Once a site for a new service facility has been chosen, the limits of its market area must be determined,4 along with the corresponding facility size. Assume (i) that the facility is to be located at the center of a circular market area and sized to cover uniform density d calls per unit area out to a radius of r; (ii) the cost of operating the facility is [f + c1size2], where f is a fixed cost and c a variable cost per unit size; and (iii)

transportation costs per call are proportional to

(straight-line, one-way) distance from the facility to the customer at t per unit distance. Formulate a 1-variable unconstrained NLP to choose a market area radius that minimizes the facility’s average total cost per call. Evaluate any calculus integrals in your objective.