation

deletes an edge between the depot and a customer from each of the two routes

that is being merged and joins the route by adding an edge between the two

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customers that are adjacent to only one edge. More advanced merging

procedures consider all the customers served by the two routes and solve a

TSP (in case of the CVRP) on these customers.

The most widely used savings heuristics for VRP is Clarke and

Wright (1964) algorithm. It is based on the notion of savings. This algorithm

naturally applies to problems for which the number of vehicles is a decision

variable and it works equally well for directed or undirected problems.

Desrochers and Verhoog (1989) and Altinkemer and Gavish (1991) described

an interesting modification to the standard savings algorithm. Van Breedam

(1994) classifies the improvement operations as “string cross”, “string

exchange”, “string relocation”, and “string mix”, which can all be viewed as

special cases of 2-cyclic exchanges. Most savings algorithms have been

proposed for the CVRP. A new variant of the savings algorithm is given by

Altinel and Öncan (2005).

Clustering algorithms are two-phase algorithms. The first phase

consists of grouping customers into subsets (clusters) where each subset

should be served by one route. The second phase then creates routes for each

subset. A third phase may be employed to repair the solution if it turns out

that some of the clusters could not be served by a single vehicle. Sweep

algorithm is a clustering approach for CVRP which is presented by Gillet and

Miller (1974). In practice, the algorithm works by sorting customers

according to their polar coordinate angle with the depot as (0, 0). The

algorithm starts from the first customer in the list and adds this customer to a

cluster. The algorithm continues to process the customers according to the

ordering and adds the customer to the current cluster as long as the cluster can

be served by a single vehicle. When it is no longer possible to add a customer

to the current cluster a new cluster is started and becomes the current cluster.

When all customers have been assigned to a cluster, a TSP tour is found for

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each cluster to produce a CVRP solution. Fisher and Jaikumar (1981)

presented a clustering heuristic for the CVRP where the number of vehicles is

fixed to K. A generalized assignment problem is then solved. This produces K

clusters that each satisfies the capacity constraint. Each cluster is turned into a

route by solving a TSP to optimality. Talliard’s (1993) algorithm defines

neighborhood using the -interchange generation mechanism. A noble feature

of Talliard’s algorithm is the decomposition of the main problems into sub

problems. Salhi and Sari (1997) proposed three level composite heuristic to

solve MDVRP.

Two-phase heuristics

It is the one in which there are two heuristics used: clustering and

routing. Based on this, there are two types of heuristics: Cluster-first and

route-second and Route-first and cluster-second. In cluster-first and route-

second heuristics, the clusters are formed and then routes are framed within

each cluster. The above mentioned clustering algorithm solves VRP in this

manner. Route-first and cluster-second heuristics is a method in which the

first giant TSP is formed without considering the side constraints and then

decomposes this into vehicle routes based on the constraints. It had been first

put forward by Beasley (1983) who observed that the second phase problem is

a standard shortest path problem on an acyclic graph and can thus be solved in

O(n2) time.

Improvement methods

Local search heuristic methods 2-opt local exchange, Or-opt

exchange etc., are generally used as an improvement method in VRP, to

improve the routes of each vehicle. Funke et al (2005) used local heuristic for

solving VRP. They have used the terms best search for a steepest descent

algorithm and first search for a descent algorithm.

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1.4.2.2 Metaheuristic methods

A special class of heuristics that has received special attention in

the last two decades is Metaheuristics. Metaheuristics provides general

frameworks for heuristics that can be applied to many problem classes. High

solution quality is often obtained using metaheuristics. Metaheuristics allow

deteriorating and even infeasible intermediary solutions in the course of the

search process. The best known metaheuristics developed for the VRP

typically identify better local optima than earlier heuristics, but they also tend

to be more time consuming. The six main types of metaheuristic methods that

have been applied to the VRP are Deterministic Annealing (DA), Simulated

Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA), Ant Colony

Optimization (ACO), and Neural Networks (NN). In recent years apart from

these metaheuristics, Particle Swarm Optimization (PSO), Bee Colony

Optimization are applied to VRP in finding near optimal solution quickly.

