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 9x1 + 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-1. What is the marginal value of a pound of pine bark? a-2. 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-1. What is the marginal value of labor? c-2. Over what range is this value in effect? d. If the manager could add an additional 60 minutes of labor, should she? Yes or No 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-1. 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? Yes or No f-2. What is the new value of the objective function, if the profit on chips increases from $6 per bag to $7 per bag? The new value is?
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 | 9x1 + 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-1. What is the marginal value of a pound of pine bark?
a-2. 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-1. What is the marginal value of labor?
c-2. Over what range is this value in effect?
d. If the manager could add an additional 60 minutes of labor, should she? Yes or No
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-1. 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? Yes or No
f-2. What is the new value of the objective function, if the profit on chips increases from $6 per bag to $7 per bag?
The new value is?
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