Minimizing conditional-value-at-risk for stochastic scheduling problems

Minimizing value-at-risk in single-machine scheduling

“…There also exist a few recent papers that propose scenario-based models for scheduling problems with multiple machines (see, e.g., Alonso-Ayuso et al, 2007;Kasperski et al, 2012); however, these also rely on risk-neutral or robust approaches. The only other works which adopt a risk-averse approach for machine scheduling problems are Beck and Wilson (2007) and Sarin et al (2014). The former paper focuses on minimizing VaR of a specific performance measure -the makespan -in job shops; the authors use the term "probabilistic minimum makespan" or "α-minimum makespan" instead of VaR at the confidence level of 1 − α.…”

“…Thus, their approaches are based on solving a particular deterministic counterpart problem and evaluating the given solutions using Monte Carlo simulation instead of solving a scenario-based optimization model. On the other hand, Sarin et al (2014) 1 focus on minimizing CVaR in single-machine scheduling as mentioned in the introduction.…”

“…Without taking a particular stance in the debate of CVaR versus VaR, we provide further viewpoints about the advantages and disadvantages of VaR and CVaR in Appendix A. Given that only a handful of papers exist on the subject (Beck and Wilson, 2007;Sarin et al, 2014), studying risk-averse machine scheduling problems involving either risk measure is a contribution to the literature.…”

“…In other words, despite the availability of linear representations of CVaR, minimizing CVaR in machine scheduling is also a tough endeavor and is far from being a well-solved problem. This is illustrated by the fact that the largest instances solved to optimality reported in the literature are still moderate-sized with 15 jobs and 400 scenarios (Sarin et al, 2014).…”

“…Various decomposition-based solution methods have been proposed to deal with such large-scale models with an emphasis on the risk-neutral two-stage stochastic programs. For example, exploiting the decomposable structure of CVaR allows for solution methods that draw upon the classical Benders decomposition (Noyan, 2012;Sarin et al, 2014). However, our formulations with the objective of minimizing VaR do not exhibit the well-known decomposable structure of traditional risk-neutral two-stage stochastic programs due to the presence of the linking constraint that reflects the fundamental structure of VaR.…”