CHAPTER 4 MULTI-DEPOT VEHICLE ROUTING...

189

CHAPTER 4

MULTI-DEPOT VEHICLE ROUTING PROBLEM

4.1 INTRODUCTION

The Vehicle Routing problem (VRP) is one of the most challenging

combinatorial optimization tasks in real time logistics applications. The 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 capacity and time constraints. In a real-world environment,

drivers choose the shortest path to reach a destination, due to this, the distance

travelled and the cost can be minimized. Collection of household garbage,

gasoline delivery trucks, goods distribution, and snow plough, street cleaning,

school bus routing, dial-a-ride systems, transportation for handicapped

persons, routing of salespeople and mail delivery are the most common

applications of the VRP. The VRP plays a vital role in distribution, logistics

and supply chain management. Huge research efforts have been devoted to

studying the VRP since 1959 where Dantzig and Ramser have described the

problem as a generalized problem of Travelling Salesman Problem (TSP).

The VRP is responsible in designing an optimal set of routes for a

fleet of vehicles in order to serve a given set of customers. The interest in

VRP is motivated by its practical relevance as well as by its considerable

difficulty. During the past few decades, an increasing number of optimization

techniques based on operations research are proposed, for the effective

management of the provision of goods and services in distribution systems.

190

The VRPs have several variants based on the operational mechanism and

mathematical modeling. Based on the constraints and characteristics of VRP,

they are classified into capacitated VRP, multi-depot VRP, periodic VRP,

split delivery VRP, stochastic VRP, VRP with backhauls, VRP with pick-up

and delivery and VRP with time window.

The Multi-Depot Vehicle Routing Problem (MDVRP), an

extension of classical VRP, is a NP-hard problem for simultaneously

determining the routes for several vehicles from multiple depots to a set of

customers and then returning to the same depot. The objective of the problem

is to find routes for vehicles such that all the customers are served at a

minimal cost in terms of total travel distance, without violating the capacity

and travel time constraints of the vehicles. During the past three decades,

classical VRPs are paid more attention in the research perspective,

comparatively, the number of research projects on MDVRP is fewer. Salhi

and Sari (1997) addressed a multi-level composite heuristic with two

reduction tests. The initial feasible solutions are constructed in the first level,

while the intra-depot and the inter-depot routes are improved in the second

and third levels. Wu et al (2002) reported a simulated annealing (SA)

heuristic for solving the multi-depot location routing problem (MDLRP). To

solve the problem on a larger scale, the original problem is divided into two

sub-problems, namely, the location-allocation problem, and the general

vehicle routing problem, respectively. Each sub-problem is then solved in a

sequential and iterative manner by the simulated annealing algorithm

embedded in the general framework for the problem-solving procedure.

Giosa et al (2002) developed a “cluster first, route second” strategy for the

MDVRP with Time Windows (MDVRPTW), an extension of the MDVRP.

Considering the operational nature of the MDVRPTW, the cluster first route

second technique, focuses more on minimizing the computational time.

191

Haghani and Jung (2005) presented a formulation for solving the dynamic

vehicle routing problem with time-dependent travel times using Genetic

Algorithm. The performance of the genetic algorithm is evaluated by

comparing its results with exact solutions and lower bounds for randomly

generated test problems. For small size problems with up to 10 demands, the

genetic algorithm provided same results as exact solutions. For the problems

with 30 demand nodes, the genetic algorithm results are found to have less

than 8% gap with lower bounds.

Lee et al (2006) handled the MDVRP by formulating the problem

as deterministic dynamic programming (DP) with finite-state and action

spaces, and then using a shortest path heuristic search algorithm. Creviera

et al (2007) proposed a heuristic combining tabu search method, and integer

programming for multi-depot vehicle routing problem in which vehicles may

be replenished at intermediate depots along their route. Jeon et al (2007)

suggested a hybrid genetic algorithm (HGA) for MDVRP, which considers

the improvement of generation for an initial solution, three different heuristic

processes, and a float mutation rate for escaping from the local solution in

order to find the best solution. Inorder to solve the MDVRP efficiently, two

hybrid genetic algorithms (HGA1 and HGA2) are developed by Ho et al

(2008). In their approach, the initial solutions are generated randomly in

HGA1. The Clarke and Wright saving method and the nearest neighbor

heuristic are incorporated into HGA2 for the initialization procedure. Results

proved that HGA2 is superior to HGA1 in terms of the total delivery time.

Chen and Xu (2008) developed a hybrid genetic algorithm (GA)

with simulated annealing for solving the MDVRP. Since the MDVRP

integrates hard optimization problems, three improvement heuristic

techniques are introduced by Mirabi et al (2010). Each hybrid heuristic

192

combines elements from both constructive heuristic search and improvement

techniques. The improvement techniques used in this method are

deterministic, stochastic and simulated annealing (SA) methods. These

techniques outperformed Giosa’s (2002) cluster first, route second method in

terms of minimum delivery time. Lau et al (2010) considered the cost due to

the total traveling distance, and the cost due to the total traveling time for

solving the MDVRP. They employed a stochastic search technique called

fuzzy logic guided genetic algorithms (FLGA) to solve the problem.

MDVRP solutions are provided by Renaud et al (1996) in three

phases: fast improvement, intensification, and diversification for 23

benchmark instances using the Tabu Search algorithm. They considered

MDVRPs with capacity and route length restrictions. GA for MDVRP

proposed by Ombuki-Berman and Hanshar (2009) employed an indirect

encoding and an adaptive inter-depot mutation exchange strategy for the

MDVRP with capacity and route-length restrictions. Thangiah and Salhi

(2001) presented a genetic clustering technique for MDVRP, where a

generalised clustering mechanism is applied to genetic algorithms as a

postoptimizer. Su (1999) presented a real time dynamic vehicle control and

scheduling system for multi-depot physical distribution. To perform the

system objectives effectively, their system had five major modules such as the

global information collection system, depot controller, route planner, vehicle

scheduler, vehicle route and time table feedback system. This method is more

suitable for scheduling based on real time status of the system.

Wenjing et al (2010) introduced a modified PSO algorithm with

mutation operator and improved inertia weight for solving MDVRP. The

simulation results showed that this modified method could not only avoid

premature automatically according to the convergence level but also get a

193

better optimal solution than the standard basic PSO. Wang et al (2008) used

particle position matrix based on goods for particle swarm optimization to

solve multi-depots single VRP model, and every matrix column corresponded

to one goods. Matrix elements are random number between 0 and 1, and

matrix elements sort rules are established to get single vehicle route thus

satisfying the objective function.

The Artificial Bee Colony algorithm (ABC) is a new

population-based meta-heuristic approach proposed by Karaboga (2005). This

approach is inspired by the intelligent foraging behavior of honeybee swarm.

Karaboga and Basturk (2007) demonstrated that ABC outperforms genetic

algorithms and particle swarm optimization in multivariable function

optimization. Brajevic (2011) presented the ABC algorithm for capacitated

vehicle routing problem. In general, ABC algorithms are applied for

continuous optimization problems. Since VRPs are combinatorial

optimization problems certain modifications are implemented by Ivona.

Though ABC is a fairly new approach introduced a few years ago, it has not

yet been applied to solve the MDVRP. Hence, it is worthwhile to evaluate the

performance of ABC algorithm for solving MDVRP.

In this thesis, the performance of the heuristics such as Genetic

Algorithms (GA) (Surekha and Sumathi Aug 2011), Modified Particle Swarm

Optimization (MPSO) (Surekha and Sumathi Oct 2011), Artificial Bee

Colony (ABC) (Surekha and Sumathi Aug 2012) optimization, Hybrid MPSO

with GA, the Genetic Swarm Optimization (GSO) and Improved Genetic

Swarm Optimization (IGSO) on MDVRP benchmark instances are addressed.

The solution to MDVRP is obtained in four stages namely grouping, routing,

scheduling and optimization. Customers are clustered based on distance

between customers and depots in the grouping phase. In routing, customers of

the same depot are assigned to several routes by Clarke and Wright saving

194

method and each route is sequenced in the scheduling phase. Better routing

and scheduling can result in shorter delivery distance, shorter time spent in

serving all customers, higher level of efficiency and lower delivery cost. The

scheduled routes are optimized using the optimization algorithms in order to

obtain a global optimum solution with optimal distance, improved robustness,

reduced computational time and better algorithmic efficiency.

A set of five different Cordeau’s benchmark instances (p01, p02,

p03, p04, p06) from the online resource of University of Malaga, Spain are

experimented using MATLAB R2008b software. Several investigations are

conducted on the benchmark instances using the proposed heuristics. The

results are evaluated in terms of depot’s route length, optimal route, optimal

distance, computational time, and number of vehicles.

The chapter is organized as follows: The basic concepts of MDVRP

along with the mathematical modeling and architecture of the proposed

approach are discussed in Section 4.2. Section 4.3 enlightens the

implementation strategies of the proposed bio-inspired intelligent heuristics

such as GA, MPSO, ABC, GSO and IGSO for solving the MDVRP. The

simulation results and experimental analysis of the MDVRP instances using

heuristic algorithms are described in Section 4.4. Section 4.5 compares and

discusses the outcome of this chapter while section 4.6 draws the conclusion

and future work of solving multi-depot vehicle routing problems.

4.2 FUNDAMENTAL CONCEPTS OF MDVRP

In MDVRP, the number and locations of the depots are

predetermined. Each depot is large enough to store all the products ordered by

the customers. Each vehicle starts and finishes at the same depot. The location

and demand of each customer is also known in advance and each customer is

195

Route 2

Route 1

Route 2

Route 1

A B

1

5 9

4

8

7

3 10

6

2

visited by a vehicle exactly once. Figure 4.1 shows an example of the

MDVRP with 2 depots and 10 customers. Since there are additional depots

for storing the products, the decision makers have to determine depots

through which the customers are served (Ho et al 2008). The decision making

stages are classified into grouping, routing, scheduling and optimization as

shown in Figure 4.2.

Figure 4.1 Example of an MDVRP with 2 depots and 10 customers

In grouping, customers are clustered based on distance between

customers and depots. In the example, customers 1,5,9,4,8 are assigned to

depot A while customers 7,10,3,6,2 are assigned to depot B. In depot A,

customers 1,5,9 are in the first route, while customers 4 and 8 are served in

the second route. The customers of the same depot are assigned to several

routes in the routing phase by Clarke and Wright saving method and each

route is sequenced in the scheduling phase. The aim of routing is to minimize

the number of routes without violating the capacity constraints. Since there

are two depots the minimum number of routes can be limited to two. More

number of routes increase the number of vehicles required thus reducing the

distance. In general, the objective of the MDVRP is to minimize the total

delivery distance or time spent in serving all customers thus utilizing efficient

amount of vehicles.

196

Figure 4.2 Decision making in MDVRP

4.2.1 Mathematical Formulation of MDVRP

The MDVRP is formulated with the objective of forming a

sequence of customers on each vehicle route. The time required to travel

between customers along with the depot and demands are known in advance.

