Two Phase Algorithm for Solving VRPTW Problem

Bhawna Minocha & Saswati Tripathi
International Journal of Artificial Intelligence and Expert Systems (IJAE), Volume (4): Issue (1): 2013 1
Two Phase Algorithm for Solving VRPTW Problem
Bhawna Minocha bminocha@amity.edu
Amity School of Computer Sciences
Noida, India
Saswati Tripathi saswati@iift.ac.in
Indian Institute of Foreign Trade
Kolkata, India
Abstract
Vehicle Routing Problem with Time Windows (VRPTW) is a well known NP hard combinatorial
scheduling optimization problem in which minimum number of routes have to be determined to
serve all the customers within their specified time windows. Different analytic and heuristic
approaches have been tried to solve such problems. In this paper we propose a two phase
method which utilizes Genetic algorithms as well as random search incorporating simulated
annealing concepts to solve VRPTW problem in various scenarios.
Keywords: Vehicle Routing Problem with Time Windows, Genetic Algorithm, Random Search
Algorithm, Simulated Annealing
1. INTRODUCTION
The Vehicle Routing Problem (VRP) is a well known NP-hard combinatorial optimization problem.
It arises in distribution systems that usually involve scheduling in constrained environment. It lies
at the heart of distribution management. It is faced each day by companies and organizations
engaged in the delivery and collection of goods and people. Much progress has been
accomplished since the publication of the first article on the “truck dispatching” problem [23].
Several variants of the basic problem have been proposed and different formulations developed.
Exact solutions techniques as well as numerous heuristics have also been developed and tried
for solving the vehicle routing problems.
In VRP a fleet of vehicles must visit a number of geographically scattered customers. All vehicles
start and end at a common home base, called the depot. Each customer must be visited exactly
once. The cost of traveling between each pair of customers and between the depot and each
customer is given. Additionally each customer demands a certain quantity of goods delivered
(known as customer demand). Each vehicle has an upper limit on the amount of goods that it can
carry (known as the capacity). All vehicles are of the same type and hence have the same
capacity. The problem is for a given scenario to plan routes for each vehicle, such that all routes
start and end at the depot, each customer is served by exactly one vehicle, the total demand at
locations enroute do not exceed the carrying capacity of the vehicle in a way that the overall cost
of the routes are minimized. The Vehicle Routing Problem with Time Windows (VRPTW) is a
generalization of VRP. VRPTW is one of its variant amongst Capacitated VRP, Multiple Depot
VRP, VRP with Backhauls and VRP with Pick-up and Deliveries.
VRPTW is VRP with additional restriction that each customer has a time window. The vehicle can
visit the customer in this specified time window only. It does not accept a vehicle after the latest
time specified in the time window. However, if a vehicle arrives at such a destination prior to
earliest specified time of the window, vehicle must wait and service will not start until the time
window of the customer actually opens. Figure 1 shows a graphical model of VRPTW and its
solution.

The objective of VRPTW is to service all the customers as per their requirement while minimizing
the number of vehicles required as well as the total travel distance by all the vehicles used
without violating capacity constraints of the vehicles and the customers’ time window
requirements such that each customer is visited only once by one of the vehicles. All the routes
are to start and ultimately end at the originating depot.
VRPTW arise in a variety of real life situations. VRPTW arises in milk float, mail delivery, school
bus routing, solid waste collection, heating oil distribution, newspaper distribution, transportation
of persons, parcel pick-up and delivery, dial-a-ride systems, airline/railway fleet routing and
several such other situations.
FIGURE 1: Typical output for VRPTW.
VRPTW is NP-hard. It has been extensively investigated in recent years using analytic
optimization techniques, heuristics and meta-heuristics approaches. The term heuristics and
meta-heuristics are usually associated with Random Search based techniques, Simulated
Annealing based techniques, Tabu Search, Genetic Algorithms, Particle Swarm Optimization,
Ant Colony Optimization, cross-entropy, stochastic approximation, multi-start clustering
algorithms.
The available literature on VRPTW can be broadly divided into two categories: exact optimization
and heuristic algorithms. Using exact optimization techniques, [10], [4], [11], [12] obtained
significant improvements in Solomon's benchmark problem. Survey of the VRPTW literature by
heuristics has been given by [3], [14]. Tabu search is used by [5], [25] to solve these problems
whereas [7], [26], [27] considered ant colony optimization approach. Large neighborhood search
(LNS) is applied by [18]. This was extended by [17] as Adaptive-LNS approach to solve VRPTW
problems. Parallelization of a two-phase metaheuristic technique proposed by [9] for solving
VRPTW. A complete survey of the VRPTW literature, which includes methods of both the
categories, is given by [6].
The Genetic Algorithm (GA) approach was proposed by [8] in 1975. It is an adaptive heuristic
search method that mimics evolution through natural selection. It works by combining selection,
crossover and mutation operations of genes. Genetic Algorithm and random search heuristics
have been frequently used for solving combinatorial optimization problems which are
nondeterministic polynomial hard or nondeterministic polynomial complete. Whereas genetic
algorithms search for the optimal solution in a large region of search space making use of genetic
operators, random search based techniques usually achieve this in conjunction with annealing
approach.
GA approach was first used by Blanton et al. [2] to solve VRPTW. They hybridized GA approach
with a greedy heuristic. A cluster-first, route-second method using genetic and local search
optimization was used by [22] and GENEROUS by [15]. A multi-objective representation of
VRPTW using pareto-ranking was used by [13]. [1], [21] and many more also used GA for solving
VRPTW problems.
1
4
7
2
3
Depot
6
5
Route2
Route1

