A New Mathematical

8/18/2019 A New Mathematical

1/10

Journal of Manufacturing Systems 30 (2011) 83–92

Contents lists available at ScienceDirect

Journal of Manufacturing Systems

journal homepage: www.elsevier .com/ locate / jmansys

Technical paper

A new mathematical model for a competitive vehicle routing problem with time

windows solved by simulated annealing

R. Tavakkoli-Moghaddama, M. Gazanfari b, M. Alinaghian b,∗, A. Salamatbakhshc, N. Norouzic

a Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iranb Department of IndustrialEngineering, Iran University of Science and Technology, Tehran, Iranc Department of Industrial Engineering, Islamic Azad University – Tehran South Branch, Tehran, Iran

a r t i c l e i n f o

Article history:Received 8 June 2010

Received in revised form 8 April 2011

Accepted 27 April 2011

Keywords:

Vehicle routing problem

Competitive situation

Transportation

Simulated annealing

a b s t r a c t

This paper presents an extension of a competitive vehicle routing problem with time windows (VRPTW)

to find short routes with the minimum travel cost and maximum sale by providing good services to

customers before delivering the products by other rival distributors. In distribution of the products with

short lifetime that customers need special device for keeping them, reaching time to customers influences

on the sales amount which the classical VRPs are unable to handle these kinds of assumptions. Hence, a

new mathematicalmodel is developed for the proposed problem and for solving the problem, a simulated

annealing (SA) approach is used. Then some small test problems are solved by the SA and the results are

compared with obtained results from Lingo 8.0. For large-scale problems, the, Solomon’s benchmark

instances with additional assumption are used. The results show that the proposed SA algorithm can find

good solutions in reasonable time.

© 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Transportationhas an importantrole in various domains,such as

enterprise, economic and service systems. By this way, researchers

are interested in improving the routes, deleting the unnecessary

travels and creating the replacement short routes. In addition,

many problems, such as traveling salesman problem (TSP), vehi-

cle routing problem (VRP) and the like, are developed by this

approach. VRPs include problems where fleet of vehicles presents

service from depot to geographically dispersed customers’ set with

specific demands so that the cost function of transportation is

minimized. The VRP was formulated mathematically by Dantzig

and Ramser [1]. Clarke and Wright [2] developed Dantzig and

Ramser’s approach by using a saving algorithm. Also, a different

approachbased on the branch-and-bound algorithmwas presented

by Loporte et al. [3]. Christian et al. [4] solved the capacitatedVRP with stochastic demands by a branch-and-price algorithm.

Since VRPs are categorized as NP-hard problems [5] that no exact

algorithm is used to solve them in large sizes due to high computa-

tional cost; thus approximate and heuristic algorithms are used in

practice. Thus, many researchers developed heuristics and meta-

heuristics, such as local search [6], minimum K-trees [7], adaptive

memory (AM) [8], column generation [9], simulated annealing (SA)

∗ Corresponding author.

E-mail address: [email protected](M. Alinaghian).

[10–12], genetic algorithms (GAs) [12,13], tabu search (TS) [13,14],

ant system [15,16]. Tavakkoli-Moghaddamet al. developed a math-ematical model forthe VRPwith backhaulsby a memetic algorithm

[17]. Furthermore, Tavakkoli-Moghaddam et al. [18] presented a

linear-integer model of a capacitated VRP with the independent

route length in order to minimize the heterogeneous fleet cost and

maximize the capacityutilization.This presented model wassolved

by a hybrid simulated annealing.

The vehicle routing problem with time windows (VRPTW) is an

extension of the VRP where delivery of goods to each customer

should be occurred in the interval [ai, bi], in which ai and bi are

the earliest and the latest allowable times that the service should

be taken place. The VRPTW is divided in two parts, namely VRP

with soft time window (VRPSTW) and VRP with hard time win-

dow (VRPHTW). The VRPSTW is a relaxation of the VRPHTW, in

which the delivery of goods is allowed outside the time windows if a penalty is paid. However in the VRPHTW, the deviation from the

time windows constraint is not allowed at all.

