New Multi-product Valid Inequalities for a Discrete Lot-sizing Problem

eline Gicquel

and Michel Minoux

Laboratoire de Recherche en Informatique, Universit

e Paris Sud, Orsay, France

Laboratoire d’Informatique de Paris 6, Universit

e Pierre et Marie Curie, Paris, France

Keywords:

Supply-chain Management, Production Planning, Lot-sizing, Mathematical Programming, Valid Inequalities.

Abstract:

We consider a problem arising in the context of industrial production planning, namely the multi-product dis-

crete lot-sizing and scheduling problem with sequence-dependent changeover costs. We aim at developping

an exact solution approach based on a standard Branch & Bound procedure for this combinatorial optimiza-

tion problem. To achieve this, we propose a new family of multi-product valid inequalities which enables

us to better take into account in the mixed-integer linear programming formulation the conﬂicts between dif-

ferent products simultaneously requiring production on the resource. We then present both an exact and a

heuristic separation algorithm in order to identify the most violated valid inequalities to be added in the initial

MILP formulation within a cutting-plane generation algorithm. We ﬁnally discuss preliminary computational

results which conﬁrm the practical usefulness of the proposed valid inequalities at strengthening the MILP

formulation and at reducing the overall computation time.

1 INTRODUCTION

Capacitated lot-sizing arises in industrial production

planning whenever changeover operations such as

preheating, tool changing or cleaning are required be-

tween production runs of different products on a ma-

chine. The amount of the related changeover costs

usually does not depend on the number of products

processed after the changeover. Thus, to minimize

changeover costs, production should be run using

large lot sizes. However, this generates inventory

holding costs as the production cannot be synchro-

nized with the actual demand pattern: products must

be held in inventory between the time they are pro-

duced and the time they are used to satisfy customer

demand. The objective of lot-sizing is thus to reach

the best possible trade-off between changeover and

inventory holding costs while taking into account both

the customer demand satisfaction and the technical

limitations of the production system.

An early attempt at modelling this trade-off can be

found in (Wagner and Whitin, 1958) for the problem

of planning production for a single product on a single

resource with an unlimited production capacity. Since

this seminal work, a large part of the research on lot-

sizing problems has focused on modelling operational

aspects in more detail to answer the growing industry

need to solve more realistic and complex production

planning problems. An overview of recent develop-

ments in the ﬁeld of modelling industrial extensions

of lot-sizing problems is provided in (Jans and De-

graeve, 2008).

In the present paper, we focus on one of the vari-

ants of lot-sizing problems mentioned in (Jans and

Degraeve, 2008), namely the multi-product single-

resource discrete lot-sizing and scheduling problem

or DLSP. As deﬁned in (Fleischmann, 1990), several

key assumptions are used in the DLSP to model the

production planning problem:

- A set of products is to be produced on a single ca-

pacitated production resource.

- A ﬁnite time horizon subdivided into discrete peri-

ods is used to plan production.

- Demand for products is time-varying (i.e. dynamic)

and deterministically known.

- At most one product can be produced per period and

the facility processes either one product at full capac-

ity or is completely idle (discrete production policy).

- Costs to be minimized are the inventory holding

costs and the changeover costs.

In the DLSP, it is assumed that a changeover be-

tween two production runs for different products re-

sults in a changeover cost. Changeover costs can

depend either on the next product only (sequence-

independent case) or on the sequence of products

(sequence-dependent case). We consider in the

355

Gicquel C. and Minoux M..

New Multi-product Valid Inequalities for a Discrete Lot-sizing Problem.

DOI: 10.5220/0004751703550362

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 355-362

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

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

the demand for product p in

period t, h

the inventory holding cost per unit per

period for product p and S

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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356

Using this notation, the DLSPSD can be seen as

the problem of assigning at most one product to each

period of the planning horizon while ensuring de-

mand satisfaction and minimizing both inventory and

changeover costs. We thus introduce the following bi-

nary decision variables:

- y

where y

= 1 if product p is assigned to period

t, 0 otherwise.

- w

pqt

where w

pqt

= 1 if there is a changeover from p

to q at the beginning of t, 0 otherwise.

