EUROPEAN
JOURNAL
OF OPERATIONAL
RESEARCH
European
Journal
of Operational
Research
106 (1998) 539-545
A tabu search heuristic for the undirected selective travelling
salesman problem
Michel Gendreau
” Centre de recherche
SW les transports.
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a3b, Gilbert
UniwrsitC
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de Montrkal,
op&-ationnelle.
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Received
Unirersitt!
‘**, FrkdCric Semet ‘A
C. P. 6128, succursalr
Uniurrsitt!
Canada
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Laporte
de Montkd,
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Mont&i.
C. P. 6128. .succursc&
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Mont&l,
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de Montrkd,
C. P. 612X. succursulr~ Cmtrr-rilic.
Montrkl.
Cunudr
H3C 3J7
I May 1996: received in revised form I April 1997
Abstract
The undirected Selective Travelling Salesman Problem (STSP) is defined on a graph G= ( V, E) with positive profits
associated with vertices, and distances associated with edges. The STSP consists of determining a maximal profit Hamiltonian cycle over a subset of V whose length does not exceed a preset limit L. We describe a tabu search (TS) heuristic
for this problem. The algorithm iteratively inserts clusters of vertices in the current tour or removes a chain of vertices.
Tests performed on randomly generated instances with up to 300 vertices show that the algorithm consistently yields
near-optimal
solutions.
0 1998 Elsevier Science B.V. All rights reserved.
Keywor&
Selective travelling
salesman
problem;
Tabu search heuristic
1. Introduction
The undirected
Selective Travelling Salesman
Problem (STSP) is defined on a graph G = ( V, E)
where V = {v~,v~,...,v,}
is a vertex set and
E = {(vi, ui): vi, vi E V, i < ,j} is an edge set. Vertex IJOrepresents a depot. With each vertex vi of
V \ { vn} is associated a positive profit pi. We assume the STSP is defined in an m-dimensional
space and each vertex t)i is represented by its coor-
*Corresponding
author.
[email protected].
0377-2217/98/$19.00
PIISO377-22
Fax: +I-514-343-7121:
e-mail:
gil-
0 1998 Elsevier Science B.V. All rights reserved.
17(97)00289-O
dinates. A distance matrix D is associated with E
and dij is defined as the Euclidean
distance between vi and vi. In what follows, dij must be interpreted as d’/i whenever j < i. The STSP consists of
determining
a Hamiltonian
tour over a subset of V
including
the depot and having a length not exceeding a preset constant L. A tour T will be denoted by an ordered set {vg? . , pt. CO} and its
length by P(T).
The STSP is often called the Orienteering Prohlem due to its connection
with orienteering,
a treasure-hunt
game in which players must collect
scores in a preset time frame by visiting control
points
(see, e.g., Hayes
and Norman,
1984;
540
M. Gendreau et al. I European Journal of’Operational Research 106 (1998) 539-545
Tsiligirides,
1984). An application
of the STSP to
the delivery of home-heating
oil is provided
in
Golden et al. (1987).
The STSP is an NP-hard combinatorial
optimization problem (see, e.g., Laporte and Martello,
1990). Exact algorithms
have been provided
by
Laporte
and Martello
(1990)
Ramesh
et al.
(1992), and Gendreau et al. (1995). The latter algorithm appears to be the best and can solve to optimality instances involving
up to 300 vertices. A
number of heuristics have also been proposed by
various
authors,
see, e.g., Hayes and Norman
(1984),
Tsiligirides
(1984),
Golden
et
al.
(1987, 1988) Keller (1989) Laporte and Martello
(1990) Ramesh and Brown (1991) Chao (1993)
and Wang et al. (1995). All these heuristics have
tested on a limited set of small size problems
(n < 33) and the last three are comparable
in quality. Recently, Gendreau
et al. (1995) have developed two new heuristics
called Hl and H2. In
Hl, an STSP solution is gradually constructed
by
inserting a pair of vertices, or a single vertex, into
the current tour. Periodic reoptimizations
of the
tour are performed
using the postoptimization
phase of GENIUS.
