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Journal ArticleDOI

Modeling Industrial Lot Sizing Problems: A Review

03 Jan 2008-International Journal of Production Research (Taylor & Francis)-Vol. 46, Iss: 6, pp 1619-1643
TL;DR: In this article, the authors give an overview of recent developments in the field of modeling deterministic single-level dynamic lot sizing problems, focusing on the modeling of various industrial extensions and not on the solution approaches.
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.

Summary (3 min read)

1. Introduction

  • The authors will discuss models that have been developed for optimizing production planning and inventory management.
  • They can be classified according to their time scale, the demand distribution and the time horizon.
  • The authors realize that a model and its solution approach are inherently linked: more complex models demand also more complex solution approaches to solve them.
  • Third, a comprehensive review further allows us to indicate new areas for further research.

2.1. The single item uncapacitated lot sizing problem

  • The objective is to minimize the total cost of production, set up and inventory (1).
  • The authors 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).
  • This problem was first discussed in the seminal paper by Wagner and Whitin (1958) .
  • In network terms the authors say that node 0 is the supply or source node, nodes 1 to 5 are the demand nodes and the demand balance equations (2) correspond to the conservation of flow constraints.

2.2. Capacitated Multi-Item Lot Sizing Problem (CLSP)

  • Of course, companies do not have an unlimited capacity and usually they make more than one product.
  • In the large bucket model, several items can be produced on the same machine in the same time period.
  • For each item the authors have the demand balance equations ( 6) and set up constraints (7).
  • Karmarkar and Schrage (1985) consider this problem without set up costs and called it the product cycling problem.
  • Jordan and Drexl (1998) showed the equivalence between DLSP for a single machine and the batch sequencing problem.

3. Further Extensions of Lot Sizing Models

  • Production planning problems are often classified according to the hierarchical framework of strategic, tactical and operational decision making (e.g. Bitran and Tirupati 1993) .
  • A yearly master production schedule at the plant level is used for tactical planning.
  • The authors organize these extensions around four topics: set ups, production, inventory and demand, but clearly some extensions relate to more than one of them.
  • Here the authors also discuss the use of these models in a rolling horizon way.
  • On the other hand, these models are incorporated into more tactical and strategic problems for which the operational lot sizing decisions are a core substructure.

3.1.1. Extension on the set ups

  • Sometimes, there are not only set ups for individual items, called minor set ups, but there is a joint or major set up as well, which is incurred when at least one product is produced.
  • Miller and Wolsey (2003) consider the discrete lot sizing problem with set ups but without start ups.
  • Computational results show that this model leads to considerable cost savings through the set up carry over (Gopalokrishnan et al. 2001) .
  • In the PLSP at most two different items can be produced in each time period.
  • Set up cost and time reduction programmes require an initial capital investment and result in a more flexible production.

3.1.2. Extensions on the Production

  • Every time production exceeds a multiple of the batch size, a new set up cost is incurred.
  • When multiple parallel machines are available, the lot sizing problem does not only include the timing and level of production, but also the allocation of production to machines.
  • Chan et al. (2002) propose a modified all-unit discount structure: if the total cost is higher than the total cost at the start of the next quantity interval, you only pay the lower cost.
  • Campbell and Mabert (1991) impose such cyclical schedules for the CLSP with set up times.

3.1.3. Extensions on the inventory

  • The inventory can also be bounded by upper and lower limits (Love 1973 , Swoveland 1975 , Erenguc and Aksoy 1990 , Sandbothe and Thompson 1993 , Gutiérrez et al. Veinott (1969) permits the proportional growth or deterioration of inventory.
  • Hsu (2000) and Chu et al. (2005) consider the uncapacitated single item lot sizing problem with an age dependent inventory cost as well as an age dependent deterioration rate where a part of the inventory is lost by carrying it to the next period.
  • Jain and Silver (1994) look at the problem with random life time perishability.
  • According to some stochastic process, the total inventory becomes either worthless or remains usable for at least the next period.

3.1.4. Extension on the demand

  • The objective function includes the backlog cost.
  • The extension with backlogging is also considered for the DLSP (Jans and Degraeve 2004 a ) and for the coordinated replenishment problem (Robinson and Gao 1996).
  • A variable l t representing the lost sales is added into the demand equation and the cost of a stockout is properly accounted for in the objective function.
  • For each demand an earliest and latest delivery date is specified and demand can be satisfied in this period without penalty.

