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

Optimality of Treating Interference as Noise: A Combinatorial Perspective

For single-antenna Gaussian interference channels, we re-formulate the problem of determining the Generalized Degrees of Freedom (GDoF) region achievable by treating interference as Gaussian noise (TIN) derived in [3] from a combinatorial perspective. We show that the TIN power control problem can be cast into an assignment problem, such that the globally optimal power allocation variables can be obtained by well-known polynomial time algorithms. Furthermore, the expression of the TIN-Achievable GDoF region (TINA region) can be substantially simplified with the aid of maximum weighted matchings. We also provide conditions under which the TINA region is a convex polytope that relax those in [3]. For these new conditions, together with a channel connectivity (i.e., interference topology) condition, we show TIN optimality for a new class of interference networks that is not included, nor includes, the class found in [3]. Building on the above insights, we consider the problem of joint link scheduling and power control in wireless networks, which has been widely studied as a basic physical layer mechanism for device-to-device (D2D) communications. Inspired by the relaxed TIN channel strength condition as well as the assignment-based power allocation, we propose a low-complexity GDoF-based distributed link scheduling and power control mechanism (ITLinQ+) that improves upon the ITLinQ scheme proposed in [4] and further improves over the heuristic approach known as FlashLinQ. It is demonstrated by simulation that ITLinQ+ provides significant average network throughput gains over both ITLinQ and FlashLinQ, and yet still maintains the same level of implementation complexity. More notably, the energy efficiency of the newly proposed ITLinQ+ is substantially larger than that of ITLinQ and FlashLinQ, which is desirable for D2D networks formed by battery-powered devices.Comment: A short version has been presented at IEEE International Symposium on Information Theory (ISIT 2015), Hong Kon

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lotsizing problem with step-wise production costs (LS-SW). We first provide a survey of different optimization methods proposed to solve LS-SW. Then, flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities. The single node flow set can be seen as a generalization of some valid relaxations of LS-SW. A new class of valid inequalities we call mixed flow cover, is derived from the integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and if V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. A cutting plane algorithmis proposed for these three classes of valid inequalities. The exact separation algorithmproposed for the constant capacitated case runs in polynomial time. Finally, some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities.single-item capacitated lot sizing problem, flow cover inequalities, cutting plane algorithm

A heuristic approach for big bucket multi-level production planning problems

Multi-level production planning problems in which multiple items compete for the same resources frequently occur in practice, yet remain daunting in their difficulty to solve. In this paper, we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot-sizing problems. In addition, unlike many other heuristics, it generates high quality lower bounds using strong formulations, and its simple scheme allows it to be easily implemented in the Xpress-Mosel modeling language. Extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic, particularly for challenging problems