Consider the following variant of the maximum expected covering model. In this case, we want to maximize the number of demands that can be covered within the coverage distance Dc, with a probability...

Consider the following variant of the maximum expected covering model. In this case, we want to maximize the number of demands that can be covered within the coverage distance Dc, with a probability of at least Q⁰, when the probability of finding an individual vehicle busy is q. In other words, the probability of at least one available vehicle being able to cover the demand at a node, given that K vehicles are located within a distance Dc, of the node, is, as before 1  qk. For the node to be covered with a probability of Q⁰, we need this quantity to be at least Q⁰.

(a) For q = 0.25 and Q⁰
= 0:98, find the minimum value of K (i.e., the minimum number of facilities needed to be located within Dc, of a node for the node to be covered with probability 0.98).

(b) Using the notation defined in Section 4.6, formulate the problem of locating P vehicles (on the demand nodes of the network) so that we maximize the number of demands that are covered with a probability of at least 0.98 (when q = 0.25). State the constraints and objective function in words as well as mathematical notation.