Extended Model Formulation of the Proportional Lot-Sizing and Scheduling Problem with Lost Demand Costs

We consider mixed-integer linear programming (MIP) models of production planning problems known as the small bucket lot-sizing and scheduling problems. We present an application of a class of valid inequalities to the case with lost demand (stock-out) costs. Presented results of numerical experiments made for the the Proportional Lot-sizing and Scheduling Problem (PLSP) confirm benefits of such extended model formulation

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

A Multi-level Capacitated Lot-sizing Problem with Safety Stock Deficit and Production Manners: A Revised Simulated Annealing

[1] Corresponding author e-mail: [email protected]   [1] Corresponding author e-mail: [email protected]   Lot-sizing problems (LSPs) belong to the class of production planning problems in which the availability quantities of the production plan are always considered as a decision variable. This paper aims at developing a new mathematical model for the multi-level capacitated LSP with setup times, safety stock deficit, shortage, and different production manners. Since the proposed linear mixed integer programming model is NP-hard, a new version of simulated annealing algorithm (SA) is developed to solve the model named revised SA algorithm (RSA). Since the performance of the meta-heuristics severely depends on their parameters, Taguchi approach is applied to tune the parameters of both SA and RSA. In order to justify the proposed mathematical model, we utilize an exact approach to compare the results. To demonstrate the efficiency of the proposed RSA, first, some test problems are generated; then, the results are statistically and graphically compared with the traditional SA algorithm

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

Mixing sets linked by bidirected paths

Recently there has been considerable research on simple mixed-integer sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot- sizing. This in turn has led to study of more general sets, called network-dual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the network-dual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a network-dual set in the original space of variables are known only for some special cases. Here we study two new cases, in which the continuous variables of the network-dual set are linked by a bi- directed path. In the first case, which is motivated by lot-sizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2^n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomial-time separation algorithm.mixing sets, extended formulations, mixed integer programming, lot-sizing with sales

The uncapacitated lot-sizing problem with sales and safety stocks

We examine a variant of the uncapacitated lot-sizing model of Wagner-Whitin involving sales instead of fixed demands, and lower bounds on stocks. Two extended formulations are presented, as well as a dynamic programming algorithm and a complete description of the convex hull of solutions. When the lower bounds on stocks are non-decreasing over time, it is possible to describe an extended formulation for the problem and a combinatorial separation algorithm for the convex hull of solutions. Finally when the lower bounds on stocks are constant, a simpler polyhedral description is obtained for the case of Wagner-Whitin costs

The uncapacitated lot-sizing problem with sales and safety stocks

We examine a variant of the uncapacitated lot-sizing model of Wagner-Whitin involving sales instead of fixed demands, and lower bounds on stocks. Two extended formulations are presented, as well as a dynamic programming algorithm and a complete description of the convex hull of solutions. When the lower bounds on stocks are non-decreasing over time, it is possible to describe an extended formulation for the problem and a combinatorial separation algorithm for the convex hull of solutions. Finally when the lower bounds on stocks are constant, a simpler polyhedral description is obtained for the case of Wagner-Whitin costs

The uncapacitated lot-sizing problem with sales and safety stocks

We examine a variant of the uncapacitated lot-sizing model of Wagner-Whitin involving sales instead of fixed demands, and lower bounds on stocks. Two extended formulations are presented, as well as a dynamic programming algorithm and a complete description of the convex hull of solutions. When the lower bounds on stocks are non-decreasing over time, it is possible to describe an extended formulation for the problem and a combinatorial separation algorithm for the convex hull of solutions. Finally when the lower bounds on stocks are constant, a simpler polyhedral description is obtained for the case of Wagner-Whitin costs.lot-sizing, production planning, mixed integer programming, integral polyhedra, extended formulations