Journal ArticleDOI

# A unified tabu search algorithm for vehicle routing problems with soft time windows

01 May 2008-Journal of the Operational Research Society (Palgrave Macmillan)-Vol. 59, Iss: 5, pp 663-673
TL;DR: A unified penalty function and a unified tabu search algorithm are presented, with which different types of VRPSTW can be solved by simply changing the values of corresponding parameters in the penalty function.
Abstract: The different ways of allowing time window violations lead to different types of the vehicle routing problems with soft time windows (VRPSTW). In this paper, different types of VRPSTW are analysed. A unified penalty function and a unified tabu search algorithm for the main types of VRPSTW are presented, with which different types of VRPSTW can be solved by simply changing the values of corresponding parameters in the penalty function. Computational results on benchmark problems are provided and compared with other methods in the literature. Some best known solutions for the benchmark problems in the literature have been improved with the proposed algorithm.

### Introduction

• The basic vehicle routing problem (VRP) calls for the determination of a set of minimum-cost routes to be performed by a fleet of vehicles to serve a given set of customers with known demands, where each route originates and terminates at a single depot.
• Therefore, the priority is given to the first objective.
• The vehicle routing problem with time windows is an extension of the basic VRP in which vehicle capacity constraints are imposed and each customer i is associated with a time interval [ai , bi ], called a time window, during which service must begin.
• In any vehicle route, the vehicle may not arrive at customer i after bi to begin service.
• (3) Setting the time windows to be soft may allow significant savings in the number of vehicles required and/or the total vehicle travel distance of all routes to be achieved.

### Notation and problem representation

• If any edges between vertices are missing in the original graph, then they are replaced by edges with an artificially high cost.
• The start time of each vehicle route is greater than or equal to a0.
• In practice, these coefficients could represent the costs of lost sales, goodwill, etc, due to the customer inconvenience for not meeting the time windows, and are therefore often called inconvenience costs.
• Chiang and Russell (2004) developed a TS solution method for the problem.
• Based on Type 2, define Pmax as the maximum allowable violation of the time windows, also known as Type 5.

### The tabu search algorithm

• The algorithm is based on tabu search, a local search metaheuristic introduced first by Fred Glover in 1986, and since then has been used to solve many practical applications.
• A thorough discussion may be found in Glover and Laguna (1997).
• The authors developed an effective TS heuristic for the open vehicle routing problem (OVRP) in a previous work (Fu et al, 2005, 2006).
• The full details of the algorithm were given in the referenced paper and so only the outline of the algorithm and the changes that have been made to apply it to the VRPSTW are described here.

### Initial solution

• An initial solution is required for any TS algorithm to start the local search process.
• With respect to the TS metaheuristic for the VRP, some researchers, such as Van Breedam (2001) and Brandão (2004), claimed that the performance of the TS heuristic was highly dependent on the quality of the initial solutions.
• In their previous work (Fu et al, 2005), it was shown that the initial solution did not have a large influence on the final solutions in the proposed TS heuristic.
• Therefore, the initial solution is generated by building up successive routes where the next customer is chosen at random and added to the end of the route unless this violates the capacity or route length constraints; in that case the route is completed back to the depot and a new route starts.

### Neighbourhood structure

• The TS algorithm presented in this paper uses the same mixed neighbourhood structure as was introduced in Fu et al (2005), but some necessary modification is made for the VRPSTW where each vehicle route must start and end at the depot.
• There are four different types of move that may be used in this mixed neighbourhood structure, and the algorithm selects a type of move randomly at each iteration.
• The four different types of move are referred to as vertex reassignment, vertex swap, 2-opt and ‘tails’ swap.
• The first and the last 0 of a solution are not allowed to be selected and removed during this 2-interchange process.

### Evaluation of solutions

• The extent of infeasibility can be measured by incorporating the vehicle capacity and maximum route length constraints into the objective function by adding a penalty if the constraints are broken.
• The tabu list and stopping criterion also follow the method described in Fu et al (2005).
• The tabu list contains the move attributes of solutions during the last 5–10 (selected randomly) iterations.
• The search is terminated if either a specified number of iterations have elapsed in total or since the last best solution was found.

### TS algorithm

• The unified TS algorithm for VRPSTW is described below.
• Generate an initial feasible solution randomly, and set this solution as the current solution and the best solution so far;.
• Set the new solution as the current solution, update the tabu list and increment iter;.
• Its simple but powerful mixed neighbourhood structure and the stochastic elements in the method allow effective diversification within a local search framework.

### Computational results and comparison

• This unified TS algorithm was coded in Delphi 7.0 and implemented on a Pentium-II PC running on 600MHz with 184MB RAM.
• In problem sets R1, C1 and RC1, the time window is narrow at the central depot so that only a few customers can be served on each route.
• The authors unified TS algorithm for three of the six main types of VRPSTW that were previously defined (Type 1, Type 2, and Type 3) was tested on the benchmark problems in the literature.

### Comparison of the results on Type 1 of VRPSTW

• The results reported were feasible solutions to the VRPHTW in each case, that is, the percentage of non-violated time windows was 100%, and the number of routes was set to the best solution reported in the literature for each problem, not minimized by the algorithm.
• The real Euclidean distances between customers are used during the computations, whereas the final results are rounded to the second decimal.
• In the table, their best solutions are compared with those produced by Taillard et al (1997) for the VRPSTW and other algorithms reported in the literature for the VRPHTW, respectively, using the format: number of routes/total travel distance.
• When the percentage of non-violated time windows is 100%, they are compared with those for the VRPHTW as well.
• An ‘H’ after ∗∗ indicates that their algorithm has improved the best published solution and after ∗ means a tie with the best published solution for the VRPHTW.