Once a solution has been determined, an attempt is made to improve it by removing in turn two of its vertices, and proceeding
as in the construction
phase. A full description
of
GENIUS
can be found
in Gendreau
et al.
(1992). Briefly, GENIUS contains a tour construction phase, called GENI, and a postoptimization
phase, called US. Starting with three arbitrary vertices, GENI inserts at each iteration an unrouted
vertex between two of its p closest neighbours
on
the partially constructed
tour, where p is a user
controlled
parameter.
While making an insertion,
GENI performs a local reoptimization
of the tour.
Once a complete tour has been obtained,
the US
postoptimization
procedure
is repeatedly
applied
to the tour until no further improvement
is possible. Here vertices are successively removed from
the tour, and reinserted,
using the same rules as
in the tour construction
phase. Heuristic H2 constructs a first tour having a cost not exceeding
0.8L, first using all vertices, and gradually removing some of them. Vertex insertions are then performed as in Hl. An attempt
to improve
the
current solution is then made, by swapping verti-
ces between the current vertex set and its complement.
Both heuristics
have been applied
to
instances involving
between 20 and 300 vertices.
Heuristic Hl is usually better than H2. In most
cases, the solution is less than 10% from the optimum, but there are a few cases where the gap is
16% or 17%.
In spite of the apparent simplicity of the STSP,
it is rather difficult to devise consistently good heuristics for this problem. Part of the trouble lies in
the fact that profits and distances are independent
and a good solution with respect to one criterion is
often unsatisfactory
with respect to the other. It is
usually hard to accurately select the vertices that
are a part of the solution. Thus, if L is relatively
small and few vertices are on the optimal tour, a
wrong choice of vertices may translate into a large
profit difference. At the other extreme, if most vertices belong to the solution, then the STSP is similar to a Travelling
Salesman Problem (TSP), but
this does not make the problem any easier since
it is complicated
to pinpoint the few vertices that
are excluded from the solution.
All heuristics suggested so far in the literature
are based on simple construction
and improvement techniques
similar to those used for the
TSP. Typically as we will see, the direct application of these techniques
to the STSP may cause
the algorithm
to move in undesirable
directions.
Moreover, while these methods are in general rather quick, they fail to explore a larger portion of the
solution space and do not correct previous wrong
decisions. With the possible exception of the neural network
based heuristic
of Wang
et al.
(1995), these methods
use classical local ascent
schemes and tend to become trapped in local optima. Our goal is to devise a tabu search (TS) heuristic for the STSP that will perform
a more
thorough search and circumvent
these difficulties.
Our computational
results indicate that solutions
produced by our method are always near-optimal.
The remainder of this paper is organized as follows. In Section 2, we describe and discuss a simple heuristic used as a starting point for the TS
algorithm. The main components
of the TS heuristic are presented in Section 3, followed by a step
by step description
in Section 4. Computational
results are provided in Section 5.
hf. Gendreuu et ul. I Europeun Journul c~fOperutionul Reseurd
541
106 (19%‘) 539 545
2. A simple heuristic
3. Main features of the tabu search algorithm
We first describe a simple heuristic, called Insert
und Shake, used as a starting point for our TS algorithm. It combines the TSP tour extension heuristic described in Rosenkrantz
et al. (1977), and
GENIUS.
It gradually extends a tour T until no
further vertex can be introduced
without violating
the distance limit L. Then an attempt to obtain a
shorter tour on the vertices of T is made by applying GENIUS.
If this fails to produce a better tour,
the algorithm
terminates.
Otherwise more vertex
insertions
are attempted
and the same process is
repeated.
Step
1 (InitiLd
tour):
Set
6 := 0 and
T := { o,j. UO}. Determine
uj E V \ T having
the
minimal
ratio
(l&i + djo)/pj
and
such
that
do, + d,(, < L. If no such c’, exists, the STSP is infeasible: stop. Otherwise, set T := (~0, u,, uo}.