It is assumed that all vehicles have the same capacity, and each vehicle starts

its travel from a depot, upon completion of service to customers, it has to

return to the depot. The notations used and the mathematical model are as

follows:

Sets

I – Set of all depots

J – Set of all customers

K – Set of all vehicles

Indices

i – depot index

j – customer index

k – route index

Grouping / Clustering

Routing

Scheduling and Optimization

197

Parameters

N – Number of vehicles

Cij – Distance between point i and j, JIji,

Vi – Maximum throughput at depot i

di – Demand of customer j

Qk – Capacity of vehicle (route) k

Decision variables

otherwisekrouteonjpreceedsyimmediateliif

xijk ,0,1

otherwiseidepottoallottedisjcustomerif

zij ,0,1

Ulk – auxiliary variable for sub-tour elimination constraints in route k

Mathematical model

The objective function is to minimize the total distance of all

vehicles given by Equation (4.1),

JIi JIj Kkijkij xCmin (4.1)

Each customer has to be assigned a single route according to

Equation (4.2),

Kk JIiijk Jjx ,1 (4.2)

The capacity constraint for a set of vehicles is given by

Equation (4.3),

198

Jj JIikijkj KkQxd , (4.3)

Equation (4.4) gives the new sub-tour elimination constraint set as,

KkJjlNNxUU ijkjklk ,,,1 (4.4)

The flow conservation constraints are expressed as in Equation (4.5),

JIiKkxx

JIj JIjjikijk ,,0 (4.5)

Each route can be served at most once according to Equation (4.6),

Ii Jjijk Kkx ,1 (4.6)

The capacity constraints for the depots are given in Equation (4.7) as,

IiVzd i

Jjiji , (4.7)

Constraints in Equation (4.8) specify that a customer can be

assigned to a depot only if there is a route from that depot going through that

customer,

KkJjIixxz

JIuujkiukij ,,,1)( (4.8)

The binary requirements on the decision variables are given by

Equations (4.9) and (4.10)

KkJjIixijk ,,},1,0{ (4.9)

JjIiz ij ,},1,0{ (4.10)

The positive values of the auxiliary variable is defined in

Equation (4.11) as,

199

KkJlUlk ,,0 (4.11)

The objective function according to Equation (4.1) minimizes the

total delivery distance. Constraint in Equation (4.2) ensures that each

customer is allotted with only one route while constraint in Equation (4.3)

guarantees the capacity limit of vehicles. Similarly, the sub-tour avoidance is

imposed by Equation (4.4) and the flow limits by Equation (4.5). The route to

be served and the limit on the depots are given by Equation (4.6) and

Equation (4.7) respectively. Thus the MDVRP aims at minimizing the total

delivery distance by satisfying the mentioned constraints.

4.2.2 Architecture of the Proposed Intelligent MDVRP System

Multi-depot vehicle routing problem is more challenging and

sophisticated than the single-depot VRPs. In addition, MDVRP is NP-hard,

which means that an efficient algorithm for solving the problem to optimality

is unavailable. Therefore, MDVRP is difficult to solve with exact procedures

such as branch and bound, branch and cut algorithms. To deal with the

problem efficiently and effectively, heuristic algorithms like GA, MPSO,

ABC, GSO and IGSO are applied in this work. Figure 4.3 illustrates the block

diagram of the proposed multi-depot vehicle routing problem.

The well known Cordeaux’s instances (p01, p02, p03, p04, p06) are

chosen as benchmark problems. The customers are clustered based on the

distance between customers and depots. The customers of the same depot are

assigned to several routes using the Clark and Wright saving method (Clarke

and Wright 1964). The routes obtained from Clarke and Wright saving

method is scheduled and optimized using heuristic algorithms such as genetic

algorithm, modified particle swarm optimization, artificial bee colony

algorithm, genetic swarm optimization and improved genetic swarm

optimization. The results of all optimization techniques such as GA, MPSO,

200

ABC, GSO and IGSO are evaluated with respect to number of customers

serviced, number of vehicles required, optimal route, optimal distance, and

computational time. A comparative analysis of the proposed heuristic

techniques is performed based on optimal distance, robustness, computational

competency, and algorithmic efficiency to identify the suitable optimization

algorithm for solving MDVRP.

Figure 4.3 Proposed block diagram for MDVRP

4.2.3 Grouping Assignment

The group assignment also referred as cyclic assignment requires

each depot to attract one customer every time until all customers are assigned.

The cyclic assignment assigns one customer at a time in a cycle based on the

location of the depot heads. First, the head of each depot is set as the depot

itself. Then, for each depot, the closest customer to the depot head is assigned

to it and the depot head is updated by the closest customer, if the vehicle of

the depot has enough serving capacity. The above procedure is repeated until

all customers are assigned.

The first step is to assign customers to each of n links, known as the

grouping problem. Because the objective here is to minimize the total delivery

Benchmark Input

Cordeaux’s instances

p01, p02, p03, p04, p06

Grouping

Grouped with respect to

distance between customers and depots

Routing

Clarke and Wright Saving method

Scheduling and Optimization

GA, MPSO, ABC, GSO and IGSO

Experimental results and comparative analysis

Optimal distance, computational

competency, algorithmic efficiency

201

time spent in distribution, customers are assigned to the nearest depot.

Grouping assignment procedure is given below:

Step 1 : Calculate the distance among each customer.

Step 2 : Calculate the distance between every customer and every depot.

Step 3 : Assign the customers to the nearest depot.

Step 4 : The customers are divided into M groups, where M is equal to the

number of depots.

For example, in a distribution network, a supplier owns two depots

(dA and dB) to deliver the products to a set of customers. Each customer, ci

should be assigned to a single depot exactly.

Equation (4.12) shows that, if the customer ci is located near to

depot A , then the customer ci is assigned to depot A .

If ),(),( dBciDdAciD , then assign ci to dA ; (4.12)

Equation (4.13) indicates that, if the customer ci is located near to

depot B , then the customer ci is assigned to depot B .

If ),(),( dBciDdAciD , then assign ci to dB ; (4.13)

Equation (4.14) ensures that, if the distance between customer ci

and depot A is equal to the distance between customer ci and depot B , then

customer ci is assigned to either depot Aor depot B .

In case ),(),( dBciDdAciD , select a depot arbitrarily (4.14)

Equation (4.15) represents the distance between customer i and

depot k .

202

22 )()(),( dkcidkci yyxxdkciD (4.15)

In the above Equation (4.15), xci and yci denote x and y coordinates of the customer ci , xdk and ydk indicates the x and y coordinates of depot k . The customers are grouped and served by either depot A or depot B .

4.2.4 Routing Algorithm

In recent years, since the evolution of supply chain management, the system design problem considering the location of distribution facilities has become more significant. In routing phase, customers in each group are divided into different routes. The aim of routing is to minimize the number of routes, or vehicles, used while not violating the vehicle capacity constraint. A key element of many distribution systems is the routing and scheduling of vehicles through a set of customers requiring service. Better routing decision can result in higher level of customer satisfaction. A wrong grouping assignment solution will result in routes of higher total cost (distance) than with a better grouping assignment. So the routing phase is strongly dependent on the grouping assignment.

In 1964 Clarke and Wright published an algorithm for solving the classical vehicle routing problem. The savings algorithm is a heuristic algorithm, and therefore it does not provide an optimal solution to the problem with certainty. The method does, however, often yield a relatively good solution, that deviates slightly from the best known solution. In the first step of the savings algorithm the savings for all pairs of customers are

calculated, and all pairs of customer points are sorted in descending order of the savings. Subsequently, from the top of the sorted list of point pairs one

pair of points is considered at a time. When a pair of points ji is

considered, the two routes that visit i and j are combined (such that j is

visited immediately after i on the resulting route), if this can be done without deleting a previously established direct connection between two customer

203

points, and if the total demand on the resulting route does not exceed the vehicle capacity. There are two versions of the savings algorithm, namely sequential and parallel. In the sequential version exactly one route is built at a time, while in the parallel version more than one route may be built at a time.

Figure 4.4 Illustration of the savings concept (a) Separate route (b) Same route

Initially in Figure 4.4(a), customers i and j are visited on separate

routes. An alternative to this is to visit the two customers on the same route, for example in the sequence ji as illustrated in Figure 4.4(b). Since the

transportation costs are given, the savings that result from driving the route in Figure 4.4(b) instead of the two routes in Figure 4.4(a) can be calculated.

Denoting the transportation cost between two given points i and j by cij, the

total transportation cost Da in Figure 4.4(a) is:

0000 jjiia ccccD (4.16)

Equivalently, the transportation cost Db in Figure 4.4(b) is:

00 jijib cccD (4.17)

By combining the two routes the savings Sij can be obtained as,

ijojibaij cccDDS 0 (4.18)

204

The customers with larger saving value are grouped in the same route without violating the vehicle capacity constraint.

4.3 BIO-INSPIRED TECHNIQUES FOR MDVRP

Due to the complexity of the problem, solving the MDVRP to optimality is extremely time-consuming. In order to tackle the problem efficiently, researchers have preferred heuristic methods over exact methods like branch and bound, branch and cut algorithms (Ho et al 2008). In this dissertation, the application of GA, MPSO, ABC, GSO and IGSO to solve MDVRPs are proposed. The step by step procedure required for implementing these intelligent algorithms is delineated in this section.

4.3.1 Solution Representation and Fitness Function

The solution representation and fitness function for all the proposed bio-inspired techniques is analogous and the feasible solution is generated based on three basic steps: Grouping, Routing and Scheduling. The individuals for the solution of the MDVRP are encoded using path representation, in which the customers are listed in the order in which they are visited. Consider an MDVRP instance with 6 customers designated 1-6. If the path representation for this instance is (0 2 4 1 0 3 6 5 0), then two routes are required by the vehicles to serve all the customers. The first route starts from the depot at 0 and travels to customers 2, 4 and 1, upon serving the vehicle

returns to the depot. Similarly, the second route starts from depot at 0, services customers 3, 6, 5 and returns to the depot. While applying heuristics based optimization techniques each individual in the initial population consists of n links for n depots in the MDVRP.

Grouping – In this stage, the customers are assigned to each of the n links. The objective of the MDVRP is to minimize the total delivery time and hence customers are assigned to the nearest depots. In the example, there are two depots A and B, each customer ci has to be assigned to a single depot exactly.

205

Depot B Depot A

Customers 0A 0B 1 2 3 4 5 6 7 8 9 10

Grouping

1 4 5 8 9 2 3 6 7 10

Routing

Depot B Depot A

0A 1 5 9 0A 0B 2 6 0B 7

Scheduling

0A 8 4 3 10 0B

Depot B Depot A

0A 9 5 1 0A 0B 2 6 0B 3 0A 8 4 7 0B 10

This process of grouping is done based on the distance computation according to the following rule:

If D(ci,A) < D(ci,B), then customer ci is assigned to depot A

If D(ci,A) > D(ci,B), then customer ci is assigned to depot B

If D(ci,A) = D(ci,B), then customer ci is assigned to a depot chosen arbitrarily between A and B

In the above cases, 22 )()(),( kckci yyxxkcDii

, represents the

distance between customer ci and depot k.

Figure 4.5 Solution representation and initial population

Routing – The customers in the same link are assigned to several routes using

Clarke and Wright Saving method. The routing is based on the distance

travelled by the vehicles for serving the customers. A saving matrix ),( ji ccS is

constructed for every two customers i and j in the same link. Further, the

206

customers with large saving value are grouped in the same route without

violating the capacity constraints. The saving matrix is constructed according

to ),(),(),(),( jijiji ccDckDckDccS .

Scheduling – Starting from the first customer, the delivery sequence is

chosen such that the next customer is as close as to the previous customer.

This process is repeated until all the unselected customers are sequenced. The

scheduling is performed by the respective optimization technique based on the

fitness function. The fitness function for the MDVRP is described in this

section. At the end of the scheduling phase, a feasible solution of the MDVRP

example problem (Figure 4.1) is constructed as shown in Figure 4.5.

