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present paper the DLSP with sequence-dependent
changeover costs (denoted DLSPSD in what fol-
lows). Sequence-dependent changeover costs are
mentioned in (Jans and Degraeve, 2008) as one of the
relevant operational aspects to be incorporated into
lot-sizing models. Moreover, a significant number
of real-life lot-sizing problems involving sequence-
dependent changeover costs have been recently re-
ported in the academic literature: see among (Silva
and Magalhaes, 2006) for a textile fibre industry or
(Ferreira et al., 2012) for soft drink production.
A wide variety of solution techniques from the
Operations Research field have been proposed to
solve lot-sizing problems: the reader is referred to
(Buschk
¨
uhl et al., 2010; Jans and Degraeve, 2007) for
recent reviews on the corresponding literature. The
present paper belongs to the line of research dealing
with exact solution approaches aiming at providing
guaranteed optimal solutions for the problem. A large
amount of existing exact solution techniques consists
in formulating the problem as a mixed-integer lin-
ear program (MILP) and in relying on a Branch &
Bound type procedure to solve the obtained MILP.
However the computational efficiency of such a pro-
cedure strongly depends on the quality of the lower
bounds used to evaluate the nodes of the search tree.
In the present paper, we seek to improve the quality
of these lower bounds so as to decrease the total com-
putation time needed to obtain guaranteed optimal so-
lutions for medium-size instances of the problem.
Within the last thirty years, much research has
been devoted to the polyhedral study of lot-sizing
problems in order to obtain tight linear relaxations
and improve the corresponding lower bounds: see e.g.
(Pochet and Wolsey, 2006) for a general overview
of the related literature and (Belvaux and Wolsey,
2001; Gicquel et al., 2009; van Eijl and van Hoesel,
1997) for contributions focusing specifically on the
DLSP. However, these procedures mainly focus on
the underlying single-product subproblems and thus
fail at capturing the conflicts between multiple prod-
ucts sharing the same resource capacity. This leads
in some cases to significant integrality gaps for multi-
product instances of the DLSPSD. In what follows,
we propose a new family of multi-product valid in-
equalities to partially remedy this difficulty and dis-
cuss both an exact and a heuristic algorithm to solve
the corresponding separation problem. To the best of
our knowledge, this is one of the first attempts focus-
ing on improving the polyhedral description of multi-
product lot-sizing problems.
The main contributions of the present paper are
thus twofold. First we introduce a new family of valid
inequalities representing conflicts on multi-period
time intervals between several products simultane-
ously requiring production on the resource. Second
we formulate the corresponding separation problem
as a quadratic binary program and propose to solve it
either exactly by relying on a quadratic programming
solver or approximately through a Kernighan-Lin
type heuristic algorithm. The results of the prelim-
inary computational results carried out on medium-
size instances show that the proposed valid inequali-
ties are efficient at strengthening the linear relaxation
of the problem and at decreasing the overall compu-
tation time needed to obtain guaranteed optimal solu-
tions of the DLSPSD.
The remainder of the paper is organized as fol-
lows. In Section 2, we recall the initial MILP formu-
lation of the multi-product DSLPSD and the previ-
ously published single-product valid inequalities. We
then present in Section 3 the proposed new multi-
product valid inequalities and discuss in Section 4
both an exact and a heuristic algorithm to solve the
corresponding separation problem. Preliminary com-
putational results are discussed in Section 5.
2 MILP FORMULATION
We first recall the initial MILP formulation of the
DLSPSD. We use the network flow representation of
changeovers between products, which was proposed
among others by (Belvaux and Wolsey, 2001), as this
leads to a tighter linear relaxation of the problem. We
then discuss the valid inequalities first proposed by
(van Eijl and van Hoesel, 1997) to strengthen the un-
derlying single-product subproblems.
2.1 Initial MILP formulation
We wish to plan production for a set of products de-
noted p = 1...P to be processed on a single production
machine over a planning horizon involving t = 1...T
periods. Product p = 0 represents the idle state of the
machine and period t = 0 is used to describe the initial
state of the production system.
Production capacity is assumed to be constant
throughout the planning horizon. We can thus w.l.o.g.
normalize the production capacity to one unit per pe-
riod and express the demands as binary numbers of
production capacity units: see e.g. (Fleischmann,
1990). We denote d
pt
the demand for product p in
period t, h
p
the inventory holding cost per unit per
period for product p and S
pq
the sequence-dependent
changeover cost to be incurred whenever the resource
setup state is changed from product p to product q.
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