Genetic Algorithms have been frequently used for solving VRPTW problem which is NP-hard.
However it has been noticed that the techniques do not guarantee a near optimal solution in each
case. In some cases the yielded solution is far from the global optimal. Keeping this in view an
attempt has been made in the present study to first search for the optimal solution of the VRPTW
problem using conventional genetic algorithm approach and once the solution is achieved to
perturb it randomly in annealing type manner to see if a still better solution can be achieved.
The present study is organized as under: In section 2, we describe the proposed two phase
optimization technique for solving VRPTW. Section 3 describes the details of genetic algorithm
approach adopted by us for solving VRPTW. Use of random search approach in annealing type
manner for further improvement of the obtained results is next explained in section 4. Application
of the technique on selected 30 problems taken from [19] benchmark set of problems is
considered in section 5. Conclusions based on present study are finally drawn in section 6.
2. PROPOSED ALGORITHM FOR VRPTW
The proposed algorithm for solving vehicle routing problem with time windows works in two
phases. In the first phase it uses a genetic search based algorithm described in section 3 to
generate an optimal solution. In the next phase it perturbs iteratively the optimal solution obtained
in Phase I in a random search manner incorporating annealing concept to see if a still better
solution is possible or not. The proposed algorithm works as under:
Phase I: Genetic Algorithm phase
i) Build an initial population and evaluate fitness value of each individual in the population.
ii) Find the individual which has best fitness value.
iii) Generate new individuals by applying genetic operators to suitably selected members
from the current population.
iv) Evaluate fitness of new individuals and place them in new population.
v) Repeat step (ii) to (iv) till specified stopping criterion satisfied.
vi) Store the individual with the best fitness value.
Phase II: Random Search Phase
i) Use the best solution obtained in phase I as current solution.
ii) Randomly perturb the current solution to obtain a new solution and evaluate the fitness.
iii) Is new solution better than the current one? If yes, replace current solution by the new
one else replace current solution by the new one in annealing type manner.
iv) Repeat step (ii) and (iii) iteratively a specified number of times.
The final solution achieved in phase II is the desired optimal solution. A special feature of the
proposed two phase algorithm is that the Phase II does not start random neighbourhood search
from an arbitrarily chosen solution but with the best solution achieved in Phase I using genetic
algorithm approach. So there are great chances of the obtained solution being better than the one
provided by genetic algorithm approach in case such a solution exists.
3. GENETIC ALGORITHM PHASE
In this phase an initial population is first generated. After building the initial population, all
individuals are evaluated according to the fitness criteria. The evolution continues with

tournament selection in which good individuals are selected for reproduction. In each generation
two best individuals are preserved for the next generation without being subjected to genetic
operations. Crossover and mutation operations are then applied to modify the suitably selected
individuals to form new feasible individuals for the population. The detailed description of genetic
algorithm used to solve vehicle routing problem with time windows is as under.
3.1 Chromosome & Individual Representation
The chromosomes in genetic algorithms are often represented as a fixed-structure bit string, for
which the bit positions are assumed to be independent and context insensitive. However, such a
representation is not suitable for VRPTW, which is an order-oriented NP-hard optimization
problem where specific sequences among customers are essential.
For solving VRPTW, the representation that we have used is same as used by [24]. The
representation is as follows: Each customer is assigned a unique integer identifier i, where i ε N.
An individual, which is a collection of chromosomes, represents a complete routing solution.
Depending on how the customers are routed and distributed, every individual can have different
number of routes for the same data set.
Each chromosome represents a route, which is variable in length. It contains a sequence of
customers in the order in which they are visited by the vehicle. A different vehicle is needed for
every chromosome of the individual. Every individual and every route ensure to be feasible, in
terms of capacity and time window constraints. The central depot is not considered in this
representation, because all routes necessarily start and end at the depot. However time from the
depot to first customer of the route and time from last customer of the route to depot is taken into
account. Figure 2 represents an illustrative complete routing solution for a problem instance with
25 customers, consisting of 3 routes which are served by 3 vehicles; genetically we call each
route a chromosomes and complete solution an individual.
FIGURE 2: An individual with 3 chromosomes (vehicles / routes). It represents a solution for a problem
instance with 25 customers.
3.2 Initial Population
An initial population is built such that each individual (which represents a complete routing
solution in our case) is a feasible solution. In other words every individual and every
chromosome/route in the selected population satisfies time window and capacity constraints. The
first feasible solution is generated using Push Forward Insertion Heuristics (PFIH) introduced by
[19]. This method has been frequently used in literature. Details of this method are available in
[22]. Rest of the solutions of initial population are generated by selecting the customers in a
random manner and inserting them in an existing route, if such a possibility exists, otherwise a
new route is created. Any customer that violates any constraint is deleted and a new route is
added to serve such a customer. This process is repeated until all the customers get served and
a feasible initial population has been generated.
3.3 Fitness
The fitness function measures the quality of the represented solutions. As soon as all the
individuals have been created, they are ranked as per their fitness. It is commonly taken as the
5 - 24 – 11 – 16 – 10 -7 – 15 – 17 – 12 – 4
20 – 3 -18 -1 – 21
14 -13 - 9 – 6 -19 – 8 -25 -23- 22 -2
Individual
Chromosome

