The price of equity in the Hazmat Transportation Problem

We consider the problem of the transportation of hazardous materials on a road network (Hazardous Materials Transportation Problem). We can figure it this way: there are N trucks which have to transport some kind of dangerous material from one or many production points to one or many garbage dumps and we have to select a set of paths which is optimal from the point of view of risk, cost and equity. The optimization of cost and risk on a network leads quite spontaneously to shortest path and flow problems which are milestones of Operational Research, but equity is somehow unusual and hard to define. We consider and compare two different ideas. The first approach simply requires that all the areas involved in the transportation network share the same level of risk. This is a fair and intuitive idea but it could also lead to “improper” solutions where risk is equal but uniformly high. The second (more interesting) definition of equity we use is inspired by the concept of fairness of J. Rawls [2,3]. Basically, in this context, the difference principle means that we may introduce disparities only if they advantage the worst-off, namely reduce the risk of the less favourite area (the most exposed to the risk).