Early studies of solution methods forthe VRPTW canbe found in

Golden and Assad [19] and Solomon [20]. Ombuki et al. [21] prop-

agated a VRPTW by a Pareto approach, weighted sum and genetic

algorithmin such a waythe total lengthand numberof thevehicles

were minimized. Geiger [22] developed a VRPTW for minimizing

the total distance, total deviations from the time window bounds,

number of violations, and number of vehicles by a Pareto approach

and genetic algorithm. An exhaustive review of the VRPHTW can

be found in [23]. Qureshi et al. [24] presented a new column gen-

0278-6125/$ – see front matter © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jmsy.2011.04.005

http://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.jmsy.2011.04.005http://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.jmsy.2011.04.005http://www.sciencedirect.com/science/journal/02786125http://www.elsevier.com/locate/jmansysmailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.jmsy.2011.04.005http://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.jmsy.2011.04.005mailto:[email protected]://www.elsevier.com/locate/jmansyshttp://www.sciencedirect.com/science/journal/02786125http://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.jmsy.2011.04.005


2/10

84 R. Tavakkoli-Moghaddam et al./ Journal of Manufacturing Systems 30 (2011) 83–92

eration based on the exact optimization approach for the VRPSTW.

Due to the high complexity level of the VRPTW a tabu search algo-

rithm was proposed by Taillard et al. [25] to solve the VRPSTW.

The proposed algorithm minimizes the total traveled distance and

total penalty that must be paid for presenting the service delay

outside time windows. Furthermore, Tavakkoli-Moghaddam et al.

[26] proposed a multi-criteria vehicle routing problem with con-

sidering soft time windows and their problem is solved by SA. In

addition a comprehensive review of heuristics methods used for

solving the VRPHTW and the VRPSTW can be found in [27,28]. Due

to the importance of service times presented by other companies

in real world, distribution companies design the routes of fleets

with respect to the condition of other competitors for obtaining

the maximum sale.

This paper proposes a new approach for VRPs, in which the cost

of routes is minimized while the amount of sale is concurrently

maximized. This approach needs to consider some other parame-

ters, such as competition between distributors, deciding factors of

the customers and service time presented by companies for cus-

tomers where the classical VRP is unable to achieve good solutions

for these kinds of assumptions.

In this paper, a novel variant of vehicle routing problem with

time windows is presented that occurs in a competitive environ-

ment.In this situation, itis very importantto attendthe service time

presented to customers in such a way that if the vehicle presents

the service to customers later than its rival, it will miss a partial

of its sale. Demand of each customer is divided of two parts, the

first part (didi), does not depend on time and should be sent to the

customer completely, the secondpart (dtdi), is time-dependent and

it would be lost if the rival’s arrival time is earlier than vehicle’s

arrival time to the customer. Therefore distributor‘s reaching time

to the customers influences on the amount of sales.

Since there are no related papers on a competitive approach on

theVRP in ourbest knowledge, in this work a new presentedmodel

forthe VRPwith competitive time windows (VRPCTW) is presented

that can be considered as a new class of the VRPTW. Despite of

similarities between the classical models of VRPTW with proposed

model there are several differences that are described as follows:

• In the classicalmodels,arriving to the customer earlier thanlower

bound of time windows is not desirable whereas in the new

model it is desirable.• In the classical models, arriving to the customer at any time in

time window bounds has same desirability, but in the proposed

model, desirability of arriving time from lower bound to upper

bound of time windows decrease.• In the new model, arriving to the customer after upper bound

of time windows makes partial of sale miss, but in the classical

models total sale is missed or only desirability decrease.

It is proven that VRPs belong to the category of NP-hard prob-

lems [5]; thereforethe model is solved by simulatedannealing (SA)

for large-scale problems. For some small cases, the related results

of the proposed SA are compared with the results obtained by the

Lingo8 software using a branch-and-bound algorithm. The related

results show the appropriate efficiency for solving the problem,

especially for large sizes.

The rest of this paper is organized as follows. Section 2 intro-

duces theproblem andSection 3 represents the model formulation.

The problem-solving methodology is described in Section 4 and

the computational results are discussed in Section 5. Finally, the

conclusion is presented in Section 6.

2. Problem definition

In a competitive environment, it is very important to attend the

service time presentedto customersin such a waythatif thevehicle

presents the service to customers later than its rival, it will miss a

partial of its sale. For this reason, distributing companies define the

route of their vehicles based upon other rival companies’ strategies

for serving customers. In other words, sometimes in competitive

situationsconsidering the competition between vehicles is needed.