Fitness Evaluation - For the MDVRP, the objective function is to minimize

the maximum delivery time spent among n depots. The delivery operations

start at the same time in every depot, while it takes different time to complete

serving the assigned customers. Some vehicles belonging to a depot may

complete the delivery faster while other vehicles may complete their task in a

longer duration. Let Dt be the total delivery time required by a depot k and let

)min( tD represent the minimum delivery time spent by all n depots, then

k cm

k

m

ict icicdcmcdD

1 1)](),1([)]0(),([ (4.19)

where V

yyxxbad abab

22 )()(),( is the travel time of a vehicle from

customer a to b. V is the speed of the vehicle, c(i) is the location of the ith

customer, c(0) is the initial position of the depot, mc is the number of

customers in route r, mk is the number of routes in depot k. Thus the fitness

function is defined as )min( tDF . This function acts as the objective

function to solve MDVRPs using the proposed intelligent techniques.

207

4.3.2 Implementation of MDVRP using GA

Genetic Algorithms (GA) is based on a parallel search mechanism

(Goldberg 1989), which makes it more efficient than other classical

optimization techniques such as branch and bound, tabu search method and

simulated annealing. The basic idea of GA is to maintain a population of

candidate solutions that evolves under selective pressure. GA imitates the

mechanism of natural selection and the survival of the fittest as witnessed in

natural evolution. The GA can avoid getting trapped in a local optimum by

tuning the genetic operators, crossover and mutation. Due to its high potential

for global optimization, GA has received great attention in solving multi-

depot vehicle routing problems. The general scheme of the genetic algorithm

for MDVRP is explained in this section.

Initial population

The initial population is formed based on the routes, depots and

customers according to the description available in section 4.3.1.

Selection

During each generation, the parents are selected for mating and

reproduction. In this MDVRP application, tournament selection is applied

(Renaud et al 1996) to generate new individuals in the population. This

selection strategy is based on fitness evaluation given in section 4.3.1. The

selection procedure is as follows:

Step 1 : Select a set of g individuals from the population in a random

manner to form the tournament set

Step 2 : Choose a random number rn in the range [0,1]

208

Step 3 : If rn < threshold select the fittest individual from the tournament

set for reproduction

Else choose any two chromosomes in random from the

tournament set for reproduction

Step 4 : Apply elitism to guarantee that the best individuals are selected

Crossover

A problem specific crossover technique, the Best Cost Route

Crossover (BCRC) developed by (Ombuki-Berman and Hanshar 2009), for

vehicle routing problem with time windows (VRPTW) is applied in this work

for MDVRP with slight improvements. The steps involved in BCRC are

shown below:

Step 1: Choose the parents from tournament selection

Step 2: Select a route from each parent in a random manner

Step 3: Remove all customers belonging to route 1 from parent 1

Step 4: For every customer belonging to route 1

Compute the cost of insertion of route 1 into each location of

parent 2 and store the costs in an ordered list.

For each insertion location, check whether the insertion is

feasible or not

Generate a random number ]1,0[nr

Choose the first feasible insertion location if rn < threshold

Else if rn > threshold choose the first entry in the ordered list,

despite of the feasibility

Step 5: Repeat Step 4 for customer belonging to route 2

209

Mutation

The flip bit mutation operator is used in finding an MDVRP

solution using GA. A substring is selected from the parent in a random

manner and flips to form an offspring. The route reversal mutation works on

only one chromosome at a time.

The design steps used for the implementation of MDVRP based on

GA are:

Step 1 : Generate randomly the initial population of GA based on routes,

depots and customers.

Step 2 : Evaluate the fitness function for each chromosome in the

population according to Equation (4.19).

Step 3 : Calculate the total fitness value, fi which is represented as the

summation of individual fitness of each chromosome

Step 4 : Repeat the following steps until new chromosomes have been

created

Select a pair chromosome from current population, based on

tournament selection.

With the crossover probability of 0.6, exchange the pair using

BCRC to form two new chromosomes.

Mutate the two new chromosomes at each row with the

mutation probability (0.02) and place the resulting

chromosomes in the new population.

Step 5 : Replace the current population with new population.

Step 6 : If the fixed number of generations is reached, stop and return the

best solution in current population else go to step 2.

210

The chromosomes in the population evolve through successive

iterations, called generations, thus evaluating the measure of fitness based on

Equation (4.19). The fitter the chromosomes, the higher the probabilities of

being selected to perform the genetic operations, such as crossover and

mutation. Selected individuals are chosen for reproduction (or crossover) at

each generation, with an appropriate mutation factor to randomly modify the

genes of an individual, in order to develop the new population. The current

population is replaced with the new population, until fixed number of

generations is reached.

4.3.3 Solving MDVRP Using MPSO

Particle Swarm Optimization (PSO) is a population based search

technique, consisting of potential solutions known as particles, duplicating a

flock of birds. The particles are initialized randomly based on the

optimization problem and they freely fly across the multidimensional search

space. Each particle updates its own velocity and position thus driving the

swarm towards the best fit regions. Eventually all the particles gather around

the point with the highest objective function. When compared with GAs, PSO

does not have any genetic operators like crossover and mutation. In GA, the

chromosomes share the information with each other and the whole population

moves towards the optimal solution, while in PSO, only the best particle

shares the information with others. Due to this one way information sharing

mechanism, the evolution of PSO is always towards the best solution.

Moreover, all particles tend to converge to the best solution quickly without

getting trapped in the local minima.

Solution representation and fitness evaluation

The particles for the solution of the MDVRP are encoded using

path representation and the fitness function is evaluated based on distance

according to the explanation given in section 4.3.1.

211

Updating Particles

In the evolutionary process, MPSO is used to generate new

offspring based on new position of the particle, thus updating the velocity

according to

itv

it

gt

it

it

itt

it

it xprcxprcvwxx

1

)()( 22111

(4.20)

The best position visited by the kth particle since the first time step

is known as the local best position denoted as pbest itp , while gbest g

tp

represents the best position that the kth particle and its neighbors have visited

from the beginning of the algorithm. In this work, since the distances are real

coded, the velocities and position update rules are slightly modified according to,

)}()(int{ 22111it

gt

it

it

itt

it xprcxprcvwv (4.21)

}{ 11it

it

it vxabsx (4.22)

where abs {} denotes the absolute value function, int {} is the integral

function, r1 and r2 are two independent random numbers uniformly distributed

in the range {0,1}, c1 and c2 are the weighting factors also known as cognitive

and social parameters respectively.

During each iteration, the pbest and gbest are updated based on the

objective (fitness) value. If the current value of pbest is better, pbest and its

objective value are updated with the current position and objective value.

Similarly, if the objective value is better than the objective value of gbest, then

gbest and its objective value are updated with the position and objective value

of the current best particle.

212

Algorithm

The step by step procedure for implementing the MDVRP using

MPSO is shown below:

Step 1 : Initialize the parameters of MPSO to suitable values. Set the

value of maximum number of iterations.

Step 2 : Initialize the population of particles according to the delivery

sequence of customers.

Step 3 : Evaluate the fitness function for all particles and choose the best

local and global positions pbest and gbest of the particles

Step 4 : Update the particles according to Equations (4.21) and (4.22)

Step 5 : Compute the fitness function of all the updated particles and

determine the best fit particle. Compare the fitness and update the

particles accordingly.

Step 6 : Test for stopping condition. If maximum number of iterations is

reached, then stop, else continue with Step 3.

4.3.4 Artificial Bee Colony Based MDVRP

The Artificial Bee Colony (ABC) optimization algorithm is

becoming more popular recently, due to the foraging behavior of honeybees.

ABC is a population based search technique, in which the individuals known

as the food positions are modified by the artificial bees during course of time.

The objective of the bees in turn is to discover the food sources with high

nectar concentration. The colony of artificial bees is grouped into employed

bees, onlooker bees and scout bees. During initialization phase, the objective

of the problem is defined along with the ABC algorithmic control parameters.

An employed bee is assigned for every food source available in the problem.

213

In employed bee phase, the employed bee stays on a food source and stores

the neighborhood of the source in its memory. During the onlooker phase,

onlooker bees watch the waggle dance of employed bees within the hive to

choose a food source. The employed bee whose food source has been

abandoned becomes the scout bee. Scout bees search for food sources

randomly during the scout phase. Thus the local search is carried out by the

employed bees and the onlooker bees while the global search is performed by

the onlooker and the scout bees, thus maintaining a balance between the

exploration and exploitation process. In this section, the step by step

procedure to implement ABC technique for MDVRP is discussed.

Generation of initial solutions

The initial solutions to the ABC problem are formed using the

grouping, and routing stages. In this mode, ABC algorithm generates

randomly distributed initial solutions, due to which the constraints satisfaction

may not be guaranteed. Therefore during initialization every solution has to

be checked according to each route as follows:

If the value of more than one position in the corresponding

positions of all routes in the solution is 1, one position is

selected in random and its value is set to 1, while the value of

others is set to 0.

If the values of the corresponding positions of all routes in the

solution are all 0, one position is selected in random and its

value is set to 1, and the others remain unchanged.

This kind of an initialization ensures for every initial solution that

each customer is served exactly once by exactly one vehicle. If the vehicle

starts and ends at different depots then the solution is infeasible.

214

Constraints handling

In order to handle the constraints, the ABC algorithm employs

Deb’s rules (Deb 2000), which are used in constrained optimization problems.

The method uses a tournament selection operator, where two solutions are

compared at a time by applying the following criteria:

Any feasible solution satisfying all constraints is preferred to

any infeasible solution violating any of the constraints.

Among two feasible solutions, the one having better fitness

value is preferred

Among two infeasible solutions, the one having the smaller

constraint violation is preferred

Because initialization with feasible solutions is a very time

consuming process and in some cases it is impossible to produce a feasible

solution randomly, the ABC algorithm does not consider the initial population

to be feasible. Structure of the algorithm already directs the solutions to

feasible region in the running process due to the Deb’s rules.

Neighborhood operators

In order to produce new solutions for the employed and onlooker

bees two neighborhood operators are applied (Eiben and Smith 2003). The

rst neighborhood operator is called SwapMutation. It is introduced as a

neighborhood operator for solving the travelling salesman problem (TSP) and

is called “2-change”. The idea of the mutation operation is to randomly

mutate the tour and hence produce a new solution g that is not very far from

the original one f. In this dissertation, the mutation operator is designed to

215

conduct customer exchanges in a random fashion. The steps for the

SwapMutation operator are as follows:

Randomly select two routes from the solution f and randomly

select two customers from each selected route.

Exchange the customers in the different routes and generate

the new solution g.

The second neighborhood operator, also based on random changes,

is called InsertMutation. The steps for the InsertMutation operator are as

follows:

Randomly select the routes from the solution f and randomly

select one customer from one selected route.

Remove the selected customer from one route to the other and

generate the new solution g.

After the new solution is produced selection process is applied

based on Deb’s method. If the new solution g is accepted instead of the

solution f, SwapMutation operator is applied in the mutated routes in order to

improve the new solution. SwapMutation is applied in the following way:

Compare all possible pairwise exchanges of customer

locations in the route to find the exchange that produces the

shortest distance.

When the pair of customers whose exchange produces the

shortest distance is found, the route is rearranged. If such pair

of customers is not determined, the route stays unchanged.

216

In employed bee phase SwapMutation operator is applied two

times. Onlooker bee phase has one difference from employed bee phase: it

applies one SwapMutation and one InsertMutation operator instead of two

SwapMutation operators.