### Comparison of the results on Type 2 of VRPSTW

• The heuristic started with low penalty coefficients, which were gradually increased.
• The comparison of the results is shown in Table 3, using the format: number of routes/total travel distance, percentage of non-violated time windows.
• Among the 12 improved solutions, eight of them require a lower number of vehicle routes and four are of shorter total travel distance (for the same number of routes required and non-violated time windows).
• For the problem sets of C1, comparing with the heuristic of Koskosidis et al (1992) as well as their algorithm for Type 1 of VRPSTW, their algorithm for Type 2 of VRPSTW takes much more CPU time to find the same best known solutions, and cannot obtain the best known solutions in two instances.

### Comparison of the results on Type 3 of VRPSTW

• For Type 3 of VRPSTW, Balakrishnan (1993) described three simple heuristics, Chiang and Russell (2004) developed a TS solution method.
• The eight problems based on the R1 and RC1 sets of the Solomon benchmark problems were used to test the algorithms.
• The hard time windows in the original benchmark data were converted to soft time windows by allowing a certain percentage of time window violation, Emax = Lmax, that is, Pmax, as in Balakrishnan (1993) and Chiang and Russell (2004).
• Pmax (Emax and Lmax) and Wmax were expressed as a percentage of the maximum allowable route time duration.

### Conclusions

• The different forms of time window violation allowed lead to different types of VRPSTW.
• The existing approaches in the literature are usually designed for a special type of VRPSTW.
• Finally, to test the computational performance of the algorithm, the authors ran it on the benchmark problems and compared the results with other methods for three types of VRPSTW in the literature.
• Acknowledgements—We are grateful to the anonymous referees for their useful comments and suggestions that helped us to improve the presentation of this paper.the authors.the authors.

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Journal of the Operational Research Society (2008) 59, 663 --673
www.palgrave-journals.com/jors
A uniﬁed tabu search algorithm for vehicle routing
problems with soft time windows
ZFu
1
, R Eglese
2
and LYO Li
3
1
Central South University, Changsha, P.R. China;
2
Lancaster University Management School, Lancaster,
UK; and
3
The Hong Kong Polytechnic University, Hong Kong, P.R. China
The different ways of allowing time window violations lead to different types of the vehicle routing problems
with soft time windows (VRPSTW). In this paper, different types of VRPSTW are analysed. A uniﬁed penalty
function and a uniﬁed tabu search algorithm for the main types of VRPSTW are presented, with which different
types of VRPSTW can be solved by simply changing the values of corresponding parameters in the penalty
function. Computational results on benchmark problems are provided and compared with other methods in
the literature. Some best known solutions for the benchmark problems in the literature have been improved
with the proposed algorithm.
Journal of the Operational Research Society (2008) 59, 663 673. doi:10.1057/palgrave.jors.2602371
Published online 7 February 2007
Keywords: distribution; vehicle routing; soft time windows; heuristics; tabu search
Introduction
The basic vehicle routing problem (VRP) calls for the
determination of a set of minimum-cost routes to be performed
by a ﬂeet of vehicles to serve a given set of customers with
known demands, where each route originates and terminates
at a single depot. The total cost consists of two parts, which
are the objectives to be minimized: (1) the total number of
routes (vehicles) required to serve all customers and (2) the
total vehicle travel distance of all routes. It is assumed that
the capital cost of an additional vehicle will always exceed
any travelling cost that could be saved by its use. Therefore,
the priority is given to the ﬁrst objective. Each customer must
be assigned to only one vehicle and the total demand of all
customers assigned to a vehicle does not exceed its capacity.
The vehicle routing problem with time windows (VRPTW)
is an extension of the basic VRP in which vehicle capacity
constraints are imposed and each customer i is associated
with a time interval [a
i
, b
i
], called a time window, during
which service must begin. In any vehicle route, the vehicle
may not arrive at customer i after b
i
to begin service. If a
vehicle arrives before a
i
, it waits. When the time window
requirements are strictly enforced, the problem is also called a
VRP with hard time windows (VRPHTW). The problem may
arise in a variety of applications, including retail distribution,
school bus routing, bank deliveries, mail and newspaper deliv-
eries, municipal waste collection, fuel oil deliveries, and more.
Correspondence: Z Fu, School of Trafﬁc and Transportation Engineering,
Central South University at Railway Campus, Changsha, Hunan 410075,
P.R. China.
E-mail: zhfu@mail.csu.edu.cn
The time windows are designed to handle issues such as
regulations, and customer preferences. For example, a store
may only accept deliveries before it opens for business at 9:00
a.m. For VRPHTW, research has ﬂourished over the last two
Cordeau et al (2002).
The vehicle routing problem with soft time windows
(VRPSTW) is an extension of VRPHTW in which some or all
customer time window requirements are not strictly enforced
and can be violated by paying appropriate penalties. For each
customer i, we can deﬁne penalty functions to calculate the
penalty payable if the vehicle arrives before a
i
or after b
i
.
If a certain customer’s time window cannot be violated, that
is, it is hard, the penalty payable to that customer for any
violation is set to inﬁnity. In that respect, the VRPHTW is
a special case of the VRPSTW in which no violations are
allowed. There are many good reasons for allowing the time
windows to be soft, as stated in Koskosidis et al (1992),
Balakrishnan (1993), Taillard et al (1997), Fagerholt (2001),
and Chiang and Russell (2004), including:
(1) Many applications do not require a time window that is
precise to exact points of time. Therefore, time windows
are usually soft by nature.
(2) It is usually impossible to determine accurate vehicle
travel times in practice.
(3) Setting the time windows to be soft may allow signiﬁcant
savings in the number of vehicles required and/or the total
vehicle travel distance of all routes to be achieved.
(4) The VRPSTW model is more general and includes
the VRPHTW. An algorithm for the VRPSTW can be