The competition in distribution of products with a short life cycle

is an example, which customers need a special device for keeping

them. Therefore reaching time of a distributor to customers influ-

ences on the amount of sales. Hence, a model presented in this

paper is to find the routing of vehicles in a competitive environ-

mentsuchthatit can be consideredas a new version of the VRPTW.

This problem is proposed under a condition that a competition is

between distributors for obtaining more sales and market shares.

Before presenting the model, these parameters are introduced

for clarification of the problem. Let G = (V , A) be a graph, where

V ={v 1, v 1, . . ., v n} is the nodes set and, A = {(v i, v j): v i, v j ∈V } is thearcsset which each arc (v i, v j) is associated with a non-negativecost

C ij.

t li lower bound of rival’s arrival time to node i

t ui upper bound of rival’s arrival time to node i

t ri rival’s arrival time to node i

t di actual distributer’s vehicle arrival time to node i

f r ( x) probability distribution function of rival’s arrival time to node

i

F r ( x) cumulativedistributionfunctionof rival’s arrival timeto node

i

dtdi time dependent demand of node i

dini time independent demand of node i

Di maximum number of customer’s demand in node i

E (Di) Expected coverage of the ith customer’s demand

Demand of each customer is divided of two parts, the first part

(dini), does not depend on time and should be sent to the customercompletely, the second part (dtdi), is time-dependent and it would

be lost if the rival’s arrival time is earlier than vehicle’s arrival time

to the customer. Eq. (1) shows the relation between components of

the demand.

Di = dini + dtdi (1)

The problem is solved under the following assumptions:

(1) Each vehicle has a fixed capacity.

(2) Each customer location is serviced from only one vehicle.

( 3) The total demands of customers must not exceed from the

capacity of the vehicle.

(4) Each vehicle has to depart the visited customer and return to

the depot.(5) A vehicle is allowed to deliver goods before the earliest allow-

able service time windows without missing any sale. Except

of the rival’s arrival time, all problem parameters (e.g., cus-

tomers’ demands and travel time) are assumed to be known

with certainty.

Arriving after the latest rival’s arrival time, t ui, the first part of

demand, Dtdi, will be lost. If the arrival time occurs in the time win-

dow, the expected value of the obtained sale before rival reduces

according to the probability distribution function of the rival’s

arrival time.

The rival’s service time distribution to customers can be deter-

mined by the use of stochastic methods. In this problem, the rival’s

arrival time distribution to customer i is attributed by the uniform


3/10


4/10


5/10

R. Tavakkoli-Moghaddam et al./ Journal of Manufacturing Systems 30 (2011) 83–92 87

Fig. 2. Pseudocode of the proposed SA.

4.1.1. Simulated annealing implementation

The parameters of the proposed SA are as follows:

R number of iterations

EL epoch length indicating the number of the accepted solutions

in each temperature to achieve the equilibrium

T 0 initial temperature

A rate of the current temperature decrease (i.e., cooling schedule)

X feasible solutionC ( X ) objective function value for X

N counter for the number of the accepted solutions in each tem-

perature

IR counter for the number of iterations

The main steps of the SA algorithm are shown in Fig. 2.

Fig. 4. Sequence of customersin their routesby vehicles.

4.2. Representation and decoding method

This section presents a decoding method for the representation

of a solution.The solution representation of thegiven problem with

n customers consists of n dimensional particles. The particles rep-

resentation consist of n numbers in the bound (0, m + 1− ε]. Theinteger part of each dimension represents the vehicle that serves

the nodes. Thus, the same integer parts represent the same vehi-

cle for serving the customers. The fractional part represents the

sequence of the customer in the route. In other words, it shows pri-

ority of serving by the vehicle. The decoding method is based on

the following steps:

Step (1) For each customer (i.e., customer i where i = 1, . . ., n),repeat Step 2.

Step (2) Fill all the dimension of representation with numbers

in the bound (0, m + 1− ε]. The ith dimension represents the ith

node. Allocate customer i to vehicle k if the integer part of the ith

dimension of representation is k.

Step (3) Sort the allocated customers of each vehicle according tothe fractional part of the related dimension of customers.