The pseudo–code of the proposed ABC algorithm is given below:

Construct initial employed bee colony solutions;

Evaluate fitness value for each solution;

iteration = 1;

While iteration

EMPLOYED BEE PHASE

For each employed bee apply two times:

SwapMutation operator

Selection process based on Deb’s method

If the new solution is accepted, improve the new solution and

evaluate them

ONLOOKER BEE PHASE

Calculate the probability values for the solutions

For the onlookers selected depending on probabilities, the

following is applied two times:

First time Swap Mutation operator, second time Insert

Mutation operator

217

Selection process based on Deb’s method

If the new solution is accepted, improve the new solution and

evaluate them

SCOUT BEE PHASE

Determine every infeasible solution for the scout and replace them

with new produced solutions

iteration = iteration + 1

Memorize the best solution achieved so far

End While

4.3.5 GSO and IGSO for MDVRP

Hybrid algorithms are used as effective methods for obtaining

optimal solution to large scale constraint based optimization problems. In this

work, solution to MDVRP is achieved using hybrid combination of GA and

MPSO, termed as GSO. In GSO, GA and MPSO are hybridized to replace the

worst particles found by MPSO by the best particles obtained by GA and vice

versa. The GSO is modified with an enhancement step using MPSO, denoted

as Improved GSO (IGSO) in order to improve the optimal distance and

computational time of MDVRP instances obtained by GSO. The basic ideas

behind the implementation of IGSO are:

The infeasible solutions produced during the algorithmic run

are adjusted by enhancing the elites in the hybrid combination

of GA and MPSO.

218

The chances of the algorithm getting trapped into the local

optimization at a very early stage can be avoided by the IGSO

algorithm.

The step by step implementation of the proposed hybrid algorithm

for solving MDVRP is described as follows:

Solution coding and fitness evaluation

The structure of the initial solutions for the MDVRP based on

IGSO is framed based on the grouping and routing techniques. The routes

obtained using Clarke and Wright saving method is further scheduled based

on fitness function using IGSO. The description of solution representation and

fitness evaluation are provided in section 4.3.1.

Enhancement

In each generation, half of the best individuals termed as elites are

selected based on the fitness evaluation. The elites are enhanced, instead of

reproducing next generation offspring directly. The enhancement of the elites

is performed by updating the velocity and position of particles in MPSO

according to Equation 4.23 and 4.24 respectively.

)}()(int{ 22111it

gt

it

it

itt

it xprcxprcvwv (4.23)

}{ 11it

it

it vxabsx (4.24)

Selection

While the elites are enhanced using MPSO, the GA operations are

performed in parallel. In order to select parents from the selected best half

219

elites, tournament selection scheme is applied. Two enhanced elites are

selected randomly, and their fitness values are compared to select the one

with better fitness as a parent and placed in the mating pool.

Offspring generation

Parents are selected randomly from the mating pool in groups of

two and two offspring are created by performing crossover on the parent

solutions. Single individual is selected based on the mutation probability and

a new genetic offspring is generated to maintain diversity in the population. In

this work, best cost route crossover (BCRC) and route reversal mutation are

applied to increase the diversity in the population such that better elites are

produced in the next generation. The offspring generated from GA and the

enhanced elites from MPSO form a new population and their fitness is

evaluated and compared in order to select the elites for the next generation.

Stopping condition

The algorithm terminates with the number of iterations and

computation of the best solution.

The flowchart of the proposed IGSO approach is shown in

Figure 4.6. The initial swarm consists of the scheduled sequence based on the

fitness function of the MDVRP. The enhanced elites using MPSO and

offspring generated using GA form a new population for further generations

and evaluated for fitness. The process is repeated for a fixed number of

iterations and the optimal results for the MDVRP instances are determined.

220

Figure 4.6 Flowchart of the proposed IGSO algorithm

4.4 SIMULATION RESULTS

In this section, the simulation results of stages involved in solving

the Multi-Depot Vehicle Routing Problem (MDVRP) such as grouping,

routing and optimization is analyzed. The main objective of the multi-depot

vehicle routing problem is to minimize the total delivery distance, number of

vehicles required and computational time. The performance of the MDVRP

using GA, MPSO, ABC, GSO and IGSO is evaluated using a set of five

Cordeau’s instances namely p01, p02, p03, p04 and p06 taken from

http://neo.lcc.uma.es/radi-aeb/WebVRP/ online resource of University of

N

Y

Initial population

Fitness Evaluation

Stop?

Tournament selection

Best Cost Route Crossover and Route

reversal mutation

Offspring generation

New population

Enhancement using MPSO

Enhanced elites

Select best half elites and discard

others

Best Individuals

221

Malaga, Spain. The simulation is done using MATLAB R2008b on Intel i3

CPU, 2.53GHz, 4GB RAM PC.

4.4.1 Experimental Benchmark Instances

The specifications of five Cordeau’s instances such as p01

(4 depots and 50 customers), p02 (4 depots and 50 customers), p03 (5 depots

and 75 customers), p04 (2 depots and 100 customers) and p06 (3 depots and

100 customers) are shown in Table 4.1. In this work, the depots and

customers are represented in the form of coordinates and each depot has a

limited number of vehicles. Each vehicle has a constant loading capacity and

each vehicle starts and finishes at the same depot. The demand of each

customer is also known in advance and each customer is visited by a vehicle

exactly once.

Table 4.1 Specifications of benchmark instances

Parameters / Instances p01 p02 p03 p04 p06 Total No. of Customers 50 50 75 100 100

Total No. of Depots 4 4 5 2 3 Total No. of Vehicles 32 20 35 24 30

No. of Vehicles in each Depot 8 5 7 12 10 Capacity of each Vehicle 80 100 140 100 100

Best known distance in km 576.87 473.53 641.19 1001.59 876.5

4.4.2 Grouping and Routing

In grouping, the customers are assigned to the adjacent depots so

that the distance travelled by the vehicle is shorter. The customers are

clustered based on the minimum distance between customers and depots. In

the grouping phase, the Euclidean distance between the customer and the

depot is computed, and based on the minimum distance, the Cordeau’s

instances p01, p02, p03, p04, p05 are grouped and the results are shown in

Table 4.2.

222

Table 4.2 Assignment of customers to depot for benchmark instances

Depot Customers Allotted No. of customers Problem instance: p01

A(20,20) 4-13-17-18-19-25-40-41-42-44-45 11 B(30,40) 5-6-7-10-12-14-15-23-24-27-33-37-38-39-43-46-47-48-49 19 C(50,30) 1-2-8-9-11-16-21-22-26-28-29-30-31-32-34-50 16 D(60,50) 3-20-35-36 4

Problem instance: p02 A(20,20) 4-13-17-18-19-25-40-41-42-44-45 11 B(30,40) 5-6-7-10-12-14-15-23-24-27-33-37-38-39-43-46-47-48-49 19 C(50,30) 1-2-8-9-11-16-21-22-26-28-29-30-31-32-34-50 16 D(60,50) 3-20-35-36 4

Problem instance: p03

A(40,40) 3-4-5-6-9-12-15-17-18-20-25-26-27-29-30-32-34-37-39-40-44-45-47-48-50-

51-55-60-67-68-70-75

32 B(50,22) 7- 8-13-35-46-52-57-58-72 9 C(55,55) 10-11-14-19-31-38-53-54-59-65-66 11 D(25,45) 2-16-21-24-28-33-36-49-62-63-69-71-73-74 14 E(20,20) 1-22-23-41-42-43-56-61-64 9


A(15,35)

2-5-6-7-8-13-14-15-16-17-18-21-22-23-36-37-38-40-41-42-43-44-45-46-47-48-52-53-56-57-58-59-60-61-72-73-74-75-82-83-84-85-86-87-89-91-92-93-

94-95-96-97-98-99-100

55

B(35,55)

1-3-4-9-10-11-12-19-20-24-25-26-27-28-29-30-31-32-33-34- 35-39-49-50-51-54-55-62-63-64-65-66-67-68-69-70-71-76-77-78-79-80-81-88-90

45


A(15,20) 5-14-15-16-17-37-38-41-42-43-44-45-46-57-59-61-84-85-86-87- 91-92-93-

95-96-97-98-99-100

29

B(50,20) 1-3-9-10-11-12-20-24-25-29-30-32-33-34-35-50-51-54-55-63- 64-65-66-68-

70-71-76-77-78-79-80-81-90

33

C(35,35) 2-4-6-7-8-13-18-19-21-22-23-26-27-28-31-36-39-40-47-48-49-52-53-56-58-

60-62-67-69-72-73-74-75-82-83-88-89-94

38

In p01, 11 customers (customer IDs: 4, 13, 17, 18, 19, 25, 40, 41,

42, 44, 45) are assigned to depot A whose location is at (20,20) indicating the

x and y coordinates, 19 customers (customer IDs: 5, 6, 7, 10, 12, 14, 15, 23, 24,

27, 33, 37, 38, 39, 43, 46, 47, 48, 49) to be served by depot B located at (x-

coordinate:30, y-coordinate:40), 16 customers (customer Ids: 1, 2, 8, 9, 11, 16,

21, 22, 26, 28, 29, 30, 31, 32, 34, 50) are assigned to depot C located at (x-

coordinate:50, y-coordinate:30) and remaining customers (customer IDs: 3, 20,

35, 36) are assigned to depot D located at (x-coordinate:60, y-coordinate:50).

Similarly in the other benchmark problems such as p02, p03, p04 and p06, the

customers are grouped to the associated depots as shown in Table 4.2.

In routing phase, the customers in each group are divided into

different routes. The aim of routing is to minimize the number of routes, or

vehicles used, without violating the vehicle capacity constraint. The Clark and

223

Wright saving method is applied to solve the routing process. There are two

versions of the savings algorithm, a sequential and a parallel version. In the

sequential version exactly one route is built at a time (excluding routes with

only one customer), while in the parallel version more than one route may be

built at a time. In this work, the parallel version is used to compute the routes

at a faster rate. It is worth noting that the number of routes may be reduced

during the process of the parallel version of the savings algorithm. For

example, the two routes 0-1-2-0 and 0-3-4-0 will be combined into one route

if the connection from depot A to B is established; in that case the resulting

route becomes 0-1-2-3-4-0. Fully loaded percentage (FLP) is defined as the

number of products loaded in each vehicle in every depot and is calculated

according to Equation (4.26),

100

QLQQFL

(4.26)

where FL indicates the fully loaded percentage of each vehicle, Q stands for

capacity of each vehicle, LQ represents the loading quantity of each vehicle

during the distribution process.

Table 4.3 Route allocation for customers in instance p01

Depot Route FLP (%) D (km) NV

A(40,40)

A-4-13-17-A 85.85 28.825

5 A-18-19-A 78.69 21.21 A-25-40-A 86.82 20.65

A-41-42-44-A 92.5 34.28 A-45-A 89.65 16.25

B(50,22)

B-5-6-7-B 75.64 43.2

6

B-10-12-14-B 74.78 47.28 B-15-23-24-27-B 93.75 56.225

B-33-37-38-B 83.5 28.425 B-39-43-46-47-B 93.5 53.82

B-48-49-B 64.85 32.7

C(55,55)

C-1-2-C 78.51 28.46

5 C-8 -9-11-C 86.84 42.3

C-16-21-22-26- 28-29-C 92.4 73.3 C-30-31-32-C 86.5 27.405

C-34-50-C 72.65 19.325

D(25,45) D-3-20-D 76.56 16.42

2 D-35-36-D 85.6 14.88

224

In the routing phase, the route of vehicles, fully loaded percentage (FLP), distance (D) and the number of vehicles (NV) served by a depot are computed and the results are tabulated. The route allocation of customers for

instance p01 using the Clark and Wright saving method is shown in Table 4.3. The depots along with their coordinates are specified, and the set of routes followed by the customers with the distance is tabulated. It can be observed that for depot B located at (50, 22), there are 6 set of routes generated for 6 different vehicles. The total distance for the assigned vehicles to start from the depot B, serve all the customers and return to the depot B is 261.65 km.