objective function of the optimization problem but it may not necessarily coincide with it. For
solving VRPTW using the proposed algorithm following fitness criteria’s have been considered:
1. Inverted Distance (INVD): Inverse of the total traveled distance is used to calculate
the fitness of the individuals.
2. Distance Traveled and Number of Routes (DR): In this case we minimize the total
distance traveled keeping at the same time the number of vehicles as low as
possible as each route requires one vehicle to operate it.
3. Number of Routes and Distance traveled (RD): Same as (2) however with the
priorities interchanged. First priority is to reduce the number of routes and second
priority is to reduce total distance traveled.
4. Weighted Sum Method (WM): In this case effort has been made to minimize
weighted sum of two objectives. For VRPTW, the weighted sum objectives which
has been minimized is:
F(x) = α · V + β · TD
where α and β are suitable weight coefficients associated with total number of
vehicles, V and total distance traveled, TD by vehicles. The weight values of the coefficients used
have to be established empirically (we have used α = 100 and β = 0.001) for this study.
3.4 Selection and Elitism
In selection, parents are selected for crossover. There are many methods proposed in the
literature for this. In this study, an x-way tournament selection procedure has been used. Here x
individuals are randomly selected and then the individual with highest fitness is declared the
winner. This process is repeated until the number of selected individuals equals the number
necessary for crossover. In this study, tournament size, i.e. x has been taken to be 3.
In the elitism process the good individuals are retained for reproduction. This ensures that the
best solution obtained from the present population is copied unaltered in the next population. We
replace the two worst individuals in the new population with the best two individuals of the parent
population.
3.5 Crossover and Mutation
The classical single/double point crossover operators are relevant to string entries that are order
less. They put two integer/binary strings side by side and make a cut point (or two cut points) on
both of them. A crossover is then completed by swapping the portions after the cut point (or
between the cut points) in both the strings. However it is not appropriate for scheduling problems
like TSP or VRP where sequence or order among the integers is very important because
duplication and omission of vertices can produce infeasible sequences in the offspring. Therefore
in present study Route-Exchange crossover is used. Earlier this crossover has been applied by
[21], [13]. However they choose the best route according to the objectives for crossover. In the
present study, once a pair of individuals is selected for crossover, efforts are made to exchange a
route that has minimum number of nodes in each of the two individuals. To ensure that all
individuals are feasible routing solutions after crossover any duplication is deleted.
Mutation is necessary for inserting new characteristics that are not present in the current
individuals. Without mutation the search gets limited to a very small area in the feasible region. In
present study effort has been made to achieve this by transferring customers from a route that
has minimum number of customers to other routes if feasible so that such a route gets deleted (if
possible) and the number of routes reduced.