Step 2 (hsert):
If T= V, stop. Determine
t‘, @ T and two consecutive
vertices u, and CA of
T such that (dfj + cl,k - dik)/p, is minimal
and
4’(T) + d,, + d,a - d,k < L. If no such v, exists, go
to Step 3 if fi = 0 or stop if 6 = 1. Otherwise, insert I’j between ~1;and CA,set (S= 0, and repeat this
step.
Stq 3 (Shke):
Attempt to determine a shorter
tour by applying GENIUS to T. If this fails, stop.
Otherwise, set rS := 1 and go to Step 2.
The first two steps of this algorithm
correspond to heuristic
H2 of Laporte and Martello
(1990). Executing
Step 3 will often enable the
process to include
more vertices in the tour.
The myopic insertion
criterion
used in Step 2
may sometimes
lead to a wrong choice. Indeed
this rule may lead to the selection of a vertex Cj
very far from the current
tour, but having a
relatively large profit pi. This vertex may have
been preferred to another vertex Us with a smaller
profit pV. However, selecting L’,.could be preferable to selecting l:, in cases where a third vertex
L’, is very close to I’~ and P,. fp, > pi. This is
why some heuristics
(see, e.g., Ramesh
and
Brown.
1991; Gendreau
et al., 1995) consider
the possibility
of inserting
two vertices
at a
time. In our TS algorithm,
we extend this idea
further by inserting clusters of vertices of any cardinality.
Before proceeding
to the step by step description of the algorithm, we outline its main features.
In what follows, a solution
T is an ordered set
c,,
Q}
whose
total
length
may possibly ex{co.. . .
ceed L. In other words, infeasible solutions
are
considered during the search process.
3.1. Proximit~~ measure cmd clustrrs
The proposed TS heuristic moves from one solution to another by operating on clusters of vertices rather than exclusively on single vertices, so as
to allow the insertion or removal of several vertices
at a time during the search process. To constitute
clusters of vertices candidates for insertion moves,
we first define the dispersion index of a non-empty
subset R of V as
if lR\ > 1.
if /RI = 1.
and the prcuimity meclsure between
subsets R and S of V as
two non-empty
r,cs
Note
that
if R = {q}
and
S = {c,},
then
A(R,S) = d,,. This proximity
measure was introduced in the context of hierarchical
classification
by the two statisticians
Tricot
and Donegani
(1989) to account for the compactness
of R and
S when measuring the proximity of these two sets.
Thus in the example of Fig. 1, R is closer to S than
it is to S’.
Using this proximity measure, we define several
partitions
of V \ { u,~} within a preprocessing
step
of the algorithm. Each of these partitions contains
several clusters of vertices. The algorithm will operate on these partitions.
To define the partitions,
we proceed as follows. Let P,. = { C,.X} denote a
partition of V \ { ~0) into clusters C,. 1~Cr.?, . .
Step
1 (First partition):
Set r := 1 and
P? := {{U,}.
, {L?,,}}.
542
M. Gendreau et al. I European Journal of Operational
Research 106 (1998) 539-545
the TSP heuristic US with p = 4 (Gendreau et al.,
1992) to the vertices of T U CLk.
3.3. Candidate moves
R
S
a) S has a large dispersion
Fig. 1. Proximity
sions.
R
s
b) S’ has a small dispersion
of R with two sets having
different
disper-
Step 2 (Next partitions): If r = n, stop. Otherwise, define P,.+I from P, by merging the two clusters C,.? and C,.k.of P, yielding min+{A( C,, C,.k)},
set r := r + 1 and repeat this step.
Note that JP,( = n + 1 - r and all clusters C,.F
for which e # i* and e # k” are included in P,+l .
We only retain partitions P, corresponding to
r = 1, [n/2], [2n/31, [3n/4], . . . , [9n/lOl. One reason for keeping at most 10 partitions is to save
memory. Moreover, removing partitions that are
very similar to one another will create a diversification effect in the search process. This will become
clearer later on.