3.1.5. Time Horizon

  • Schedules are usually implemented in a rolling horizon fashion.
  • New research (Simpson 2001) , however, indicates that the Wagner-Whitin rule still outperforms all the other heuristics in a wide variety of cases.
  • The rolling horizon approach can result in nervousness of the planning in the sense that schedules must be frequently changed.
  • The last set up may be advantageous beyond the planning horizon, so only a proportion of the set up cost has to be borne within the planning horizon.

3.2. Tactical and strategic models

  • Hierarchical production planning (Hax and Meal 1975 , Bitran, Haas and Hax 1981 , Graves 1982 , Bitran and Tirupati 1993 ) is a sequential procedure for solving production planning at different levels of aggregation.
  • Items are aggregated into families and families into types.
  • There is a lot of interaction between the different levels and the sequential optimization does usually not result in a global optimum.
  • Rajagopalan and Swaminathan (2001) optimize the capacity acquisition, production and inventory decisions over time in an environment with increasing demand.

4. Conclusions and New Research Directions

  • While some extensions are motivated from the literature or general practical observations (e.g. Jaruphongsa et al. 2005, Van Vyve and Ortega 2004) , many of the extensions discussed in Section 3 are inspired by a specific real life problem.
  • Some distinguishing characteristics, such as the use of flexible recipes, the existence of by-products, the integration of lot sizing and scheduling, storage constraints and a focus on profit maximization, affect the planning and scheduling.
  • One of the major limitations of the lot sizing models that the authors discuss in this review is the assumption of deterministic demand and processing times.
  • The inclusion of industrial concerns lead to larger and more complex models and consequently more complex algorithms are needed to solve them.

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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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For Peer Review Only
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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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For Peer Review Only
3
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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References
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Book
01 Jan 1998
TL;DR: In this paper, the authors present a framework for inventory management and production planning and scheduling with a focus on the most important (Class A) and routine (Class C) items.
Abstract: THE CONTEXT AND IMPORTANCE OF INVENTORY MANAGEMENT AND PRODUCTION PLANNING AND SCHEDULING. The Importance of Inventory Management and Production Planning and Scheduling. Strategic Issues. Frameworks for Inventory Management and Production Planning and Scheduling. Forecasting. TRADITIONAL REPLENISHMENT SYSTEMS FOR MANAGING INDIVIDUAL--ITEM INVENTORIES. Order Quantities When Demand is Approximately Level. Lot Sizing for Individual Items with Time--Varying Demand. Individual Items with Probabilistic Demand. SPECIAL CLASSES OF ITEMS. Managing the Most Important (Class A) Inventories. Managing Routine (Class C) Inventories. Style Goods and Perishable Items. THE COMPLEXITIES OF MULTIPLE ITEMS AND MULTIPLE LOCATIONS. Coorinated Replenishments at a Single Stocking Point. Supply Chain Management and Multiechelon Inventories. PRODUCTION PLANNING AND SCHEDULING. An Overall Framework for Production Planning and Scheduling. Medium--Range Aggregate Production Planning. Material Requirements Planning and its Extensions. Just--in--Time and Optimized Production Technology. Short--Range Production Scheduling. Summary. Appendices. Indexes.

2,739 citations

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TL;DR: Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods and desire a minimum total cost inventory management scheme which satisfies known demand in every period.
Abstract: (This article originally appeared in Management Science, October 1958, Volume 5, Number 1, pp. 89-96, published by The Institute of Management Sciences.) A forward algorithm for a solution to the following dynamic version of the economic lot size model is given: allowing the possibility of demands for a single item, inventory holding charges, and setup costs to vary over N periods, we desire a minimum total cost inventory management scheme which satisfies known demand in every period. Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods.

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Book
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TL;DR: In this article, one item with a constant demand rate and time-varying demands is described. But, the model is based on a single item with constant lead times.
Abstract: 1 General Introduction2 Systems and Models3 One Item with a Constant Demand Rate4 Time-Varying Demands5 Several Products and Locations6 Stochastic Demand: One Item with Constant Leadtimes7 Stochastic Leadtimes: The Structure of the Supply System8 Several Items with Stochastic Demands9 Time-Varying, Stochastic Demand: Policy Optimization Bibliography Appendix A: Optimization and Convexity Appendix B: Dynamical Systems Appendix C: Probability and Stochastic Processes Appendix D: Notational Conventions

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