4. RANDOM SEARCH PHASE
The application of random search concept incorporating simulated annealing in Phase II of the
algorithm to solve the VRPTW is as follows: Initially the best solution, say S of the Phase I of the
genetic algorithm is taken as the starting current solution. From this a new solution S′ is obtained
by randomly moving a customer of the current solution from one route to another route where it is
feasible. The new solution if it decreases the objective function value (or leaves it unchanged) is
accepted to replace the current solution; else it is accepted to replace the current solution in
annealing type manner. More precisely, the new solution S′ is accepted as the new current
solution if ∆E ≤0, (where ∆E = Fitness (S′) - Fitness(S)) else to ensure the search to escape a
local optimum, solutions that increases the objective function value are accepted if
exp (-∆E /Tk) > θ (4.1)
where θ is a randomly selected number between [0, 1], if ∆E > 0, where Tk is a parameter called
the “temperature”. (The value of Tk is gradually decreased from a relatively large value to a small
value close to zero. These values are controlled by a cooling schedule which specifies the initial
and temperature values at each stage of the algorithm. Eqn. (4.1) implies that in a minimization
problem large increases in objective function, so called uphill moves, are more likely to be
accepted when Tk is high. As Tk approaches zero most uphill moves are rejected.
In order to achieve good optimization results the simulated annealing metaheuristic has to be
adjusted specific to the problem. With respect to the VRPTW following parameters have been
taken into account:
a) Initial temperature of annealing T0: If T0 is too high then almost all new solutions are
accepted and the search produces a series of random solutions. When T0 is too low, very
few movements are allowed which reduce the scope of the search. After a series of
experiments we decided to set constant initial temperature T0 =100.
b) Cooling schedule: One of the most popular temperature reduction functions is by
[28]. It is based on geometric reduction Tk+1 = Tk* γ. In the present work we have set
parameter γ as 0.96.
c) Number of annealing steps executed in each temperature: It is usually related to
the size of a solution neighbourhood. In our computational experiments the numbers of
annealing steps are set as n
2
, where n is the number of customers in the problem
considered.
d) Termination condition: We set the termination criteria as T0 =0.
Contrary to the classical approach in which a solution to the problem is taken as the last solution
obtained in the annealing process, we memorize the best solution found during the whole
annealing process and record it as the best solution found. The algorithm has been coded in C++
and run on an Intel(R) Core(TM) 2 Duo 2.0 GHz for solving chosen test problems.
5. NUMERICAL RESULTS
In this section we present our computational experience of using the proposed algorithm in
solving a set of benchmark test problems selected from Solomon’s [19] set of problems.
In the present study 30 problems have been chosen from Solomon’s set of problems. Of these 15
problems are of 25 customers, 10 problems of 50 customers and 5 problems of 100 customers.
Each problem was solved ten times using developed algorithm. The best and the worst results
obtained in each problem in ten trials are listed in the tables. We first applied the two phase
algorithm considering the vehicle routing problem as a single objective problem, thus applied the
fitness criteria INVD. Results obtained in phase I and their refinement after the phase II is shown in
TABLE 1 for 25 customer problems, TABLE 2 for 50 customer problems and TABLE 3 for 100

customer problems. Out of ten solutions obtained in each case the Best solution and Worst
solution of each problem after completion of phase I and phase II are listed in these tables.
Stopping criterion used for phase I is a maximum of 1000 generations or when no improvement is
observed in the objective function value of the best individual in consecutive 100 generations. In
the Phase I, the population size was taken as 100; crossover rate was taken to be 0.80 and
mutation rate 0.20.
However VRPTW is in reality a multi objective problem in which we went to minimize the number
of routes used as well as the total distance traveled. Thus we have used the same two phase
algorithm to solve VRPTW considering it as a multi-objective problem in order to check if
performance in both objectives can be simultaneously improved. The performance of proposed
algorithm is tested on the same set of 30 benchmark problems. TABLE 4, 5 and 6 shows the
results obtained after phase II for the three multiobjective criteria’s DR, RD and WM for 25
customer, 50 customer and 100 customer problems respectively.
Tables also present a summary of the results obtained and their comparison with the best-known
solutions available to us from literature. Bold numbers in tables indicate that the obtained solution
are same as the best-known or has yielded an improvement on the currently best known solution
in the literature.
5.1 Discussion on the Results
Results presented in the TABLE 1, 2 and 3 shows the results after completion of genetic algorithm
Phase I and random search based Phase II. For 25 customer problems there are only four cases
out of fifteen in which no further refinement in the solutions is seen after Phase I. For 50 customer
problems there are two such cases out of ten and for 100 customer problems there is one such
case out of five. In all out of thirty problems, in twenty three cases Phase II refines solution
obtained after Phase I.
For 25 customer problem the proposed algorithm has produced new improved results in six
problems out of fifteen problems investigated as shown in TABLE 1. The obtained results require
less number of vehicles, however at the expense of additional distance to be traveled. In TABLE 2
for 50 customer problems; five problems from set R2 show improvement while one problem from
each C1 and C2 set have identical solution with the best-known solutions. In R201 problem
obtained results reflect significant decrease in the requirement of number of vehicles, (maximum
reduction is two and minimum is one) than the number of vehicles used in best-known solutions.
In case of 100 customer problems proposed algorithms provide improved results in one problem
out of five considered as shown in TABLE 3 while another problem provide solution identical with
the best-known solution available in the literature. In this case the improvement is in terms of less
total distance traveled than the best-known solution. However, it needs one extra vehicle than the
number of vehicles used in the best-known. In the remaining problems solutions obtained are
either identical to the best known solutions or within 10% of the best-known solution. Figure 3
shows the geographical representation of solution obtained for problem C101 for 100 customers.
TABLE 4, 5 and 6 shows the results obtained after phase II for the three multiobjective criteria’s
DR, RD and WM for 25 customer, 50 customer and 100 customer problems respectively. In
TABLE 4 for 25 customer problems the proposed algorithm has yielded either improved results in
terms of lesser number of vehicles required than the best known solutions or identical solutions
with the best-known solutions in ten problems out of selected fifteen problems. In some cases
such as RC203 and RC206 the improvement is quite significant as it decreases the need of
number of vehicles by one to two vehicles compared to the number of vehicles used in the best-
known solutions. However this improvement in terms of lesser number of vehicles required is at
cost of more distance to be traveled. In case of remaining five problems obtained solutions are
within 10% of the best-known solutions. A comparison of the results obtained with three
objectives shows that objectives DR and WM provide better results than objective RD.

