A garden store prepares various grades of pine bark for mulch: nuggets (x1), mini-nuggets (x2), and chips (x3). The process requires pine bark, machine time, labor time, and storage space. The follow- ing model has been developed.
Maximize 9x1 + 9x2 + 6x3 (profit)
Subject to Bark Machine Labor Storage
5x1 + 6 x2 + 3x3 <>600 poounds 2 x1 + 4 x2 + 5x3 <>600 minutes 2 x1 + 4 x2 + 3x3 < >480 hours 1x1 + 1x2 + 1x3 <>150 bags
x1, x2 , x3 > 0
a. What is the marginal value of a pound of pine bark? Over what range is this price value appropriate?
b. What is the maximum price the store would be justified in paying for additional pine bark?
c. What is the marginal value of labor? Over what range is this value in effect?
d. The manager obtained additional machine time through better scheduling. How much additional machine time can be effectively used for this operation? Why?
e. If the manager can obtain either additional pine bark or additional storage space, which one should she choose and how much (assuming additional quantities cost the same as usual)?
f. If a change in the chip operation increased the profit on chips from $6 per bag to $7 per bag, would the optimal quantities change? Would the value of the objective function change? If so, what would the new value(s) be?
g. If profits on chips increased to $7 per bag and profits on nuggets decreased by $.60, would the optimal quantities change? Would the value of the objective function change? If so, what would the new value(s) be?
h. If the amount of pine bark available decreased by 15 pounds, machine time decreased by 27 minutes, and storage capacity increased by five bags, would this fall in the range of feasibility for multiple changes? If so, what would the value of the objective function be?