Extended formulations for stochastic lot-sizing problems

Abstract

In this paper, extended formulations for stochastic uncapacitated lot-sizing problems with and without backlogging are developed in higher dimensional spaces that provide integral solutions. Moreover, physical meanings of the decision variables in the extended formulations are explored and special cases with more efficient formulations are studied.

Introduction

The lot-sizing (LS) problem, determining when to produce and how much to produce at each time period so as to minimize the total production cost, is fundamental in production and inventory management. Meanwhile, significant research results have been derived for this problem and its deterministic variants, including polynomial time algorithms  [8], [3], [7], cutting planes describing the convex hull in the original space  [2], and extended formulations for the problems with Wagner–Whitin costs  [6].

Recently, with the consideration of cost and demand uncertainties, as well as dependency among different time periods, scenario-tree based stochastic lot-sizing is introduced in  [4]. Under this setting, the uncertain problem parameters are assumed to follow a discrete-time stochastic process with finite probability space and a scenario tree $\mathcal{T}=(\mathcal{V},\mathcal{E})$ is utilized to describe the resulting information structure. Each node $i\in \mathcal{V}$ corresponds to a possible realization of uncertain problem parameters up to the time period this node belongs to. We denote the corresponding probability as $p_i$. Except the root node, i.e., node 1, each node $i\in \mathcal{V}$ has a unique parent $i^{-}$. In addition, we let $\mathcal{C}\left(i\right)$ represent the set of children of node $i$, and $\mathcal{V}\left(i\right)$ represent the set of descendants of node $i$ (including itself). Moreover, corresponding to each node $i\in \mathcal{V}$, we let ${\alpha }_i,h_i,{\beta }_i$, and $d_i$ represent unit production cost, holding cost, fixed setup cost, and demand respectively. All these parameters are assumed nonnegative with $p_i$ included. The corresponding stochastic uncapacitated lot-sizing problem (SULS) can be formulated as follows:

$$min\sum _{i\in \mathcal{V}}({\alpha }_i{x}_i+{\beta }_i{y}_i+h_i{s}_i)$$$$\text{s.t. }{s}_{i^{-}}+x_i=d_i+s_i\text{,}$$$$x_i\le M{y}_i\text{,}$$$$x_i\ge 0,s_i\ge 0,y_i\in \{0,1\},\forall i\in \mathcal{V}\text{,}$$

where $x_i,y_i$, and $s_i$ represent the production level, the setup decision, and the inventory left at the end of the time period corresponding to the state defined by node $i$. Constraints (1), (2)indicate inventory balance and production capacity.

The extended formulation for SULS is first attempted in  [1], in which a reformulation is introduced to reduce the LP relaxation gap. Later on, in  [9], an extended formulation of SULS is provided for a special case in which demands are deterministic (although costs are uncertain) and Wagner–Whitin costs are assumed. In this paper, we derive extended formulations for general SULS problems in which both demands and costs are uncertain.