664 Journal of the Operational Research Society Vol. 59, No. 5
extended to solve the VRPHTW by appropriately raising
the penalty coefﬁcients.
(5) VRPSTW can always generate feasible solutions in
instances where a hard time window approach would have
failed. Problems with hard time windows and a small
ﬂeet might not have a solution satisfying all customers.
In this case, the VRPSTW model would yield usable
solutions where some of the customers would not be
serviced within the desired time windows.
(6) The VRPSTW model can be used to ﬁnd a good trade-off
between ﬂeet size and service quality to customers.
Note that the objectives to be minimized for the VRPSTW
become three: (1) the total number of routes (vehicles)
required to serve all customers, (2) the total penalty of time
window violations, and (3) the total vehicle travel distance
of all routes.
The VRPSTW has received some attention over the last
decade. Koskosidis et al (1992) presented an optimization-
based heuristic to solve the problem. Balakrishnan (1993)
described three simple heuristics for the VRPSTW, which
were based on nearest neighbour, Clarke–Wright savings, and
space-time rules, respectively. Besides these classical heuris-
tics, Taillard et al (1997) designed a tabu search (TS) heuristic
for a special case of the VRPSTW, in which the total number
of vehicles required was given and ﬁxed. This meant that the
ﬂeet size was not a decision variable. Therefore, the objective
(1) mentioned above did not need to be minimized. Fagerholt
(2001) described a real ship scheduling problem with soft
time windows and proposed an optimization-based approach
based on a set partitioning formulation to solve the problem.
Recently, Chiang and Russell (2004) developed a TS solution
method for the type of VRPSTW that Balakrishnan (1993)
studied.
In this paper, we present a uniﬁed penalty function and
a uniﬁed TS algorithm for the different types of VRPSTW.
Experimental results are reported and compared with other
approaches.
Notation and problem representation
Let G = (V , E) be a given undirected network,
where V ={0,...,n} is the vertex set and E is the edge set.
Vertices i = 1,...,n correspond to the customers, whereas
vertex 0 corresponds to the depot.
A non-negative cost, c
ij
, is associated with each edge
(i, j ) E and represents the travel time (distance) spent
from vertex i to vertex j. If vehicle k travels directly from
vertex i to vertex j,thenx
ijk
= 1, otherwise x
ijk
= 0. A
complete network is assumed. If any edges between vertices
are missing in the original graph, then they are replaced by
edges with an artiﬁcially high cost.
Each customer i (i = 1,...,n) is associated with a known
non-negative demand, d
i
, to be delivered or picked up. A set
of identical vehicles, each with capacity C, is available at the
depot. To ensure feasibility, we assume that d
i
C for each
i = 1,...,n. Each vehicle may serve at most one route. Each
vehicle route must start and end at the depot.
Furthermore, each customer i (i = 1,...,n) is associated
with a time window [a
i
, b
i
] during which service should
ideally begin and a service time s
i
goods. The depot has a service time s
0
=0, and a time window
[a
0
, b
0
]. Normally, a
0
= 0. The start time of each vehicle
route is greater than or equal to a
0
. Moreover, observe that
the time window requirements induce an implicit orientation
of each route even if the original matrices are symmetric, and
an implicit route length constraint (ie a latest time instant to
complete the route) where the maximum route length L = b
0
.
Each route length consists of the vehicle travel time, waiting
time at some customers, and time to serve all customers on
the route. Therefore, the length of route k is
n
i=0
n
j=0
c
ij
x
ijk
+
iN
k
w
i
+
iN
k
s
i
where w
i
is the waiting time at customer i and N
k
is the set
of customers assigned to route k.
In the hard time window case, a vehicle is not allowed
to arrive at customer i after b
i
to begin service. If a vehicle
arrives before a
i
, it waits. However, in the soft time window
case, each customer time window can be violated by paying
an appropriate penalty. Let t
i
denote the arrival time instant
at customer i ; the penalty payable P(t
i
) can be represented
as a linear function of the amount of time window violation.
For customer i, penalty coefﬁcients
i
and
i
are deﬁned to
denote the penalty payable for each time unit of service begin-
ning before a
i
and after b
i
, respectively. In practice, these
coefﬁcients could represent the costs of lost sales, goodwill,
etc, due to the customer inconvenience for not meeting the
time windows, and are therefore often called inconvenience
costs. It is not necessary for
i
and
i
to be equal, or for
these penalty coefﬁcients to be equal across customers. For
some important customers or customers whose time window
requirements are very strict, the value of
i
and
i
can be
greater, even set to inﬁnity which, in effect, converts this time
window to a hard time window.
The different ways of allowing time window violations lead
to different types of VRPSTW.
Type 1: If a vehicle arrives before a
i
, it waits, as in the hard
time window case. But a vehicle is allowed to arrive
at customer i after b
i
to begin service by paying
an appropriate penalty, see Figure 1(a). Taillard et
al (1997) proposed a TS heuristic for this type of
problem.
Type 2: Both early and late service at customer locations
are allowed by paying appropriate penalties, see
Figure 1(b). Koskosidis et al (1992) presented
an optimization-based heuristic for this type of
problem.

