Estimating the Optimal Probability of a Candidate Basic Solution in Stochastic Linear Programming

In this paper, we study the linear programming problem with random objective coefficients that have specific probability distributions. Since a candidate basic solution is hard to maintain optimal in all conditions, it is essential to estimate its probability of being optimal. To accomplish the goal, we assume one can get the probability distribution of coefficients, which forms the corresponding stochastic problem. Aided by the optimality's characteristics of a feasible basic solution in linear programming, we propose the probability estimation approach by utilising the probability density functions of coefficients. Furthermore, we show that the approach can estimate multiple candidate basic solutions, enabling a decision-maker to choose the one with the most significant probability. For generality, we test the approach with coefficients in uniform distribution and normal distribution, where we also simulate the process and make a verification. Moreover, since one can regard the uniform distribution as equivalent to the interval one, we compare both results and show that our estimation is more convincible.