An exact scenario-independent deterministic equivalent form of stochastic programs embedding Multivariate Extreme Value discrete choice problems

We address the class of two-stage Stochastic Programs embedding, in their second stage, a set of Discrete Choice Problems (tsSP-DCPs), one independent from the other, but all linked by the first-stage decisions This decisional structure can be found within many managerial and organizational contexts in relation to several applications such as location–allocation, routing, scheduling, and sequencing. Generally, solving a two-stage stochastic program requires the analytical derivation of the second-stage problem's expected optimum, which in turn implies calculating a multidimensional integral. Therefore, a common practice is approximating the random variables involved through a finite set of scenarios and solving a huge scenario-dependent program, which affects the scalability of making optimal decisions under uncertainty. However, under some assumptions commonly adopted in the discrete choice context, we can prove that a closed-form analytical expression of the expected second-stage optimum of a tsSP-DCP can be derived, and an exact scenario-independent equivalent deterministic program can be obtained. Through a numerical showcase, we validate our approach in terms of efficiency and effectiveness. Our equivalent deterministic form, which only requires estimating a few parameters in practice, is far less computationally demanding than any scenario-based deterministic equivalent forms, thereby simplifying the decision-making process. Finally, we show that our methodology can be generalized to address a larger class of two-stage stochastic programs, i.e., those in which the second-stage expected optimum is decomposable into a finite number of expectations of Extreme Values and in which second-stage utilities may also depend on first-stage decisions.