Equitable Sequencing of a Given Set of Hazardous Materials Shipments

In a recent paper, R. Gopalan, K. Kolluri, R. Batta and M. Karwan (1990) consider a model to route a set of hazardous materials shipments from an origin to a destination, so as to minimize the global risk to the community while simultaneously maintaining a desired level of equity between zones within the community. If one follows the routes produced via their solution methodology, the overall risk is small and equity between zones is achieved after all the shipments are over. However, equity may be severely violated at an intermediate stage of the shipment process. Since an accident can occur at any stage, this is not a desirable situation. Motivated by this, in this paper we consider the problem of equitably sequencing a given set of hazardous materials shipments. We presume, of course, that the set of routes are such that they engender low overall risk to the community as a whole, and once they are all traversed the risk is equitably distributed among the zones of the community—Gopalan, Kolluri, Batta, and Karwan's paper provides such routes for the case of a single origin and destination; their procedure is easily adaptable for the case of multiple origins and destinations. The objective function considered in this paper is to minimize the sum of the maximum differences in risk that exist between any two zones, where the sum is taken over the trips made. We formulate the resulting equitable sequencing problem as an integer programming problem and as a dynamic programming problem. Optimal solution strategies are examined for small-sized problems. Several heuristic solution strategies are proposed to obtain the upper bounds needed for dynamic programming fathoming and for obtaining reasonable solutions to large-sized problems. The proposed solution methods are tested on a real data set from the City and County of Albany, New York, as well as on a randomly generated data set.