The capacitated lot sizing problem: a review of models and algorithms (original) (raw)
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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. This review has a threefold contribution. Since the excellent reviews of and 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 this paper from other lot sizing reviews. A recent review of solution approaches can be found in Jans and Degraeve . 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. 1620 R. Jans and Z. Degraeve 1624 R. Jans and Z. Degraeve
Modeling of Multi-Level Capacitated Lot-Size Scheduling Problem
American Journal of Applied Sciences, 2011
Problem statement: Lot-size is the clustering of items for transportation or manufacturing processes occurring at the same time. The issue in lot-size problem is to design production processes so that the feasible production quantities are equal to customer demand quantities and the timing of production is such that inventory positions are almost zero. Approach: In this study, we explore the multi-level lot-size and scheduling problem. It is on a multi-level capacitated lot-size problem or known as the multi-level lot-size problem with bottlenecks. Two models were introduced to solve the multi-level lot-size problem namely the Billington model and Alf Kimms model. Using these models, a simple heuristic method was designed to solve a multi level capacitated lot sizing scheduling problem. Results: In this study, we showed that Alf Kimms model is more efficient than Billington model. The result given by Alf Kimms model is always feasible without further modification unlike the Billington model. This is due to the way the constraint is devised to ensure the inventory balance. The constraint used in Alf Kimms model ensures that inventory in hand is always sufficient to fulfill the demand occurred in each period. However, the use of this constraint in Billington model is to ensure that the total production for each item is always greater than or equal to the total external demand in the time horizon. Therefore, without some form of modification, the result given by Billington model will be infeasible production plan. Conclusion: A comparative study between these models shows that both models were successfully devised to solve capacitated multi-level lot sizing problem with the objective function to minimize the total holding costs and setup-cost. This study also shows that the production schedule will always start at the last period because this will give the lowest costs and it also shows that Alf Kimms model gives a set of feasible sub-optimal schedule compare to Billington model.
An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging
European Journal of Operational Research, 2011
This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions. manuscript no. (Please, provide the mansucript number!) capacitated single level multi-item lot-sizing problem with backlogging. examined the uncapacitated single item lot-sizing problem with backlogging and start-up costs, when Wagner-Whitin costs are assumed. Cheng et al. (2001) formulated single-level lot-sizing problems with provisions for backorders using a fixed-charge transportation model and proposed a heuristic solution method. Ganas and Papachristos (2005) proposed a polynomial-time algorithm for the single-item lot-sizing problem with backlogging. Song and Chan (2005) proposed a dynamic programming algorithm for solving a single-item lot-sizing problem with backlogging on a single machine at a finite production rate. Mathieu (2006) proposed two extended linear programming (LP) reformulations of single-item lot-sizing problems with backlogging and constant capacities. In a recent study, Küçükyavuz and Pochet (2009) provided the full description of the convex hull for the single-level uncapacitated problem with backlogging. Wu and Shi (2009b) proposed a heuristic that combines domain knowledge from the different strategies of relax-and-fix effectively for the capacitated multi-level lot sizing problem with the consideration of backlogging. We refer the interested reader to Pochet and Wolsey (2006) for a detailed general review of different lot-sizing problems. We note that the term backlog is used interchangeably with backorder in the lot-sizing literature, referring to any demand that is not satisfied on time but in a later time period, no matter what type of manufacturing environment. In our context, we consider a model that is flexible enough to apply to both MTO (Make-To-Order) and MTS (Make-To-Stock) environments when production is planned based on fixed demands or forecasts. The past research has also considered other classes of lot sizing problems. For example, Thizy and van Wassenhove (1985) designed a Lagrangian relaxation (LR) approach, in which capacity constraints are relaxed, in an attempt to decompose the problem into N uncapacitated single item lot-sizing subproblems, solvable by the Wagner-Whitin algorithm. Trigeiro (1987) developed a similar approach for solving the deterministic, multi-item, single-operation lot-sizing problem. Trigeiro et al. (1989) also proposed LR based methods for large-scale lot-sizing problems. Kuik and Salomon (1990) evaluated a simulated annealing heuristic for solving multi-level lot-sizing problem. Pochet and Wolsey (1991) applied strong cutting planes
A survey of lot-sizing and scheduling models
2001
Abstract: This paper surveys Lot-Sizing and Scheduling Models emphasising single-stage cases. The objective here is to present different aspects of such models in the operational research area and notes the most common modern methods to solve them. Metaheuristic methods feature heavily in the research literature. In this work the reader will find many references, though it must be noted that the speed with which the publications in the artificial intelligence and operational research areas is increasing significantly.
A heuristic solution of multi-item single level capacitated dynamic lot-sizing problem
Journal of Mechanical Engineering, 2008
The multi-item single level capacitated dynamic lot-sizing problem consists of scheduling N items over a horizon of T periods. The objective is to minimize the sum of setup and inventory holding costs over the horizon subject to a constraint on total capacity in each period. No backlogging is allowed. Only one machine is available with a fixed capacity in each period. In case of a single item production, an optimal solution algorithm exists. But for multi-item problems, optimal solution algorithms are not available. It has been proved that even the two-item problem with constant capacity is NP (nondeterministic polynomial)-hard. That is, it is in a class of problems that are extremely difficult to solve in a reasonable amount of time. This has called for searching good heuristic solutions. For a multi-item problem, it would be more realistic to consider an upper limit on the lot-size per setup for each item and this could be a very important parameter from practical point of view. T...
Three Mathematical Models for a Integrated Lot Sizing and Scheduling Problem
2014
The objective of this work is to propose mathematical models for the Integrated Lot Sizing and Scheduling Problem (ILSP) considering a production process involving one stage, one machine and considering sequence dependent set up times and costs. An ilustrative example is used to study the computational behavior of the models when the instances are solved by a general purpose software.
An improved model and a heuristic for capacitated lot sizing and scheduling in job shop problems
Scientia Iranica, 2017
This paper studies the problem of capacitated lot-sizing and scheduling in job shops with a carryover setup and a general product structure. After analyzing the literature, the shortcomings are easily realized; for example, the available mathematical model is unfortunately not only non-linear but also incorrect. No lower bound and heuristic is developed for the problem. Therefore, we first develop a linear model for the problem on-hand. Then, we adapt an available lower bound in the literature to the problem studied here. Since the problem is NP-hard, a heuristics based on production shifting concept is also proposed. Numerical experiments are used to evaluate the proposed model and algorithm. The proposed heuristic is assessed by comparing it against other algorithms in the literature. The computational results demonstrate that our algorithm has an outstanding performance in solving the problem.
Facets and algorithms for capacitated lot sizing
Mathematical Programming, 1989
The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a single-item capacitated version of this problem, and use
A Lagrangean relaxation approach for capacitated lot sizing problem with setup times
International Journal of Operational Research, 2013
In this research, we consider a production planning problem that determines the production timings and quantities for multiple products over a finite number of periods without violating capacity constraints. This problem is commonly referred to as the capacitated lot sizing (CLS) problem. We develop a model that explicitly considers setup times for products and different types of production capacities such as regular time and overtime. We develop a heuristic based on Lagrangian relaxation to solve this CLS problem. Computational results show that our algorithm gives reliable results while comparing solution values to lower bounds.