Stochastic Programming Models

In this chapter, we present formulations of stochastic programs involving risk aversion and describe their properties. Risk aversion is very important in many decision-making problems in operations research and engineering involving risk. We introduce risk functions in Sect. 2.1 and the notion of risk measures, describing axioms that define a coherent risk measureCoherent risk measure. We consider two main classes of stochastic programming: mean-risk stochastic programming (MR-SP) and probabilistically constrained stochastic programming (PC-SP). MR-SP problems have a weighted mean and dispersion statistic in the objective to capture risk averseness and include risk-neutral SP. Unlike MR-SP, PC-SP allows for a subset of constraints in the problem to hold for a chosen probability or reliability level. We present formulations of MR-SP in Sect. 2.3 and give examples of quantile and deviation risk measures that can be employed in this setting. We demonstrate how to prove coherence properties for different mean-risk measures in Sect. 2.4. MR-SP problems can be difficult to solve, especially using standard optimization solvers. Therefore, we define deterministic equivalent problem (DEP) formulations for different risk measures in Sect. 2.5. The DEPs are useful for devising practical decompositionDecomposition algorithms toward solving these problems. This is covered in Chap. 6. We address formulations of PC-SP in Sect. 2.6. We describe their properties and state the DEPs. We end the chapter with a review of some examples of other classes of risk-averse SP found in the literature in Sect. 2.7.