In Chapter 4, we discussed the problem of maximizing coverage subject to a constraint that we locate exactly (or no more than) P facilities. This model implicitly assumes that the cost of locating at each candidate location is approximately the same. This, however, may not be the case. Thus, in many cases, it is important to be able to analyze the trade off between the fixed facility location costs and the percentage of the demands that are covered.
(a) Formulate the problem of (i) maximizing the number of covered demands and (ii) minimizing the total fixed costs of the selected facilities as a two-objective problem. (Note that you will not have an explicit constraint limiting the number of facilities being located.) Clearly define all sets, all inputs and all decision variables and state the objective functions and constraints in both words and using notation.
(b) Reformulate the two-objective problem outlined in part (a) but now minimize the number of uncovered demands instead of maximizing the number of covered demands.
(c) Using a weight of a for the objective of minimizing the number of uncovered demands and 1ð Þ a for the weight on the total cost, restructure the problem of part (b) as a weighted objective problem.
(d) Show that the problem you formulated in part (c) can be thought of as an uncapacitated fixed charge facility location problem. What decision variable redefinitions and input transformations must you do to obtain this formulation?
(e) Using SORTCAP.GRT, the first demand data set (representing the state populations) for the 49-node problem and a coverage distance of 450 miles, use SITATION to find at least four noninferior points on the (approximate) trade off curve of uncovered demand versus total facility location cost. Be sure that two of the points correspond to (i) the point that covers all demand at minimum total fixed cost and (ii) the point that has the minimum total cost (and, from among all of the solutions with minimum cost, is the one that minimizes the uncovered demand).
(f) Clearly discuss how you obtain the inputs for this analysis.
(g) For each solution, give the value of a that you used. Also give the total fixed cost, the number of uncovered demands, and the number of facilities that are located, and the locations of the facilities.
(h) How does the solution that covers all demand at minimum cost compare with the traditional maximum cover solution in terms of (i) the total cost of the two solutions and (ii) the number of facilities located?