Clusters of vertices candidate for removal from
a solution are defined as sets Hij where ai and Ujare
the two end vertices of a (possibly degenerate)
chain on the current tour T = (~0, . . , v,, . . . , Uj,
...I 00).
3.2. Neighbourhood structure
The neighbours of a solution T = {VO,. . . ,
vt, vg} are other solutions obtained either by removing a cluster HU from T or by inserting a cluster C,, into T. Removing Hij from T is
straightforward. The resulting tour T \ Hij is obtained by taking away the ordered subset of vertices {Vi,. . . , Vi} from T and by connecting the
predecessor of ai to the successor of Uj. Inserting
a cluster is more involved. First observe that a
tabu status will be attributed to individual vertices
in the course of the algorithm, as explained in Section 3.5. To insert a cluster C,.kinto T, first remove
all tabu vertices from C,.k,and all vertices of C,k already belonging to T to obtain CLk. Assuming
Cik # 0 a neighbour of T is obtained by applying
The sets of vertices Hij candidate for removal are
defined as follows. Consider a solution T = {VO,
where
. 1 Vjo 7 Vi, I ’ . . 1 Vjl 1 Viz, . . . ) uj;.-l , Vi0 7 .
1 VO}t
are
the
J
longest
edges
of
(Vjo, vil), . . 3 @j,-l> vio)
the tour and 2 is an input parameter randomly selected in the interval [3, J/2], and 6 is the maximum
between 6 and the number of vertices appearing on
the initial tour. Then the sets Hij are simply
K,jj 1.. . ,Hi,-lj,-l*
The sets of vertices C& candidate for inclusion
are Cii . , Cl,,, and Cl, Cs2.. . , where s is randomly selected in {[n/21,. . . , [9n/lOl}, according
to a discrete uniform distribution.
3.4. Evaluation of candidate moves
The value of a move associated with the removal of a chain Hij is measured by the ratio of saved
distance over lost profit, and is computed as
I
f(Kj) = de(T)- e(T\ Hij)]
C
Pkl
P. EM,
where a is a parameter dynamically adjusted
throughout the algorithm (see Section 3.6).
We now compute the value of a move associated with the insertion of a cluster Cik. Denote by
T U {uk} the tour obtained by inserting vk into T
in its best position. In what follows, we use
l(T U {ok}) - f?(T) in the denominator of some expressions. If this denominator is zero, then the expression is set equal to an arbitrary large value.
The value of an insertion is measured by the ratio
of added profit over added distance. We distinguish two cases. (1) If r = 1, vertex Uk is inserted
into T where the smallest distance increase is obtained. The value of the corresponding move is
then
&k)
=pk/[a(e(T
” {Vk))
-
e(T))1.
M. Gendreau et al. I European Journal
(2) If Y = s > 1, the gravity centre i7kof C:, is first
computed for all clusters of P, and a preliminary
move evaluation is made according to the formula
qf Operational
Research 106 (1998) 539-545
543
as W is non-empty. In practice this procedure allows for the possible insertion of some tabu vertices of C.&’\ C.& whenever s > 1.
3.6. Parameter cI
The cluster Ci.P corresponding to maxk {g(C$ } is
then selected, and the US algorithm is applied to
T u C,& to yield a tour of length .!(T U Cl,,). The
exact move evaluation associated with Cik. is
The cluster to be removed or inserted corresponds
to
The computational complexity of the chain removal evaluation is easily determined as O(n’). The
complexity of the chain insertion evaluation is
the same as that of US: each application can be executed in O(np” + n2) time (see Gendreau et al.,
1992), but the maximum number of applications
cannot be expressed as a function of n.
3.5. Tabu status and aspiration criterion
When a vertex is removed from the current tour
it is assigned a tabu status for t!Iiterations, where 8
is randomly selected in the interval [5,25] according to a discrete uniform distribution. No tabu status is assigned to a vertex when it is included in the
solution, This was tested and found to be inefficient: it is better to allow trial insertions of vertices
in the tour even if these are of short duration.
