Piecewise linear approximations for the static–dynamic uncertainty strategy in stochastic lot-sizing

• Published in: Omega (Oxford) Vol. 50; pp. 126 - 140
• Main Authors: Rossi, Roberto, Kilic, Onur A., Tarim, S. Armagan
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01-01-2015; Pergamon Press Inc
• Subjects: Approximation; Fill rate; First order loss function; Integer programming; Mathematical functions; Non-stockout probability; Parameter estimation; Penalty cost; Piecewise linearisation; Quality of service; Static–dynamic uncertainty; Stochastic lot sizing; Stochastic models; Studies; Non-stockout probability; Piecewise linearisation; Fill rate; Static–dynamic uncertainty; Penalty cost; First order loss function; Stochastic lot sizing
• ISSN: 0305-0483; 1873-5274
• DOI: 10.1016/j.omega.2014.08.003

In this paper, we develop a unified mixed integer linear modelling approach to compute near-optimal policy parameters for the non-stationary stochastic lot sizing problem under static–dynamic uncertainty strategy. The proposed approach applies to settings in which unmet demand is backordered or lost; and it can accommodate variants of the problem for which the quality of service is captured by means of backorder penalty costs, non-stockout probabilities, or fill rate constraints. This approach has a number of advantages with respect to existing methods in the literature: it enables seamless modelling of different variants of the stochastic lot sizing problem, some of which have been previously tackled via ad hoc solution methods and some others that have not yet been addressed in the literature; and it produces an accurate estimation of the expected total cost, expressed in terms of upper and lower bounds based on piecewise linearisation of the first order loss function. We illustrate the effectiveness and flexibility of the proposed approach by means of a computational study.
•We developed a unified MILP modelling approach for the nonstationary stochastic lot sizing problem.•Our approach is based on a piecewise linearisation of the first order loss function.•Our approach applies to settings in which unmet demand is backordered or lost.•We considered backorder penalty costs, non-stockout probabilities, or fill rate constraints.•Our computational analysis demonstrates that our approach features a low optimality gap.