Forexample, suppose that 7 nodes and3 vehicles areavailable.A

simple representation is shown in Fig.3. In this example, customers

7 and2 areallocatedto vehicle 1.The sequenceof visitingcustomers

is shown in Fig. 4.

Fig. 3. Example of a problem representation and decoding.


6/10


Fig. 5. 2-Opt exchange operation.

4.3. Initial solution generation

To generate initial solutions, which is the first step in any meta-

heuristic approach, a creative approach is applied as follows:

(1) Sort the customers according to t li none descending.

(2) Allocate the first customer (i.e., r 1) to the first vehicle (i.e., v 1).

(3) Allocate the second customer (r 2) to the current vehicle (v 1) if

its distance to depot is more than the first customer allocated

(r 1). It is necessaryto consider thelimitation of thecapacityand

travel time of the vehicle.

(4) Repeat the following steps for the reminding customers till allcustomers are serviced by all vehicles.

(5) Select the sequence of the reminding customers.

(6) Estimate the new customer’s distance with the last customer

allocatedof each route anddepotas well. Allocate thecustomer

to that route where the minimum distance is achieved while

the capacity of vehicles and the travel time limitations are not

violated.

4.4. Local search methods

After generating initial solutions, one of the following steps is

repeated for improving the initial solution.

4.4.1. 2-Opt exchange operator

The2-optoperator improves singleroute. It exchangesthe route

directionbetween twoconsecutivenodes. If thecostfunction of the

route has been improved, then the modified route is kept; other-

wise, the route returns to the last condition. The 2-opt exchange

operation is shown in Fig. 5, in which the edge (i, i + 1) and ( j,

j +1) are replaced by edge (i, j) and (i + 1, j + 1), thus reversing the

direction of customers between i + 1 and j.

4.4.2. Insertion method

Select a route at random, then select as the max [0.1× (length

of the route), 2] nodes in each route, and change the integer num-

ber of selected nodes in the bound of [0, m] randomly to change

the vehicle that services the selected node. In other words, a nodeis departed from one route and it is added to another route using

insertionheuristics.Anewinsertionisacceptedifthevehiclecapac-

ity and maximum time, in which vehicle v can be used, are not

violated.

4.5. Parameter setting

The performance of any meta-heuristic algorithmis usually sen-

sitive to the settings of the parameters that influence its search

behavior. In this section, the parameters for the proposed meta-

heuristics are explained and introduced experimentally according

to Table 1. In this table, a is the cooling rate, EL is the number of

the accepted solution in each temperature for achieving the equi-

librium, and T 0 represents the initial temperature.

To set T 0, an initial solution is first generated and then 20 neigh-

bors of theinitial solution arefound. Finally, theobjective functions

of them are calculated. f min and f max are set for the minimum

and maximum changes on the objective function obtained from

neighbors, respectively. T 0 is set based on Eq. (25):

f min + 0.1(f max −f min) (25)

5. Computational results

Computational results are shown in small and large-scale cases

for the model verification and the SA results, respectively. For themodel verification, 10 test problems in small sizes are solved by

the Lingo 8.0 software. The coordination of nodes is generated

randomly in the uniform distribution [0,100] and the customer’s

demand is generated randomly in the uniform distribution [0,10].

The total vehicles capacity is considered as to 1.2 times of the total

demands of customers. To determine the time windows bound-

aries, the given problem is solved without considering customers’

time windows boundaries the maximum travel time of vehicles is

considered as U . then the value of time windows boundaries are

calculated in a = Rand (0, U −U /n) and b = a + U /n for each customer

where a and b are the lower and upper bounds of time windows,

respectively. Also, allthe solution procedures arecoded in theMAT-

LAB7 program andall thetest problems are run using the Intel Dual

Core, 2.5 GHz compiler and 1 GB of RAM.

The comparison of Lingo with the proposed algorithm shows

that our proposed algorithm can obtain approximately an optimal

solution in less time than Lingo as shown in Table 2. The average

gap between the optimal and the SA solutions is 1.288% showing

the efficiency of the proposed SA. Furthermore, increasing the size

of the problem increases the solution time of Lingo exponentially

whileit does nottangible effecton thesolution time ofthe proposed

algorithm.