We use the following aspiration criterion. If removing vertices from the current tour or adding
new vertices to it yields a feasible tour T, then an
attempt is made to insert in vertex not belonging
to T, irrespective of its tabu status. We select the
vertex
Up
yielding
max,,,i&k/[QT
U {ok})
-e(r)]},
where w = {Vk$! T: [(Z-U {IQ}) <L}.
The US algorithm (with p=4) is then applied to
T U {VA.} and this operation is repeated as long
The parameter c(reflects the relative importance
attributed to length and profit of a tour. Initially, CI
is equal to 0.0 1. At every iteration, CIis modified as
follows. If e(r) > L, set, ~1:= c(p, where p is randomly selected in [ 1.5,2] according to a continuous
uniform distribution. If P(T) < L, set CI:= LX/~.
4. Step by step description of the algorithm
We can now proceed to the step by step description of the TS algorithm.
Step 1 (Initial solution): Construct an initial
tour T by means of Insert and Shake heuristic described in Section 2. Let z( 7’) be the profit of T and
e(T) its length. Record the best tour T’ := T. Set
c( := 0.01 and 6 = max(6, IT]}.
Step 2 (Partitions): Construct all partitions C,k
by means of the procedure described in Section 3.2, and retain 10 of these partitions as explained previously.
Repeat Steps 3-7 jbr 1000 iterations.
Step 3 (Best move determination):
Select
s E {m/2], . . , [9n/lO]} and 1 E [3, a/2]. Consider all moves producing a neighbour of T by inclusion of vertices of a cluster of PI or P,,, or by
exclusion of vertices of a chain Hii of T. Determine
the best move as described in Section 3.5 and implement it. If the best move consists of removing
from T a chain Hij, declare all vertices of H;j tabu
for 0 iterations, where 0 E [5,25].
Step 4 (Aspiration move): If l(T) < L, attempt
to insert the vertices of W = {vk $! T: P(TU
{ck}) < L}, as described in Section 3.5.
Step 5 (a update): Select p E [1.5,2]. If
e(T) > L, set CI:= czp”;if e(T) < L, set r := c(/p.
Step 6 (Tour improvement): If the iteration
count is a multiple of 5, apply GENIUS to T.
Step 7 (Best solution update): If k’(T) <L and
.z(T) > z( T*), set T* := T.
544
M. Genrireou et (II. I European Journul of’Operrrtiond
5. Computational
results
Our algorithm was tested on randomly generated instances. First, n vertices ui were generated in
the [0, lOOI square according to a continuous uniform distribution, and each d;j was set equal to the
Euclidean distance between vi and vj. An integer
profit was assigned to each vertex of V \ {ug} according to a discrete uniform distribution on
[l,lOO]. To determine L, we first solved a TSP to
optimality on G and we then set L := Pt*, where
t* is the length of the optimal TSP tour, and, B is
a tightness parameter 0 < p 6 1. A small value of
p, means that few vertices will appear in the final
solution. Conversely, a large value of B implies
that few vertices will be excluded.
The program was coded in C and run on a Sun
Sparcstation 1000. We only attempted those instances for which an exact solution was determined
by means of the exact algorithm of Gendreau et al.
(1995). In these, /? took the values 0.1, 0.3,0.5, 0.7,
0.9, and IZtook the values 20,40, . . ,300. Up to five
instances were considered for each combination of
j? and n, but the number of instances that could be
solved to optimality varied between one and five
for each combination. In what follows, all statistics
are average values over the number of instances out
of five that could be solved optimally.
We provide computational results on our TS
heuristic, as well as comparison with the HI heuristic presented in Gendreau et al. (1995) and with
the Insert and Shake heuristic described in Section 2, and used as a starting point for the TS algorithm.