This leads to the following MILP formulation de-

noted DLSPSD0 for the problem.

LS0

=min

∑

p=1

∑

t=1

∑

τ=1

pτ

− d

pτ

)

∑

p,q=0

p,q

T −1

∑

t=1

p,q,t

(1)

∑

τ=1

pτ

≥

∑

τ=1

pτ

∀p,∀t (2)

∑

p=0

= 1, ∀t (3)

p,t

∑

q=0

q,p,t

∀p,∀t (4)

p,t

∑

q=0

p,q,t+1

∀p,∀t (5)

∈ {0, 1} ∀p,∀t (6)

p,q,t

∈ {0, 1} ∀p,∀q, ∀t (7)

The objective function (1) corresponds to the min-

imization of the inventory holding and changeover

costs over the planning horizon.

∑

τ=1

pτ

− d

pτ

) is

the inventory level of product p at the end of pe-

riod t. Constraints (2) impose that the cumulated de-

mand over interval [1,t] is satisﬁed by the cumulated

production over the same time interval. Constraints

(3) ensure that, in each period, the resource is either

producing a single product or idle. Constraints (4)-

(5) link setup variables y

with changeover variables

pqt

through equalities which can be seen as ﬂow

conservation constraints in a network. They ensure

that in case product p is setup in period t, there is a

changeover from another product q (possible q = p)

to product p to at the beginning of period t and a

changeover from product p to another product q (pos-

sible q = p) at the end of period t.

2.2 Single-product Valid Inequalities

We now recall the expression of the valid inequali-

ties proposed by (van Eijl and van Hoesel, 1997) for

the single product DLSP. We denote d

p,t,τ

the cumu-

lated demand for product p in the interval {t,...,τ}

and ∆

p,v

the v

positive demand period for product p.

∆

p,d

p,1,t

is thus the period with the v

positive unit

demand for product p after period t occurs.

∑

τ=1

pτ

− d

pτ

) +

∑

v=1

p,t+v

∆

p,d

p,1,t

∑

τ=t+v+1

∑

q6=p

q,p,τ

≥ w

∀p,∀t, ∀w ∈ [1, d

p,t+1,T

] (8)

The idea underlying valid inequalities (8) is to

compute a lower bound on the inventory level of a

product p at the end of a period t,

∑

τ=1

pτ

− d

pτ

by considering both the demands and the resource

setup states for this product in the forthcoming pe-

riods τ = t + 1...∆

p,d

p,1,t

. The reader is refered to

(van Eijl and van Hoesel, 1997) for a full proof of

validity for these inequalities. In the computation ex-

periments to be presented in Section 5, we use a stan-

dard cutting-plane generation algorithm to strengthen

the formulation DLSPSD0 by adding violated valid

inequalities of family (8). The resulting improved for-

mulation is denoted DLSPSD1.

Constraints (8) can be understood as a way to

strengthen the demand satisfaction constraints (2) by

expressing in a more detailed way the need for each

individual product to access the resource in order

to satisfy its own demand on a given subinterval of

the planning horizon. However, in the resulting DL-

SPSD1 formulation, the conﬂicts between different

products simultaneously requiring production on the

resource will only be handled by the single-period ca-

pacity constraints (3). In what follows, we propose to

improve this representation of the conﬂicts between

different products by considering multi-period multi-

product valid inequalities.

3 NEW MULTI-PRODUCT VALID

INEQUALITIES

We now present the multi-period multi-product valid

inequalities we propose to strengthen the linear

relaxation of the multi-product DLSPSD.

Proposition 1.

Let SP ⊂ {0...P} be a subset of products.

Let t ∈ [1,T ] be a period within the planning horizon.

Let (θ

,...,θ

) ∈ [0,T]

be a set of periods

such that θ

< t if p ∈ SP. For each period τ ∈ [1, T ],

we denote SD

= {p = 1...P|θ

> τ}.

The following inequalities are valid for the multi-

product DLSPSD.

