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. Thefollowing model has been developed.Maximize Z = 9 x1 + 9x2 + 6x3 (profit)Subject toBark 5x1 + 6x2 + 3x3 ≤ 600 pounds Machine 2x1 + 4x2 + 5x3 ≤ 600 minutesLabor 2x1 + 4x2 + 3x3 ≤ 480 hoursStorage 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 valueappropriate?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 oneshould 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, wouldthe optimal quantities change? Would the value of the objective function change? If so, whatwould the new value(s) be?h. If the amount of pine bark available decreased by 15 pounds, machine time decreased by 27minutes, and storage capacity increased by five bags, would this fall in the range of feasibilityfor multiple changes? If so, what would the value of the objective function be?
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
following model has been developed.
Maximize Z = 9 x1 + 9x2 + 6x3 (profit)
Subject to
Bark 5x1 + 6x2 + 3x3 ≤ 600 pounds
Machine 2x1 + 4x2 + 5x3 ≤ 600 minutes
Labor 2x1 + 4x2 + 3x3 ≤ 480 hours
Storage 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
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?
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