An Efficient Transformation Of The Generalized Traveling Salesman Problem

Abstract

The Generalized Traveling Salesman Problem (GTSP) is a useful model for problems involving decision of selection and sequence. The problem is defined on a directed graph in which the nodes have been pregrouped into m mutually exclusive and exhaustive nodesets. Arcs are defined only between nodes belonging to different nodesets with each arc having an associated cost. The GTSP is the problem of finding a minimum cost m- arc directed cycle which includes exactly one node from each nodeset. We show how to efficiently transform a GTSP into a standard asymmetric Traveling Salesman Problem (TSP) over the same number of nodes. The transformation allows certain routing problems which involve discrete alternatives to be modeled using the TSP framework. One such problem is the heterogeneous Multiple Traveling Salesman Problem (MTSP) for which we provide a GTSP formulation.

Full text

AN EEFICIENT TRANSFORMATION OF THE
GENERALIZED TRAVELING SALESMAN PROBLEM*
CHARLES E. NOON
Management Science
Program,
The
University
of
Tennessee,
Knoxville 37996.
JAMES C. BEAN
Department of Industrial
and
Operations
Engineering,
The
University
of Michigan,
Ann
Arbor 48109.
ABSTRACT
The Generalized Traveling Salesman Problem (GTSP)
is a
useful model
for
problems involving
decisions of selection and
sequence.
The problem
is
defined on a directed graph
in which
the nodes
have been pregrouped into
m
mutually exclusive
and
ejdiaustive nodesets. Arcs
are
defined only
between nodes belonging to different nodesets with each arc having an associated cost. The GTSP
is
the
problem
of
finding
a
minimum cost m-aic directed cycle which includes exactly
one
node
from each nodeset.
In
ttiis paper, we show how
to
efficiently transform
a
GTSF into
a
standard
asymmetric
Traveling
Salesman Problem (TSP) over
the same
number of
nodes.
The
transformation
allows certain routing problems which involve discrete alternatives
to be
modeled using
the
TSF
framework. One such problem is the heterogenous Multiple Traveling Salesman Problem (MTSP)
for which we provide
a
GTSP formulation.
Keywords
:
Traveling salesman problem, vehicle routing problem, integer programming.
RESUME
Le probleme du voyageur de commerce generalise (PVCG) s'applique
a
de nombreuses situations
ou Ton doit prendre
des
decisions simultanees
de
selection
et
d'ordonnancement.
On
definit
le
probleme
sur un
graphe oriente dans lequel les sommets sont partitionnes en
m
ensembles.
On
definit des arcs avec co'ts entre toute paire de sommets appartenant
a
deux ensembles differents.
Le PVCG consiste
a
determiner un circuit de moindre co't possedant exaetement un sommet par
ensemble. Dans cet article, on propose une transformation efficace
du
PVCG en
un
probleme
de
voyageur de commerce classique (PVC) sur un sommet. Cette transformation permet la modelisa-
tion
de
certains problemes
de
toumees avec choix discrets comme
des
PVC.
Un
exemple
de ce
type de probleme est le probleme du m-TSP avec co"ts heterogenes, pour lequel
on
propose
une
formulation de type PVCG.
Mots clefs: probleme du voyageur
de
commerce,
probleme
de
confection
de
toumees
de
vehicules,
programmation en nombres entiers.
1.
INTRODUCTION AND PROBLEM STATEMENT
The Traveling Salesman Problem (TSP) is one of the oldest and most widely studied optimization
problems in the
field
of operations research (see
[9]).
For the standard
TSP, we
assume that a single
salesman, based at a home city, must visit a number of
cities
(or customers).
TTie
problem is to find
the salesman's minimum cost cycle (or
tour)
which leaves the home city, visits each customer city
once and returns to the home city.
Real-life routing or sequencing problems often require additional decisions or considerations.
Often, the standard TSP is used only after a number of discrete decision have been made. For
example, in vehicle routing, the standard TSP is often apphed after the customer to vehicle assign-
ments have been made. Preliminary decisions may also include choice of ffeet vs. contract carder,
depot locations, backhauling, and alternative vehicles.
The
Generalized Traveling Salesman Problem
(GTSP) is a useful model for problems involving
decisions of selection and
sequence.
The asymmetric version of
the
problem
is
defined on a directed
graph with nodes Af, connecting arcs A and a vector of corresponding arc costs c. The nodes are
pregrouped into m mutually exclusive and exhaustive
nodesets,
i.e.,
Af = Si
U S2
U
• • •
LI
S^ with
S/ n 5/ = 0, for all /,/,/ ^ J. Connecting arcs are defined only between nodes belonging to
*
Revd. Aug.
1991;
Revd. Oct.
1991;
Ace. Oct. 1991 INFOR
vol. 31,
no. 1, Feb. 1993
39