To evaluate the proposed SA on large-scale problems,it is tested

on a set of 56 Solomon’s test cases [29]. In these problems and for

allof thetest problems in theoriginaldata, there are100 customers

thatare divided intosix setsof R1, R2, C1, C2, RC1 and RC2.In setsof

R1 and R2, the customer locations, demands and time windows are

randomlyand uniformlydistributed; however, theyare clusteredin

sets C1and C2.SetsRC1 andRC2 display a combinationof uniformlydistributed and their clusters.

Because of thedifference between thegiven problem (VRPCTW)

and the classical VRPTW, some assumptions are inserted into the

Solomon’s test problems for adjusting the standard test problems

with the given problem. Hence, all of the customers’ demands

are considered as time dependent demands. Also, in the classical

Table 1

SA parameters.

SA parameter settings

No. of iterations 100, 200, 500

a 0.99

EL 50


7/10


Table 2

Results in small-sized problems.

No. of problems No. of customers nv Lingo 8 SA (200 iterations)

Best solution Time (s) Best solution Time (s) Gap (%)

1 8 2 17.71 12 17.71 12.528 0.000

2 8 2 22.40 12 22.40 13.188 0.000

3 8 3 15.68 78 15.96 15.081 1.754

4 10 2 22.26 127 22.26 14.781 0.000

5 10 3 29.37 320 29.68 15.016 1.082

6 10 4 21.11 430 21.45 16.494 1.619

7 12 2 24.78 489 25.32 17.543 2.196

8 12 2 29.04 542 29.40 15.467 1.237

9 12 3 28.20 1245 28.85 18.769 2.288

10 12 4 27.08 2530 27.81 18.653 2.708

Average 23.763 578.5 24.084 15.752 1.288

Table 3

Comparison of theperformanceof theproposed SA algorithmfor a setof C .

Problem nv 100 iterations 200 iterations 500 iterations

Best solution Time (s) Gap (%) Best solution Time (s) Gap (%) Best solution Time (s)

C101 10 34.83 253.67 3.2 34.01 756.57 0.8 33.74 1373.37

C102 10 78.14 320.71 58.1 65.35 649.36 32.2 49.41 1066.51

C103 10 89.65 225.56 76.7 69.13 506.18 36.3 50.73 1292.34C104 10 93.31 291.24 50.0 80.24 626.01 28.9 62.23 1051.22

C105 10 37.35 214.26 4.8 35.80 534.22 4.8 35.65 1055.68

C106 10 33.57 225.32 0.1 34.59 628.06 3.1 33.56 1497.61

C107 10 39.88 213.77 4.1 39.03 544.60 1.9 38.30 1000.58

C108 10 50.67 228.82 32.6 43.79 476.25 14.6 38.21 938.61

C109 10 54.47 221.55 47.3 40.40 532.90 9.3 36.97 1127.72

C201 3 24.49 320.56 1.5 24.31 814.36 0.7 24.14 1786.14

C202 3 67.23 337.71 39.4 51.76 788.76 7.3 48.22 1688.32

C203 3 69.19 349.91 59.8 62.89 840.94 45.2 43.31 1909.99

C204 3 81.02 362.07 55.1 73.20 886.64 40.1 52.25 1817.24

C205 3 24.90 354.64 2.8 24.46 746.88 1.0 24.23 1785.99

C206 3 25.69 345.12 3.1 25.08 745.45 0.6 24.92 1766.83

C207 3 25.48 334.64 4.8 24.97 679.57 2.7 24.31 1761.36

C208 3 25.58 355.16 5.4 24.34 689.22 0.3 24.27 1760.13

Average 50.32 291.45 26.4 44.32 673.29 13.3 37.91 1451.74

Table 4

Comparison of theperformanceof theproposed SA algorithmfor a setof R.