Table 4.4 shows the results obtained for the benchmark instance p02. From the experiments carried out it is seen that 4 vehicles are used by depot A, 5 by depot B, 4 by depot C and 1 by depot D, respectively. The distances are computed using Euclidean distance and do not guarantee the optimal value.



A(40,40)

A-4-13-17-A 85.85 28.825

4 A-18-19-A 78.69 21.21 A-25-40-41-A 86.82 20.65 A-42-44-45A 92.5 34.28

B(50,22)

B-5-6-7-10-B 75.64 43.2

5 B-12-14-15-B 74.78 47.28

B-23-24-27-33-B 93.75 56.225 B-37-38-39-43-46-B 83.5 28.425

B-47-48-49-B 93.5 53.82

C(55,55)

C-1-2-8-C 78.51 28.46

4 C-9-11-16-21-22-26-C 86.84 42.3 C-28-29-30-31-32C 92.4 73.3

C-34-50-C 86.5 27.405 D(25,45) D-3-20-35-36-D 76.56 16.42 1

The routing algorithm is run for the p03 benchmark instance consisting of 5 depots, 75 customers and the results are shown in Table 4.5.

The highest fully loaded percentage of depot A is 96.82%, which implies that the vehicle serves customers through route A-20-25-26-27-29-A in an efficient manner. Each route is followed by a vehicle, 7 vehicles serve the customer requirements in depot A, while 2 vehicles serve depot B, 3 vehicles in depot C and D, and 2 vehicles serve depot E following the routes as shown in the Table 4.5.

225



A(40,40)

A-3-4-5-6-A 85.85 43.49

7

A-9-12-15-17-18-A 78.69 52.21 A-20-25-26-27-29-A 96.82 74.2 A-30-32-34-37-39-A 93.5 68.4

A-40-44-45-47-A 89.65 53.065 A-48-50-51-55-60-A 93.65 79.27

A-67-68-70-75-A 68.51 46.77

B(50,22) B-7-8-13-35-46-52-B 95.64 54.29 2 B-57-58-72-B 74.78 29.82

C(55,55) C-10-11-14-C 78.51 34.475

3 C-19-31-38-53-C 86.84 48.32 C-54-59-65-66-C 90.4 41.79

D(25,45) D-2-16-21-D 76.56 48.34

3 D-24-28-33-36-D 85.6 54.87 D-49-62-63-69-71-73-74-D 94.65 74.165

E(20,20) E-21- 22-23-41-42-43-E 91.84 50.78 2 E-56-61-64-E 81.35 23.445

The benchmark instance p04 consisting of 2 depots and 100

customers, considered as one of the large size problems is run using the

routing algorithm and the results are shown in Table 4.6. It is observed that

the total distance required by depot A with 11 vehicles to serve customers is

708.51 km, and similarly depot B used 10 vehicles with a total distance of

733.94 km, respectively such that all customers are served.



A (15,35)

A-2-5-6-7-8-13-A 90.62 67.835

11

A-14-15-16-17-18-21-A 93.68 100.99 A-22-23-36-37-38-A 89.37 83.725

A-40-41-42-43-44-45-46-A 93.5 99.28 A-47-48-52-A 76.7 49.325

A-53-56-57-58-59-60-A 92.5 81.15 A-61-72-73-74-75-A 85.55 50.1

A-82-83-84-85-A 76.5 46.725 A-86-87-89-91-92-A 87.6 44.945

A-93-94-95-A 78.96 39.24 A-96-97-98-99-100-A 81.5 53.39

B (35,55)

B-1-3-4-9-10-B 90.5 48.965

10

B-11-12-19-20-24-25-B 94.5 94.885 B-26-27-28-29-30-B 93.5 104.395

B-31-32-33-34-B 88.6 72.38 B-35-39-49-B 87.6 43.325

B-50-51-54-55-62-63-B 92.5 117.46 B-64-65-66-67-B 76.5 52.435

B-68-69-70-71-76-B 96.61 78.085 B-77-78-79-80-81-B 95.5 86.585

B-88-90-B 58.67 34.37

226

The route allocation for customers in p06 instance with 3 depots

and 100 customers is shown in Table 4.7. Depot A is allocated with a set of 6

routes with a total distance of 216.49 km with loading capacities of 93.75%,

95%, 68.75%, 95%, 88.75%, and 73.75%, respectively. Similarly, depot B is

allocated with 7 routes whose total distance is 397.22 km and depot C is

allocated with 9 routes with a total distance of 476.36 km.



A (15,20)

A-5-14-15-16-17-A 93.75 44.905

6

A-37-38-41-42-43-44-45-46-A 95 72.725 A-57-59-61-84-A 68.75 39.605

A-85-86-A 95 19.46 A-87-91-92-93-95-A 88.75 26.77

A-96-97-98-99-100-A 73.75 13.025

B (50,20)

B-1-3-9-10-11-B 83.75 54.055

7

B-12-20-24-25-29-30-B 83.75 88.05 B-32-33-34-35-50-51-B 98.75 54.06

B-54-55-63-64-65-B 73.75 83.555 B-66-68-70-B 82.5 37

B-71-76-77-78-79-80-B 92.5 58.5 B-81-90-B 56.25 22

C (35,35)

C-2-4-6-7-8-13-18-C 97.5 108.86

9

C-19-21-22-23-C 93.75 57.965 C-26-27-28-31-C 95 69.2 C-36-39-40-47-C 90 24.805

C-48-49-52-C 93.75 32.23 C-53-56-58-60-62-C 75 79.6 C-67-69-72-73-74-C 91.25 47.1 C-75-82-83-88-89-C 86.25 47.6

C-94-C 53.25 9

4.4.3 Impact of GA on MDVRP

In GA, each chromosome is represented by the route of each

vehicle. The set of chromosomes form the initial population thus forming the

search space. The population size decides the number of chromosomes in a

single generation. A larger population size slows down the GA run, while a

smaller value leads to exploration of a small search space. A reasonable range

of the population size is between [50,100]. Based on the encoding of

chromosomes in this work, the population size is set to 50. Genetic algorithms

generate a new route sequence by selecting two individuals in the population

227

to which the genetic operators crossover and mutation are applied. Best cost

route crossover technique is applied to exploit a better solution for MDVRP.

The choice of mutation is route reversal type, which explores a wider search

space for the MDVRP. The parameters used in solving MDVRP using GA for

five benchmark problems are shown in Table 4.8.

Table 4.8 Parameter settings for GA based MDVRP

Parameters Settings Population size Based on the number of customers

Selection Tournament selection Crossover Best Cost Route crossover Mutation Route reversal mutation

Crossover probability 0.6 Mutation probability 0.02

Elite 4

The routes are scheduled and optimized using GA to determine the

number of customers serviced (NC), number of vehicles required (NVR), and

optimal route. NVA indicates the total number of vehicles available in each

depot for the corresponding benchmark instance. These parameters are

computed and presented in Table 4.9 for benchmark instances p01, p02, p03,

p04 and p06, respectively. A total of 11 customers are serviced by depot A in

instance p01 with 4 optimal routes such as A-17-4-13-A, A-19-18-A, A-25-

41-40-A, A-44-42-45-A. Likewise depot B serves 19 customers, depot C

serves 16 customers and depot D serves 4 customers with the optimal routes

as shown in table. Instance p02 has four depots, each of which serves 11, 19,

16 and 4 customers with 3, 5, 3 and 1 optimal route respectively. A set of 16

optimal routes is generated by instance p03 using GA to serve 75 customers

from 5 depots. Though instance p04 has only 2 depots, the number of vehicles

required is too large since there are 100 customers to be served from these

depots. Depot A has 10 optimal routes to serve 55 customers while depot B with

8 optimal routes to supply 45 customers. Instance p06 has 3 depots each

generating 5, 5 and 7 optimal routes to serve 29, 33 and 38 customers respectively.

228

Table 4.9 Optimal route using GA

Depot NVA NC NVR Optimal Route Instance p01

A(20,20) 8 11 4 A-17-4-13-A, A-19-18-A, A-25-41-40-A, A-44-42-45-A B(30,40) 8 19 5 B-7-10-5-6-B, B-12-14-15-B, B-23-27-33-37-24, B-39-38-46-43-47-B, B-48-49-B C(50,30) 8 16 4 C- 8-2-9-1-C, C-28-26-11-21-16-22-C, C-29-31-32-30-C, C-50-34-C D(60,50) 8 4 1 D-35-20-3-36-D

Instance p02 A(20,20) 5 11 3 A-13-18-4-17-A, A-25-41-40-19-A, A-44-45-42-A B(30,40) 5 19 5 B-6-7-10-5-B, B- 15-12-23-14-B, B-27-24-37-33-38-B, B- 46-39-47-43-48-B, B-49-B C(50,30) 5 16 3 C-9-2-8-1-C, C- 11-16-21-29-28-26-22-C, C-34-50-31-32-30-C, D(60,50) 5 4 1 D-3-20-35-36-D

Instance p03

A(40,40)

7

32

6 A-9-3-6-5-4-A, A-18-17-20-15-26-12-25-A, A-34-27-37-29-30-32-A, A- 40-39-44-47-

45-A, A-48-60-67-55-50-51-68-A, A-75-70-A B(50,22) 7 9 2 B-52-46-7-35-8-13-57-B, B-72-58-B C(55,55) 7 11 3 C-19-14-11-10-C, C-59-65-31-38-53-54-C, C-66-C D(25,45) 7 14 3 D-2-24-16-21-D, D-49-36-71-69-28-62-73-33-63-D, D-74-D E(20,20) 7 9 2 E-61-22-42-41-56-23-43-1-E, E-64-E

Instance p04

A(15,35) 12 55 10 A-8-7-6-13-2-14-5-A, A-22-15-16-17-18-21-A, A-38-43-41-23-40-36-37-42-A, A-45-46-47-44-A, A-57-56-58-53-52-48-A, A-73-72-74-59-61-60-A, A-82-83-84-85-75-A,

A-91-86-89-87-92-A, A-97-95-94-96-93-A, A-99-98-100-A

B(35,55) 12 45 8 B-1-4-3-9-11-10-B, B-27-19-20-24-25-12-26-B, B-29-30-31-28-B, B-33-32-35-34-39-B, B-49-51-55-54-50-62-B, B-67-65-66-64-63-B, B-69-70-71-78-77-68-76-B, B-88-

90-81-79-80-B Instance p06

A(15,20) 10 29 5 A-17-5-37-15-14-38-16-A, A-41-59-45-46-44-43-42-57-A, A-61-84-86-85-A, A-96-95-87-97-92-91-93-A, A-100-98-99-A

B(50,20) 10 33 5 B-24-12-3-20-9-11-10-1-B, B-34-29-25-30-32-33-35-B, B-55-65-64-63-51-50-54-B, B-70-66-71-68-76-B, B-81-79-78-80-77-90-B

C(35,35) 10 38 7 C-7-19-8-18-6-13-4-2-C, C-26-23-22-21-27-C, C-31-36-40-39-28-C, C-49-47-48-C,

C-52-53-56-67-58-60-62-C, C-69-82-83-72-75-74-73-C, C-88-94-89-C

The computational results obtained through GA for the five

benchmark instances are shown in Table 4.10 in terms of computed optimal

distance, computational time and gap between the best known and optimal

distance. The gap between the best known distance and optimal

distances )( dP , is calculated using the following Equation (4.27),

100

bk

bkod V

VVP (4.27)

where oV is the optimal distance obtained by the optimization technique and

bkV is the best known distance of the instance. From the table, it is observed

that the optimal distances computed by GA for all the instances are near

229

optimal to the best known solution. These distances are computed based on a

total of 14 routes for instance p01, 12 routes for instance p02, 16 routes for

instance p03, 18 routes for instance p04 and 17 routes for instance p06. The

differences between the best known and optimal distances interpret near

optimal solutions for instances p04 (0.975449%), and p06 (0.682259%).