40
CE.
NOON AND J.C. BEAN
Figure 1. Example GTSP with Feasible Tour in Bold
different sets, that is, there are no intraset arcs. Each defined arc (i,;) e A has a corresponding
nonegative cost
c,y
> 0. The GTSP can be stated as the
problem
of
finding a
minimum
cost
m-arc
cycle which includes exactly
one node from
each
nodeset.
The standard asymmetric TSP is a special
case of the GTSP with nodesets of cardinalify
one.
Figure
1
displays an example GTSP defined on
a directed graph. The bold lines illustrate a feasible
cycle
with a total cost of
59.
Early applications using GTSP formulations are given for sequencing computer files [3] and
for routing welfare clients through governmental agencies [13]. More recently, Laporte, Nobert
and Mercure [8] discuss GTSP applications in the routing of mail vans. Several proposed appli-
cations are given in [10] including warehouse order-picking with multiple stock locations, airport
selection/routing for courier
planes,
and certain
fypes
of flexible manxifacturing scheduling.
The main focus of this paper
is
to show how any problem formulated as a GTSP can be trans-
formed into a standard asymmetric
TSP.
One benefit of
this
type of transformation is that it allows
existing asymmetric TSP approaches, both heuristic and optimal, to be applied to the transformed
GTSP problems. Such an approach may not necessarily outperform specialized algorithms for the
GTSP
fype
problems (see
[3], [7], [8],
[11],
and [13]), however, it does provide researchers a means
for pursuing or verifying optimalify on smaller problems.
Another, perhaps more significant, benefit is that the transformation may allow us to relate
polyhedral results for the well-studied TSP to more realistic, complex routing problems. Facets
of the asymmetric TSP polytope could be examined under an inverse transformation to possibly
yield strong cuts for the GTSP polytope. Many important problems in the area of vehicle rout-
ing have relaxations which can be modeled as GTSP's and could benefit from this new source of
valid inequalities. One such problem is the basic single-depot capacitated vehicle routing problem
(VRP).
When the vehicle capacify constraints of the VRP are dualized, the relaxed problem be-
comes a heterogeneous Multiple Traveling Salesman Problem
(MTSP).
After presenting details of
the transformation in Section 2, we provide a GTSP formulation for the heterogeneous MTSP in
Section 3.
2.
TRANSFORMATION OF GTSP TO TSP
In this section we show that any asymmetric GTSP can be transformed to a standard asymmet-
ric
TSP.
The fact that this is possible is not surprising since complexify theory assures us that any

TRAVELING SALESMAN PROBLEM 41
NP-hard problem can be polynomially transformed to any other NP-hard problem. Such trans-
formations, however, are often accompanied by a significant increase in the size of the problem
instance. For the transformation we present, there is no increase in the number of nodes and only
a slight increase in the number of
arcs.
We begin by formally defining an instance of the asjf^mmet-
ric GTSP which we refer to as V. The transformation consists of two stages. In the first stage, we
transform V into a clustered TSP which we denote as V. In the second stage, V is transformed
into V", a standard asymmetric TSP.
Deiinition
V is an Instance of the asymmetric GTSP defined over nodes M, connecting arcs A and a vector of
nonnegative arc costs c. In addition, M is the union ofm mutually exclusive and exhaustive nodesets
5*1
U
52 U
• • • U
^m =
A^
and A contains only arcs which connect nodes of different nodesets.
Although the problem form presented assumes a rigid structure, it should be noted that we are not
narrowly focusing on a special case of the GTSP. In [10], it is shown that problems with overlapping
node sets, intraset arcs, and more general costs, can be transformed to a problem in the form of V.
In the first stage of the transformation, we construct an instance of iJie clustered TSP by re-
defining arcs and arc costs. A clustered TSP is similar to the standard asymmetric TSP except that
the nodes are pregrouped into mutually exclusive and ejdiaustive node clusters. A feasible tour for
the clustered TSP must visit all nodes of a cluster before visiting the nodes of any other cluster.
Definition
V' is a clustered TSP defined over nodes
JV',
connecting
arcs
A' and a vector of corresponding
arc
costs
c'. Inaddition, M'
is
the union of m mutually
exclusive
and exhaustive node
clusters
Ci\JC2^-
•
-UCm =
J\f'. The data for V
is
constructed as follows.
SetN'
=
J\f
andleteachsetof nodesin M,Si, correspond
to a cluster of nodes in A/"', Q. For each nodeset (or
cluster),
we assume its member nodes have a given
arbitrary
ordering.
Let
i^,i^,
...,V be the nodes of nodeset Si (or cluster Ci) with r = 15,1 =
|C,
|.
Thearcset A' is constructed as follows. Within each cluster
Ci
with r = \Ci\ > 1,
create
intracluster
arcs
forming a single directed cycle according to its given
ordering.
Hence, create arcs (i^,i^),(i^,i^),
{i^,i*),.
..,(j''"^\j'''),(f,j^) in A'. Assign to each of these intracluster
arcs
a cost of
zero,
that
is,
c'^^^^
=
c'lifi
= ... =
c\,_nr
—
c'-r^i
= 0. For each arc {i',k^) e A, with] > 1, create an arc {i'~^,k^) G A' with
identical
cost,
hence,
c^-ij^.^
—
Cj/ke.
For each arc {i^,k^) € A with r = |C,|, create an arc
(i^
,k^) e J^
with cost
c'.,j^e
=
Problem V is a clustered TSP with a special structure. Since the only intracluster arcs are the zero
cost arcs which form a directed cycle, it is easy to see that any feasible solution to the clustered TSP
will traverse |C/| - 1 of the intracluster arcs of C,. Figure 2 displays V constructed from the GTSP
given in Figure 1. The following lemma establishes the relationship between V and V.
Lemma 1
Problem V is equivalent
to
problem V.
Proof
Given any feasible solution to V,x, expressed as a sequence of nodes
jc
= {i^
fc^,...
,f^,i'},
we can
construct a solution3; to V asy = {i^,P+^,... ,i'~^,k^,k^+^,...
,k^~'^,...
,p'',p''^^,...,
p''~^,i'}.
The solutiony is feasible for V by definition of a clustered TSP by the construction of A'. Since
each intercluster arc of y has cost equal to a corresponding interset arc of x, the costs of the two
solutions are the same.
Any feasible solution to V which enters cluster C, through node
i^
with;' / 1, must leave the
cluster from node
i^'~^.
Any feasible solution to V which enters cluster Q through node
j\
must
leave the cluster from node i' where r = |C;|. This means if we are given a feasible solution to
T" which uses intercluster arc
(i^,
k^), then the solution must necessarily include an intercluster arc
into node
P'^^
and an intercluster arc out of node k^^^. Therefore, given any
>;
feasible for V we