Simulated Annealing

Teodorovic and Pavkovic (1992) build a SA-based algorithm for

VRP for stochastic case. Osman’s (1993) implementation of simulated

annealing is much more involved, and also more successful. It uses a better

starting solution, some parameters of the algorithm are adjusted in a trial

phase, richer solution neighborhoods are explored, and the cooling schedule is

more sophisticated. The neighborhood structure of this algorithm uses a -

interchange generation mechanism in which two routes p and q are first

selected, together with two subsets of customers Sp and Sq, one from each

route, satisfying |Sp and |Sq . The operation swaps the customers of Sp

with those of Sq as long as this is feasible. The algorithm that is implemented

is tested on symmetric VRPs with an unspecified number of vehicles.

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Van Breedam (1995) compared and tested several version of SA

using different neighborhood structures. Tests are conducted on the 14

Christofides, Mingozzi, and Toth instances. These experiments are useful to

identify best SA strategies, but overall they confirm the superiority of tabu

search-based heuristics.

Tabu Search

One of the first attempt to apply tabu search to the VRP is due to

Willard (1989). The solution is first transformed into a giant tour by

replication of the depot, and neighborhoods are defined as all feasible

solutions that can be reached from the current solution by means of 2-opt or

3-opt exchanges. The next solution is determined by the best non-tabu move.

In Osman (1993), neighborhoods are again defined by means of -interchange

generation mechanism, with =2. This includes a combination of 2-opt

moves, vertex reassignments to different routes, and vertex interchanges

between two routes. In one version of the algorithm called BA (Best

Admissible), the whole neighborhood is explored and the best non-tabu

feasible move is selected. In the other version, FBA (First Best Admissible),

the first admissible improving move is selected if one exists; otherwise, the

best admissible move is implemented. The different versions of TS applied to

VRP are adaptative memory procedure (Rochat and Taillard 1995), network

flow TS (Xu and Kelly 1996). Gendreau et al (1996) developed a tabu search

heuristic using a proxy for the objective function in order to efficiently

evaluate potential moves for VRP with stochastic demand and stochastic

customer. Granular Tabu for VRP had been proposed by Toth and Vigo

(2003) to VRP and its application.

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Genetic Algorithm

Thangiah and Gubbi (1993) used genetic algorithm to find good

cluster of customers for VRP with ‘cluster first and route second’. Another

GA application for the time-constrained CVRP may be found in Schmitt

(1994, 1995). An interesting feature of this work is that a route-first, cluster-

second approach is used, thus allowing the classical path representation

(without separators) to be used. The implementation proposed by Schmitt

(1995) uses the Order Crossover operator and a Swap mutation operator,

where two randomly selected vertices exchange their position. To improve the

performance of the GA, this mutation operator is then replaced by a 2-opt

local search method. GA with neighborhood search method is adopted by

Baker et al (2003). They proved that the results are competitive with TS and

SA. Yi-Liang Xu et al (2005) in their paper, proposed three genetic

representations for VRP. They used fuel truck dispatch system data set for

their investigation with hybrid GA and 2-opt routing optimization technique.

A hybrid algorithm based on GA and TS for solving CVRP is studied by

Guido et al (2008), where they described two simple heuristics and introduced

a new genetic operator namely chain mutation. The work done on the CVRP,

including its distance or time-constrained variant, is mostly aimed at

evaluating the impact of different components or parameters of a GA on the

efficiency of the search. Many researches have been done in VRPSD using

GA. Ismail and Irhamah (2008) proposed a new scheme based on a hybrid

GA and TS heuristic for VRPSD with preventive restocking. The Breeder

Genetic Algorithm (BGA) is also proposed to solve VRPSD by Irhamah and

Ismail (2009). BGA is powerful and reliable in global searching than GA.

Ant Systems

Bullnheimer et al (1997), in their paper developed a hybrid ant

systems in which each vehicle route produced in a given iteration is improved

by the 2-opt heuristic before trail update. This algorithm also uses terms

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related to vehicle capacity and distance savings with respect to the depot

when selecting the next vertex to be visited. In the trail update step, they use a

number of “elitist ants” to account for the best solution found so far

(these ants are assumed to always travel on this best solution). Their

computational experiments on the 14 problems of Christofides, Mingozzi, and

Toth indicate that the addition of a 2-opt step and the use of elitist ants are

clearly beneficial. Kawamura et al (1998) proposed a complex hybrid variant

of AS that involves 2-opt improvement procedures and probabilistic

acceptance rules reminiscent of simulated annealing. The method is applied to

two geometric 30 and 60 customer instances and it identified the optimal

solution in both cases. No other tests are performed, which makes it difficult

to assess the effectiveness of this procedure.