FIGURE 3: Geographical representation of solution obtained for problem C101 for 100 customers.
In case of 50 customer problems for multiobjective criteria’s DR, RD and WM results obtained are
presented in Table 5. The use of the two phase algorithm has provided six improved solutions
with all the three multi-objective criteria’s. Also DR criterion has produced solutions identical with
the best-known in two cases. Thus in all out of ten selected problems, eight problems either
yielded improved results in terms of lesser number of vehicles required than the best known
solutions. However this improvement is at the expense of extra distance to be traveled.
(Maximum reduction in the number of vehicles required is three and minimum one.) In remaining
two problems solutions are within 10% of the best-known solution. Performance wise the three
objectives may be ranked as WM, DR and RD, in that order. In the case of 100 customer
problems for multiobjective criteria’s results are shown in TABLE 6; proposed algorithms
produced improved results in one problem out of five considered while one problem yielded
identical solution as the best-known solution available in literature. In the present case the
improvement is in terms of less total distance traveled than the best-known solution. However, it
needs one more vehicle than the number of vehicles used in the best-known. In remaining three
problems solutions are within 10% of the best-known solution. Performance wise WM objective is
again superior to the remaining two. Figure 4 shows the geographical representation of solution
obtained for problem R105 for 100 customers.
Lesser number of vehicles means saving in terms of needed man power requirement and vehicle
maintenance cost. It also means less chaos at the depot. However, solutions which reported
lesser number of vehicles reported more distance to travel, adding to fuel cost. However fuel cost
is comparatively less expensive than arranging additional vehicles and man power needed to
operate them. Moreover, it is easy for the management to handle the things with less number of
vehicles.
6. CONCLUSIONS
Vehicle routing problem with time windows involves the optimization of routes for multiple
vehicles so as to meet all constraints and to minimize the number of vehicles needed and total
distance traveled. The proposed two phase algorithm is based on genetic algorithm and random
search incorporating simulated annealing concept. Performance of proposed algorithms is
comparable to those available in literature and in some cases even better in terms of number of
vehicles which means less fuel, manpower and vehicle maintenance cost with more distance to
travel. We solved each problem ten times and have presented the best and worst achieved
solutions to indicate the range of variations in the solution obtained. The variation of the best
solution from worst solution is generally in the range of 20 to 40 percent indicating thereby that it
is advisable to solve a problem more than once to achieve the best results. This is due to the
probabilistic nature of the techniques which search a near optimal solution. As for future work, it
may be interesting to test proposed algorithm on some application of VRPTW.

FIGURE 4: Geographical representation of solution obtained for problem R105 for 100 customers.
INVD
After Genetic Algorithm Phase After Refining the best
obtained from GA with SA in
Phase II
Proble
m
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles /
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
Known
[Ref.]
(No. of
Vehicles/
distance
traveled)
C201 2/
237.15
2/
282.25
2/
214.7
2/
250.56
2/ 214.7
[11]
R101 8/
625.03
9/
661.34
8/
623.52
9/
645.20
8/ 617.1
[11]
R102 7/
591.51
7/
628.93
7/
586.05
7/
610.46
7/ 547.1
[11]
R105 6/
569.70
6/
587.76
5/
559.84
6/
575.00
6/ 530.5
[11]
R109 5/
522.05
5/
558.03
5/
480.23
5/
554.98
5/ 441.3
[11]
RC105 4/
412.38
4/
506.66
4/
412.38
4/
495.34
4/ 411.3
[11]
RC106 3/
364.97
4/
459.10
3/
364.97
4/
431.43
3/ 345.5
[11]
RC201 3/
385.81
3/
454.50
3/
385.81
3/
452.94
3/ 360.2
[12]