Our
computational
results
are
summarized in Table 1. The meanings of the column headings are as follows:
n
N
Hl
IS
tightness parameter influencing the
number of vertices in the solution
number of vertices excluding the depot
number of instances for which an
optimal solution is available
average ratio of the heuristic solution
value over the optimum for the Hl
heuristic
average ratio of the heuristic solution
value over the optimum for the Insert
and Shake heuristic
Reserrrch 106 (I998)
TS
average ratio of the TS heuristic over
the optimum for the TS heuristic
CPU time in seconds to run the TS
heuristic
Seconds
Table 1
Summarv
539-545
of Comoutational
results
/J
n
N
Hl
IS
TS
0. I
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
5
5
5
5
5
5
5
5
5
5
5
5
4
3
1
1.000
0.959
0.894
0.980
0.877
0.916
0.900
0.880
0.916
0.929
0.875
0.846
0.884
0.824
0.945
0.905
0.880
0.945
0.933
0.874
0.841
0.823
0.804
0.806
0.807
0.786
0.793
0.828
0.755
I .ooo
1.000
1.000
1.000
0.998
0.998
0.994
0.983
0.998
0.998
0.999
0.975
0.995
0.977
1
2
15
33
57
78
92
117
124
142
160
201
222
252
298
0.3
20
40
60
80
100
120
140
160
5
5
5
5
5
4
3
3
0.951
0.937
0.918
0.906
0.940
0.902
0.891
0.805
0.923
0.938
0.863
0.849
0.893
0.855
0.868
0.789
1.000
0.994
0.996
0.989
0.994
0.991
0.979
0.971
14
64
116
183
267
308
426
493
0.5
20
40
60
80
100
120
140
160
5
5
5
5
5
4
3
1
0.955
0.914
0.873
0.930
0.938
0.927
0.906
0.898
0.918
0.866
0.873
0.849
0.888
0.855
0.857
0.894
0.993
0.991
0.982
0.994
0.974
0.964
0.977
18
96
197
338
505
638
864
1125
0.7
20
40
60
80
100
120
140
5
5
5
5
5
3
I
0.946
0.976
0.965
0.968
0.935
0.939
0.939
0.879
0.873
0.922
0.885
0.884
0.894
0.836
1.000
0.997
0.996
0.994
0.981
0.993
0.980
17
120
310
559
752
1192
1373
0.9
20
40
60
80
100
5
5
5
4
1
0.998
0.986
0.983
0.989
0.974
0.981
0.962
0.949
0.972
0.917
0.998
I .ooo
0.997
0.995
0.982
12
48
149
441
978
1.000
1.000
I.000
Seconds
M. Gendreau et al. I European Journal oj’Operational Reseurch 106 (1998) 539-545
Our results indicate that our TS heuristics always yields optimal
or near-optimal
solutions.
The gap between the optimum
and the heuristic
value is typically less than 1% for all combinations
of, [j and n that were attempted.
Our results are
better and more stable than those observed for less
sophisticated
insertions heuristics such as Hl, IS,
and the two Laporte and Martello (1990) heuristics. The values in the Hl column were obtained
in the course of experiments carried out in another
context
(Gendreau
et al., 1995). Computation
times are not always available as Hl was sometimes followed by H2, but it is safe to say that
HI takes about just as much time as our TS heuristic. This is why we did not use it as a starting
point for our method, but we preferred the faster
IS procedure instead. In any case, our TS heuristic
is so robust that it performs well even if it is initiated from a poor solution.
Direct comparisons
with other heuristics
described
in the literature
are not possible since these heuristics were tested
on the problem of determining
a chain of maximal
profit, and these heuristics
were tested on very
small instances (typically n < 30) for which our algorithm always obtains quasi-optimal
solutions.
Acknowledgements
This work was partially supported by the Natural Sciences and Engineering
Research Council
of Canada (NSERC) under grants OGP0038816,
0GP0039682
and 0GP0005392.
Frederic
Semet
also benefitted
from an NSERC
International
postdoctoral
fellowship and was partially funded
by Strategic grant STR0149269.
545
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