Extended formulations for stochastic lot-sizing problems
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 is utilized to describe the resulting information structure. Each node corresponds to a possible realization of uncertain problem parameters up to the time period this node belongs to. We denote the corresponding probability as . Except the root node, i.e., node 1, each node has a unique parent . In addition, we let represent the set of children of node , and represent the set of descendants of node (including itself). Moreover, corresponding to each node , we let , and represent unit production cost, holding cost, fixed setup cost, and demand respectively. All these parameters are assumed nonnegative with included. The corresponding stochastic uncapacitated lot-sizing problem (SULS) can be formulated as follows: where , and 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 . 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.
Section snippets
Extended formulation for SULS
We first review the optimality conditions and the dynamic programming algorithm for SULS (as described in [5]). Then, we derive the corresponding dual formulation in the dual space. Finally, we develop the extended formulation of SULS in a higher dimensional space. Without loss of generality, we assume the initial inventory level is zero. For notation brevity, we add a dummy node 0 which is the parent of node 1 and define . We let represent the cumulative demand from the root node
SULS with backlogging
Compared with SULS, is allowed negative here to capture the backlogging level at node . Accordingly, the objective function of SULS-B is updated as , where represents the unit backlogging cost. Following a similar logic as described in Section 2, we can derive optimality conditions for SULS-B. In particular, the optimality condition for the production level is the same as the one without backlogging as described in Section 2. The optimality condition for
Special cases and extensions
SULS with Wagner–Whitin costs and deterministic demands: For this case, we discuss SULS with deterministic demands and uncertain costs, which follow the Wagner–Whitin property (denoted as SULS-DW). Since the demands are deterministic, for notation brevity, we let . Accordingly, we let as the cumulative demand from time period to time period . Under this setting, if , then and for some . Accordingly, it is sufficient to
Acknowledgments
The authors would like to thank Andrew Miller for insightful discussions and suggestions on this paper. This research was partially supported by NSF Career AwardCMMI-0942156.
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