As spring approaches, the campus grounds staff is preparing to buy 500 truckloads of new soil to add around buildings and in gulleys where winter has worn away the surface. Three sources are available at $220, $270, and $290 per truckload, respectively, but the soils vary in nitrogen and clay composition. Soil from source 1 is 50%

nitrogen and 40% clay; soil from source 2 is 65%

nitrogen and 30% clay; and soil from source 3 is 80% nitrogen and 10% clay. The staff wants the total mix of soil to contain at least 350 truckloads of nitrogen, and at most 75 truckloads of clay.

Within the constraints on their needs, they wish to find a minimum cost plan to purchase the soil.

(a) Define decision variables and annotate objective and constraints to justify why this problem can be modeled as the linear program min 220t1 + 270t2 + 290t3 s.t. t1 + t2 + t3 Ú 500

.50t1 + .65t2 + .80t3 Ú 350

.40t1 + .30t2 + .10t3 … 75 t1, t2, t3 Ú 0 NOTE: This LP has optimal solution t* = 183.33, 0.00, 416.672, and the optimal shadow price on the first constraint is v1 * = 313.333

(b) Using only material given, compute the optimal solution value of the dual corresponding to the LP of part (a), and justify your computation.

(c) Using only material given, would a unit increase in coefficient/parameter 500 of the LP of part

(a) increase or decrease the optimal solution value, and by how much? Also would the rate of change lessen or steepen with the magnitude of the increase? Explain.

(d) Assuming it has some impact, would an decrease in coefficient/parameter 270 of the LP of part

(a) increase or decrease the optimal solution value, and would the rate of change lessen or steepen with the magnitude of the decrease? Explain.

(e) Assuming it has some impact, would an increase in coefficient/parameter 0.65 of the LP of part

(a) increase or decrease the optimal solution value, and would the rate of change lessen or steepen with the magnitude of the increase? Explain.