Bilevel stochastic transportation problem with exponentially distributed demand

Abstract

In this paper, we consider a bilevel stochastic transportation problem (BSTP) which is a two level hierarchical program to determine optimal transportation plan for a single product assuming that customers’ demands for the product are stochastic, in particular, exponentially distributed random variables. In our model, we suppose that the leader and the follower operate two separate groups of plants in a decentralized firm. The leader, who moves first, determines quantities shipped to customers, and then, the follower decides his own quantities rationally. There are holding and shortage costs at the customer zones. The leader’s objective is to minimize the sum of corresponding total transportation costs and the total expected holding cost. Holding costs can be negative which implies that the leader can sell excess quantities at some prices. Similarly, the follower’s objective is to minimize the sum of the corresponding total transportation costs and the total expected shortage cost. Our proposed model is transformed into a single level nonlinear programming by using its Karush-Kuhn-Tucker (KKT) conditions, and then, it is applied with a branch and bound algorithm to obtain noncooperative solutions. A small numerical example is also given to illustrate our model. Keywords: Bilevel programming, stochastic programming, stochastic transportation problem, exponentially distributed demand