1.5 PARTICLE SWARM OPTIMIZATION

Particle swarm optimization (PSO) is a population based

evolutionary computation technique developed by Eberhart and Kennedy

(Eberhart and Kennedy 1995; Kennedy and Eberhart 1995), inspired by social

behavior of bird flocking or fish schooling. The PSO concept originated as a

simulation of a simplified social system. The original intent is to graphically

simulate the graceful, unpredictable choreography of collision-proof birds in a

flock. Then, this simulation is modified by incorporating nearest neighbor

velocity, multidimensional search and acceleration, eliminating ancillary

variables. During the evolution of algorithms, it is realized at some point that

the model is an optimizer. A simple original algorithm (Eberhart et al 1996)

is the result after eliminating number of parameters during the process of trial

and error.

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1.5.1 Components of PSO

The main components of PSO are

Swarm of particles that represent the population of solutions

Each particle represents a candidate solution

Elements of a particle represent parameters to be optimized

The PSO is initialized with a population of random

solutions/particles. Each particle is treated as a point and has two attribute

values: position and velocity. The particle keeps track of its coordinates in

the problem space which are associated with the best solution (fitness) it has

achieved so far and is called as pbest. Another “best” value that is tracked by

the particle swarm optimizer is the best value, obtained so far by any particle

in the neighbors of the particle. This location is called lbest. When a particle

takes all the population as its topological neighbors, the best value is a global

best and is called gbest. The particles fly through the problem space in each

generation by following pbest and gbest values.

1.5.2 Procedure for PSO

The size of swarm is I and each particle is of dimension n. The ith

particle is represented by Xi=(xi1,xi2,…xin). The gbest and pbest solutions are

represented Pg=(pg1,pg2,pg3…pgn) and Pi=(pi1,pi2,pi3…pin) respectively. The

velocity rate of ith particle is Vi=(vi1,vi2,vi3…vin).

The implementation of PSO is outlined as follows (Eberhart and

Shi 2001).

1. Initialize the swarm of particles with random positions and

velocities on n dimensions in the problem space

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2. Repeat the steps 3 – 5, until maximum iterations T or

minimum error criteria is attained

3. For each particle,

a. Evaluate the desired optimization fitness function in n

variables

b. Compare particles fitness with particles pbest. If the

current value is better, then store pbest value equal to

current value using Equation (1.1) and (1.2)

=) , = 1

) , ( ) > 1

(1.1)

=, = 1

, ( ) > 1 (1.2)

4. Compare fitness with the populations overall previous best. If

current value is better, then store the current value as the gbest

value using Equation (1.3) and (1.4)

= max( ) = 1 (1.3)

( ) = 1 (1.4)

5. For each particle,

a. Change the velocity and position of the particle

according to the Equations (1.5) and (1.6), respectively

( + 1) ( ) . ( ) ( ) + . ( ) ( )

(1.5)

( + 1) ( ) ( + 1) (1.6)

where 1 i I, 1 j n, cp, cg are learning factors / acceleration

constants, r1 and r2 is a random number between 0 and 1, w is

a inertia factor, vij(t) is the velocity of the ith particle in the tth

iteration of jth dimension, xij(t) is the jth position of the ith

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particle in the tth iteration. cp and cg represent the cognitive

and social parameters.

The equation (1.2) and (1.4) stores the pbest and gbest of particles.

The equation (1.5) helps each particle to move towards the best of the

obtained solution by calculating the difference between its current position

value and the pbest or gbest value. This difference value guides the particle to

move towards the optimal or near optimal solution in due course. Searching in

PSO by using cognitive and social component for dth position of ith particle is

shown in Figure 1.3.

Figure 1.3 Searching Diagram of PSO

The velocity updates are done in two ways:

Synchronous mode in which best position is updated only

after all particles positions and personal best positions have

been updated.

Asynchronous mode in which the best position found is

updated immediately after each particle’s position update.

They lead to faster propagation of best solution.