ZFu
et al
Tabu search algorithm 665
a
i
b
i
a
i
b
i
a
i
-P
max
-W
max
a
i
-P
max
a
i
b
i
b
i
+P
max
a
b
c
d
e
f
a
i
b
i
b
i
+P
max
a
i
-P
max
a
i
b
i
b
i
+P
max
a
i
-P
max
a
i
b
i
b
i
+P
max
Figure 1 Main types of VRPSTW. In each diagram, the horizontal axis represents time and the vertical axis represents the penalty cost.
Type 3: Assume the maximum allowable violation of the time
windows to be P
max
. Then the outer time window of
each customer i may be enlarged to [a
i
P
max
, b
i
+
P
max
]. Furthermore, assume that the maximum
waiting time allowed before the outer time window is
W
max
(see Figure 1(c)). This is the type of the prob-
lem that Balakrishnan (1993) and Chiang and Russell
(2004) studied. Balakrishnan (1993) described three
simple heuristics for it. Chiang and Russell (2004)
developed a TS solution method for the problem.
Type 4: Based on Type 1, dene P
max
as the maximum allow-
able lateness at customer locations, see Figure 1(d).
Type 5: Based on Type 2, dene P
max
as the maximum allow-
able violation of the time windows. Then the outer
time window of each customer i may be enlarged to
[a
i
P
max
, b
i
+ P
max
], see Figure 1(e). The time for
the start of service at customer i must be within the
outer time window [a
i
P
max
, b
i
+ P
max
] and prefer-
ably within the inner time window [a
i
, b
i
]. The case
study by Fagerholt (2001) belongs to this type of
problem, in which an optimization-based approach
based on a set partitioning formulation was proposed
to solve the problem.
Type 6: Based on Type 3, remove the limitation of the max-
imum allowable waiting time W
max
, see Figure 1(f).
Some other variants can be dened on the base of the above
main types of VRPSTW. If we let E
max
and L
max
represent
the maximum allowable violation of time windows before
a
i
and after b
i
max
, then a unied
penalty function for the above main types of VRPSTW can
be dened as
P(t
i
) =
(infeasible) if t
i
< a
i
E
max
W
max
i
E
max
if a
i
E
max
W
max
t
i
< a
i
E
max
i
(a
i
t
i
) if a
i
E
max
t
i
< a
i
0ifa
i
t
i
b
i
i
(t
i
b
i
) if b
i
< t
i
b
i
+ L
max
(infeasible) if t
i
> b
i
+ L
max
where
i
and
i
are the penalty coefcients. P(t
i
) can repre-
sent the penalty function for different types of VRPSTW
as different values are assigned to W
max
, E
max
,andL
max
.
When W
max
=∞, E
max
= 0, and L
max
=∞, it is for Type 1.
When W
max
= 0, E
max
=∞,andL
max
=∞, it is for Type 2.
When 0 < W
max
and E
max
, L
max
< , it is for Type 3. When
W
max
=∞, E
max
= 0, and 0 < L
max
< , it is for Type 4.
When W
max
= 0, 0 < E
max
,andL
max
< , it is for Type 5.
When W
max
=∞,0< E
max
,andL
max
< , it is for Type 6.
Furthermore, when W
max
=∞and E
max
= L
max
= 0, it is for
the VRPHTW.
As mentioned in the Introduction, the VRPSTW has
three objectives to be minimized. Wishing to nd a good
trade-off between cost and service quality to customers, like
Balakrishnan (1993), we judge the quality of the solution
obtained using the following criteria in decreasing order of
importance: (1) the total number of vehicles used, (2) the
total deviation of time window to start service, and (3) the
total vehicle travel distance. Therefore, a feasible solution
with a certain number of vehicles always dominates over any
other feasible solutions requiring more vehicles. This in fact
introduces a hierarchical objective function: rst, minimize
objective (1) and then, for the same number of vehicles,
minimize objectives (2) and (3). The objective function can
be expressed in pre-emptive goal-programming notation
as
Min z = P
1
K + P
2
n
i=1
P(t
i
) +
n
i=0
n
j=0
K
k=1
c
ij
x
ijk
where K represents the total number of vehicles used, and
P
1
and P
2
denote priority levels ( P
1
>>P
2
). Since objective
(2), as stated above, is more important than objective (3), then
the level of the importance can be adjusted by the penalty
(weight) coefcients
i
and
i
.
The tabu search algorithm
Our algorithm is based on tabu search, a local search meta-
heuristic introduced rst by Fred Glover in 1986, and since
then has been used to solve many practical applications. It is
designed to guide the solution process to escape local optima.
A thorough discussion may be found in Glover and Laguna
(1997).

