Median location problem with two probabilistic line barriers: extending the hook and jeeves algorithm

We consider a median location problem in the presence of two probabilistic line barriers on the plane under rectilinear distance. It is as-sumed that the two line barriers move on their corresponding horizontal routes uniformly. We first investigate different scenarios for the position of the line barriers on the plane and their corresponding routes, and then define the visibil-ity and invisibility conditions along with their corresponding expected barrier distance functions. The proposed problem is formulated as a mixed-integer nonlinear programming model. Our aim is to locate a new facility on the plane so that the total weighted expected rectilinear barrier distance is minimized. We present efficient lower and upper bounds using the forbidden location prob-lem for the proposed problem. To solve the proposed model, the Hooke and Jeeves algorithm (HJA) is extended. We investigate various sample problems to test the performance of the proposed algorithm and appropriateness of the bounds. Also, an empirical study in Kingston-upon-Thames, England, is con-ducted to illustrate the behavior and applicability of the proposed model.