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1.5.3 Variants of PSO

PSO has undergone a number of changes from its beginning. Most

of these changes affect the velocity update of the particle. Kennedy and

Eberhart (2001), Engelbrecht (2005), Clerc (2006), and Poli et al (2007),

detailed the description of many PSO variants.

Most particle swarm optimization algorithms are designed to search

in continuous domains. However, there are a number of variants that operate

in discrete spaces. It is of two types: Binary PSO (Kennedy and Eberhart

1997) and integer PSO (Parsopoulos and Vrahatis 2002).

The constriction coefficient is introduced as an outcome of a

theoretical analysis of swarm dynamics. It is the first method to control the

“explosion” of the swarm and developed by Clerc and Kennedy (2002).

The bare-bones particle swarm (Kennedy 2003) is a version of the

particle swarm optimization algorithm in which the velocity- and position-

update rules are substituted by a procedure that samples a parametric

probability density function.

In the standard particle swarm optimization algorithm, a particle is

attracted towards its best neighbor. A variant in which a particle uses the

information provided by all its neighbors in order to update its velocity is

called the fully informed particle swarm (FIPS) (Mendes et al 2004).

1.5.4 Hybrid PSO

Other evolutionary computation capabilities are incorporated to

PSO as hybrid PSO or the adaptation of PSO parameters for better

performance as adaptive PSO. PSO also required to work under complex

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environments such as multi-objective or constraint optimization problem or

tracking dynamic systems.

Many researchers incorporated selection, mutation, crossover as

well as Differential Evolution (DE) into PSO algorithm. The main goal is to

increase the diversity of the population by:

1. Preventing the particles to move close to each other and

collide (Krink et al 2002).

2. Self-adapt parameters such as the constriction factor,

acceleration constants (Miranda and Fonseca 2002) or inertia

weight (Lovbjerg and Krink 2002)

Selection procedure and self-adapting properties for its parameters

are incorporated to PSO algorithm. Angeline (1998) incorporated tournament

selection used in EP to PSO. Miranda and Fonseca (2002) introduced self

adaption capabilities to swarm by modifying the concept of a particle to

include, not only the objective parameters, but also set of strategic parameters.

PSO algorithm is included with differential evolution operator to

improve its performance in two ways:

1. To avoid falling into local minima (Zhang and Xie 2003).

2. To find the optimal parameters such as inertia and acceleration

constants for canonical PSO (Kannan et al 2004).

Gaussian mutation is combined for updating the velocities and

position of particles (Higashi and Iba 2003). Unimodal and multimodal

functions are tested. The improvement of standard PSO is done by inclusion

of mutation or a simple replacement of the best fitted value. El-Dib et al

(2004) applied the reproduction system of GA to PSO that modifies both the

position and velocity vectors of randomly selected particles in order to reach

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the optimum. Along with mutation, crossover and elitism are incorporated by

Juang (2004). The new generation of populations are generated half by elites

and remaining by crossover and mutation.

The SA-PSO is presented by Chaojun and Zulian (2006). The

disadvantage of getting into the local best point of standard PSO is overcome

effectively and the ability of global optimality are toned up.

A novel Condensed Hybrid Optimization algorithm using Enhanced

Continuous Tabu Search (ECTS) and PSO is proposed by Chen-Hung (2009). The

ECTS is modified TS, whichhas good search capabilities on large search spaces.

Turbulent PSO (TPSO) is hybridized with fuzzy logic controller

(Liu and Abraham 2005) to produce Fuzzy Adaptive TPSO (FATPSO) to

adaptively regulate the velocity parameters during an optimization run, thus

enabling coarse-grained explosive searches to occur before being replaced by

fine-grained exploitation later. The problem with both high and low

dimensions performs well with TPSO and FATPSO.

1.5.5 Applications of PSO

Like any other evolutionary algorithms, PSO can be applied to solve

most of the optimization problems. The potential application areas of PSO are:

System design (Eberhart and Shi 1998; Kennedy 2001;

Eberhart and shi 2001)

Multi-objective optimization (Tandon 2000)

Power system (Yoshida et al 1999)

Classification (Wang et al 2007; Chia-chong 2006; Alba et al 2007)

Pattern recognition (Sugisaka and Fan 2005)

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Biological system modeling and diagnosis (Eberhart and Hu

1999)

Dynamic System (Eberhart and Shi 2001)

Scheduling (planning)

Signal processing

Games

Robotic applications

Decision making

Simulation and identification, etc.