666 Journal of the Operational Research Society Vol. 59, No. 5
We developed an effective TS heuristic for the open vehicle
routing problem (OVRP) in a previous work (Fu et al, 2005,
2006). The full details of the algorithm were given in the
referenced paper and so only the outline of the algorithm and
the changes that have been made to apply it to the VRPSTW
are described here.
Initial solution
An initial solution is required for any TS algorithm to start
the local search process. With respect to the TS metaheuristic
for the VRP, some researchers, such as Van Breedam (2001)
and Brand
˜
ao (2004), claimed that the performance of the TS
heuristic was highly dependent on the quality of the initial
solutions. However, in our previous work (Fu et al, 2005),
it was shown that the initial solution did not have a large
inuence on the nal solutions in the proposed TS heuristic.
In this paper, therefore, the initial solution is generated by
building up successive routes where the next customer is
chosen at random and added to the end of the route unless
this violates the capacity or route length constraints; in that
case the route is completed back to the depot and a new route
starts.
Neighbourhood structure
Some discussion of different neighbourhood structures for
VRP was presented in Fu et al (2005), including proposals
from Pureza and Franc¸a (1991), Osman (1993), Gendreau et al
(1994), Duhamel et al (1997), Cordeau et al (2001), Brand
˜
ao
(2004) and Taillard et al (1997).
The TS algorithm presented in this paper uses the same
mixed neighbourhood structure as was introduced in Fu et al
(2005), but some necessary modication is made for the
VRPSTW where each vehicle route must start and end at
the depot. There are four different types of move that may
be used in this mixed neighbourhood structure, and the algo-
rithm selects a type of move randomly at each iteration. The
four different types of move are based on the 2-interchange
mechanism, but differ in the details of the moves that are
implemented. The four different types of move are referred to
as vertex reassignment, vertex swap, 2-opt and tails swap.
Some of the possible transformations are described in the
following paragraphs.
A solution of the problem can be denoted by a permuta-
tion (0, i
1
, i
2
, i
3
, 0, i
4
,...,0, i
n1
, i
n
, 0) of (0,...,n).Only
0 is allowed to appear more than once, each time for a new
route. The rst 0 and the following vertices before the second
0 consist of the rst delivery route, the second 0 and the
following vertices before the third 0 consist of the second
delivery route, and so on. In consideration of feasibility, the
rst as well as the last item of a solution must be 0. If one
0 is adjacent to another, that means one route contains no
customer, so it can be eliminated.
Select two different vertices (customer or depot, within the
same route or different ones) randomly. Examples are shown
underlined below. Perform one of the following four moves
randomly.
(a) Vertex reassignment:Removetherst selected vertex
from its current position on the route and insert it into
the position after the second selected vertex, that is,
X
1
= (0135604790280) X
2
= (0156047902830),
X
1
= (0135064709280) X
2
= (0135647092800).
(b) Vertex swap: Exchange the positions of two
selected vertices, that is, X
1
= (0135064790280)
X
2
= (0135460790280), X
1
= (0135604790280)
X
2
= (0135004796280).
(c) 2-opt: Reverse the order of all elements between two
selected vertices like the standard 2-opt move in TSP,
if two selected vertices are within the same route, that
is, X
1
= (0135640790280) X
2
= (0146530790280),
X
1
= (0135604790280) X
2
= (0135697400280).
(d) Tails swap: Exchange the tails after two selected
vertices (from the selected vertex to the end of the route;
both vertices must be customers), if two selected vertices
are in different routes, that is, X
1
= (0135604790280)
X
2
= (0137904560280).
The rst and the last 0 of a solution are not allowed to
be selected and removed during this 2-interchange process.
Note that some moves only make small changes to the current
solution and carry on the search within a restricted part of the
solution space, facilitating the algorithm to converge; some
moves make larger changes to the current solution and guide
the search to different areas of the solution space.
(a)(c) are the moves that can possibly reduce the number
of routes. The tails swap or move (d) was originally intro-
duced because of the structure of the OVRP, where we wished
to preserve the tails or nal parts of routes which would
typically include customers far from the depot, while allowing
the set of customers included in the early part of the route to
be changed. However, this type of move has been retained for
the VRPSTW since it allows changes while preserving the
sequence of customers in the earlier and later parts of routes,
a feature that may be benecial when time window require-
ments are an important feature.
To test the inuence of the neighbourhood structure and
the search strategy on the performance of the algorithm, we
compared the following three cases:
(a) Single type of neighbourhood structure (that is, with only
one type of move).
(b) Mixed neighbourhood structure with the above four types
of move, and perform all of the four moves one by one
for the selected two different vertices.
(c) Mixed neighbourhood structure with the above four types
of move, but perform one of the four moves randomly for
the selected two different vertices.
It showed that case (3) usually produced better solutions
than the other cases did. Therefore, the mixed neighbourhood

