Yves Pochet

Lot-Sizing with Constant Batches: Formulation and Valid Inequalities

We consider the classical lot-sizing problem with constant production capacities LCC and a variant in which the capacity in each period is an integer multiple of some basic batch size LCB. We first show that the classical dynamic program for LCC simplifies for LCB leading to an On2 min{n, C} algorithm where n is the number of periods and C the batch size. Using this new algorithm, we show how to formulate both problems as linear programs with On3 variables and constraints. A class of so-called k, l, S, I inequalities are described for LCB which capture both the dynamic nature of the problem as well as the capacity aspects. For LCB, we prove that these inequalities are the only facet-defining inequalities of a certain form. For LCC, we show that these inequalities include all the known classes of valid inequalities. Finally, we discuss several open questions and possible extensions.

Mixing mixed-integer inequalities

Abstract.Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.¶Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.¶We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure.

Polyhedra for lot-sizing with Wagner-Whitin costs

We examine the single-item lot-sizing problem with Wagner—Whitin costs over ann period horizon, i.e.pt+ht⩾pt+1 fort = 1, ⋯,n−1, wherept, ht are the unit production and storage costs in periodt respectively, so it always pays to produce as late as possible.We describe integral polyhedra whose solution as linear programs solve the uncapacitated problem (ULS), the uncapacitated problem with backlogging (BLS), the uncapacitated problem with startup costs (ULSS) and the constant capacity problem (CLS), respectively. The polyhedra, extended formulations and separation algorithms are much simpler than in the general cost case. In particular for models ULS and ULSS the polyhedra in the original space have only O(n2) facets as opposed to O(2n) in the general case. For CLS and BLS no explicit polyhedral descriptions are known for the general case in the original space. Here we exhibit polyhedra with O(2n) facets having an O(n2 logn) separation algorithm for CLS and O(n3) for BLS, as well as extended formulations with O(n2) constraints in both cases, O(n2) variables for CLS and O(n) variables for BLS.

A branch-and-bound algorithm for the hybrid flowshop

This paper introduces a branch-and-bound algorithm for the hybrid flowshop scheduling problem to minimize makespan. The algorithm can also cope with problems with release dates and tails. Several heuristics are used to compute upper bounds. Lower bounds are based upon the single-stage subproblem relaxation. Several upper and lower bounding strategies are considered. Numerical tests show that, in a few minutes of running time, and even for the hardest (i.e. without a bottleneck stage) and mid-size problems, the algorithm has reduced the initial gap between upper and lower bounds by 50% on average.

Capacitated facility location: valid inequalities and facets

We examine the polyhedral structure of the convex hull of feasible solutions of the capacitated facility location problem. In particular we derive necessary and sufficient conditions for a family of “effective capacity” inequalities to be facet-defining, and further results on a more general family called “submodular” inequalities.

Forward-reserve allocation in a warehouse with unit-load replenishments

Many companies configure their warehouse with a forward area and a reserve area. The former is used for efficient order-picking, the latter for replenishing the forward area. We consider a situation in which orders are picked during a certain time period, referred to as the picking period. Prior to the picking period there is sufficient time to replenish the forward area. Our objective is to determine which replenishments minimize the expected amount of labor during the picking period. Further, we present a second model with a constraint on the replenishment activity. We model the problem as a binary programming problem and present efficient heuristics that provide tight performance guarantees. We compare the heuristics with procedures that are popular in practice and show that significant labor-savings are possible.

Valid inequalities and separation for capacitated economic lot sizing

A family of valid inequalities for the capacitated economic lotsizing problem is given. In the case of equal capacities, studied in more detail, a large subclass of the inequalities defines facets. A heuristic for the separation problem, based on these inequalities, is defined for use in a cutting plane algorithm. We give computational results for 12 and 24 periods test problems and for both the equal and different capacity cases. We also indicate how to extend this class of inequalities for more general capacitated fixed charge networks.

The uncapacitated lot-sizing problem with sales and safety stocks

Abstract.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.

Mathematical Programming Models and Formulations for Deterministic Production Planning Problems

We study in this lecture the literature on mixed integer programming models and formulations for a specific problem class, namely deterministic production planning problems. The objective is to present the classical optimization approaches used, and the known models, for dealing with such management problems.We describe first production planning models in the general context of manufacturing planning and control systems, and explain in which sense most optimization solution approaches are based on the decomposition of the problem into single-item subproblems.Then we study in detail the reformulations for the core or simplest subproblem in production planning, the single-item uncapacitated lot-sizing problem, and some of its variants. Such reformulations are either obtained by adding variables - to obtain so called extended reformulations - or by adding constraints to the initial formulation. This typically allows one to obtain a linear description of the convexh ull of the feasible solutions of the subproblem. Such tight reformulations for the subproblems play an important role in solving the original planning problem to optimality.We then review two important classes of extensions for the production planning models, capacitated models and multi-stage or multi-level models. For each, we describe the classical modeling approaches used. Finally, we conclude by giving our personal view on some new directions to be investigated in modeling production planning problems. These include better models for capacity utilization and setup times, new models to represent the product structure - or recipes - in process industries, and the study of continuous time planning and scheduling models as opposed to the discrete time models studied in this review.

An integrated model for warehouse and inventory planning

The purpose of this article is to evaluate the value of integrating tactical warehouse and inventory decisions. Therefore, a global warehouse and inventory model is presented and solved. In order to solve this mathematical model, two solution methodologies are developed which offer different level of integration of warehouse and inventory decisions. Computational tests are performed on a real world database using multiple scenarios differing by the warehouse capacity limits and the warehouse and inventory costs. Our observation is that the total cost of the inventory and warehouse systems can be reduced drastically by taking into account the warehouse capacity restrictions in the inventory planning decisions, in an aggregate way. Moreover additional inventory and warehouse savings can be achieved by using more sophisticated integration methods for inventory and warehouse decisions.