Mathematical programming heuristics for production planning problems under stochastic demand

Abstract

The Stochastic Programming (SP) can be defined as a mathematical programming technique that deals with uncertain data, which can be represented by random variables. Such technique is specially used when future events are not perfectly anticipated. The stochastic terminology is antagonist to the deterministic one, which reflects some data randomness. Generally, stochastic programming problems include two different components: a structural one (first stage) which is fixed and uncertainty free and a control one (second stage) which is affected by the uncertainty in the imput data. The decision variables are divided in two sets: * X is the first stage variables vector, which must be fixed before the random values become known. The optimal value for these variables relies only on the currently available data and is not conditioned to the random variable outcomes. * Y is the second stage variables vector that are also known as corrective actions or control decisions. These variables are determined after the random variables outcome. The optimal value for y depends both on the first stage decisions and the random variables outcome. To a more detailed review on Two-Stage Linear Stochastic Programming, the reader may refer to Birge & Louveaux (1997). Generally, stochastic problems are hard to solve, mainly when the number of scenarios - used to represent the possible outcomes for the Random Variables - is too big. Some researchers use decomposition methods and columns generation in order to solve them in several contexts (Alem et al., 2008). This project motivation is to apply mathematical programming based heuristics, Fix-and-Optimize and Relax-and-Fix, to production planning problems with stochastic demand. These problems are essentially multi-stage and feature a set of integer and binary variables related to production lots to be produced and machine/operations set up, configuration that makes the problem of hard exact solution. Moreover, when there is stochastic demand, the problems get bigger dimension, and the task of finding a feasible solution using commercial optimization packages may be unreliable even for small and medium instances.