PSO can solve variety of optimization problems, especially in the

field of multi-dimensional continuous space optimization problems.

Expressing the particle position is not easy, hence it is restricted in the

application of COP. Many recent researches are mostly in the application of

PSO to COP. The main COPs addressed by PSO are:

Scheduling

o Job shop scheduling

o Flow shop scheduling, etc.

Routing

o Traveling Salesman Problem

o Vehicle Routing Problem

1.5.5.1 Scheduling

PSO is applied for solving task assignment problem by Salman et al

(2002). PSO is hybrid with Hill Climbing (HC) algorithm as a local search

24

method for solving task assignment in distributed systems (Peng-Yeng et al

2006). Bo Liua et al (2008) implemented the flow shop scheduling with

limited buffers. They proposed a hybrid PSO for solving the flow shop

scheduling, where permutation encoding is used for representing the particles.

1.5.5.2 Routing

The fuzzy discrete PSO (Pang et al 2004) is a matrix based

approach where the TSP is represented as a matrix containing the set of cities

and the TSP sequences. The particle position is a fuzzy matrix, the values of

which represent the degree of membership to the corresponding elements of

the TSP matrix. PSO is combined with Fast Local Search and GA concepts to

guide the particle movement at macro level (exploration) and to improve the

solution locally (exploitation). Lopes and Coelho, (2005) and Habibi et al

(2006) also developed a hybrid PSO, Ant Colony Optimization (ACO) and

SA are incorporated by them. The ACO algorithm replaces the individual best

element of PSO, whilst the cooling process of SA is used to control the

exploration of the group best element, both of which are then applied within

the PSO framework. There are many researches work done in the area of VRP

using PSO. The following section explains in detail.

1.5.6 Literature Survey for VRP using PSO

Ning et al (2004), Luo and Shi (2006) and Shao et al (2009) adopted

a mixed approach of Global and Local version of PSO (GL-PSO), for solving

VRP, Vehicle Routing Problem with stochastic travel time, Vehicle Routing

Problem with non-full load. The PSO is hybrid with SA for solving VRP

(Liu and Shi 2006; Zhang et al 2006; Chen et al 2006; Ma et al 2007). In

SA-PSO, (Yang and Qian 2008) adopted 2-opt exchange method to construct

neighborhood solution.

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Zhang and Wu (2008) applied Fitness-Distance- Ratio based PSO

(FDR-PSO) to facility VRP. Social learning behaviors of a particle in

FDR-PSO include not only the gbest but also the nbest. The nbest is the best

positions visited by the near neighbors of a particle, and accomplished by

selecting a particle that maximizes the ratio of the fitness difference to the

one-dimensional distance. GLNPSO is a multiple social structure technique

that is built by combining the three well-known social structures: gbest, lbest,

and nbest.

Multi-depot VRP is solved by Wang et al (2007) by proposing a

hybrid optimization algorithm which is composed of PSO and improved

ACO. Order numbers for vehicles to freight are got by particle position

vector, single vehicle route is got by ACO, and then evaluated and filtered

particles according to optimal vehicle routes, circulated until terminate

qualification.

A hybrid algorithm of PSO with TS algorithm is proposed by Zhao

et al (2008) for solving multi-depots single vehicle routing problem. They

used particle position matrix expression. The results showed that VRP model

and algorithm are effective and offerd a way to multi-depots single. Ai, and

Kachitvichyanukul (2008) proposed an adaptive version of PSO based on the

social structure of GL-PSO. This method has additional capability to self-

adapt its inertia weight (w). The w is controlled so that the balance between

exploration and exploitation phases of the swarm is maintained. The

performance of this adaptive PSO is evaluated for solving VRP instances.

The recent publication of GLNPSO algorithm is resolving CVRP

(Ai and Kachitvichyanukul 2007) and VRPSPD (Ai and Kachitvichyanukul

2009). It is tested on VRPSPD and CVRP using benchmark data sets

available from literatures and it is competitive with other published results.