ZFu
et al
Tabu search algorithm 667
structure and the search strategy in which the current move is
randomly selected among the four moves are adopted in this
TS algorithm.
Evaluation of solutions
To facilitate the exploration of the search space, a move is
allowed even if it results in an infeasible solution in terms of
the vehicle capacity or route length. The extent of infeasibility
can be measured by incorporating the vehicle capacity and
maximum route length constraints into the objective func-
tion by adding a penalty if the constraints are broken. So a
solution can be evaluated by the following objective function
Min z = P
1
K + P
2
n
i=1
P(t
i
) +
K
k=1
n
i=0
n
j=0
c
ij
x
ijk
+ p(E
c
(k) + E
t
(k))
where E
c
(k) and E
t
(k) are the excess of load and dura-
tion in route k, respectively, and p is the infeasible penalty
coefcient. E
c
(k) and E
t
(k) are equal to zero for all routes
if a solution is feasible. p ∈[0.00001, 200000] is initially
equal to 1 and weighted by a self-adjusting parameter: every
10 iterations, it is divided by 2 if all 10 previous solutions
were feasible or multiplied by 2 if all were infeasible. This
mechanism was also used in Fu et al (2005), and is based on
the method used by Gendreau et al (1994) in the Tabu route
algorithm for the VRP.
Tabu list and stopping criterion
The tabu list and stopping criterion also follow the method
described in Fu et al (2005). The tabu list contains the
move attributes of solutions during the last 510 (selected
randomly) iterations. The search is terminated if either a
specied number of iterations have elapsed in total or since
the last best solution was found.
TS algorithm
The following variables are used in the description of the TS
algorithm:
iter current number of iterations;
max
iter maximum number of iterations;
cons
iter current number of consecutive iterations
without any improvement to the best solu-
tion so far;
max
cons iter maximum number of consecutive iterations
without any improvement to the best solu-
tion so far;
cand
list current number of candidate moves on the
list;
max
cand list maximum number o f candidate moves on
the list.
The unied TS algorithm for VRPSTW is described below.
Set W
max
, E
max
, L
max
,
i
and
i
to the corresponding value
according to the type of VRPSTW to be solved and the level
of importance between objectives (2) and (3).
Generate an initial feasible solution randomly, and set this
solution as the current solution and the best solution so far;
Set iter and cons
iter to 0;
While (iter
max iter)and(cons iter max cons iter)
do
Begin
While (cand
list max cand list) do
begin
Select two vertices randomly;
Perform one of the four types of neighbourhood move
randomly;
Add the solution produced by the selected move to the
candidate list;
end;
Select the best solution in the candidate list if it is not tabu,
or it produces a solution strictly
better than the best solution so far;
Set the new solution as the current solution, update the
tabu list and increment iter;
If the new solution improves the best solution so far, update
the best solution so far, and set
cons
iter to 0; Otherwise, increment cons iter;
End.
This style of TS algorithm was found to be effective for the
OVRP in Fu et al (2005), which is why it has been used as the
basis for solving the VRPSTW in this paper. Its simple but
powerful mixed neighbourhood structure and the stochastic
elements in the method allow effective diversication within
a local search framework.
Computational results and comparison
This unied TS algorithm was coded in Delphi 7.0 and
implemented on a Pentium-II PC running on 600 MHz with
184 MB RAM. To show the computational performance of
the algorithm, it was tested on the 56 benchmark prob-
from: http://w.cba.neu.edu/msolomon/problems.htm) and
the results were compared with others in the literature.
These test problems were extensively used to test algo-
rithms both for the VRPHTW and for the VRPSTW. In these
100-customer Euclidean problems, the travel times are equiv-
alent to the corresponding Euclidean distances. Six different
sets of problems are dened, namely R1, C1, RC1, R2, C2,
and RC2. In problem sets R1 and R2, customers are randomly
distributed, whereas in sets C1 and C2, they are clustered;
and a mix of random and clustered structure is contained

##### Citations
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TL;DR: The proposed Unified Hybrid Genetic Search metaheuristic relies on problem-independent unified local search, genetic operators, and advanced diversity management methods and shows remarkable performance, which matches or outperforms the current state-of-the-art problem-tailored algorithms.

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### Cites background from "A unified tabu search algorithm for..."

• ...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al. (2011) PR08 Pirkwieser and Raidl (2008) ZK11 Zachariadis and Kiranoudis (2011) HDH09 Hemmelmayr et al. (2009) RP06 Ropke and Pisinger (2006a) ZK12 Zachariadis and Kiranoudis (2012) ISW09 Imran et al. (2009) RL12 Ribeiro and Laporte (2012)...

[...]

• ...State-of-the-art algorithms: B10 Belhaiza (2010) KTDHS12 Kritzinger et al. (2012) RT10 Repoussis and Tarantilis (2010) BDHMG08 Bräysy et al. (2008) MB07 Mester and Bräysy (2007) RTBI10 Repoussis et al. (2010) BER11 Bektas et al. (2011) MCR12 Moccia et al. (2012) RTI09a Repoussis et al. (2009a) BLR11 Balseiro et al. (2011) NB09 Nagata and Bräysy (2009) RTI09b Repoussis et al. (2009b) BPDRT09 Bräysy et al. (2009) NBD10 Nagata et al. (2010) SDBOF10 Subramanian et al. (2010) CM12 Cordeau and M. (2012) NPW10 Ngueveu et al. (2010) SPUO12 Subramanian et al. (2012) F10 Figliozzi (2010) P09 Prins (2009) SUO13 Subramanian et al. (2013) FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al. (2011) PR08 Pirkwieser and Raidl (2008) ZK11 Zachariadis and Kiranoudis (2011) HDH09 Hemmelmayr et al. (2009) RP06 Ropke and Pisinger (2006a) ZK12 Zachariadis and Kiranoudis (2012) ISW09 Imran et al. (2009) RL12 Ribeiro and Laporte (2012) Table 3 displays the list of acronyms for the benchmark instances and methods used in the comparative analysis....

