A novel approach for solving decision-making problems with stochastic linear-fractional models

Ali Habeeb,Rasheed Al-Salih,Watheq Laith

doi:10.15587/1729-4061.2021.241916

Abstract

Stochastic chance-constrained optimization has a wide range of real-world applications. In some real-world applications, the decision-maker has to formulate the problem as a fractional model where some or all of the coefficients are random variables with joint probability distribution. Therefore, these types of problems can deal with bi-objective problems and reflect system efficiency. In this paper, we present a novel approach to formulate and solve stochastic chance-constrained linear fractional programming models. This approach is an extension of the deterministic fractional model. The proposed approach, for solving these types of stochastic decision-making problems with the fractional objective function, is constructed using the following two-step procedure. In the first stage, we transform the stochastic linear fractional model into two stochastic linear models using the goal programming approach, where the first goal represents the numerator and the second goal represents the denominator for the stochastic fractional model. The resulting stochastic goal programming problem is formulated. The second stage implies solving stochastic goal programming problem, by replacing the stochastic parameters of the model with their expectations. The resulting deterministic goal programming problem is built and solved using Win QSB solver. Then, using the optimal value for the first and second goals, the optimal solution for the fractional model is obtained. An example is presented to illustrate our approach, where we assume the stochastic parameters have a uniform distribution. Hence, the proposed approach for solving the stochastic linear fractional model is efficient and easy to implement. The advantage of the proposed approach is the ability to use it for formulating and solving any decision-making problems with the stochastic linear fractional model based on transforming the stochastic linear model to a deterministic linear model, by replacing the stochastic parameters with their corresponding expectations and transforming the deterministic linear fractional model to a deterministic linear model using the goal programming approach

Highlights

Linear programming problems have a wide range of real-world applications
If some or all of the coefficients of decision variables are random variables with joint probability distribution, the problem is known as stochastic linear programming (SLP)
In some real-world applications, the decision-maker has to formulate the problem as a ratio between two linear objective functions, and the problem is known as a fractional linear programming problem

Summary

Introduction

Linear programming problems have a wide range of real-world applications. The linear programming model consists of the linear objective function and linear constraints. If some or all of the coefficients of decision variables are random variables with joint probability distribution, the problem is known as stochastic linear programming (SLP). In some real-world applications, the decision-maker has to formulate the problem as a ratio between two linear objective functions, and the problem is known as a fractional linear programming problem. If some or all of the coefficients are random variables with joint probability distribution, the problem is known as stochastic linear fractional programming (SLFP). The fundamental drawbacks of such solution are the difficult-to-predict uncertainty of the result as well as the complexity of transforming the stochastic linear fractional model into a stochastic linear model To solve this optimization problem, one needs to establish a computational procedure for solving these kinds of problems

Objectives

Methods

Conclusion