42C.E. NOON AND
J.C.
BEAN
Figure 2. Problem V Constructed from Problem V given in Figure 1
(broken line arcs have zero cost)
can construct a feasible solution to 'P,x, as follows. For every intercluster arc (P,k^) iny, include
arc (t'+^,^^) injc. Since
cn+ikt —
c'^jj^e,
the costs of
the
two solutions will be the same. •
Once the problem is in the form of V, it can be solved directly as a clustered TSP as in [5] or we
can use an approach equivalent to that of Chisman
[2]
to transform a clustered TSP into a standard
asymmetric TSP by simply adding a large cost to all the intercluster arcs, as follows.
Definition:
V" is a
standard asymmetric
TSP
defined over nodes
U",
connecting arcs
A', and a
corresponding
vector
of
arc costs
c".
The
data for V"
is constructed
as follows.
Set
H" s M'and A" = A.
The arc
costs for V"
are computed
as
follows.
Let
c'-j
— c[j
+
(3
ifi e M' and] e W
belong to different
clusters,
and
let
c\j
—
c\j
ifi € M' and] € M'
belong to the same
cluster,
where
+oo > /3 > Y. '^'u-
Our main result for transforming the GTSP to a standard asymmetric TSP can easily be stated.
Theorem 2:
Given
V,
an asymmetric
GTSP
with
m
nodesets,
we can transform it to
V",
a standard asymmetric
TSP
over the same number
of nodes.
Given
an
optimal solution
to V"
with cost strictly less than
(m + l)/3,
we can construct an optimal solution
to V.
3.
A
GTSP FORMULATION OF THE HETEROGENEOUS MTSP
For the Multiple Traveling Salesmen Problem (MTSP), each of
n
customer cities must be visited
by one of
v
salesmen. The salesmen's travel costs between cities are considered
homogeneous
if
they are the same for all salesmen, or
heterogeneous
if they are different. For the single-depot
homogeneous MTSP, a number of transformations to the standard TSP have been provided (see
[1],[4],
[6], and [12]). For the heterogeneous MTSP, no such transformations are given. In this
section we show how a single-depot heterogeneous MTSP can be modeled as an asymmetric GTSP.
By using the results in Section
2,
the problem can then be transformed to an asymmetric
TSP.