[...]

• ...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al. (2011) PR08 Pirkwieser and Raidl (2008) ZK11 Zachariadis and Kiranoudis (2011) HDH09 Hemmelmayr et al. (2009) RP06 Ropke and Pisinger (2006a) ZK12 Zachariadis and Kiranoudis (2012) ISW09 Imran et al....

[...]

• ...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al....

[...]

• ...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al....

[...]

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TL;DR: This paper investigates a practical variant of the vehicle routing problem (VRP), called VRP with simultaneous delivery and pickup and time windows (VRPSDPTW), in the logistics industry and shows that MOLS outperforms MOMA in most of instances.
Abstract: This paper investigates a practical variant of the vehicle routing problem (VRP), called VRP with simultaneous delivery and pickup and time windows (VRPSDPTW), in the logistics industry. VRPSDPTW is an important logistics problem in closed-loop supply chain network optimization. VRPSDPTW exhibits multiobjective properties in real-world applications. In this paper, a general multiobjective VRPSDPTW (MO-VRPSDPTW) with five objectives is first defined, and then a set of MO-VRPSDPTW instances based on data from the real-world are introduced. These instances represent more realistic multiobjective nature and more challenging MO-VRPSDPTW cases. Finally, two algorithms, multiobjective local search (MOLS) and multiobjective memetic algorithm (MOMA), are designed, implemented and compared for solving MO-VRPSDPTW. The simulation results on the proposed real-world instances and traditional instances show that MOLS outperforms MOMA in most of instances. However, the superiority of MOLS over MOMA in real-world instances is not so obvious as in traditional instances.

165 citations

Journal ArticleDOI
TL;DR: This paper develops a solution procedure, in which feasible vehicle routes are constructed via a tabu search algorithm, and proposes a linear programming model to handle the detailed scheduling of customer visits for given routes.

145 citations

Journal ArticleDOI
TL;DR: A new iterative route construction and improvement algorithm to solve vehicle routing problems with soft time windows that is intuitive and able to accommodate general cost and penalty functions is proposed.
Abstract: The solution of routing problems with soft time windows has valuable practical applications. Soft time window solutions are needed when: (a) the number of routes needed for hard time windows exceeds the number of available vehicles, (b) a study of cost-service tradeoffs is required, or (c) the dispatcher has qualitative information regarding the relative importance of hard time-window constraints across customers. This paper proposes a new iterative route construction and improvement algorithm to solve vehicle routing problems with soft time windows. Due to its modular and hierarchical design, the solution algorithm is intuitive and able to accommodate general cost and penalty functions. Experimental results indicate that the average run time performance is of order O(n2). The solution quality and computational time of the new algorithm has been compared against existing results on benchmark problems. The presented algorithm has improved thirty benchmark problem solutions for the vehicle routing problems with soft time windows.

133 citations

### Cites background or methods from "A unified tabu search algorithm for..."

• ...Hashimoto et al. (2006) proposes an algorithm for flexible time windows (hard and soft) and travel times using local search; soft time window and soft traveling time constraints are treated as part of the objective function and the authors deal with a generalized VRP. Fu et al. (2008) adapted a tabu search algorithm, previously used in the open vehicle routing problem (Fu et al., 2005), for the VRPSTW....

[...]

• ...Construction heuristics include the work of Solomon (1987), Potvin and Rousseau (1993), and Ioannou et al. (2001)....

[...]

• ...The solution results presented by Fu et al. (2008) using the unified tabu search method are denoted (UTS)....

[...]

• ...Balakrishnan (1993), Chiang and Russell (2004), and Fu et al. (2008) also set a maximum vehicle waiting time limit Wmax....

[...]

• ...To the best of the author’s knowledge, the only three journal publications that include results for VRPSTW benchmark problems and comply with prerequisites (a)–(c) are: Balakrishnan (1993), Chiang and Russell (2004), and Fu et al. (2008)....

[...]

Journal ArticleDOI
TL;DR: A two-stage multiobjective multidepot vehicle routing problem with time windows is proposed and a hybrid neighborhood structure is designed for solution improvement, which significantly outperforms two other representative algorithms.
Abstract: This paper proposes a multiobjective multidepot vehicle routing problem with time windows and designs some real-world test instances. It develops a two-stage multiobjective evolutionary algorithm (TS-MOEA) for dealing with the problem. Stage I of our proposed algorithm focuses on finding extreme solutions, and forms a coarse Pareto front, while stage II extends the found extreme solutions for approximating the whole Pareto front. The two-stage strategy provides a new method to balance convergence and diversity. Moreover, a hybrid neighborhood structure is designed for solution improvement. Experimental result shows that TS-MOEA significantly outperforms two other representative algorithms.

93 citations

### Cites background from "A unified tabu search algorithm for..."

• ...3(ei − bi) as in [47], where [bi, ei] denotes the time window of i....

[...]

