Chang 1982

Citation

Chang, S. Y., E. D. Brill Jr., & L. D. Hopkins. 1982. Use of mathematical models to generate alternative solutions to water resources planning problems. Water Resources Research, 18(1), 58-64.

Summary

Due to the complexity of many water resources planning problems, it is often impossible to fully know all the attributes of the objectives, or the trade-offs between attributes. Problems often have a large set of feasible solutions, and many of these are nearly as optimal as the ideal solution. Un-modeled considerations in problem formulations may influence the results of the optimal solution. When bearing this in mind, it is often prudent to generate multiple alternatives. The article presents three methods for alternative generation (AG): 1) Hop, Skip, and Jump (HSJ), 2) random method, and 3) branch and bound screening (BBS).

The AG methods were tested on a case study of a wastewater treatment system with 15 waste sources, 12 potential plant sites, and 15 potential pipe links. The problem was solved using a mixed integer programming model.

The HSJ method then generated multiple solutions by minimizing the sum of the non-zero variables in the original answer, subject to a limit on total annual cost. The cost constraint was set 10% higher than the optimal cost. The random method works by randomly selecting decision variables and maximizing the sum. Again, feasible solutions are limited by a total annual cost constraint. The BBS method generates a set of feasible solutions meeting the constraints of the objective function, and then screens them to maximize difference in solutions.

Three criteria are used to evaluate the AG methods: 1) performance against the objective function standard (i.e. total annual cost), 2) difference between alternatives produced, and 3) computation time. The results demonstrated that BBS was best at meeting the cost constraint, followed by the random method, and HSJ. On the other hand, BBS generates the smallest amount of difference between alternatives. Each type of method produced distinct types of difference. HSJ varied in degree of centralization of plants. Random tended to produce solutions with different locations, but the same number of plants and interceptors. BBS resulted in solutions where some parts remained constant, but others varied in arrangement. BBS required the least computing time, followed by the random method, and then HSJ. However, computing time was very small overall. Required time will be highly problem dependent.

Considering that each method can be computed quickly, and each result in solutions that are distinctly different, it may be best in practice to use all three methods in alternative exploration.

Discussion

I think that this is an important article with more technical suggestions on AG methods than the Brill 1979 article. I agree that a pure economic efficiency objective is limited in public-sector planning. As a future topic of research, I am curious how sensitive the solutions are to changes in the limit of the cost constraint. In this case, the author used 10%, but he stated that setting the number is a matter of judgment. I wonder how much variation in this number affects the results.