TRAVELING SALESNL^JSr PROBLEM43
In the single-depot heterogeneous
MTSP,
the salesmen begin and end their tours from a com-
mon depot (cify
0).
Let
\d^j\
< -t-oo be given as the cost for salesman
r
to travel from cify
i
to cify;.
For notational
convenience,
we assume
d,y
is
given for all r
=
1,...,
v,
i =
0,...,
n,
and; =0,...,n,
with /
^j.
The problem is to determine a set of v distinct salesman routes such that each customer
is
visited by exactly one salesman and the total travel costs are minimized.
To model the problem as
a
GTSP, we create v -\-n mutually exclusive nodesets. The first
v
nodesets, Sr,r = 1,2, ...,v, each contain exactly one node. Let the node in nodeset
Sr
be denoted
as
0^,for/-
=
1,2,...
,v. These
v
nodes represent copies of the depot
with
each node corresponding
to a particular salesman. Hence, node
(f e
Sr represents "salesman r's depot". The remaining n
nodesets.
Si,
i =
v-{-l,...
,v
+
n,
each correspond to a particular customer. Each of these nodesets
contains
v
nodes with each node corresponding to a visit by one of the salesmen. Let
f
denote the
node in the nodeset of customer i corresponding to a visit by salesman r.
For the preceding GTSP node and nodeset structure, we define the arcs and their costs as
shown in Table 1.
Table
1.
Arcset for GTSP model of heterogeneous MTSP
Define arc
i^,neA
(i^ne^
(j-^,(r-^)eA
ij^,O^)eA
With cost
COr.=dr,
I'f
y
C;.0^+:-^;0
For all
rill".::;"
„
19 y 1
7-1,2,...,«
Comment
salesman r ventures out from
"his"
depot
to the
r"*
node of any customer nodeset
salesman r can travel among
the nodesets using only the r*
node in each customer nodeset
from the
r'"^
node of any customer
nodeset to "salesman (r
-F
l)'s depot"
salesman
v
completes the cycle by
returning to "salesman l's depot"
After venturing from his depot, node 0^, salesman r may pass only through the r* node of
each customer nodeset he
visits.
After salesman r's
visits
are complete, he returns to the depot into
node 0^+', thus launching salesman
r
-f-1 on his
way.
A feasible GTSP tour can be likened to the
movements of a single driver assigned to
a
fleet of heterogeneous vehicles. The driver ventures
from the depot in vehicle 1, performs the vehicle
1
deliveries, returns vehicle
1
to the depot, then
ventures from the depot in vehicle
2,
performs vehicle
2
deliveries, and so on.
The GTSP formulation for the heterogeneous MTSP has at most (n
-t-
l)v nodes and
(n
-I-
l)vn
arcs.
Since there is exactly one GTSP arc created for each
d\j
given, degeneracy is not built into
the model. As constructed, the model requires each salesman to visit at least one customer. This
requirement can be relaxed by adding zero cost arcs
(0",
0^)
and (0^+^) for r
=
1,...,
v
-
1. The
formulation can also be easily modified to accommodate fixed salesman charges, multiple depots
and salesman/customer incompatibilities.
4.
CONCLUDING REMARKS
The results presented in Section 2 allow us to transform any problem modeled as
a
GTSP into
a standard asymmetric
TSP.
One practical compHcation with this transformation is that arc costs
can become very large. This is of no theoretical importance, but may cause stabilify problems
in some solution techniques. In particular, the resulting TSP has an arc and arc cost structure
that will cause severe difficulties for traditional assignment-based methods for solving asymmetric

44 C.E. NOON AND J.C. BEAN
TSP's.
Specifically, the directed zero cost intracluster cycle will cause a simple subtour elimination
algorithm to branch m levels before the first non-zero lower boimd might be reached.
The approaches for the TSP based on cutting-plane methods may, however, be suitable for
solving the transformed problem. Such methods, however, could be successfully applied only if
the large costs are avoided. Within a cutting-plane approach, the presence of large costs would
pose numerical stability problems for the LP-solver and would inhibit variable elimination. The
large costs could, however, be avoided by modifying the transformation in the construction of P".
Instead of adding the large constant to intercluster arcs, constraints could be added which would
explicitly serve the same purpose as the added constant. The final product of the transformation
would be an asymmetric TSP with no large costs but with an additional set of constraints.
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••'•
•
-cVsil^ Charles Noon is an Assistant Professor
of
Management Science
at
the University of
"
•*
Tennessee
at
Knoxville. He joined
the
University
of
Tennessee faculty
in
1987 after
receiving
his
Ph.D.
in
Industrial
and
Operations Engineering from
the
University
of
Michigan. Dr. Noon's research interests include vehicle routing, production scheduling
and discrete optimization. His published
works
have appeared in Operations Research
and Interfaces.
James Bean is an Associate Professor in the Department of Industrial and Operations
Engineering at
the
University of
Michigan.
He has earned
a
Master's Degree and Ph.D.
from Stanford University in operations research and
a
B.S.
in mathematics from Harvey
Mudd College. Professor Bean's research interests are in infinite horizon optimization
and integer programming
as
applied
to
equipment replacement, capacity expansion,
production, scheduling and
biomechanics.
Professor Bean
is an
Associate Editor for the
journal Management Science and Editor
of
the ORSA/TIMS Annual Comprehensive
Index.