Relatively less difference is obtained for instances p01 and p02 with values

3.740877% and 1.081241% respectively, while for instance p03 GA is not

capable of finding the optimal distance since the difference

9.051919% is quite large.

Table 4.10 Computational results for benchmark instances using GA

Benchmark Instance Type

No. of customers

No. of Depots

Best known distance (km)

Optimal distance

(km)

Computational time (seconds)

Gap between best known and

optimal distance (%)

p01 50 4 576.87 598.45 4.0692 3.740877 p02 50 4 473.53 478.65 3.4207 1.081241 p03 75 5 641.19 699.23 6.8128 9.051919 p04 100 2 1001.59 1011.36 10.1081 0.975449 p06 100 3 876.50 882.48 9.3177 0.682259

4.4.4 Evaluation of MPSO for MDVRP

The parameters of MPSO are set as shown in Table 4.11 based on

empirical study and studies from literature. The acceleration constants c1 and

c2 are set to 1.28 and initial population of swarm varies with respect to the

customers and depots in each instance. Inertia weight, w, determines the

search behavior of the algorithm. Large values for w facilitate searching new

locations whereas small values provide a finer search in the current area. A

balance can be established between global and local exploration by decreasing

the inertia weight during the execution of the algorithm.

230

Table 4.11 Parameter settings for MPSO based MDVRP

Parameters Settings

Swarm size Based on the number of customers

Cognitive factor, c1 1.28

Social coefficient, c2 1.28

First inertia weight, wint 0.9

Last inertia weight, wend 0.4

No. of iterations 300

The routes for each vehicle are taken as the swarm particles and the

MPSO algorithm iterates to determine the pbest and gbest to optimize the

delivery distance. The scheduled routes for the five benchmark instances p01,

p02, p03, p04 and p06 are optimized using MPSO, and the route length,

optimal route, number of customers serviced and number of vehicles required

are evaluated and the results are shown in Table 4.12.

Instance p01 has generated a total of 13 routes from 4 depots thus

servicing all the 50 customers. In instance p02 with 4 depots, depot A

produces 3 optimal routes, depot B generates 4 optimal routes, while depot C

framed 3 optimal routes and depot D with one optimal route. A total of 75

customers are served in instance p03 with 5 depots requiring 14 vehicles

among the available 35 vehicles. Similarly, instance p04 has generated 15

optimal routes with 2 depots and instance p06 constructed 17 optimal routes

with 3 depots.

231

Table 4.12 Optimal route using MPSO


A(20,20) 8 11 3 A-13-18-4-17-A, A-40-41-25-19-A, A-44-45-42-A B(30,40) 8 19 5 B-7-6-5-10-B, B-15-12-23-14-B, B-33-37-24-27-38-B, B-39-46-48-43-47-B, B-49-B C(50,30) 8 16 4 C-9-1-8-2-C, C-26-28-29-21-C, C-16-11-22-C, C-30-34-50-31-32-C D(60,50) 8 4 1 D-20-3-36-35-D

Instance p02 A(20,20) 5 11 3 A-13-19-17-4-18-A, A-25-41-40-42-A, A-45-44-A B(30,40) 5 19 4 B-6-7-10-5-B, B-12-15-14-24-23-B, B-39-38-27-43-37-33-B, A-47-48-46-49-A C(50,30) 5 16 3 C-9-1-8-2-C, C-26-28-29-21-16-11-22-C, C-30-34-50-31-32-C D(60,50) 5 4 1 D-3-20-35-36-D

Instance p03

A(40,40) 7 32 5 A-12-9-3-6-5-4, A-25-18-17-29-20-15-27-26-A, A-39-34-37-30-32-A,A- 40-44-47-48-45-A,A-67-75-70-60-68-51-50-55-A,

B(50,22) 7 9 2 B-52-46-7-35-8-13-57-B, B-72-58-B C(55,55) 7 11 3 C-11-14-19-10-C, C-59-54-53-38-31-65-C, C-66-C D(25,45) 7 14 2 D-21-2-24-16-28-D, D-74-36-71-69-62-73-33-63-49-D E(20,20) 7 9 2 E-56-41-42-22-61-1-43-23-E, E-64-E

Instance p04

A(15,35) 12 55 8 A-8-5-14-15-2-13-6-7-A,A-37-16-17-36-18-21-23-22-A,A-38-44-45-46-47-40-41-42-43-A, A-52-48-57-56-58-53-A, A-60-73-72-75-74-59-61-A, A-84-85-82-83-A, A-93-

92-87-89-86-91-A, A-94-95-97-100-98-99-96-A

B(35,55) 12 33 7 B-11-10-1-12-4-3-9-B, B-26-28-27-19-20-29-24-25-B, B-30-31-33-34-35-32-B, B-51-50-54-55-39-49-B, B-67-62-64-63-66-65-B, B-78-71-70-69-76-68-77-B, B-79-81-90-

88-80-B Instance p06

A(15,20) 10 29 5 A-5-37-15-14-38-16-17-A,A- 45-61-44-43-41-57-42-59-46-A, A-86-84-85-A,A- 91-92-97-87-95-96-93-A, A-100-98-99-A

B(50,20) 10 33 5 B-20-11-10-1-12-24-3-9-B, B-29-25-33-30-32-35-34-B, B-55-54-50-51-65-64-63-B, B-68-71-66-70-76-B,B- 80-77-79-78-81-90-B

C(35,35) 10 38 7 C-6-13-2-4-19-7-8-18-C, C-26-23-22-21-27-C, C-31-36-40-39-28-C, C-49-47-48-52-

C, C-67-56-58-60-62-69-53-C, C-88-82-83-73-74-75-72-C, C-94-89-C

The results obtained through MPSO are shown in Table 4.13 in

terms of optimal distance, computational time, and gap between best known

distance and optimal distance. The gap between the best known distance and

optimal distances obtained by MPSO is computed according to Equation

(4.27). It can be inferred from the results that the MPSO algorithm is capable

of obtaining near optimal solutions when compared with the best known

distance. For instance, the best known distance of p02 is 473.53 km while the

optimal value is 475.47 km with a gap of 0.41%. It can also be observed that

the computational time increase with the size of the problem.

232

Table 4.13 Computational results for benchmark instances using MPSO


No. of customers

No. of Depots

Best known distance (km)

Optimal distance (km)


Gap between best known and optimal distance (%)

p01 50 4 576.87 580.35 3.6319 0.603255 p02 50 4 473.53 475.47 2.9434 0.409689 p03 75 5 641.19 687.65 5.1899 7.245902 p04 100 2 1001.59 1008.55 7.7211 0.694895 p06 100 3 876.50 880.35 8.1418 0.439247

4.4.5 Solution of MDVRP based on ABC

In ABC, a colony of artificial forager bees act as agents searching

for artificially rich food sources analogous to good solutions for a given

problem. In this work, the position of a food source represents a set of

possible routes to the multi-depot vehicle routing problem. The parameters

that govern the ABC algorithm are colony size, number of food sources, limit,

number of employed bees, number of onlooker bees and maximum number of

iterations. The colony size is based on the number of customers in each

MDVRP instance. Smaller the colony size faster the convergence rate and

vice versa. The number of employed bees and the onlooker bees are usually

set to one half of the colony size for best results. The number of food sources

is always chosen to be equal to the number of employed bees. The ABC

algorithm is run for a maximum of 800 iterations irrespective of the size of

the problem. The parameters and their values used for running the ABC

algorithm for MDVRP are shown in Table 4.14.

Table 4.14 ABC parameters for MDVRP

S.No Parameters Value

1 Colony size Based on No. of customers

2 No. of food sources No. of employed bees

3 Food source limit No. of onlooker bees x dimension of the problem

4 No. of employed bees No. of customers/2

5 No. of onlooker bees No. of customers/2

6 Maximum No. of iterations 800

233

The optimal route evaluated using ABC along with the number of

vehicles required and the number of customers serviced for the problem

instances is presented in Table 4.15. In instance p01, Depot A requires 3

vehicles for generating three optimal routes A-13-17-4-18-A, A-40-41-25-19-

A, and A-42-45-44-A, respectively. In the same way, from the table it can be

seen that depot B requires 5 vehicles for generating the optimal routes B-6-7-

10-5-B, B-15-12-23-14-B, B-33-37-24-27-38-B, B-39-46-48-43-47-B, and B-

49-B. Likewise depots C and D produce optimal routes with 3 and 1 vehicles

respectively. Similarly, for instance p02, ABC locates the optimal routes for

depots A, B, C and D each serving 11, 19, 16 and 4 customers with

constructive number of vehicles such as 3, 4, 3 and 1 respectively. ABC on

instance p03 with 5 depots produces 5 optimal routes from depot A, 2 from

depot B, 3 from depot C, 2 from depot D and 2 from depot E.

Table 4.15 Optimal route using ABC


A(20,20) 8 11 3 A-13-17-4-18-A, A-40-41-25-19-A, A-42-45-44-A B(30,40) 8 19 5 B-6-7-10-5-B, B-15-12-23-14-B, B-33-37-24-27-38-B, B-39-46-48-43-47-B, B-49-B C(50,30) 8 16 3 C-2-9-1-8-C, C-11-22-26-28-29-21-16-C,C-30-32-31-50-34-C, D(60,50) 8 4 1 D-20-3-36-35-D

Instance p02 A(20,20) 5 11 3 A-19-13-18-4-17-A, A-42-40-41-25-44-A, A-45-A B(30,40) 5 19 4 B-7-5-10-12-6-B, B-27-33-15-14-24-23-B, B-47-43-48-46-38-39-37-B, B-49-B C(50,30) 5 16 3 C-1-11-9-2-8-C, C-28-31-26-22-16-30-21-29-C, C-50-32-34-C D(60,50) 5 4 1 D-36-35-20-3-D

Instance p03

A(40,40) 7 32 5 A-6-3-9-12-4-5-A, A-25-18-17-29-20-15-27-26-A, A-32-39-34-37-30-40-A, A-45-60-47-48-51-44-50-55-A, A-67-75-68-70-A

B(50,22) 7 9 2 B-52-46-7-35-8-13-57-B, B-72-58-B C(55,55) 7 11 3 C-10-19-14-11-C, C-65-31-38-53-54-59-C, C-66-C D(25,45) 7 14 2 D-2-24-16-28-21-D, D-49-74-36-71-69-62-73-33-63-D E(20,20) 7 9 2 E-22-42-41-56-23-43-1-61-E, E-64-E

Instance p04

A(15,35) 12 55 8 A-8-7-6-13-2-15-14-5-A,

A-22-23-21-18-36-17-16-37-A, A-46-45-44-38-43-42-41-40-47-A, A-52-48-57-56-58-53-A, A-73-72-75-74-59-61-60-A, A-85-84-83-82-A, A-89-86-91-93-92-87-A, A-96-99-98-100-97-95-94-A

B(35,55) 12 33 7 B-3-9-11-10-1-12-4-B, B-24-25-26-28-27-19-20-29-B, B-31-30-32-35-34-33-B, B-51-50-54-55-39-49-B,B- 67-65-66-63-64-62-B, B-78-77-68-76-69-70-71-B, B-80-88-90-81-79-B

Instance p06

A(15,20) 10 29 4 A-17-16-38-14-15-41-42-37-5-A, A-59-46-45-84-61-44-43-57-A, A-91-86-85-92-87-A, A-95-96-99-93-100-98-97-A

B(50,20) 10 33 5 B-20-11-10-1-12-25-24-3-9-B, B-29-50-33-51-30-32-35-34-B, B-63-55-54-65-66-64-B, B-68-76-70-71-78-79-77-B, B-90-81-80-B

C(35,35) 10 38 6 C-19-7-8-18-6-13-2-21-4-C, C-27-22-23-26-28-C, C-47-40-39-31-36-C,C- 52-48-49-56-53-C, C-67-69-62-60-58-73-72-C,C- 89-83-82-88-75-74-94-C

234

Instance p04 has a total of 24 vehicles available from 2 depots, and

the optimal routes framed require only 15 vehicles shared as 8 and 7 among

depots A and B correspondingly. It is also observed from the table that

problem instance p06 with 3 depots has 30 vehicles available, among which

50% of the vehicles are only used to serve all the customers.