NewMulti-productValidInequalitiesforaDiscreteLot-sizingProblem

357

∑

q=1

q,1,θ

∑

p∈SP

≤

∑

τ=1

(9)

where

is deﬁned by:

=min



∑

q∈SD

q,τ

∑

p∈SP

p,t



if τ /∈ [t − 1;t + 1]

t−1

∑

q∈SD

t−1

,p∈SP

qpt

= 0

t+1

∑

p∈SP,q∈SD

t+1

pq,t+1

Before providing the proof for Proposition 1, we

brieﬂy explain the idea underlying valid inequalities

(9). We choose a subset SP of products. If none of

these products is assigned for production in period t

(i.e.

∑

p∈SP

= 0), all corresponding valid inequal-

ities are trivially respected. But if one of these prod-

ucts is produced in period t (i.e.

∑

p∈SP

= 1), then

we have to make sure that we are able to satisfy the

total cumulated demand

∑

q=1

q,1,θ

on the remaining

periods 1...t − 1,t + 1...T . In this case, the right hand

side of inequalities (9) computes a tight upper bound

(

∑

τ=1

) of the total production capacity remaining

to satisfy this cumulated demand.

Proof. Let (y, w) be a feasible solution of the DL-

SPSD. We arbitrarily choose a subset of products SP,

a period t and a vector of periods (θ

,...,θ

)

such that θ

< t if p ∈ SP and show that all proposed

inequalities (9) are valid for the considered feasible

solution.

We distinguish two main cases:

- Case 1:

∑

p∈SP

= 0

In this case, the left hand side of the inequalities is

equal to 0 whereas the right hand side is nonnegative.

All inequalities (9) are thus trivially true.

- Case 2:

∑

p∈SP

= 1

In this case, the left hand side of inequalities (9)

is equal to the total cumulated demand over intervals

[1,θ

] for products q = 1..P, i.e. to

∑

q=1

q,1,θ

∑

p∈SP

= 1 means that period t is devoted to

the production of one of the products in SP. As we

have θ

< t for each product p ∈ SP, period t can-

not be used to satisfy the cumulated demand d

p,1,θ

of any product in SP. Hence (y,w) can be a feasible

solution of the DLSPSD if and only if the remaining

total cumulated production capacity over the periods

1...t −1,t +1...T is sufﬁcient to satisfy the cumulated

demand

∑

q=1

q,1,θ

We now seek to compute a tight upper bound for

the production capacity C

available in each period

τ ∈ [1,t − 1] ∩ [t + 1, T ] to satisfy the cumulated de-

mand

∑

q=1

q,1,θ

- By capacity constraints (3), we have C

≤ 1, i.e.

≤

∑

p∈SP

- Moreover, the cumulated demand d

q,1,θ

for a prod-

uct q can only be satisﬁed by a production for q in

period τ if τ ≤ θ

as demand backlogging is not al-

lowed here. Hence period τ can be used to satisfy part

of demand

∑

q=1

q,1,θ

only if the resource is setup

for one of products q = 1..P such that τ ≤ θ

. This

gives C

≤

∑

q∈SD

q,τ

We thus obtain C

≤ min(

∑

q∈SD

q,τ

∑

p∈SP

)

∀τ ∈ [1,t − 1] ∩ [t + 1,θ].

Now, we can exploit our knowledge of the setup

state of the resource in period t to further strengthen

these inequalities. Namely, we know that a product p

belonging to SP is produced in period t. A changeover

to (resp. from) this product p thus has to take place

at the beginning (resp. at the end) of period t. This

means that:

- If period t − 1 is to be used to satisfy the demand

of one of the products belonging to SD

t−1

, there

must be a changeover from this product q ∈ SD

t−1

to the product p ∈ SP at the beginning of period t.

The production capacity available in period τ = t − 1

for the products in SD

t−1

is thus limited by C

t−1

≤

∑

p∈SP,q∈SD

t−1

q,p,t

- Similarly, if period t + 1 is to be used to sat-

isfy the demand of one of the products belonging to

t+1

, there must be a changeover from the prod-

uct p ∈ SP to this product at the end of period t.

The production capacity available in period τ = t + 1

for the products in SD

t+1

is thus limited by C

t+1

≤

∑

p∈SP,q∈SD

t+1

p,q,t+1

We can thus strengthen the upper bound

of C

t−1

(resp C

t+1

) by replacing the term

min(

∑

q∈SD

q,τ

∑

p∈SP

) by

∑

p∈SP,q∈SD

t−1

q,p,t

(resp.

