A new freight airline22 is designing a huband-

spoke system for its operations. From a total of 34 airports to be served, 3 will be selected as hubs. Then (one-way) airport-to-airport freight quantities fi,j will be routed via the hubs (fi,i = 0 for all i). That is, flow from i to j will begin at i, go to the unique hub k for i, then pass to the (possibly same) hub / for j before being shipped on to j.

The goal is to minimize the total of flow time unit transportation costs ci,j taking into account a 30%

savings for flows between hubs that results from economies of scale (ci,i = 0 for all i).

(a) Explain why appropriate decision variables for an integer programming model of this hub design problem are (i, k = 1,

c, 34)

xi,k ! e 1 if airport i is assigned to a hub at k 0 otherwise yk ! e 1 if a hub opens at k 0 otherwise

(b) Use only the xi,k (i.e., disregard whether hubs are open) to formulate a quadratic objective function summing origin-to-hub, hub-to-hub, and hub-to-destination transportation costs for each pair of nodes.

(c) Complete an INLP model of the problem by adding linear main constraints and appropriate variable-type constraints.