Decision-dependent probabilities in stochastic programs with recourse

Abstract

Stochastic programming with recourse usually assumes uncertainty to be exogenous. Our work presents modelling and application of decision-dependent uncertainty in mathematical programming including a taxonomy of stochastic programming recourse models with decision-dependent uncertainty. The work includes several ways of incorporating direct or indirect manipulation of underlying probability distributions through decision variables in two-stage stochastic programming problems. Two-stage models are formulated where prior probabilities are distorted through an affine transformation or combined using a convex combination of several probability distributions. Additionally, we present models where the parameters of the probability distribution are first-stage decision variables. The probability distributions are either incorporated in the model using the exact expression or by using a rational approximation. Test instances for each formulation are solved with a commercial solver, BARON, using selective branching.

Chance-Constrained Programming with Decision-Dependent Uncertainty

“…Let n = n 1 + n 2 + n 3 , B = {0, 1}, R + be the set of non-negative real numbers, and Z + be the set of non- (see, e.g., [19,28,31,54]) a decision-dependent or endogenous source of uncertainty as it is impacted by and depends on decision variables. Often, decision-dependent uncertainties are modelled as fixed parameters (see Section 3.1).…”

“…The literature on SP models with decision-dependent uncertainty is very sparse (see [28] for a recent review and taxonomy). The first formulation stems from Jonsbraten et al [31].…”

“…The two main types -Type 1 and Type 2 -of decision-dependent uncertainty [4,21] characterize two distinct ways in which decisions affect the probability space of stochastic variables [37]. In problems with Type 1 decision-dependent uncertainty [28], decision variables do not affect the non-anticipativity constraints in stochastic problems with recourse, but do impact the probability distribution of (some) random variables.…”

“…Problems with Type 2 decision-dependent uncertainty [28] are two-or multi-stage stochastic programs with recourse in which decisions affect the time at which information is revealed and uncertainty gets resolved, and more precisely, which non-anticipativity conditions are decision-dependent. The class of decision-dependent chance-constrained problems studied here belong to the Type 1 decision-dependent uncertainty category.…”

“…Therefore, (9) represents the expected number of requests carried out without delay. Note that other functional formslinear or not -satisfying the properties in Proposition 3 can be used to model the busy probabilities and the dependence between decision-dependent random variables and decision variables (see, e.g., [21,28]).…”