The Study Of Distributionally Robust Weber Problem Based On A New Probability Distribution Set

In this paper, we mainly use the distributionally robust optimization method based on a new probability distribution set to solve the Weber location problem under uncertainty, and we show that our method is better than the well-known Min Max-Regret method and when aincreases, the number of samples required decreases by numerical experiments. The details of the paper is as follows:In chapter 1, we introduce the background and current work for the study of facility location under uncertainty, and we also briefly introduce the work of our paper.In chapter 2, we introduce the new probability distribution set, as the covariance matrix of stochastic parameters in facility location problems is not always definite. The new probability distribution set is formed using a new virtual random vector.In chapter 3, we first introduce some concepts of variational inequalities and stochastic variational inequalities, then we form the residual function of Weber location problem. When the weights are stochastic and the location of the customers are certain, we verify that the residual function satisfy the assumptions of the distributionally robust optimization method in every rectangle obtained by cutting the plane, thus the problem can be transformed to a semidefinite optimization problem by duality theory. We reduce the quantity of the constraints of the problem from exponential complexity to polynomial complexity.In chapter 4,we introduce the algorithm of our problem and draw some conclusions. Overall speaking,the classical Min Max-Regret method only considers the performance of the system at the worst scenario which happens with very low probability, so the decision made by this method is probably bad at most scenarios, And furthermore, this method doesn't exploit the probability and statistics information of the samples. The distributionally robust optimization method with the new probability distribution set overcomes the deficiency of the Min Max-Regret method, and the decision from this method is better than that from the Min Max-Regret method. In addition this method makes use of the probability and statistics information hidden in the samples. Finally we verify the efficiency and effectiveness of our method by solving a number of Weber problem under uncertainty. We also find that in our experiments when the ain S+aI increases, the number of samples required decreases and it doesn't affect the performance of the solution. In chapter 5, we draw some conclusions and give some future research directions.