Optimal Multi-Level Lot Sizing for Requirements Planning Systems

Steinberg, Earle; Napier, H. Albert

doi:10.1287/mnsc.26.12.1258

Cited by 121 publications

(33 citation statements)

References 25 publications

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“…First proposed as a means of creating pure and generalized network structure for netforms [17,18,19,22], the layering strategy, as elaborated in succeeding sections, is not confined in principle to this setting, but continues to find its chief practical application in the domain of network-related formulations [7,16,20,31,32]. Complementing this strategy, interest has also emerged in identifying pre-existing embedded pure and generalized network structure or its equivalent [2,3,4,5,12,33,34].…”

Section: Introductionmentioning

confidence: 99%

Layering strategies for creating exploitable structure in linear and integer programs

Glover

Klingman

1988

Mathematical Programming

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The strategy of subdividing optimization problems into layers by splitting variables into multiple copies has proved useful as a method for inducing exploitable structure in a variety of applications, particularly those involving embedded pure and generalized networks. A framework is proposed in this paper which leads to new relaxation and restriction methods for linear and integer programming based on our extension of this strategy. This framework underscores the use of constructions that lead to stronger relaxations and more flexible strategies than previous applications. Our results establish the equivalence of all layered Lagrangeans formed by parameterizing the equal value requirement of copied variables for different choices of the principal layers. It is further shown that these Lagrangeans dominate traditional Lagrangeans based on incorporating non-principal layers into the objective function. In addition a means for exploiting the layered Lagrangeans is provided by generating subgradients based on a simple averaging calculation. Finally, we show how this new layering strategy can be augmented by an integrated relaxation/ restriction procedure, and indicate variations that can be employed to particular advantage in a parallel processing environment. Preliminary computational results on fifteen real world zero-one personnel assignment problems, comparing two layering approaches with five procedures previously found best for those problems, are encouraging. One of the layering strategies tested dominated all non-layering procedures in terms of both quality and solution time.

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Section: Introductionmentioning

confidence: 99%

Layering strategies for creating exploitable structure in linear and integer programs

Glover

Klingman

1988

Mathematical Programming

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“…A dynamic programming formulation has been used for the multiitem system when the end item has a periodic demand [Crowston et al 1973]. Other authors adopt echelon, commonality concepts into lotsizing problems with general product structures [Steinberg and Napier 1980, Afentakis et al 1984, Afentakis and Gavish 1986, McKnew et al 1991. Due to the computational complexity in such multi-item systems, in particular when capacity constraints are considered, heuristic approaches have been developed such as simulated annealing and genetic algorithms [Kuik and Salomon 1990, Kim and Kim 1996, Kimms 1999, Delleart et al 2000, Tang 2004, Kaku et al 2009], among others.…”

Section: Introductionmentioning

confidence: 99%

The space of solution alternatives in the optimal lotsizing problem for general assembly systems applying MRP theory

Grubbström

Tang

2012

International Journal of Production Economics

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MRP Theory combines the use of Input-Output Analysis and Laplace transforms, enabling the development of a theoretical background for multi-level, multi-stage production-inventory systems together with their economic evaluation, in particular applying the Net Present Value principle (NPV). less thanbrgreater than less thanbrgreater thanIn a recent paper (Grubbstrom et al., 2010), a general method for solving the dynamic lotsizing problem for a general assembly system was presented. It was shown there that the optimal production (completion) times had to be chosen from the set of times generated by the Lot-For-Lot (L4L) solution. Thereby, the problem could be stated in binary form by which the values of the binary decision variables represented either to make a production batch, or not, at each such time. Based on these potential times for production, the problem of maximising the Net Present Value or minimising the average cost could be solved, applying a single-item optimal dynamic lotsizing method, such as the Wagner-Whitin algorithm or the Triple Algorithm, combined with dynamic programming. less thanbrgreater than less thanbrgreater thanThis current paper follows up the former paper by investigating the complexity defined as the number of possible feasible solutions (production plans) to compare. We therefore investigate how properties of external demand timing and properties of requirements (Bill-of-Materials) have consequences on the size of this solution space. Explicit expressions are developed for how the total number of feasible production plans depends on numbers of external demand events on different levels for, in particular, the two extreme cases of a serial system and a full system (the latter, in which items have requirements of all existing types of subordinate items). A formula is also suggested for general systems falling in between these two extremes. For the most complex full system, it is shown that the number of feasible plans will be the product of elements taken from Sylvesters sequence (an instance of doubly exponential sequences) raised to powers depending on numbers of external demand events

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“…Multilevel lot sizing problems associated with requirements planning sys-tems have been modelled as xed charge generalized network ow problems (see, e.g., Steinberg and Napier, 1980;Rao and McGinnis, 1983). The bill of material structure combines dierent amounts of raw materials in the assembly or production of other components, which in turn may be combined in diering amounts, in order to satisfy the demands for one or more nal, or nished, products.…”

Section: Introductionmentioning

confidence: 99%

Lot sizing with inventory gains

Waterer

2007

Operations Research Letters

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This paper introduces the single item lot sizing problem with inventory gains. This problem is a generalization of the classical single item capacitated lot sizing problem to one in which stock is not conserved. That is, the stock in inventory undergoes a transformation in each period that is independent of the period in which the item was produced. A 01 mixed integer programming formulation of the problem is given. It is observed, that by projecting the demand in each period to a distinguished period, that an instance of this problem can be polynomially transformed into an instance of the classical problem. As a result, existing results in the literature can be applied to the problem with inventory gains. The implications of this transformation for problems involving dierent production capacity limitations as well as backlogging and multilevel production are discussed. In particular, it is shown that the polynomially solvable classical constant capacity problems become NP-hard when stock is not conserved.

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Optimal Multi-Level Lot Sizing for Requirements Planning Systems

Cited by 121 publications

References 25 publications

Layering strategies for creating exploitable structure in linear and integer programs

Layering strategies for creating exploitable structure in linear and integer programs

The space of solution alternatives in the optimal lotsizing problem for general assembly systems applying MRP theory

Lot sizing with inventory gains

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