RC202 3/
361.64
3/
465.04
3/
361.64
3/
433.73
3/ 338.0
[10]
RC203 2/
468.98
3/
482.96
2/
411.47
3/
441.22
3/ 326.9
[4]
RC204 2/
438.45
2/
475.28
2/
401.05
2/
452.17
3/ 299.7
[4]
RC205 3/
399.10
3/
482.34
3/
363.46
3/
432.66
3/ 338.0
[12]
RC206 2/
526.33
2/
548.06
2/
503.28
2/
528.46
3/ 324.0
[10]
RC207 2/
477.63
2/
562.91
2/
433.23
2/
533.65
3/ 298.3
[10]
RC208 1/
440.80
1/
500.75
1/
427.26
1/
483.79
2/ 269.1
[4]
TABLE 1: Results generated by Two Phase Algorithm for Solomon’s 25 customers set Problems
(considering VRPTW as single-objective optimization problem).
INVD
After Genetic Algorithm Phase After Refining the best obtained
from GA with SA in Phase II
Proble
m
Best
(No. of Vehicles/
distance
traveled)
Worst
(No. of Vehicles/
distance
traveled)
Best
(No. of Vehicles/
distance traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
Known
[Ref.]
(No. of
Vehicles/
distance
traveled)
C101 5/
385.90
6/
444.27
5/
362.5
6/
411.30
5/ 362.5
[11]
C201 3/
408.73
3/
390.26
3/
390.26
3/
434.72
3/ 360.2
[12]
C205 3/
389.83
3/
459.36
3/
360.2
3/
435.24
3/ 360.2
[12]
R101 12/
1097.97
12/
1190.42
12/
1055.56
12/
1178.27
12/ 1044
[11]
R201 5/
1108.55
5/
1262.96
5/
1099.81
5/
1206.59
6/ 791.9
[10]
R202 4/
1127.16
4/
1150.84
4/
1059.12
4/
1149.09
5/ 698.5
[10]
R203 4/
1097.14
4/
1172.92
4/
1047.22
4/
1125.25
5/ 605.3
[4]

R206 3/
1023.04
3/
1179.46
3/
949.08
3/
1167.75
4/ 632.4
[4]
R209 3/
1165.67
3/
1208.34
3/
1145.18
3/
1198.22
4/ 600.6
[10]
RC101 8/
981.96
9/
1024.66
8/
981.96
9/
1056.65
8/ 944
[11]
INVD
After Genetic Algorithm Phase After Refining the best
obtained from GA with SA in
Phase II
Problem
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
Known
[Ref.]
(No. of
Vehicles/
distance
traveled)
C101 10/
838.07
11/
1054.11
10/
828.94
11/
1022.68
10/ 828.94
[16]
R101 21/
1788.7
21/
1877.92
20/
1736.60
21/
1869.44
19/ 1650.8
[16]
R102 18/
1516.67
19/
1664.11
18/
1516.67
19/
1661.76
17/ 1486.12
[16]
R105 15/
1391.85
17/
1546.56
15/
1380.05
17/
1539.5
14/ 1377.11
[18]
RC101 15/
1775.79
18/
1960.92
15/
1640.98
18/
1953.77
14/ 1696.94
[20]
DR RD WM
Proble
m
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
Known
[Ref.]
C201 2/
214.7
2/
250.56
2/
214.7
2/
250.56
2/
214.7
2/
282.25
2/ 214.7
[11]

R101 8/
618.33
8/
665.83
8/
623.52
8/
677.80
8/
629.95
8/
651.44
8/ 617.1
[11]
R102 7/
586.00
7/
620.21
7/
590.41
7/
668.86
7/
584.20
7/
628.93
7/ 547.1
[11]
R105 5/
588.57
6/
628.72
5/
591.89
6/
609.90
5/
556.72
6/
612.57
6/ 530.5
[11]
R109 5/
480.09
5/
575.80
5/
483.94
5/
557.77
5/
461.71
5/
566.09
5/ 441.3
[11]
RC105 4/
419.72
4/
508.64
4/
421.73
4/
507.84
4/
419.72
4/
506.66
4/ 411.3
[11]
RC106 3/
363.82
4/
443.43
3/
371.21
4/
455.30
3/
375.74
4/
463.63
3/ 345.5
[11]
RC201 2/
519.35
2/
621.95
2/
519.35
2/
612.85
2/
496.37
2/
608.03
3/ 360.2
[12]
RC202 2/
478.37
2/
560.42
2/
509.95
2/
597.46
2/
478.84
2/
554.53
3/ 338.0
[10]
RC203 2/
419.02
1/
524.38
2/
568.73
2/
427.57
1/
524.38
2/
572.42
2/
441.76
1/
521.53
2/
525.48
3/ 326.9
[4]
RC204 2/
382.41
2/
485.68
2/
391.66
2/
483.27
2/
386.53
2/
485.59
3/ 299.7
[4]
RC205 2/
549.29
2/
697.92
2/
537.83
2/
681.46
2/
546.23
2/
653.75
3/ 338.0
[12]
RC206 2/
497.02
1/
565.59
1/
648.04
2/
510.77
1/
566.86
1/
664.53
2/
483.93
1/
565.59
1/
653.26
3/ 324.0
[10]
RC207 2/
433.76
2/
558.87
2/
435.32
2/
560.72
2/
424.39
2/
562.91
3/ 298.3
[10]
RC208 1/
420.53
1/
483.79
1/
425.94
1/
500.75
1/
427.78
1/
504.51
2/ 269.1
[4]
TABLE 4: Results generated by Two Phase Algorithm for Solomon’s 25 customers set Problems (considering
VRPTW as multi-objective optimization problem).