R101 19 69.92 51.96 18.7 62.24 90.23 5.7 58.90 349.09

R102 17 68.57 46.23 24.3 69.05 89.48 25.2 55.17 326.19

R103 13 119.96 49.48 55.5 104.85 96.13 35.9 77.14 319.03

R104 10 171.43 62.20 46.1 145.24 120.46 23.8 117.31 356.37

R105 14 49.61 44.90 7.6 49.34 98.16 7.0 46.11 242.26

R106 12 98.36 51.33 54.0 97.78 84.84 53.1 63.87 309.63

R107 10 124.76 59.56 47.3 102.60 109.59 21.2 84.67 283.68

R108 9 167.07 63.97 59.4 148.21 120.40 41.5 104.78 737.77

R109 11 48.08 48.54 13.4 45.75 102.37 7.9 42.40 438.31

R110 10 47.49 54.33 6.4 44.76 107.88 0.3 44.62 514.65

R111 10 56.88 53.39 40.9 48.88 115.26 21.1 40.37 508.49R112 9 50.31 57.57 22.4 44.64 118.61 8.6 41.10 712.90

R201 4 46.61 152.85 39.9 37.23 325.59 11.8 33.31 692.93

R202 3 68.08 351.28 68.2 64.68 784.42 59.8 40.48 1901.95

R203 2 87.81 389.05 52.7 75.27 888.67 30.9 57.50 1660.74

R204 3 129.73 1173.53 74.6 99.01 2453.90 33.2 74.32 5522.43

R205 3 52.17 342.97 56.8 40.65 770.71 22.2 33.27 1899.95

R206 3 69.86 368.53 66.7 52.06 930.32 24.2 41.90 1691.04

R207 2 103.13 1193.48 71.9 79.91 2411.69 33.2 60.01 5384.14

R208 3 134.83 1111.53 69.4 114.32 2510.75 43.6 79.59 5612.23

R209 3 44.61 358.55 33.0 41.85 741.87 24.8 33.53 1791.94

R210 3 49.44 365.44 53.6 43.69 730.40 35.8 32.18 1868.25

R211 2 59.02 1599.35 59.9 49.60 2598.85 34.4 36.92 5515.25

Average 83.38 350.00 45.3 72.24 713.07 26.3 56.50 1679.97


8/10


Table 5

Comparison of the performanceof theproposedSA algorithmfor a setof RC .



RC101 14 63.24 95.72 10.3 59.94 208.69 4.5 57.34 300.41

RC102 12 117.86 232.49 28.4 95.43 326.71 4.0 91.77 408.58

RC103 11 158.38 178.91 35.1 134.88 433.66 15.1 117.21 508.68

RC104 10 211.79 247.03 44.9 163.53 557.77 11.9 146.11 562.94

RC105 13 56.76 120.26 11.7 51.30 202.61 0.9 50.82 370.16

RC106 11 61.31 173.38 15.1 54.35 304.10 2.1 53.25 493.11

RC107 11 65.28 172.94 23.7 54.83 317.27 3.9 52.78 542.65

RC108 10 55.75 233.54 12.2 52.40 426.42 5.5 49.69 277.23

RC201 4 48.14 181.37 35.5 39.99 383.54 12.5 35.54 940.65

RC202 3 92.68 404.86 57.4 72.52 729.11 23.1 58.89 1867.27

RC203 3 111.23 382.96 46.8 97.61 747.02 28.9 75.74 1786.43

RC204 3 135.58 371.83 47.4 116.32 719.33 26.5 91.96 2000.79

RC205 4 49.93 177.45 55.4 39.78 364.43 23.8 32.12 984.50

RC206 3 62.84 359.15 81.2 51.67 806.13 49.0 34.68 1623.57

RC207 3 51.12 357.09 26.1 47.50 748.87 17.2 40.53 1858.59

RC208 3 54.46 358.92 42.1 43.05 717.19 12.3 38.33 1915.30

Average 87.27 252.99 35.8 73.44 499.55 15.1 64.17 1027.55

Fig. 6. Quality vs iteration.

VRPTW, two considered objectives are to minimize the number of

vehicles and the travel distance. However, in our presented prob-

lem, minimizing the number of the vehicles is not the problem

objective. Hence, the number of vehicles of the test problems is set

with the best solution reported in the literature that can be found

at: http://www.crt.umontreal.ca/cordeau/tabu/.

The objective function is presented in two terms. The first term

minimizes the total travel cost and the second one maximizes the

sale in a rival situation. In each test problem, two objective func-

tions arechanged into one objective function by weighting to each

term. The weights of two terms are considered 1 and 5, respec-

tively.

