On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

Methodological framework for an integrated multi-scale vulnerability and resilience assessment

The deliverable illustrates the methodological framework to assess vulnerability and resilience across different temporal and spatial scales, acknowledging the different domains where the latter may manifest, and in particular in the natural and the built environment, allocating a large importance to the so called “critical infrastructures”, in social and economic systems. A set of four matrices has been developed to identify what aspects should be looked at before the impact, that is to say what shows the potential ability or inability to cope with an extreme; at the impact, addressing in particular the capacity (or incapacity) to sustain various types of stresses (in the form of acceleration, pressure, heat…); in the time immediately after the impact, as the ability (or inability) to suffer losses and still continue functioning; and in the longer term of recovery, as the capacity to find a new state of equilibrium in which the fragilities manifested during and after the impact are addressed. Developing the framework, a particular attention has been paid to the relationships among systems within the same matrix and among matrices, across spatial and temporal scales. A set of matrices has been developed for different natural hazards, including in particular landslides and floods, trying to include as much as possible what past cases, the international literature and prior experience of involved partners have indicated as relevant parameters and factors to look at. In this regard, the project builds on the state of the art, embedding what has been learned until now in terms of response capacity to a variety of stresses and in the meantime identifying gaps to be addressed by future research

A computational analysis of lower bounds for big bucket production planning problems

In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research

Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms

This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time

Integrer les activites pastorales et forestieres dans la gestion de l'espace mediterraneen

National audienc

Stochastic lot-sizing with backlogging: computational complexity analysis

Dynamic programming, Integer programming, stochastic programming, Lot-sizing,