The evaluated results in terms of optimal distance and

computational time are shown in Table 4.16. The optimal distance obtained

by instance p01 for serving all customers from 4 depots with a total of 12

vehicles is 584.72 km. Instance p02 has generated 11 optimal routes with near

optimal distance of 477.91 km. For instance p03 with 75 customers and 5

depots, the best known distance reported is 641.19 km while the proposed

ABC is capable of achieving an optimal distance of 668.84 km. For instances

p04 and p06, the gap between the best known distance and the obtained

optimal is found to be 0.9065586% and 0.635482% respectively.

Comparatively, ABC takes a longer time to converge in spite of the less

tuning parameters. This fact is evident from the very high computational time

tabulated for the instances considered in this work.

Table 4.16 Computational results for benchmark instances using ABC


No. of customers

No. of Depots

Best known distance

(km)

Optimal distance

(km)


Gap between best known and optimal

distance (%) p01 50 4 576.87 584.72 8.1456 1.3607919 p02 50 4 473.53 477.91 8.9547 0.9249678 p03 75 5 641.19 668.84 12.6472 1.1930941 p04 100 2 1001.59 1010.67 17.5541 0.9065586 p06 100 3 876.50 882.07 18.0546 0.635482

4.4.6 MDVRP based on GSO and IGSO

The hybrid combination of GA and MPSO delivered better optimal

distance when compared to the individual performance of both GA and

MPSO for the p01, p02, p03, p04 and p06 benchmark instances. The

235

parameters of GSO and IGSO are set as shown in Table 4.17. The size of

population is based on the customers in each MDVRP instance. The social

and cognitive factors are set according to 40 21 cc . The inertia weight is

varied in order to maintain a proper balance between the exploration and

exploitation of the algorithm and is set such that 112

c+c 21 . In the GA

constituent of IGSO, best cost route crossover with a crossover rate of 0.7 and

route reversal mutation with a rate of 0.028 are chosen as the optimal setting.

The hybridization parameter applicable in GSO is set to 0.5 to ensure that

individuals of GA and MPSO participate equally in the hybrid combination.

Table 4.17 Parameter settings for GSO and IGSO

Parameters GSO IGSO Population size Instance dependant Instance dependant

Maximum inertia weight 0.9 0.9 Minimum inertia weight 0.4 0.4

Initial velocity 0 0 Initial position Random Random

Cognitive factor (c1) 1.28 1.24 Social factor (c2) 1.28 1.24

Error gradient 1e-25 1e-25 Type of Crossover Best cost route crossover Best cost route crossover Type of Mutation Flip bit Route reversal mutation

Crossover rate 0.6 0.7 Mutation rate 0.02 0.028

Maximum number of iterations Instance dependant Instance dependant Hybridization factor 0.5 -

For the problem instances, the number of vehicles required and the

optimal route are computed by GSO and IGSO and the results are presented

in Table 4.18 (a) and Table 4.18 (b). From Table 4.18 (a), for instance p03,

Depot A requires 5 vehicles for generating five optimal routes such as A-5-6-

3-9-12-4-15-A, A-20-29-30-17-18-25-26-27-A, A-40-44-32-39-34-37-A, A-

47-60-45-48-55-50-51-A, A-70-68-75-67-A respectively. In the same way,

from the table it can be seen that depot B requires one vehicle for generating

the optimal route B-13-8-35-7-58-72-46-52-57-B. Likewise, depot C, D and E

require 2 vehicles each for finding the optimal route.

236

Table 4.18(a) Optimal route using GSO


A(20,20) 8 11 3 A-13-19-17-4-18-A, A-44-42-40-41-25-A, A-45-A B(30,40) 8 19 4 B-7-6-12-10-5-B, B-15-14-24-23-27-B,B-46-38-39-33-37-43-B,B- 48-47-49-B C(50,30) 8 16 3 C-9-11-1-8-2-C, C-28-31-26-22-16-30-21-29-C, C-32-34-50-C D(60,50) 8 4 1 D-20-3-36-35-D

Instance p02 A(20,20) 5 11 2 A-13-19-17-4-18-A, A-40-42-45-44-25-41-A B(30,40) 5 19 3 B-7-5-10-12-6-B, B-33-15-37-14-24-23-27-B, B-46-48-43-47-39-49-38-B C(50,30) 5 16 3 C-11-1-8-2-16-9-C, C-29-22-28-31-26-32-30-21-C, C-34-50-C D(60,50) 5 4 1 D-3-20-35-36-D

Instance p03

A(40,40) 7 32 5 A-5-6-3-9-12-4-15-A, A-20-29-30-17-18-25-26-27-A, A-40-44-32-39-34-37-A, A-47-

60-45-48-55-50-51-A, A-70-68-75-67-A B(50,22) 7 9 1 B-13-8-35-7-58-72-46-52-57-B C(55,55) 7 11 2 C-11-14-19-10-31-C, C-38-65-66-59-54-53-C D(25,45) 7 14 2 D-2-21-28-16-24-D, D-69-71-36-74-49-63-33-73-62-D E(20,20) 7 9 2 E-41-42-22-61-1-43-23-56-E, E-64-E

Instance p04

A(15,35) 12 55 8 A-13-6-7-8-5-14-15-2-A, A-18-21-23-22-37-16-17-36-A, A-43-42-41-40-47-46-45-44-38-A, A-57-56-58-53-52-48-59-A, A-72-75-74-61-84-83-82-60-73-A, A-87-85-91-86-

89-A, A-98-93-99-96-94-95-97-92-A, A-100-A

B(35,55) 12 45 6 B-3-9-11-10-1-12-4-B, B-30-19-27-28-26-25-24-29-20-B, B-33-34-35-32-31-39-B,B-51-

55-54-50-62-49-64-63-B, B-69-67-68-65-66-70-B, B-80-76-88-90-71-81-78-79-77-B Instance p06

A(15,20) 10 29 4 A-14-38-43-15-41-42-37-5-17-16-A, A-61-44-57-59-46-45-84-A, A-91-87-92-85-86-A

B(50,20) 10 33 4 B-11-10-1-12-25-24-29-3-9-20-B, B-32-30-51-33-50-54-34-35-B, B-65-68-55-70-63-

64-66-B, B-77-79-78-81-71-90-76-80-B

C(35,35) 10 38 6 C-19-7-8-18-6-13-2-21-4-C, C-27-22-23-26-28-C,C- 47-40-39-31-36-C,C-56-58-60-

48-49-52-53-C, C-67-69-62-73-72-74-75-C, C-83-94-89-88-82-C

Based on the results obtained in Table 4.18 (b), to serve all the

customers, instance p01 with 50 customers and 4 depots requires 11 vehicles,

while instance p02 requires 9 vehicles. Instance p03 has 5 depots and the

optimal number of routes from each depot A, B, C, D and E is 5, 1, 2, 2, and 1

respectively. The number of optimal routes generated by instance p04 is 7 and

5 corresponding to depots A and B. Instance p06 serves 29 customers from

depot A with 4 vehicles, 33 customers from depot B with 4 vehicles and 38

customers from depot C with 6 vehicles.

237

Table 4.18(b) Optimal route using IGSO


A(20,20) 8 11 3 A-19-13-18-4-17-A, A-40-41-25-44-42-A, A-45-A B(30,40) 8 19 4 B-7-5-10-12-6-B, B-15-27-23-24-14-B, B-46-38-39-33-37-43-B, B-49-47-48-B C(50,30) 8 16 3 C-2-8-1-11-9-C, C-29-28-31-26-22-16-30-21-C, C32-34-50-C D(60,50) 8 4 1 D-3-36-35-20-D

Instance p02 A(20,20) 5 11 2 A-19-13-18-4-17-A, A-40-41-25-44-45-42-A B(30,40) 5 19 3 B-7-5-10-12-6-B, B-37-15-33-27-23-24-14-B, B-43-48-46-38-49-39-47-B C(50,30) 5 16 3 C-9-16-2-8-1-11-C, C-31-26-32-30-21-29-22-28-C, C-50-34-C D(60,50) 5 4 1 D-3-36-35-20-D

Instance p03

A(40,40) 7 32 5 A-12-9-3-6-5-15-4-A, A-17-18-25-26-27-20-29-30-A, A-40-44-32-39-34-37-A, A-45-60-47-48-51-50-55-A, A-75-68-70-67-A

B(50,22) 7 9 1 B-72-46-52-57-13-8-35-7-58-B C(55,55) 7 11 2 C-14-19-10-31-11-C, C-53-38-65-66-59-54-C D(25,45) 7 14 2 D-21-2-24-16-28-D, D-69-71-36-74-49-63-33-73-62-D E(20,20) 7 9 1 E-41-42-22-61-1-43-23-56-64-E

Instance p04

A(15,35) 12 55 7 A-5-8-7-6-13-2-15-14-A, A-21-23-22-37-16-17-36-18-A, A-45-44-38-43-42-41-40-47-46-A, A-57-56-58-53-52-48-59-A, A-74-75-72-73-60-82-83-84-61-A, A-86-91-85-87-

89-A, A-98-93-99-100-96-94-95-97-92-A

B(35,55) 12 45 5 B-11-9-3-4-12-1-10-25-26-28-27-19-30-20-29-24-B, B-33-34-35-32-31-39-B, B-51-55-54-50-62-49-64-63-B, B-67-69-70-66-65-68-B, B-79-78-81-71-90-88-76-80-77-B

Instance p06

A(15,20) 10 29 4 A-41-15-43-14-38-16-17-5-37-42-A, A-44-57-59-46-45-84-61-A, A-91-87-92-85-86-A, A-95-96-99-93-100-98-97-A

B(50,20) 10 33 4 B-20-11-10-1-12-25-24-29-3-9-B, B-32-30-51-33-50-54-34-35-B, B-65-68-55-70-63-64-66-B, B-76-90-71-81-78-79-77-80-B

C(35,35) 10 38 6 C-6-18-8-7-19-4-21-2-13-C, C-27-22-23-26-28-C, C-36-31-39-40-47-C, C-58-56-53-52-49-48-60-C, C-75-67-69-62-73-72-74-C, C-88-82-83-94-89- C

The experimental results obtained through GSO and IGSO for the

benchmark instances considered in this work are shown in Table 4.19 in terms

of optimal distance, computational time and gap between best known distance

and optimal distance. The gap between the best known distance (BKD) and

optimal distance obtained by IGSO is calculated according to Equation (4.27).

