Algorithm for solving optimization problems with Interval Valued Probability Measure

TL;DR

This paper introduces an algorithm that transforms complex optimization problems with probabilistic, possibilitistic, and interval uncertainties into standard linear programming problems using interval expected values and specific functions.

Contribution

It presents a novel method to handle mixed uncertainties in optimization by converting them into linear programming problems through interval expected values and polynomial possibility distributions.

Findings

Interval expected values are computed using polynomial possibility distributions.

Optimization problems with mixed uncertainties can be solved as ordinary linear programs.

The method applies to problems with linear constraints and multiple uncertainty types.

Abstract

We are concerned with three types of uncertainties: probabilistic, possibilitistic and interval. By using possibility and necessity measures as an Interval Valued Probability Measure (IVPM), we present IVPM's interval expected values whose possibility distributions are in the form of polynomials. By working with interval expected values of independent uncertainty coefficients in a linear optimization problem together with operations suggested in Lodwick and Jamison (2007), the problem after applying these operations becomes a linear programming problem with constant coefficients. This is achieved by the application of two functions. The first is applied to the interval coefficients, v: I -> R^k, where I= {[a,b] | a <= b}. The second is u: R^k -> R, applied to the product we got from a previous function. Similar concepts hold for any types of optimization problems with linear constraints. Moreover, it implied that optimization problems containing all three types of uncertainties in one problem can be solved as ordinary optimization problems.