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Modeling Industrial Lot Sizing Problems: A Review
Raf Jans, Zeger Degraeve
To cite this version:
Raf Jans, Zeger Degraeve. Modeling Industrial Lot Sizing Problems: A Review. International Journal
of Production Research, Taylor & Francis, 2008, 46 (06), pp.1619-1643. �10.1080/00207540600902262�.
�hal-00512940�
For Peer Review Only
Modeling Industrial Lot Sizing Problems: A Review
Journal:
International Journal of Production Research
Manuscript ID:
TPRS-2005-IJPR-0281.R1
Manuscript Type:
State-of-the-Art Review
Date Submitted by the
Author:
27-Apr-2006
Complete List of Authors:
Jans, Raf; RSM Erasmus University, School of Management
Degraeve, Zeger; London Business School, Decision Sciences
Keywords:
MODELLING, LOT SIZING
Keywords (user):
http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.uk
International Journal of Production Research
For Peer Review Only
1
MODELING INDUSTRIAL LOT SIZING PROBLEMS: A REVIEW
Raf Jans
RSM Erasmus University
PO Box 1738, 3000 DR Rotterdam, The Netherlands
rjans@rsm.nl
Zeger Degraeve
London Business School
Regent’s Park, London NW1 4SA, U.K.
zdegraeve@london.edu
Accepted July 5, 2006
_____________________________________________________________________
Abstract
In this paper we give an overview of recent developments in the field of modeling
deterministic single-level dynamic lot sizing problems. The focus of this paper is on
the modeling of various industrial extensions and not on the solution approaches. The
timeliness of such a review stems from the growing industry need to solve more
realistic and comprehensive production planning problems. First, several different
basic lot sizing problems are defined. Many extensions of these problems have been
proposed and the research basically expands in two opposite directions. The first line
of research focuses on modeling the operational aspects in more detail. The discussion
is organized around five aspects: the set ups, the characteristics of the production
process, the inventory, demand side and rolling horizon. The second direction is
towards more tactical and strategic models in which the lot sizing problem is a core
substructure, such as integrated production-distribution planning or supplier selection.
Recent advances in both directions are discussed. Finally, we give some concluding
remarks and point out interesting areas for future research.
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1. Introduction
In this review, we will discuss models that have been developed for optimizing
production planning and inventory management. Lot sizing models determine the
optimal timing and level of production. They can be classified according to their time
scale, the demand distribution and the time horizon. The famous Economic Order
Quantity model (EOQ) assumes a continuous time scale, constant demand rate and
infinite time horizon. The extension to multiple items and constant production rates is
known as the Economic Lot Scheduling Problem (ELSP) (Elmaghraby 1978, Zipkin
1991). The subject of this review is the dynamic lot sizing problem with a discrete
time scale, deterministic dynamic demand and finite time horizon. We will see that lot
sizing models will incorporate more and more scheduling aspects. These scheduling
models essentially determine the start and finish times of jobs (scheduling), the order
in which jobs are processed (sequencing) and the assignment of jobs to machines
(loading). Lawler et al. (1993) give an extensive overview of models and algorithms
for these problems.
A general overview of many different aspects of production planning and inventory
management can be found in Graves et al. (1993) and in standard textbooks such as
Silver et al. (1998), Hopp and Spearman (2000) or Vollmann et al. (1997). Several
studies focus specifically on the dynamic lot sizing problem (De Bodt et al. 1984,
Bahl et al. 1987, Kuik et al. 1994, Wolsey 1995, Drexl and Kimms 1997, Belvaux and
Wolsey 2001, Karimi et al. 2003, Brahimi et al. 2006, Jans and Degraeve 2006).
This review has a threefold contribution. Since the excellent reviews of Kuik et al.
(1994) and Drexl and Kimms (1997) the research on dynamic lot sizing has further
grown substantially. First of all, this paper fills a gap by providing a comprehensive
overview of the latest literature in this field. Second, this paper aims to provide a
general review and an extensive list of references for researchers in the field.
Although this literature review is very extensive, we realize that it is impossible to be
exhaustive. We realize that a model and its solution approach are inherently linked:
more complex models demand also more complex solution approaches to solve them.
However, in this paper we focus on the modeling aspect as much as possible in order
to create some structure in the ever growing literature. This focus also distinguishes
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this paper from other lot sizing reviews. A recent review of solution approaches can
be found in Jans and Degraeve (2006). We show that the lot sizing problem is a core
substructure in many applications by reviewing both more operational and tactical or
strategic problems. Third, a comprehensive review further allows us to indicate new
areas for further research. The power of production planning theory comes from the
ability to solve more and more complex industrial problems. Whereas the early
models where usually more compact, capturing the main trade-off, the extensions
focus more and more on incorporating relevant industrial concerns. Therefore, this
review is also very timely.
2. Lot Sizing Models
2.1. The single item uncapacitated lot sizing problem
The simplest form of the dynamic lot sizing problem is the single item uncapacitated
problem:
( )
∑
=
++
m
t
tttttt
shcyscxvcMin
1
(1)
s.t.
tttt
sdxs +=+
−1
Tt
∈
∀
(2)
ttmt
ysdx ≤
Tt
∈
∀
(3)
{
}
1,0;0, ∈≥
ttt
ysx
Tt
∈
∀
(4)
We have three key variables in each period t: the production level (x
t
), the set up
decision (y
t
) and the inventory variable (s
t
). With each of these key variables is a cost
associated: vc
t
, sc
t
and hc
t
are respectively the variable production cost, set up cost
and holding cost in period t. T is the set of all periods in the planning horizon and m is
the last period. Demand for each period, d
t
, is known and sd
tk
is the cumulative
demand for period t until k. The objective is to minimize the total cost of production,
set up and inventory (1). We find here the same basic trade-off between set ups and
inventory which is also present in the EOQ formula. Demand can be met from
production in the current period or inventory left over from the previous period (2).
Any excess is carried over as inventory to the next period. In each period we need a
set up if we want to produce anything (3). As there is no ending inventory in an
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