INFORMS Journal on Computing
Articles in Advance, pp. 1–11
issn 1091-9856 eissn 1526-5528
http://dx.doi.org/10.1287/ijoc.1110.0476
© 2011 INFORMS
A Generic Branch-and-Cut Algorithm for
Multiobjective Optimization Problems:
Application to the Multilabel Traveling
Salesman Problem
Nicolas Jozefowiez
CNRS, LAAS, F-31077 Toulouse, France; and Université de Toulouse, UPS, INSA, INP, ISAE, LAAS, F-31077 Toulouse, France,
nicolas.jozefowiez@laas.fr
Gilbert Laporte
CIRRELT, HEC Montréal, Montréal, Québec H3T 2A7, Canada, gilbert.laporte@cirrelt.ca
Frédéric Semet
Laboratoire d’Automatique, Génie Informatique et Signal, École Centrale de Lille, Cité Scientifique,
59651 Villeneuve d’Ascq Cedex, France, frederic.semet@ec-lille.fr
T
his paper describes a generic branch-and-cut algorithm applicable to the solution of multiobjective optimiza-
tion problems for which a lower bound can be defined as a polynomially solvable multiobjective problem.
The algorithm closely follows standard branch and cut except for the definition of the lower and upper bounds
and some optional speed-up mechanisms. It is applied to a routing problem called the multilabel traveling
salesman problem, a variant of the traveling salesman problem in which labels are attributed to the edges. The
goal is to find a Hamiltonian cycle that minimizes the tour length and the number of labels in the tour. Imple-
mentations of the generic multiobjective branch-and-cut algorithm and speed-up mechanisms are described.
Computational experiments are conducted, and the method is compared to the classical -constraint method.
Key words : decision analysis, multiple criteria; programming, multiple criteria; programming, integer,
algorithm, cutting plane; networks–graphs, traveling salesman
History : Accepted by Karen Aardal, Area Editor for Design and Analysis of Algorithms; received September
2009; revised August 2010, April 2011; accepted June 2011. Published online in Articles in Advance.
1. Introduction
The purpose of this paper is to introduce a generic
branch-and-cut methodology applicable to the solu-
tion of multiobjective integer linear programs; we will
refer to this method by the abbreviation MOB&C.
The main difference between MOB&C and standard
branch and cut lies in the definition of the lower
and upper bounds. The introduction of cuts is not
an essential feature of the method, which works
just as well in a standard branch-and-bound set-
ting. As in classical branch-and-cut algorithms, cuts
can strengthen the linear relaxation of the initial
model and may increase the lower bound values. Our
method applies to any problem for which the lower
bound can be defined as a polynomially (or pseudo-
polynomially) solvable problem. This is the case of
several routing problems such as the multilabel trav-
eling salesman problem (MLTSP), introduced later in
this paper. Although the method is generic, it must
be tailored to the problem at hand to achieve good
results.
A multiobjective problem can be stated formally as
follows:
MOP minimize
x∈D
F x =
(
f
1
x f
2
xf
n
x
)
where n ≥ 2 is the number of objective functions, x =
x
1
x
2
x
r
is the decision variable vector or solu-
tion, D is the feasible solution set, and F x is the
objective vector. The set O = F D corresponds to the
images of the feasible solutions in the objective space,
and y = y
1
y
2
y
n
, where y
i
= f
i
x, is a point of
the objective space. The Pareto dominance between
the solutions is defined as follows.
Definition 1. A solution x dominates () a solu-
tion z if and only if ∀ i ∈ 1n, f
i
x ≤ f
i
z and
∃i ∈ 1n, such that f
i
x < f
i
z.
1
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Published online ahead of print September 15, 2011