On formulations of the stochastic uncapacitated lot-sizing problem (original) (raw)

Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

Operations Research, 2008

In 1958, Wagner and Whitin published a seminal paper on the deterministic uncapacitated lot-sizing problem, a fundamental model that is embedded in many practical production planning problems. In this paper we consider a basic version of this model in which demand (and other problem parameters) are stochastic: the stochastic uncapacitated lot-sizing problem. We define the production path property of an optimal solution for our model and use this property to develop a backward dynamic programming recursion. This approach allows us to show that the value function is piecewise linear and right continuous. We then use these results to show that the dynamic programming approach yields an O(n 3 ) algorithm for the problem. In addition, we show that the value function for the problem without setup costs is continuous, piecewise linear, and convex, and therefore an even more efficient O(n 2 ) dynamic programming algorithm can be developed for this special case.

A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

Mathematical Programming, 2006

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical ( , S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the ( , S) inequalities to a general class of valid inequalities, called the (Q, S Q ) inequalities, and we establish necessary and sufficient conditions which guarantee that the (Q, S Q ) inequalities are facet-defining. A separation heuristic for (Q, S Q ) inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the (Q, S Q ) inequalities as cuts.

Stochastic lot-sizing problem with deterministic demands and Wagner–Whitin costs

In this paper, we consider a two-stage stochastic uncapacitated lot-sizing problem with deterministic demands and Wagner-Whitin costs. We develop an extended formulation in the higher dimensional space that provides integral solutions by showing that its constraint matrix is totally unimodular. We also provide the integral polyhedron of the problem in the original space by projecting the extended formulation to the original space.

Analysis of bounds for a capacitated single-item lot-sizing problem

Computers & Operations Research, 2007

Lot-sizing problems are cornerstone optimization problems for production planning with time varying demand. We analyze the quality of bounds, both lower and upper, provided by a range of fast algorithms. Special attention is given to LP-based rounding algorithms. † Corresponding author program to solve it. Polyhedral results for these problems are especially plentiful. Notable results are found in, among others, [9], , and [13] (see for a more comprehensive review). The abundance of polyhedral results has resulted in more effective integer programming based solution approaches, but many instances still require prohibitive amounts of computation time. Van Hoesel & Wagelmans [17] approach the problem differently and present a fully polynomial approximation scheme for single-item capacitated lot-sizing, with the time required to find a solution depending on the desired nearness of the value of the resulting solution to the optimal value. Unfortunately, the practical value of this approach is limited as the computational requirements are high even for moderate approximation factors.

On stochastic lot-sizing problems with random lead times

Operations Research Letters, 2008

We give multi-stage stochastic programming formulations for lot-sizing problems where costs, demands and order lead times follow a general discrete-time stochastic process with finite support. We characterize the properties of an optimal solution and give a dynamic programming algorithm, polynomial in input size, when orders do not cross in time.

A polyhedral study of the static probabilistic lot-sizing problem

Annals of Operations Research

We study the polyhedral structure of the static probabilistic lot-sizing problem and propose valid inequalities that integrate information from the chance constraint and the binary setup variables. We prove that the proposed inequalities subsume existing inequalities for this problem, and they are facet-defining under certain conditions. In addition, we show that they give the convex hull description of a related stochastic lot-sizing problem. We propose a new formulation that exploits the simple recourse structure, which significantly reduces the number of variables and constraints of the deterministic equivalent program. This reformulation can be applied to general chanceconstrained programs with simple recourse. The computational results show that the proposed inequalities and the new formulation are effective for the static probabilistic lot-sizing problems.

Dynamic-programming-based inequalities for the capacitated lot-sizing problem

IIE Transactions, 2010

Iterative solutions of forward dynamic programming formulations for the capacitated lot-sizing problem are used to generate inequalities for an equivalent integer programming formulation. The inequalities capture convex and concave envelopes of intermediate-stage value functions, and can be lifted by examining potential state information at future stages. We test several possible implementations that employ these inequalities, and demonstrate that our approach is more efficient than alternative integer programming based algorithms. For certain datasets, our algorithm also outperforms a pure dynamic programming algorithm for the problem.

Stochastic Capacitated Lot Sizing Subject to Maximum Acceptable Risk Level of Overutilization

Stochastic capacitated lot sizing problem is considered in the presence of probabilistic processing times and demand in this paper. A two-step hierarchical methodology is developed. First, stochastic capacity requirements are determined with statistical analysis & Monte-Carlo simulation. Second, a stochastic nonlinear mixed integer mathematical model is developed to solve the problem.

Uncapacitated lot sizing with backlogging: the convex hull

Mathematical Programming, 2009

An explicit description of the convex hull of solutions to the uncapacitated lotsizing problem with backlogging, in its natural space of production, setup, inventory and backlogging variables, has been an open question for many years. In this paper, we identify valid inequalities that subsume all previously known valid inequalities for this problem. We show that these inequalities are enough to describe the convex hull of solutions. We give polynomial separation algorithms for some special cases. Finally, we report a summary of computational experiments with our inequalities that illustrates their effectiveness.

A reformulation for the stochastic lot sizing problem with service-level constraints

Operations Research Letters, 2014

We study the stochastic lot-sizing problem with service level constraints and propose an efficient mixed integer reformulation thereof. We use the formulation of the problem present in the literature as a benchmark, and prove that the reformulation has a stronger linear relaxation. Also, we numerically illustrate that it yields a superior computational performance. The results of our numerical study reveals that the reformulation can optimally solve problem instances with planning horizons over 200 periods in less than a minute.