(considering VRPTW as multi- objective optimization problem).
DR RD WM
Problem
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
Known
[Ref.]
C101 5/
362.5
6/
443.36
5/
372.42
6/
458.45
5/
386.27
6/
469.57
5/ 362.5
[11]
C201 2/
501.13
2/
561.43
2/
546.73
2/
561.43
2/
501.13
2/
561.43
3/ 360.2
[12]
C205 3/
360.2
3/
471.24
3/
389.83
3/
476.49
3/
385.30
3/
473.31
3/ 360.2
[12]
R101 12/
1080.71
12/
1205.93
12/
1093.91
12/
1178.27
12/
1088.71
12/
1200.00
12/ 1044
[11]
R201 3/
1220.29
3/
1337.42
3/
1222.49
3/
1344.47
3/
1208.38
3/
1338.00
6/ 791.9
[10]
R202 3/
1054.11
3/
1231.76
3/
1057.32
3/
1252.27
3/
1041.92
3/
1205.38
5/ 698.5
[10]
R203 4/
1025.96
4/
1146.74
4/
1039.82
4/
1173.38
3/
1233.18
3/
1336.97
5/ 605.3
[4]
R206 3/
959.32
3/
1197.85
3/
964.95
3/
1138.22
3/
897.43
3/
1181.75
4/ 632.4
[4]
R209 2/
1187.20
3/
1208.34
2/
1182.49
3/
1198.22
2/
1157.29
3/
1196.76
4/ 600.6
[10]
RC101 8/
982.62
10/
1043.85
8/
986.68
10/
1037.84
8/
977.51
10/
1039.26
8/ 944
[11]

(considering VRPTW as multi-objective optimization problem).
7. REFERENCES
[1] J. Berger and M. Barkaoui, “A parallel hybrid genetic algorithm for the vehicle routing
problem with time windows” Computer Operation Research, vol. 31, pp. 2037-2053, 2004.
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constraints using genetic algorithms”. Proceedings of the 5th International Conference on
Genetic Algorithms, USA, pp. 452-459, June 1993.
[3] O. Bräysy and M. Gendreau, “Vehicle routing problem with time windows, Part II:
Metaheuristics “, Transportation Science, vol. 39, pp. 119–139, 2005
[4] A. Chabrier, “Vehicle Routing Problem with Elementary Shortest Path based Column
Generation.” Computers and Operations Research, vol. 33(10), pp. 2972-2990, 2006.
[5] J. F. Cordeau, G. Laporte and A., Mercier, “A Unified Tabu Search for Vehicle Routing
Problem with Time Windows”, Journal of the Operational Research Society vol. 52 pp. 928-
936, 2001
[6] J. F. Cordeau, G. Desaulniers, J. Desrosiers, M.M. Solomon and F. Soumis, “The VRP with
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Mathematics and Applications, Toth, P. and D. Vigo (Eds.), Philadelphia, USA, pp. 157-193,
2001
[7] L. M. Gambardella, E. Taillard, G. Agazzi, " MACS-VRPTW: A Multiple Ant Colony System
for Vehicle Routing Problems with Time Windows", New Ideas in Optimization, London:
McGraw-Hill, pp. 63-76, 1999
DR RD WM
Proble
m Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
(No. of
Vehicles/
distance
traveled)
Worst
(No. of
Vehicles/
distance
traveled)
Best
Known
[Ref.]
C101 10/
829.70
11/
1021.22
10/
838.07
11/
1035.71
10/
828.94
12/
1062.10
10/ 828.94
[16]
R101 20/
1751.74
21/
1891.10
20/
1760.08
21/
1873.43
20/
1740.18
21/
1874.66
19/ 1650.8
[16]
R102 18/
1503.71
19/
1723.94
18/
1519.92
20/
1747.98
18/
1487.29
20/
1718.42
17/ 1486.12
[16]
R105 15/
1381.23
17/
1567.47
15/
1391.85
17/
1554.88
15/
1370.02
17/
1558.35
14/ 1377.11
[18]
RC101 15/
1723.18
18/
1998.63
15/
1738.55
18/
1983.91
15/
1717.31
18/
1979.89
14/ 1696.94
[20]

