A polyhedral study of the static probabilistic lot-sizing problem

Abstract

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 chance-constrained programs with simple recourse. The computational results show that the proposed inequalities and the new formulation are effective for the the static probabilistic lot-sizing problems.

Two-stage Stochastic Lot-sizing Problem with Chance-constrained Condition in the Second Stage

“…Let s ji be the inventory at the end of time period i ∈ N in scenario j ∈ Ω, which incurs a unit holding cost h i . Then we can transform the formulation of SLSCC into a deterministic equivalent formulation as (refer to Liu and Küçükyavuz (2018)):…”

“…Similar algorithms can be generalized to SULS with random lead times (Huang and Kkyavuz (2008) and Jiang and Guan (2011)). Liu and Küçükyavuz (2018) consider that the stochastic lot-sizing model may lead to an overconservative solution with excessive inventory, because the uncertain demand in each time period has to be satisfied. Thereby, a chance-constrained lot-sizing formulation is introduced, which is referred to as the static probabilistic lot-sizing problem (SPLS).…”

“…Another dynamic variant of SPLS that updates the production schedule after the scenario realization of the former time periods is studied by Zhang et al (2014). For the polyhedral study, Liu and Küçükyavuz (2018) give the first relevant result that exploits the lot-sizing structure into the construction of valid inequalities and facet-defining inequalities for SPLS, however, the study is under a equiprobable condition.…”

“…Briefly, the demands are fixed in the first stage, and random variables in the second stage, which is likely to occur in the real life when a long-term production planning is going to be made. In addition, we assume the second stage random demands obey a finite distribution, and introduce a chance-constrained condition to avoid over-conservative solutions, like Liu and Küçükyavuz (2018) do, but without the limitation of equiprobable condition. We expect to provide a polyhedral study of our proposed two-stage stochastic lot-sizing problem with chance-constrained condition in the second stage (SLSCC).…”

A multistage stochastic lot-sizing problem with cancellation and postponement under uncertain demands

“…Later, Jiang and Guan [15] established a quadratic polinomial time algorithm for this variant. Liu and Küçükyavuz [18] proposed valid inequalities for the static probabilistic lot-sizing problem. Hosseini and MirHassani [13] generated valid inequalities for tightening a refueling station location model.…”