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Ai, and Kachitvichyanukul, (2009) proposed two solution

representations for solving the CVRP using GL-PSO: SR-1 and SR-2. The

computational result shows that SR-1 and SR-2 is competitive with other

methods for solving CVRP in terms of solution quality and computational

time. The solution representation of SR-1 with n customers and m vehicles

requires a particle with (n+2m) dimension. Each particle dimension is

encoded as a real number. The first n dimensions represent priorities of

customers; each customer is represented by one dimension. The other 2m

dimensions are related to vehicles, each vehicle is represented by two

dimensions. The solution representation of SR-2 consists of 3m dimensional

particle and each particle dimension is encoded as a real number. All

dimensions are related to vehicles, each vehicle is represented by three

dimensions: two dimensions for the reference point and one dimension for the

vehicle coverage radius. SR-2 starts with the transformation of particle to the

vehicle orientation points and the vehicle coverage radius. The vehicle routes

are then constructed based on these points and radius. A feasible route

consisting of customers that are located inside its coverage area and have not

been assigned to other vehicle is constructed for each vehicle. Vehicle

coverage area is defined as an area inside a circle centered at its reference

point within its coverage radius.

In (Sombuntham and Kachitvichayanukul 2010) improved PSO is

applied for solving MDVRP in which mutation and improved inertia are used.

MDVRP with pickup and delivery is solved using PSO with multiple social

learning structures in (Zhang and Ye 2010).

1.6 MOTIVATION FOR THE RESEARCH

The PSO is an emerging metaheuristic technique, and considered to

be very simple and consistent than other popular metaheuristic and

evolutionary methods. It has wide variety of application areas as discussed in

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section 1.5. Particle encoding for COP are difficult and complex, because of

which there is a large scope of research in applying PSO to solve COP. This

motivated to perform research in the area of PSO to COP like VRP. The

efficient and effective representation of particles to solve COP is the objective

of this research. The PSO particles represent the customer sequence. These

customers are to be serviced by a vehicle with minimum total cost of

servicing. The space and time complexity are to be reduced.

The implementation of VRP is considered as they become

ubiquitous today. VRP span a wide variety of industries and involve the

commercial distribution of many products that range from newspapers to soft

drinks to groceries to milk on a daily basis. Beyond the commercial

distribution setting, there are applications that involve waste collection, street

sweeping, and delivery of mail. Even after fifty decade, there is more research

undergoing in this area, because of its complexity as explained in section 1.1.

1.7 OBJECTIVES OF THE RESEARCH

The main objective of this research is to propose hybrid PSO for

solving VRP. The proposed hybrid PSO can be applied for solving different

types of VRP and scheduling problems. Four different hybrid PSO algorithms

are proposed to minimize the route cost and also to reduce the time and space

complexity of hybrid PSO algorithms to solve VRP. The algorithm suites well

for any type of combinatorial problem like assignment, partitioning, routing,

and scheduling. VRP is the COP considered to perform posteriori testing of

the proposed algorithm and it is compared with other existing State-of-the-Art

methods. The variants of VRP that are considered in this research for testing

the PSO algorithm are

Capacitated VRP, and

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VRP with Stochastic Demands,

Multi-depot VRP

The main objectives of this research are

1. To propose a new Hybrid PSO (HPSO) with GA operators to

minimize the route cost of vehicle of CVRP. Discrete PSO with

integer coding is used to represent the particles of PSO. It makes

the algorithm easy to model and implement. It also reduces the

space and time when compared to other existing PSO methods. To

minimize the route cost of each vehicle hill-climbing with 2-opt

local exchange is incorporated.

2. To extend the proposed HPSO to work for stochastic nature of

VRP. The Dynamic Programming (DP) is included to handle this

stochastic nature where probability distribution is known rather

than the actual value.

3. To propose a new algorithm named Modified k-means with HPSO

(MK-HPSO). It divides the customer into groups based on their

distance and demand and then actual route is formed within each

group by HPSO, thus minimizing the route cost.

4. To propose a Nested PSO (NPSO) to solve CVRP that consists of

master PSO and slave PSO. Master PSO partition the customer into

groups and slave PSO forms the route such that the cost is

minimized.

Proposed algorithms such as HPSO, MK-HPSO and NPSO are also

applied to a variant of VRP, i.e., MDVRP. A small modification is made in all

proposed algorithms. A general k-means algorithm is first applied to

preprocess the MDVRP data thus reducing it to multiple VRPs. Then, each

VRP is handled with the proposed hybrid algorithms.