∑

p∈SP,q∈SD

t+1

p,q,t+1

) and obtain the inequali-

ties (9) discussed in Proposition 1.

4 SEPARATION PROBLEM

The number of valid inequalities (9) grows very fast

with the problem size. It it therefore not possible to

include them a priori in the MILP formulation of the

problem. This is why we use a cutting-plane genera-

tion strategy to add to the MILP formulation only the

most violated valid inequalities of the family. This re-

quires solving the corresponding separation algorithm

which, given a fractional solution (

y,w) of the DL-

SPSD, will either identify a violated valid inequality

or prove that no such inequality exists.

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358

4.1 Exact Separation Algorithm

We ﬁrst discuss an exact separation algorithm, i.e.

an algorithm which is guaranteed to ﬁnd an inequal-

ity violated by a fractional solution (y, w) of the

DLSPSD if one exists. We consider each period t

and seek to identify the subset SP and the vector

(θ

,...,θ

) which provide the largest violation

of inequalities (9). To achieve this, we formulate the

separation problem for a given t as follows.

We introduce the following decision variables:

- α

= 1 if product p ∈ SP, 0 otherwise.

- β

q,θ

= 1 if θ

= θ, 0 otherwise.

- γ

= 1 if capacity C

is limited by

∑

p=0

, 0 if

is limited by

∑

q=0

∑

θ=τ

q,τ

q,θ

With this notation, the separation problem QBP

for a given t and a solution (y, w) is formulated as:

max

∑

p=0

∑

q=1

∑

θ=1

q,1,θ

q,θ

−

∑

p=0

∑

q=1

∑

θ=t−1

q,p,t

q,θ

−

∑

p=0

∑

q=1

∑

θ=t+1

p,q,t+1

q,θ

−

∑

τ=1...t−2

t+2...θ

∑

p=0

∑

q=1

∑

θ=τ

q,τ

q,θ

(1 − γ

)

(10)

∑

θ=t

p,θ

≤ 1 ∀p (11)

∑

θ=0

p,θ

= 1 ∀p (12)

∈ {0, 1} ∀p (13)

p,θ

∈ {0, 1} ∀p,∀θ (14)

∈ {0, 1} ∀τ (15)

The objective function (10) corresponds to the

maximimization of the violation of the inequalities,

i.e. we seek to identify SP and(θ

,...,θ

) so

as to maximize the difference between the left and

the right hand side of the inequality. If this value is

strictly positive, we obtain a violated valid inequal-

ity. In case this value is less than or equal to 0, it

means that all valid inequalities for period t are satis-

ﬁed by the fractional solution (

y,w). Constraints (11)

state that for a given product p, we cannot simulta-

neously include it in SP and choose a period θ

such

that θ

≥ t. Constraints (12) guarantee that for each

product p, exactly one value of θ

is chosen .

Problem QBP

is a binary program with a

quadratic objective function and a series of linear con-

straints. It can be solved to optimality by a quadratic

binary programming solver such as the one embedded

in CPLEX 12.5.

4.2 Heuristic Separation Algorithm

As can be seen from the computational experiments

to be presented in Section 5, solving to optimality a

sequence of quadratic binary programs QBP

leads to

prohibitively long computation times for the cutting-

plane generation algorithm, even for small-size in-

stances. We are thus currently investigating the devel-

opment of a heuristic separation algorithm capable of

identifying violated valid inequalities more quickly.

We discuss here a ﬁrst version of this separation

algorithm which focuses on a special case of the pro-

posed multi-product valid inequalities. This special

case consists in choosing a period θ such that θ ≥ t,

in restricting the possible values for periods θ

,...,

,...,θ

to the set {0,θ} and in imposing θ

= 0 if

p ∈ SP.

