An air courier company operates a single depot in a large city and wants to decentralize by establishing several depots, each covering the same delivery radius r. The total area to be covered is A, and the area covered by each depot is πr^2 . Thus, if n depots are installed, we must have A = nπr^2 , so that n = A/(πr^2). Expenses and revenues break down as follows:

• Trucking expenses: The number of trucks required at each depot is απr^3 , so that the total number of trucks required is nαπr^3 = A/(πr^2) απr^3 = Aαr. Each truck costs $K per day, so that total expense for trucks is $KAαr.

• Overhead: Each depot incurs an overhead (fixed) expense of $F, for a total overhead of nF = F A/(πr^2 ).

• Revenue from customers: Since the total area is fixed, the total revenue from customers is independent of the delivery radius.

1. Formulate the problem of finding the delivery radius r that minimizes total cost. Indicate clearly the decision or optimization variables of your formulation.

2. What is the optimal delivery radius? What is the corresponding optimal cost? (answers should be expressed in dependence of the problem parameters).

3. Suppose that the problem parameters are F = 100, A = 200, K = 20, α = 10 and answer the following questions:

(a) What is the optimal radius r ? ? What is the corresponding optimal cost?

(b) Determine the sensitivity of the optimal solution r ? to the overhead expense F ($100) and to the daily truck cost K ($20). Which of the two prices has the greater impact on r ? ?

(c) Determine the sensitivity of the optimal cost to the daily truck cost K ($20).