Parametric Distributionally Robust Optimisation Models for Resource and Inventory Planning Problems

Black, Ben and Kirkbride, Christopher and Dokka Venkata Satyanaraya, Trivikram (2023) Parametric Distributionally Robust Optimisation Models for Resource and Inventory Planning Problems. PhD thesis, Lancaster University.

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Abstract

Parametric probability distributions are commonly used for modelling uncertain demand and other random elements in stochastic optimisation models. However, when the distribution is not known exactly, it is more common that the distribution is either replaced by an empirical estimate or a non-parametric ambiguity set is built around this estimated distribution. In the latter case, we can then hedge against distributional ambiguity by optimising against the worst-case objective value over all distributions in the ambiguity set. This methodology is referred to as distributionally robust optimisation. When applying this approach, the ambiguity set necessarily contains non-parametric distributions. Therefore, applying this approach often means that any information about the true distribution’s parametric family is lost. This thesis introduces a novel framework for building and solving optimisation models under ambiguous parametric probability distributions. Instead of building an ambiguity set for the true distribution, we build an ambiguity set for its parameters. Every distribution considered by the model is then a member of the same parametric family as the true distribution. We reformulate the model using discretisation of the ambiguity set, which can result in a large, complex problem that is slow to solve. We first develop the parametric distributionally robust optimisation framework for a workforce planning problem under binomial demands. We then study a budgeted, multi-period new svendor model under Poisson and normal demands. In these first two cases, we develop fast heuristic cutting surface algorithms using theoretical properties of the cost function. Finally, we extend the framework into the dynamic decision making space via robust Markov decision processes. We develop a novel projectionbased bisection search algorithm that completely eliminates the need for discretisation of the ambiguity set. In each case, we perform extensive computational experiments to show that our algorithms offer significant reductions in run times with only negligible losses in solution quality.