As indicated in Section 4.4, and as should be evident from Exercise 4.8, there are often multiple alternate optima for the set covering location problem. This suggests that we can append secondary...

As indicated in Section 4.4, and as should be evident from Exercise 4.8, there are often multiple alternate optima for the set covering location problem. This suggests that we can append secondary objectives to the problem to select from among the alternate optima to the primary objective problem (that of minimizing the number of facilities needed to cover all demands) a solution that best attains some secondary objective.

(a) For the network shown in Exercise 4.8, write out as many alternate optima as possible to the set covering problem with a coverage distance of 18.

(b) Formulate the following problem:

Primary objective:

MINIMIZE the number of facilities needed to cover all demand nodes at least once with a coverage distance D¹
_c.

Secondary objective:

MAXIMIZE the number of demands (as opposed to nodes) that are covered at least twice with a coverage distance D²
_c
(which may be different from D¹
_c).

Be sure to define all inputs and decision variables clearly. Separate inputs from decision variables. State the objectives and constraints in both words and using notation.

(c) Suggest a means of solving this problem.

(d) Solve the problem for the network shown in Exercise 4.8 using coverage distances of D¹
_c
= 18 and D²
_c
= 18.