[8] J. H. Holland, “Adaptation in Natural and Artificial System”. Ann Arbor, Michigan: The
University of Michigan Press, 1975.
[9] J. Hömberger, H. Gehring, “A Two-Phase Hybrid Metaheuristic for the Vehicle Routing
Problem with Time Windows”, European Journal of Operations Research, vol. 162(1), pp.
220-238, 2005.
[10] B. Kallehauge, J. Larsen, and O. B. G. Madsen, "Lagrangean duality and non-differentiable
optimization applied on routing with time windows ." Computers and Operations Research,
vol. 33(5),pp. 1464-1487,2006.
[11] N. Kohl, J. Desrosiers, O. B. G. Madsen, M.M. Solomon and F. Soumis, "2-Path Cuts for
the Vehicle Routing Problem with Time Windows," Transportation Science, vol. 33 (1) , pp.
101-116 ,1999.
[12] J. Larsen "Parallelization of the vehicle routing problem with time windows." Ph.D. Thesis
IMM-PHD-1999-62, Department of Mathematical Modelling, Technical University of
Denmark,Lyngby, Denmark, 1999.
[13] B. Ombuki, B.J. Ross, and F. Hansher, “Multi-objective Genetic Algorithm for Vehicle
Routing Problem with Time Windows”, Applied Intelligence, vol. 24(1), pp. 17-30, 2006.
[14] B. Minocha and S. Tripathi, “Vehicle Routing Problem with Time Windows: An Evolutionary
Algorithmic Approach”, Algorithmic Operations Research, vol. 1(2), pp. 1-15, 2006.
[15] J. Potvin and S. Bengio, “The Vehicle Routing Problem with Time Windows part II: Genetic
Search. INFORMS Journal on Computing, vol. 8(2), pp. 165-172, 1996.
[16] Y. Rochat and E. Taillard, “Probabilistic Diversification and Intensification in Local Search
for Vehicle Routing” Journal of Heuristics vol. 1, pp. 147−167, 1995.
[17] S. Ropke and D. Pisinger, “A General Heuristics for Vehicle Routing Problems”, Computers
& Operations Research, vol. 34(8), pp. 2403-2435, 2007
[18] P. Shaw, “Using Constraint Programming and Local Search Methods to solve Vehicle
Routing Problems”, Principles and Practice of Constraint Programming, Springer-Verlag ,
pp. 417-431, 1998
[19] M. M. Solomon, “Algorithms for Vehicle Routing and Scheduling Problems with Time
Window Constraints”, Operations Research, vol. 35(2), pp. 254-265, 1987
[20] E. D. Taillard, P. Badeau, M. Gendreau, F. Guertin and J. Potvin, “A Tabu Search Heuristics
for the Vehicle Routing Problem with Soft Time Windows”. Transportation Science vol. 31,
pp. 170-186, 1997.
[21] K. C. Tan, L. H. Lee, K. Ou, “Hybrid Genetic Algorithms in Solving Vehicle Routing Problems
with Time Window Constraint”, Asia-Pacific Journal of Operation Research 18:121-130,
2001
[22] S. Thangiah, “Vehicle Routing with Time Windows using Genetic Algorithms”, In Application
Handbook of Genetic Algorithms: New Frontiers, vol. 2, Lance Chambers (Ed.): CRC Press,
Boca Raton, 1995, pp. 253-277.
[23] G.B. Dantzig and J. H.Ramser, “The Truck Dispatching Problem”, Management Science,
vol. 6 (1), pp. 80–91, 1959.
[24] G.B. Alvarenga, G.R. Mateus and G. De. Tomi, “Finding near optimal Solutions for Vehicle
Routing Problems with Time Windows using Hybrid Genetic Algorithm”, Internet:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.11.46&rep=rep1&type=pdf , 2003,[
Jan. 25, 2005]

[25] Moccia, L., Cordeau, J.F. and Laporte, G., 2011. “An incremental tabu search heuristic for
the generalized vehicle routing problem with time windows” Journal of the Operational
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[26] Chen, C. H., Ting, C.J. “ A Hybrid Ant Colony System for Vehicle Routing Problem with Time
Windows.” Journal of the Eastern Asia Society for Transportation Studies, 2005, 6:2822-
2836.
[27] Zhang, Q., Zhen, T., Zhu, Y., Zhang, W. and Ma, Z., 2010. A Hybrid Intelligent Algorithm for
the Vehicle Routing with Time Windows. 2010 International Forum on Information
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[28] P.J. van Laarhoven and E.H. Aarts, “Simulated Annealing” U.S.A : Dordrecht and Boston
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