Abstract
We address a variant of the single item lot sizing problem affected by proportional storage (or inventory) losses and uncertainty in the product demand. The problem has applications in, among others, the energy sector, where storage losses (or storage deteriorations) are often unavoidable and, due to the need for planning ahead, the demands can be largely uncertain. We first propose a two-stage robust optimization approach with second-stage storage variables, showing how the arising robust problem can be solved as an instance of the deterministic one. We then consider a two-stage approach where not only the storage but also the production variables are determined in the second stage. After showing that, in the general case, solutions to this problem can suffer from acausality (or anticipativity), we introduce a flexible affine rule approach which, albeit restricting the solution set, allows for causal production plans. A hybrid robust-stochastic approach where the objective function is optimized in expectation, as opposed to in the worst-case, while retaining robust optimization guarantees of feasibility in the worst-case, is also discussed. We conclude with an application to heat production, in the context of which we compare the different approaches via computational experiments on real-world data.
Highlights
Lot Sizing (LS) is a fundamental problem in a large part of modern production planning systems
We have considered a variant of the lot sizing problem with storage losses subject to demand uncertainty
We have developed two two-stage robust optimization approaches, both with first-stage set-up variables, one with second-stage storage variables but first-stage production variables, and another one in which production and storage variables are both second-stage
Summary
Lot Sizing (LS) is a fundamental problem in a large part of modern production planning systems. We focus on a variant of LS where the storage suffers from proportional losses and the product demands are subject to uncertainty.1 This variant suits the case of many applications in the energy sector where a small portion of the energy that is stored is lost over time and the demands (of, e.g., heat, as in the application that we will consider) are often not known in advance. After showing that solutions to this problem may be acausal (or anticipative), i.e., that they may contain production variables whose value, at time t, can only be determined by knowing the realization of the demand at time t > t, we introduce an affine rule approach This technique, restricting the solution set, allows for causal production plans.
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