To evaluate the quality of the solutions, the relative gap (RG)

between the best solution (BR) obtained from the proposed SAwith the maximum iterations and the best solution (RA) ofthe pro-

posed SA within other iterations for each test problem is computed

by:

RG =RA− BR

BR × 100

To evaluate the solution quality versus the run time of the

algorithm, each test problem is solved for R = 1 00, 200 and 500

iterations.

Tables 3–5 compare the performance produced by the proposed

SA in the set of C1, C2, R1, R2, RC1 and RC2 for 100, 200 and

500 iterations. In this table, nv is the number of the considered

vehicles.

Fig. 7. Convergence rate for Problem C108.

The average gaps of the proposed SA algorithm for three sets

of problems C , R and RC in 100 iterations are 26.4, 45.3 and 35.8%,

respectively. In addition, the average gaps of the SA algorithm for

these sets of problems in 200iterations are 13.3%, 26.3% and 15.1%,

respectively. In the set of RC , the coverage speed to a good solu-

tion is faster than other sets. The best results are obtained for 500

iterations in allthe sets. The relative gap for 500 iterations are zero,

because the best solutionsobtained within 500 iterations aremuchbetter than the best solutions obtained for 100 and 200 iterations.

Thus, there is no need to illustrate the “Gap” column for 500 itera-

tions. The relative gaps for 100 and 200 iterations are computed

according to the comparisons results with 500 iterations. Fig. 6

shows the convergence rate of the proposed algorithm in different

sets. The algorithm run time is acceptable for solving test prob-

lems and the maximum run time in 100, 200 and 500 iterations

are 1599.35 and 2598.85 and 5612.23, respectively for Problem

R211.

Also, convergence rate for Problem C108 is shown in Fig. 7. The

percentage of the improved resultfrom 100iterations to 200% iter-

ationsis 15.7%.Finally,the problem is solvedfor Problem C108 with

100, 200 and 500 iterations, and a schematic route is also shown in

Fig. 8.

http://www.crt.umontreal.ca/cordeau/tabu/http://www.crt.umontreal.ca/cordeau/tabu/


9/10


Fig. 8. Schematic routesfor Problem C108 with 100, 200and 500 iterations.

6. Conclusion

This paper has presented a special type of the vehicle routing

problem with time windows (VRPTW), in which a competition

exists between distributors. In distribution of the products with

short life time that customersneed special devicefor keeping them,

reaching time to customers influences on the sales amount which

the classical VRPs are unable to handle these kinds of assumptions.

In addition, a new mathematical model related to this problem

has been presented to find the short routes with the minimum

travel cost of fleet, traveled distancesand maximum sale fora com-

pany. To verify the solution technique, 10 test problems have been

solved by the Lingo 8.0 software and the related results obtained

by the proposed simulated annealing (SA) have been very efficient

approaching to the optimal solution. For small sizes, the average

gap between the proposed SA and Lingo solutions has been equal

to 1.288% showing an acceptable result. The proposed SA has been

used for solving the presented model for large-scale instances.

These solutions have shown that the presented model has been

verified andthe proposed SA has considered as a suitable approach

to obtain highqualitysolutionswith reasonablycomputational cost

and time.

Acknowledgement

The first author is grateful for the partially financial sup-

port from the University of Tehran under the research Grant No.

8106043/1/14.

References

[1] Dantzig G, Ramser JH. The truck dispatching problem. Management Science1959;6:80–91.

[2] Clarke G, Wright JW. Scheduling of vehicles from a central depot to a numberof delivery points. Operations Research 1964;12:568–89.

[3] Laporte G, Mercure H, Nobert Y. A branch and bound algorithm for a classof asymmetrical vehicle routing problems. Journal of Operational Research

Society 1992;43:469–81.[4] Christian HC, Lysgaard J. A branch-and-price algorithm for the capacitatedvehicle routing problem withstochastic demands.OperationsResearch Letters2007;35(6):773–81.

[5] Lenstra JK, Rinnooy Kan AHG. Complexity of vehicle routing and schedulingproblems. Networks 1981;11:221–7.

[6] Zachariadis EE, Kiranoudis CT. A strategy for reducing the computationalcomplexity of local search-based methods for the vehicle routing problem.Computers & Operations Research 2010;37(12):2089–105.