In this case, for a given pair of periods (t, θ),

the separation problem amounts to ﬁnding a tripar-

tition of the set of products {0...P} into 3 subsets:

SP, SDem

= {q = 1..P|θ

= θ} and SDem

= {q =

1..P|θ

= 0} such that the quadratic expression (10) is

maximized. This problem shares some common fea-

tures with graph partitioning problems. We therefore

propose to solve it using the following Kernighan-Lin

type heuristic as this type of algorithm is known to be

rather efﬁcient at solving graph partitioning problems.

Choose a tripartition of {0...P}, Π

re f

, and

compute its violation V

re f

While (test =0):

Let test = 1, PossMove = P + 1 and

cur

= Π

re f

Allow all possible moves to explore the

neighbourhood of Π

cur

While (PossMove > 0):

Evaluate all partitions obtained by car-

rying out each of the allowed moves in

the neighbourhood of Π

cur

Select the best partition obtained in this

neighbourhood of Π

cur

, Π

best

, forbid the

move used to obtain Π

best

from Π

cur

, de-

crease PossMove by 1 and set Π

cur

best

If V

best

re f

, test = 0 and Π

re f

= Π

best

The neighbourhood of a tripartition Π of {0...P} is

deﬁned as the set of tripartitions obtained by moving

NewMulti-productValidInequalitiesforaDiscreteLot-sizingProblem

359

a single product from its current subset in Π to one

of the two other subsets. Moreover, in the computa-

tional experiments to be presented in Section 5, ﬁve

different types of partitions are used to initialize the

heuristic.

4.3 Cutting-plane Generation

Algorithm

We now brieﬂy describe the cutting-plane generation

used to strengthen formulation DLSPSD1 by adding

to it some multi-product valid inequalities (9).

Compute the initial LP relaxation of the

DLSPSD using formulation DLSPSD1.

While (test = 0):

Denote (y, w) the solution of the current

linear relaxation.

For t=1...T such that ∃p such that

0.0001 < y

< 0.9999;

Let θ = t and f ound =0.

While (θ ≤ T ) and ( f ound == 0),

Solve the separation problem for

periods (t, θ) using either the exact

or the heuristic algorithm.

If a violated valid inequality has

been found, let f ound = 1.

θ = θ + 1.

If at least one violated valid inequality is

found, add all the found violated valid in-

equalities to the current formulation and

compute its LP relaxation.

Else set test = 1 to stop the cutting-plane

generation.

5 COMPUTATIONAL RESULTS

We now discuss the results of some preliminary com-

putational experiments carried out to evaluate the ef-

fectiveness of the proposed multi-product valid in-

equalities at strengthening the formulation of the

multi-product DLSPSD and to assess their impact on

the total computation time.

We randomly generated instances of the problem

using a procedure similar to the one described in (Sa-

lomon et al., 1997) for the DLSP with sequence-

dependent change-over costs and times. More pre-

cisely, the various instances tested have the following

characteristics:

- Problem dimension. The problem dimension is rep-

resented by the number of products P and the num-

ber of periods T: we solved medium-size instances

involving 4 to 10 products and 15 to 75 periods.

- Inventory holding costs. For each product, inventory

holding costs have been randomly generated from a

discrete uniform DU (5, 10) distribution.

- Changeover costs. We used two different types

of structure for the changeover cost matrix S. In-

stances of sets A1-A7 have a general cost structure:

the cost of a changeover from product p to prod-

uct q, S

, was randomly generated from a discrete

uniform DU(100, 200) distribution. Instances of sets

B1-B7 correspond to the frequently encountered case

where products can be grouped into product families:

there is a high changeover cost between products of

different families and a smaller changeover cost be-

tween products belonging to the same family. In this

case, for products p and q belonging to different prod-

uct families, S

was randomly generated from a dis-

crete uniform DU(100,200) distribution; for prod-

ucts p and q belonging to the same product family,

was randomly generated from a discrete uniform

DU (0, 100) distribution.

- Production capacity utilization. Production capacity

utilization ρ is deﬁned as the ratio between the total

cumulated demand (

∑

p=1

∑

t=1

) and the total cu-

mulated available capacity (T ). We set ρ = 0.95 for

all instances.

- Demand pattern. Binary demands d

∈ {0,1} for

each product have been randomly generated accord-

ing to the a procedure similare to the used by (Sa-

lomon et al., 1997).