The optimal distances computed through IGSO for instances p01, p02, p03,

p04 and p06 are 576.94 km, 474.03 km, 644.29 km, 1001.94 km, and 876.68

km respectively. The gap between the best known distance and the computed

optimal distance for IGSO is 0.012134% in the case of instance p01,

0.10559% in p02, 0.483476% in p03, 0.034944% in p04, and 0.020536% in

p06. The reduction in gap for instances in IGSO show that the enhancement

step introduced in the hybrid combination has improved the optimal distance

when compared to GSO. The computational time obtained shows that the

238

IGSO algorithm is capable of converging at a faster rate towards the optimal

solution than the GSO algorithm.

Table 4.19 Computational results for MDVRP instances using GSO and IGSO

Instance No. of customers

No. of Depots

BKD (km) GSO IGSO

Optimal distance

(km) CT (s) Gap (%)

Optimal distance

(km) CT (s) Gap (%)

p01 50 4 576.87 578.52 1.5316 0.286026 576.94 1.3725 0.012134 p02 50 4 473.53 474.85 1.4259 0.278757 474.03 0.9750 0.10559 p03 75 5 641.19 653.78 2.3659 1.963537 644.29 2.2773 0.483476 p04 100 2 1001.59 1005.73 3.0191 0.413343 1001.94 2.3918 0.034944 p06 100 3 876.50 877.19 3.0976 0.078722 876.68 2.4094 0.020536

4.5 COMPARATIVE ANALYSIS AND DISCUSSIONS

The experimental results obtained on MDVRP instances using the

heuristic optimization techniques such as GA, MPSO, ABC, GSO and IGSO

are compared and discussed in terms of optimal distance, robustness,

computational efficiency and algorithmic efficiency as follows:.

4.5.1 Optimal Distance

The optimal distance obtained for the benchmark instances using GA,

MPSO, ABC, GSO and IGSO algorithms are compared with the existing

approaches in literature such as genetic clustering (GC) (Thangiah and Salhi

2001) and GA (Ombuki-Berman and Hanshar 2009). The optimal distance

obtained for the five MDVRP instances using the existing approaches and

proposed algorithms are presented in Table 4.20. For instance p03, the distance

obtained by proposed GA, MPSO, ABC, GSO and IGSO are comparatively less

than the GA (Ombuki-Berman and Hanshar 2009) in literature by 1.08%,

2.72%, 5.38%, 7.51% and 8.85% respectively. MPSO yields minimum distance

for instances p01, p02, p03, p04 and p06 when compared with GA with a

difference of 3.02%, 0.66%, 1.66%, 0.28% and 0.24% respectively. Though

239

ABC took more time to generate the optimal route, the distances are reduced

when compared with the proposed GA. For instance p01, the distance obtained

by ABC is 2.29% lesser than that obtained by GA. In the same way, the

optimal distance obtained by ABC is 0.15% (instance p02), 4.34% (instance

p03), 0.07% (instance p04), and 0.05% (instance p06) lesser than GA. The

improvement in optimal distance obtained by IGSO over GSO is given by

0.273% in p01, 0.173% in p02, 1.45% in p03, 0.377% in p04, and 0.058% in

p06. The enhancement step introduced in IGSO has shown a significant

improvement in terms of optimal distance specifically in instance p03. Thus

from the comparative analysis in terms of the optimal distance, it can be

revealed that IGSO is a better approach to solve the chosen MDVRP

instances.

Table 4.20 Comparative analysis in terms of optimal distance

Instance Type

No. of customers

No. of Depots

Best known

distance (km)

Distance reported in literature (km)

Distance obtained by the proposed approaches (km)

GC (2001)

GA (2009) GA MPSO ABC GSO IGSO

p01 50 4 576.87 591.73 622.18 598.45 580.35 584.72 578.52 576.94 p02 50 4 473.53 463.15 480.04 478.65 475.47 477.91 474.85 474.03 p03 75 5 641.19 694.49 706.88 699.23 687.65 668.84 653.78 644.29 p04 100 2 1001.59 1062.38 1024.78 1011.36 1008.55 1010.67 1005.73 1001.94 p06 100 3 876.50 976.02 908.88 882.48 880.35 882.07 877.19 876.68

4.5.2 Robustness

The robustness of the heuristic techniques applied in this work is

evaluated based on the effective vehicle management without violating the

capacity limit of the vehicles. Computational results in terms of vehicles

required (NVR) and total number of vehicles available (NVA) is shown in

Table 4.21. Among the proposed intelligent algorithms, it is found from the

table that GA utilized more number of vehicles for all the instances to

determine the optimal distance, thus increasing the vehicle utilization cost.

For test instances p02, p03, p04, it is observed that MPSO and ABC use the

240

required vehicles in a similar manner without any difference. ABC exploits

less number of vehicles in a more efficient way in the case of instance p01

and p06 with 37.5% and 50% respectively, while MPSO utilizes 40.63% and

56.67% of the vehicles available. This fact implies that ABC is robust against

MPSO, in spite of MPSO achieving the optimal distance.

While comparing GSO and IGSO, no significant difference in

vehicle utilization efficiency is found in instances p01, p02 and p06. IGSO

has shown significant improvement over GSO in utilizing vehicles

specifically for instances p03 (31.43%) and p04 (50%). Overall, for all the

test instances GSO and IGSO used less number of vehicles when compared

with GA, MPSO and ABC algorithms. Considerably, from the results based

on the vehicle utilization efficiency for instances p03 and p04, it can be

demonstrated that IGSO is robust over GA, MPSO, ABC and GSO

algorithms.

Table 4.21 Comparative analysis of vehicle utilization efficiency

Instance Type

No. of customers

No. of Depots NVA NVR (%)

GA MPSO ABC GSO IGSO p01 50 4 32 43.75 40.63 37.50 34.38 34.38 p02 50 4 20 60.00 55.00 55.00 45.00 45.00 p03 75 5 35 45.71 40.00 40.00 34.29 31.43 p04 100 2 24 75.00 62.50 62.50 58.33 50.00 p06 100 3 30 56.67 56.67 50.00 46.67 46.67

4.5.3 Computational Time

The computational efficiency of the bio-inspired heuristics is

evaluated in terms of the time taken for the algorithm to converge. The

computational time of the proposed techniques for the benchmark instances

are presented in Table 4.22. It can be observed from the table that

computational time of IGSO (2.3918s) for instance p04 is comparatively less

than GA (10.1081s), MPSO (7.7211s), ABC (17.5541s) and GSO (3.0191s).

241

Overall for all the instances it is found that ABC consumes more time to

determine the optimal route and distance. On the average, the computational

time of ABC is 48.39% higher than GA, 57.73% higher than MPSO, 85.58%

higher than GSO and 82.5% higher than IGSO. The longer time for

convergence required by ABC leads to a setback in applying the ABC

technique to solve practical MDVRP instances. Computational results

indicate that the time taken by IGSO is comparatively less since the hybrid

combination of GA and MPSO minimizes the chances of the algorithm

getting trapped in the local minima.

Table 4.22 Comparative analysis of computational time

Instance Type

No. of customers

No. of Depots

Computational time (s) GA MPSO ABC GSO IGSO

p01 50 4 4.0692 3.6319 8.1456 1.5316 1.3725 p02 50 4 3.4207 2.9434 8.9547 1.4259 0.9750 p03 75 5 6.8128 5.1899 12.6472 2.3659 2.2773 p04 100 2 10.1081 7.7211 17.5541 3.0191 2.3918 p06 100 3 9.3177 8.1418 18.0546 3.0976 2.4094

4.5.4 Algorithmic Efficiency

The algorithmic efficiency of the heuristic algorithms used to solve

MDVRP in this work is calculated using BigO notation in terms of the

estimated time and calculated time of the algorithm. The algorithmic

efficiency of the proposed intelligent heuristics is evaluated for the problem

instances and the results are presented in Table 4.23. Based on the size of the

code the efficiency varies significantly. High amount of programming lines in

the code leads to a decrease in the efficiency of the algorithm. The

algorithmic efficiency of instance p02 for GA is 88.21%, for MPSO is

90.27%, for ABC is 95.88%, for GSO is 93.11% and for IGSO is 92.81%. On

an average, the algorithmic efficiency for all the MDVRP instances in this

242

work, using GA is 89.15%, 90.84% for MPSO, 95.19% for ABC, 93.22% for

GSO and 92.71% for IGSO. It is observed that due to the less number of

control parameters in ABC, the algorithmic efficiency is improved over GA,

MPSO, GSO and IGSO algorithms.

Table 4.23 Comparison of algorithmic efficiency

Instance Type

No. of customers

No. of Depots

Algorithmic efficiency (%) GA MPSO ABC GSO IGSO

p01 50 4 89.77 90.48 94.52 92.56 92.27 p02 50 4 88.21 90.27 95.88 93.11 92.81 p03 75 5 89.09 91.12 94.37 93.71 92.64 p04 100 2 89.56 90.55 95.06 93.29 92.98 p06 100 3 89.14 91.79 96.11 93.43 92.86

4.6 SUMMARY

The intelligent heuristics such as GA, MPSO, ABC, GSO and

IGSO are proposed in this chapter for solving multi-depot vehicle routing

problem. Initially the customers are grouped based on its distance from the

depot. Further the customers of the same depot are assigned to routes using

Clarke and Wright saving method. This procedure also ensures that two routes

can be easily merged into a single route. The routes are later scheduled and

optimized by the proposed intelligent heuristics. The objective of multi-depot

vehicle routing problem is to find routes for vehicles to service all the

customers at a minimal cost in terms of number of routes and total travel

distance, without violating the capacity and depot limits.

The effectiveness of the proposed techniques is tested on a set of

five different Cordeau’s benchmark instances namely p01, p02, p03, p04 and

p06 in the MATLAB R2008b environment. The intelligent heuristics are

applied to evaluate the optimal routes, number of vehicles required, and

optimal distance of the MDVRP instances. In addition to the problem

243

dependant evaluations such as optimal distance and robustness, additional

assessments in terms of computational time and algorithmic efficiency of the

proposed algorithms are also estimated.

Considering the optimal distance, it is observed from Table 4.20

that IGSO showed improvements among the other proposed algorithms such

as GA, ABC, MPSO and GSO. From Table 4.21, it can be seen that IGSO

utilizes relatively less number of available vehicles when compared with GA,

MPSO, ABC and GSO, thus implying robustness. Likewise, the

computational efficiency of IGSO is improved thus demonstrating earlier

convergence rate, which is evident from the results presented in Table 4.22.

The algorithmic efficiency of ABC (Table 4.23) is reasonably higher due to

the less number of control parameters in the algorithm. Thus based on the

investigations carried out on the proposed intelligent algorithms, it is

observed that IGSO is an appropriate approach for solving practical MDVRP

instances based on the optimal solution, robustness and computational

efficiency.

In future, it would be of considerable interest to incorporate several

practical constraints such as urgency delivery, random serve time and non-

uniform vehicle capacity to the MDVRP. This work may be extended with

new optimization techniques like Bacterial Foraging (BFO), Biogeography-

based optimization (BBO) and Artificial Immune Systems (AIS). The

experimental results may be used to analyze, compare and determine better

optimization technique in forthcoming prospects.

CHAPTER 4 MULTI-DEPOT VEHICLE ROUTING...

Documents

Transcript of CHAPTER 4 MULTI-DEPOT VEHICLE ROUTING...