Dominance rules for the optimal placement of a finite-size facility in an existing layout

In this paper, we consider the problem of optimal placement of a single finite-size, rectangular facility in the presence of other rectangular facilities. There is a non-negative interaction between the new and existing facilities as well as pairs of existing facilities. All interactions are serviced through a finite number of input/output points located on the boundary of each facility. We assume that the travel occurs according to the rectilinear metric and the travel through the facilities is not permitted. It has been established in the previous works that the optimal placement of the new facility belongs to a finite set of candidate points, whose cardinality is polynomially bounded in the number of existing facilities. The optimal placement of the new facility can be found by evaluating the objective function value at each and every candidate point. This explicit enumeration guarantees the optimal solution, however it might become time consuming for a large number of existing facilities. We propose a new procedure based on the lower bounding technique, which can effectively cut down the number of candidate points that need to be evaluated, resulting in significant reduction in the computing time. The procedure was tested on a large number of randomly generated layouts with varying congestion factors (ratio of area occupied by the existing facilities to the total layout area). These extensive numerical tests reveal that, for a moderately congested layout, there is more than 70% reduction in both the number of evaluated candidates and the computing time.