For each considered problem dimension, we gen-

erated 10 instances, leading to a total of 140 instances.

All tests were run on an Intel Core i5 (2.7 GHz)

with 4 GB of RAM, running under Windows 7. We

used a standard MILP software (CPLEX 12.5) with

the solver default settings to solve the problems with

one of the following formulations:

- DLPSD1: initial MILP formulation DLSPSD0, i.e.

formulation (1)-(7), strengthened by single-product

valid inequalities (8). We used a standard cutting-

plane generation strategy based on a complete enu-

meration of all possible valid inequalities to add them

into the formulation.

- DLSPSD2e: formulation DLSPSD1 strengthened

by multi-product valid inequalities (9). We used the

cutting-plane generation algorithm presented in Sec-

tion 4.3 to add only the most violated valid inequali-

ties and relied on the exact separation algorithm dis-

cussed in Section 4.1.

- DLSPSD2h: formulation DLSPSD1 strengthened

by multi-product valid inequalities (9). We used the

cutting-plane generation algorithm presented in Sec-

tion 4.3 to add only the most violated valid inequal-

ities and relied on the heuristic separation algorithm

discussed in Section 4.2.

Tables 1 and 2 display the computational results.

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Table 1: Preliminary computational results: exact separation algorithm.

DLSPSD1 DLSPSD2e

P T V Cst SP Gap

LP1

IP1

MPe Gap

LP2e

IP2e

A1 4 15 425 250 106 2.6% 2 0.3s 9 0.0% 0 38.5s

A2 6 15 840 315 108 0.9% 0 0.3s 3 0.1% 0 50.2s

A3 4 20 600 300 193 2.6% 5 0.4s 13 0.1% 0 2386.0s

B1 4 15 425 250 105 11.5% 6 0.3s 12 0.02% 0 51.2s

B2 6 15 840 315 107 5.3% 1 0.3s 17 1.3% 0 273.0s

B3 4 20 600 300 192 8.3% 9 0.5s 20 0.3% 2 3609.9s

Table 2: Preliminary computational results: heuristic separation algorithm.

DLSPSD1 DLSPSD2h

P T V Cst SP Gap

LP1

IP1

MPh Gap

LP2h

IP2h

A1 4 15 425 250 106 2.6% 2 0.3s 9 0.0% 0 0.1s

A2 6 15 840 315 108 0.9% 0 0.3s 3 0.2% 0 0.2s

A3 4 20 600 300 193 2.6% 5 0.4s 15 0.2% 0 0.3s

A4 6 25 1400 625 315 4.3% 9 1.0s 27 0.7% 4 1.0s

A5 6 50 2800 1050 1153 1.6% 32 6.7s 20 0.9% 11 4.7s

A6 10 50 6600 1650 1949 2.1% 99 21.0s 51 1.1% 30 22.7s

A7 8 75 6750 2025 2776 2.7% 856 151.9s 23 2.5% 660 147.5s

B1 4 15 425 250 105 11.5% 6 0.3s 16 0.1% 0 0.1s

B2 6 15 840 315 107 5.3% 1 0.3s 10 2.1% 1 0.3s

B3 4 20 600 300 192 8.3% 9 0.5s 21 0.4% 0 0.4s

B4 6 25 1400 625 307 9.2% 13 1.2s 30 0.8% 1 0.7s

B5 6 50 2800 1050 1248 12.2% 1753 47.7s 48 9.5% 983 37.6s

B6 10 50 6600 1650 1274 15.7% 25937 901.0s 97 11.9% 11284 496.0s

B7 8 75 6750 2015 2681 15.3% 25015 1961.9s 53 10.7% 22323 1904.7.0s

We provide for each set of 10 instances:

- P and T : the number of products and planning peri-

ods involved in the production planning problem.

- V and Cst : the number of variables and constraints

in the initial formulation DLSPSD0.

- SP: the number of single-product violated valid in-

equalities (8) added in the three formulations.

- MPe and MPh: the number of multi-product violated

valid inequalities added in formulation DLSPSD2e by

the exact separation algorithm and in formulation DL-

SPSD2h by the heuristic separation algorithm.