##### References
More filters
Book
Fred Glover
31 Jul 1997
TL;DR: This book explores the meta-heuristics approach called tabu search, which is dramatically changing the authors' ability to solve a host of problems that stretch over the realms of resource planning, telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics, pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservation and scores of other problems.
Abstract: From the Publisher: This book explores the meta-heuristics approach called tabu search, which is dramatically changing our ability to solve a hostof problems that stretch over the realms of resource planning,telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics,pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservationand scores of other problems. The major ideas of tabu search arepresented with examples that show their relevance to multipleapplications. Numerous illustrations and diagrams are used to clarifyprinciples that deserve emphasis, and that have not always been wellunderstood or applied. The book's goal is to provide ''hands-on' knowledge and insight alike, rather than to focus exclusively eitheron computational recipes or on abstract themes. This book is designedto be useful and accessible to researchers and practitioners inmanagement science, industrial engineering, economics, and computerscience. It can appropriately be used as a textbook in a masterscourse or in a doctoral seminar. Because of its emphasis on presentingideas through illustrations and diagrams, and on identifyingassociated practical applications, it can also be used as asupplementary text in upper division undergraduate courses. Finally, there are many more applications of tabu search than canpossibly be covered in a single book, and new ones are emerging everyday. The book's goal is to provide a grounding in the essential ideasof tabu search that will allow readers to create successfulapplications of their own. Along with the essentialideas,understanding of advanced issues is provided, enabling researchers togo beyond today's developments and create the methods of tomorrow.

6,373 citations

MonographDOI
01 Jan 2001
TL;DR: In this paper, the authors present a comprehensive overview of the most important techniques proposed for the solution of hard combinatorial problems in the area of vehicle routing problems, focusing on a specific family of problems.
Abstract: The Vehicle Routing Problem covers both exact and heuristic methods developed for the VRP and some of its main variants, emphasizing the practical issues common to VRP. The book is composed of three parts containing contributions from well-known experts. The first part covers basic VRP, known more commonly as capacitated VRP. The second part covers three main variants of VRP with time windows, backhauls, and pickup and delivery. The third part covers issues arising in real-world VRP applications and includes both case studies and references to software packages. The book will be of interest to both researchers and graduate-level students in the communities of operations research and matematical sciences. It focuses on a specific family of problems while offering a complete overview of the effective use of the most important techniques proposed for the solution of hard combinatorial problems. Practitioners will find this book particularly usef

3,395 citations

Journal ArticleDOI
TL;DR: This paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints and finds that several heuristics performed well in different problem environments; in particular an insertion-type heuristic consistently gave very good results.
Abstract: This paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints. Given the intrinsic difficulty of this problem class, approximation methods seem to offer the most promise for practical size problems. After describing a variety of heuristics, we conduct an extensive computational study of their performance. The problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems, the percentage of customers with time windows, their tightness and positioning, and the scheduling horizon. We found that several heuristics performed well in different problem environments; in particular an insertion-type heuristic consistently gave very good results.

3,211 citations

### "A unified tabu search algorithm for..." refers background or result in this paper

[...]

• ...To show the computational performance of the algorithm, it was tested on the 56 benchmark problems generated by Solomon (1987) (can be downloaded from: http://w.cba.neu.edu/∼msolomon/problems.htm) and the results were compared with others in the literature....

[...]

• ...Koskosidis et al (1992) developed an optimization-based heuristic for Type 2 of VRPSTW and tested it on the five sets of randomly generated problems and 21 instances of 56 Solomon’s benchmark problems. The heuristic started with low penalty coefficients, which were gradually increased. In our algorithm, we suppose the penalty coefficients i = i = 100 for all i and run it for the 21 instances listed. The comparison of the results is shown in Table 3, using the format: number of routes/total travel distance, percentage of non-violated time windows. The comparison of the CPU time in seconds is in Table 4. Our algorithm has improved 12 and tied seven solutions on the 21 instances listed, indicated by ∗∗ and ∗, respectively. Among the 12 improved solutions, eight of them require a lower number of vehicle routes and four are of shorter total travel distance (for the same number of routes required and non-violated time windows). For the problem sets of C1, comparing with the heuristic of Koskosidis et al (1992) as well as our algorithm for Type 1 of VRPSTW, our algorithm for Type 2 of VRPSTW takes much more CPU time to find the same best known solutions, and cannot obtain the best known solutions in two instances....

[...]

Book

01 Oct 1999
TL;DR: The techniques treated in this text represent research as elucidated by the leaders in the field and are applied to real problems, such as hilllclimbing, simulated annealing, and tabu search.
Abstract: Optimization is a pivotal aspect of software design. The techniques treated in this text represent research as elucidated by the leaders in the field. The optimization methods are applied to real problems, such as hilllclimbing, simulated annealing, and tabu search.

1,461 citations

Book ChapterDOI
26 Oct 1998
TL;DR: In this paper, a local search method called Large Neighbourhood Search (LNS) is used to solve vehicle routing problems, analogous to the shuffling technique of job shop scheduling.
Abstract: We use a local search method we term Large Neighbourhood Search (LNS) to solve vehicle routing problems. LNS is analogous to the shuffling technique of job-shop scheduling, and so meshes well with constraint programming technology. LNS explores a large neighbourhood of the current solution by selecting a number of "related" customer visits to remove from the set of planned routes, and re-inserting these visits using a constraint-based tree search. Unlike similar methods, we use Limited Discrepancy Search during the tree search to re-insert visits. We analyse the performance of our method on benchmark problems. We demonstrate that results produced are competitive with Operations Research meta-heuristic methods, indicating that constraint-based technology is directly applicable to vehicle routing problems.

1,207 citations