Personal Mini Warehouses is planning to expand its successful Nashua business into Lowell. In doing so, the company must determine how many storage rooms of each size to build. The problem has been formulated as a linear program (LP), as follows: maximize Z= 100 x1 + 80 x2 (Monthly earnings) subject to: 2 x1 + 4 x2 <= 400 (advertising budget) 100 x1 + 45 x2 <= 8000 (square footage required) x1 <= 60 (rental footage required) x1 & x2 >= 0 where x1 is the number of large spaces developed, and x2 is the number of small spaces developed.
a). The constraint boundaries are shown in the graph below. Different regions are labeled A through G. Which region or regions correspond to the feasible region of this problem?
b). Possible corner points for this problem are labeled [1] through [8] in the graph below. Which one is the optimal corner point? Justify you response with calculations and/or pictures. 200 180 160 140 120 100 80 60 40 20 D x1 20 40 60
c). What is the optimal solution for this problem, i.e., what are the optimal x1 and x2 values and the corresponding earnings level? (Hint: Don't "eyeball" it ? use algebra!)
d). Briefly explain how you would determine the shadow price of the advertising budget constraint. (Alternatively, calculate the shadow price and show the steps taken.)
e). If the rental limit constraint is decreased from 60 to 50, by how much will the monthly earnings change?
f). Several assumptions must implicitly be made to formulate and solve this problem using a linear program. Identify at least one LP assumption which is violated in this context, i.e., which may not be appropriate when the decision variables relate to the number of storage units developed.