- Gap

LP1

(resp. Gap

LP2e

, Gap

LP2h

): the average per-

centage gap between the linear relaxation of formula-

tion DLSPSD1 (resp. DLSPSD2e, DLSPSD2h) and

the value of an optimal integer solution.

- N

IP1

(resp. N

IP2e

, N

IP2h

): the average number of

nodes explored by the Branch & Bound procedure be-

fore a guaranteed optimal integer solution is found or

the computation time limit of 2700s is reached.

- T

IP1

(resp. T

IP2e

, T

IP2h

): the total computation

time (cutting-plane generation and Branch & Bound

search) needed to ﬁnd a guaranteed optimal integer

solution (we used the value of 2700s in case a guar-

anteed optimal integer solution could not be found

within the computation time limit).

Results from Table 1 show that the proposed valid

inequalities (9) are efﬁcient at strengthening formu-

lation DLSPSD1. Namely, the integrality gap is re-

duced from an average of 5.3% with formulation DL-

SPSD1 (see Gap

LP1

) to an average of 0.3% with for-

mulation DLSPSD2e (see Gap

LP2e

). We note that this

reduction is particularly signiﬁcant for instances B1-

B3 featuring a product family changeover cost struc-

ture. Moreover this formulation strengthening is ob-

tained thanks to a relatively small number of multi-

product inequalities as can be seen from the aver-

age value of MPe (12). However, even if the num-

ber of nodes needed by the Branch & Bound proce-

dure to ﬁnd a guaranteed optimal solution is slightly

reduced when using formulation DLSPSD2e, it does

not lead to an overall reduction of the computation

time. This is mainly explained by the fact that the

cutting-plane generation algorithm based on an exact

separation algorithm requires prohibitively long com-

putation times to identify the violated multi-product

valid inequalities to be added to the formulation. It is

thus necessary to resort to a heuristic separation algo-

rithm such as the one proposed in Section 4.2.

Comparison of the results obtained with the ex-

act and the heuristic separation algorithm for the in-

NewMulti-productValidInequalitiesforaDiscreteLot-sizingProblem

361

stances A1-A3 and B1-B3 (Tables 1 and 2) shows that

the proposed heuristic is efﬁcient at ﬁnding violated

valid inequalities for small size instances. Namely,

the average integrality gap for these 60 instances

when using the heuristic algorithm is the Gap

LP2h

0.5% which is close to the one obtained when us-

ing the exact algorithm (Gap

LP2e

= 0.3%). Moreover,

the number of violated valid inequalities found by the

heuristic algorithm is nearly the same as the number

of violated valid inequalities found by the exact algo-

rithm.

Results from Table 2 also conﬁrm that the pro-

posed heuristic is rather efﬁcient at ﬁnding violated

valid inequalities for larger instances. This can be

seen by looking at the results for instances A4-A7 and

B4-B7. We ﬁrst note that, for these instances, the inte-

grality gap is reduced from an average of 7.9% while

using formulation DLSPSD1 to an average of 4.7%

while using formulation DLSPSD2h. Moreover a sig-

niﬁcant decrease in the overall computation time is

obtained for instances B4-B7 when using formulation

DLSPSD2h.

6 CONCLUSIONS

We considered the multi-product discrete lot-sizing

and scheduling problem with sequence-dependent

changeover costs and proposed a new family of multi-

product valid inequalities for this problem. This en-

abled us to better take into account in the MILP for-

mulation the conﬂicts between different products si-

multaneously requiring production on the resource.

We then presented both an exact and a heuristic sepa-

ration algorithm in order to identify the most violated

valid inequalities to be added in the initial MILP for-

mulation within a cutting-plane generation algorithm.

Our preliminary results show that the proposed valid

inequalities are efﬁcient at strengthening the MILP

formulation and that their use leads to a signiﬁcant re-

duction of the overall computation time for instances

featuring a product family changeover cost structure.

Research work is currently ongoing in order to ex-

tend the proposed heuristic separation algorithm so as

to identify violated valid inequalities from the whole

family.

ACKNOWLEDGEMENTS

This work was funded by the French National Re-

search Agency (ANR) through its program for young

researchers (project ANR JCJC LotRelax).

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