[7] Fisher ML. Optimal solution of vehicle routing problems using minimum K-trees. Operations Research 1994;42:626–42.

[8] Golden B, Laporte G, Taillard E. An adaptive memory heuristic for a class of vehicle routing problems with minmax objective. Computers and OperationsResearch 1999;24:445–52.

[9] TaillardED. A heuristiccolumn generation method for the heterogeneous fleetVRP. Rairo OR 1999;33(1):1–14.

[10] Chiang WC, Russell R. Simulated annealing meta-heuristics for the vehiclerouting problem with time windows. Annals of Operations Research 1996;93:3–27.

[11] Koulamas C, Antony S, Jaen R. A survey of simulated annealingapplications tooperations research problems. Omega 1994;22(1):41–56.

[12] Thangiah SR,Osman IH, Sun T, Hybrid genetic algorithm, simulated annealingand tabu search methods for vehicle routing problems with time windows,TechnicalReport 27, ComputerScience Department, Slippery Rock University;1994.

[13] Gendreau M, Hertz A, Laporte G. A tabu search heuristic forthe vehicle routingproblem. Management Science 1994;40:1276–90.

[14] GendreauM, Laporte G, Musaraganyi C, TaillardED. A tabu searchheuristic forthe heterogeneous fleet vehicle routing problem. Computers and OperationsResearch 1999;26:1153–73.

[15] Yu B,YangZ-Z, Yao B. An improved ant colonyoptimizationfor vehicle routingproblem. European Journal of Operational Research 2009;196(1):171–6.

[16] Gajpal Y, Abad P. An ant colony system (ACS) for vehicle routing problemwith simultaneous delivery and pickup. Computers and Operations Research2009;36(12):3215–23.

[17] Tavakkoli-MoghaddamR, Saremi AR,ZiaeeMS. A memetic algorithmfor a vehi-cle routing problem with backhauls. Applied Mathematics and Computation2006;181:1049–60.

[18] Tavakkoli-Moghaddam R, Safaei N, Gholipour Y. A hybrid simulated annealingfor capacitated vehicle routing problems with the independent route length.Applied Mathematics and Computation 2006;176:445–54.

[19] Golden BL, Assad AA, editors. Vehicle routing: methods and studies. Amster-dam, The Netherlands: Elsevier; 1988.

[20] Solomon MM. On theworst-case performance of some heuristics forthe vehi-cle routing and scheduling problem with time window constraints. Networks1986;16:161–74.

[21] Ombuki B, Ross BJ, Hanshar F. Multi-objective genetic algorithm for vehiclerouting problem with time windows. Applied Intelligence 2006;24:17–30.

[22] GeigerMJ. Multi-criteria und Fuzzy Systeme in Theorie und Praxis. In:A com-putational study of genetic crossover operators for multi-objective vehicleroutingproblemwith softtime windows.Deutscher Universities-Verlag;2003.p. 191–207.

[23] Cordeau JF, Desaulniers G, Desrosiers J, Solomon MM, Soumis F, The VRP withtime windows. In: Toth P, Vigo D, editors. The vehicle routing problem, SIAMMonographs on Discrete Mathematics and Applications, Vol. 9, Philadelphia,PA;2002. p. 157–194.


10/10


[24] Qureshi AG, Taniguchi E, Yamada T. An exact solution approach for vehi-cle routing and scheduling problems with soft time windows. TransportationResearch—Part E 2009;45:960–77.

[25] TaillardE, Badeau P,GuertinF, GendreauM, PotvinJ. A tabu searchheuristic forthe vehicle routing problem with soft time windows. Transportation Science1997;31:170–86.

[26] Tavakkoli-MoghaddamR, Safaei N, Shariat MA. A multi-criteriavehicle routingproblem with soft time windows by simulated annealing. Journal of IndustrialEngineering—International 2005;1(1):28–36.

[27] Braysy O, Gendreau M. Vehicle routing problem with time windows, partI: route construction and local search algorithms. Transportation Science2005;39:104–18.

[28] Solomon MM, Desrosiers J. Time window constrained routing and schedulingproblems. Transportation Science 1988;22(1):1–13.

[29] Solomon MM. Algorithms for the vehicle routing and scheduling prob-lem with time windows constraints. Operations Research 1987;35:254–65.

Documents

A New Mathematical