Stage-t scenario dominance for risk-averse multi-stage stochastic mixed-integer programs

Abstract

This paper presents a new and general approach, named “Stage-t Scenario Dominance,” to solve the risk-averse multi-stage stochastic mixed-integer programs (M-SMIPs). Given a monotonic objective function, our method derives a partial ordering of scenarios by pairwise comparing the realization of uncertain parameters at each time stage under each scenario. Specifically, we derive bounds and implications from the “Stage-t Scenario Dominance” by using the partial ordering of scenarios and solving a subset of individual scenario sub-problems up to stage t. Using these inferences, we generate new cutting planes to tackle the computational difficulty of risk-averse M-SMIPs. We also derive results on the minimum number of scenario-dominance relations generated. We demonstrate the use of this methodology on a stochastic version of the mean-conditional value-at-risk (CVaR) dynamic knapsack problem. Our computational experiments address those instances that have uncertainty, which correspond to the objective, left-hand side, and right-hand side parameters. Computational results show that our “scenario dominance"-based method can reduce the solution time for mean-risk, stochastic, multi-stage, and multi-dimensional knapsack problems with both integer and continuous variables, whose structure is similar to the mean-risk M-SMIPs, with varying risk characteristics by one-to-two orders of magnitude for varying numbers of random variables in each stage. Computational results also demonstrate that strong dominance cuts perform well for those instances with ten random variables in each stage, and ninety random variables in total. The proposed scenario dominance framework is general and can be applied to a wide range of risk-averse and risk-neutral M-SMIP problems.

An expandable machine learning-optimization framework to sequential decision-making

“…Similarly, knapsack problems with a large number of periods are used in practical applications, including advertising, container loading, airline ticketing, and processor scheduling (Papastavrou et al, 1996;Hao et al, 2020). We generate instances for both MCLSP and MSMK based on realistic data generation techniques (Büyüktahtakın, 2022) rather than using them from the literature since training requires many instances that such benchmark datasets do not contain.…”

“…Examples of such problems include MSMK, the core resource allocation problem with stability constraints (Bampis et al, 2019), the multistage facility location problem (Eisenstat et al, 2014), and multistage prize collecting traveling salesperson (Bampis et al, 2021). In this study, we demonstrate our framework through knapsack and lot-sizing, since both are considered as general multi-stage problems and are commonly used to test improvements in the solution algorithms for multi-stage problems (Guan et al, 2009;Büyüktahtakın, 2023Büyüktahtakın, , 2022.…”

A K-means Supported Reinforcement Learning Framework to Multi-dimensional Knapsack

“…Among recent DP attempts, there are methods proposed by Pisinger [57], Bertsimas and Demir [7], and Balev et al [4]. Others, such as Büyüktahtakın [12,13], present theoretical contributions to aid solving large stochastic multi-dimensional knapsack problems. Finding an optimal solution is computationally very expensive, which motivates researchers to investigate approximation algorithms.…”

COVID-19: Optimal Allocation of Ventilator Supply under Uncertainty and Risk

“…We implement the scenario sub-problems and lower and upper bounds proposed by Büyüktahtakın (2020) to reduce the optimality gap of solving our risk-averse multi-stage stochastic programming problem (5a)-(5u), while referring to Büyüktahtakın (2020) for the proofs of those bounds originally driven for the general multi-stage stochastic programs. The scenario sub-problem and bounds are described below.…”