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All algorithms are implemented in MatLab 7.0.1 with Pentium 4.0, 2.3

GHz processor and tested with Capacitated Vehicle Routing Problem (CVRP)

Bench mark problem instances of Augerat et al set A, Augerat et al set B,

Augerat et al set P, Christofides and Eilon. MDVRP problem instances from

Christofides and Eilon, Gillett and Johnson, Chao et al are used to validate the

performance of the proposed algorithms. The data set along with the Best

Known Solution (BKS) is available at http://www.branchandcut.org and

http://neo.lcc.uma.es/radi-aeb/WebVRP/.

1.8 ORGANIZATION OF THE THESIS

Chapter 1 introduces the PSO and VRP. It describes about the PSO,

its variants and application. This chapter also discusses the evolution of VRP, its

application, variants and solution methodologies. It also covers the motivation

and objective of research and gives the outline of the research work.

Chapter 2 deals with the first work of research. The particle swarm

optimization is applied along with genetic operators (HPSO) to solve

deterministic COP such as CVRP. A two-phase heuristic technique route-first

and cluster-second is adapted. The giant sequences of customers are the

particles of HPSO. They are represented as permutation of integers. HPSO

generate the initial particles using Nearest Neighbor Heuristic (NNH) and

random permutation. Steepest descent Hill-Climbing (HC) with 2-opt

operator is incorporated to improve the quality of particles. The particles are

converted to particle position before applying the velocity and GA operator.

The varying inertia is used for updating the velocities to avoid trapping into

local optima. GA operator’s two-point crossover with gbest and pbest and

swap or inverse mutation is performed for randomly selected particles. While

generating the swarm for next iteration, elites are considered to preserve the

best particles in further generation. The computational complexity is also

analyzed. This HPSO is tested with CVRP dataset and compared with other

existing PSO works. The proposed HPSO algorithm is also used in solving a

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Multiple Depot Vehicle Routing Problem. The objective of MDVRP is to

minimize the total travel length of vehicles stationed at multiple depots.The

MDVRP may be split into multiple single VRP making it easy to solve. The

splitting of MDVRP into sub problems is taken care by general k-means

algorithm that assigns the customers around the depot. This reduces the

MDVRP to multiple single-depot VRP. The HPSO algorithm is then used for

solving each VRP. It is tested with MDVRP bench mark problem instances

and results are compared with other existing metaheuristic works.

Chapter 3 extends HPSO for stochastic VRP such as VRPSD. The

requirement/demand of customers is considered to be random in this

stochastic VRP. HPSO is incorporated with Dynamic Programming (DP) to

provide solution for such VRP. Apriori sequences of customers are generated

by HPSO and dynamic programming is incorporated within HPSO to

calculate the cost of generated apriori sequence. Backward recursion is used

in DP to solve objective function of stochastic VRP where only the

probabilities of customer’s demands are known. The HPSO-DP is tested with

VRPSD random data set and it is compared with other metaheuristic

techniques such as GA and PSO.

Chapter 4 deals with solving VRP by cluster-first and route-second

approach. It results in dividing the problem into a number of sub problems to

handle them easily. The modified k-means algorithm is proposed for

clustering and a HPSO is used for sequencing the customers within the

clusters such that their cost of servicing is minimized. The modified k-means

algorithm is improved by adding a priority measure. This also selects the

initial centroids based on the distance of customer from the depot. The

number of clusters formed is equal to the number of vehicles available. This

algorithm is tested with CVRP data set. It is also applied for solving MDVRP

and tested with benchmark problem instances of MDVRP. The results are also

compared with other existing metaheuristic methods and HPSO.

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Chapter 5 provides the Nested PSO (NPSO) for solving CVRP.

NPSO consists of Master PSO and Slave PSO. Master PSO deals with

assigning the customers to a vehicle. While assigning the customers, it deals

with the capacity constraint of a vehicle. This master PSO uses redundant

representation as its particle. Then the slave PSO deals with forming the

sequence of customers serviced by the respective vehicle. It minimizes the

routing cost of a vehicle. The proposed NPSO is tested with CVRP data set

and is also compared with other proposed and existing State-of-the-Art

methods. The NPSO is also tested with MDVRP data set and compared with

other HPSO and MK-HPS methods.

Chapter 6 summarizes the work, draws the conclusions and

provides suggestion for future work.