Question 19.SP.2: This LP model was solved by computer: Maximize 15x1 + 20x2 +...
This LP model was solved by computer:
Maximize 15x1 + 20x2 + 14x3 where x1 = quantity of product 1
x2 = quantity of product 2
x3 = quantity of product 3
Subject to
Labor 5x1 + 6x2 + 4x3 ≤ 210 hours
Material 10x1 + 8x2 + 5x3 ≤ 200 pounds
Machine 4x1 + 2x2 + 5x3 ≤ 170 minutes
x1, x2, x3 ≥ 0
The following information was obtained from the output. The ranges were also computed based on the output, and they are shown as well.
| TOTAL PROFIT = 548.00 | |||
| Variable | Value | Reduced Cost | Range of Optimality |
| Product 1 | 0 | 10.6 | unlimited to 25.60 |
| Product 2 | 5 | 0 | 9.40 to 22.40 |
| Product 3 | 32 | 0 | 12.50 to 50.00 |
| Constraint | Slack | Shadow Price | Range of Feasibility |
| Labor | 52 | 0 | 158.00 to unlimited |
| Material | 0 | 2.4 | 170.00 to 270.91 |
| Machine | 0 | 0.4 | 50.00 to 200.00 |
a. Which decision variables are basic (i.e., in solution)?
b. By how much would the profit per unit of product 1 have to increase in order for it to have a nonzero value (i.e., for it to become a basic variable)?
c. If the profit per unit of product 2 increased by $2 to $22, would the optimal production quantities of products 2 and 3 change? Would the optimal value of the objective function change?
d. If the available amount of labor decreased by 12 hours, would that cause a change in the optimal values of the decision variables or the optimal value of the objective function? Would anything change?
e. If the available amount of material increased by 10 pounds to 210 pounds, how would that affect the optimal value of the objective function?
f. If profit per unit on product 2 increased by $1 and profit per unit on product 3 decreased by $.50, would that fall within the range of multiple changes? Would the values of the decision variables change? What would be the revised value of the objective function?
a. Products 2 and 3 are in solution (i.e., have nonzero values); the optimal value of product 2 is 5 units, and the optimal value of product 3 is 32 units.
b. The amount of increase would have to equal its reduced cost of $10.60.
c. No, because the change would be within its range of optimality, which has an upper limit of $22.40. The objective function value would increase by an amount equal to the quantity of product 2 and its increased unit profit. Hence, it would increase by 5($2) = $10 to $558.
d. Labor has a slack of 52 hours. Consequently, the only effect would be to decrease the slack to 40 hours.
e. The change is within the range of feasibility. The objective function value will increase by the amount of change multiplied by material’s shadow price of $2.40. Hence, the objective function value would increase by 10($2.40) = $24.00. (Note: If the change had been a decrease of 10 pounds, which is also within the range of feasibility, the value of the objective function would have decreased by this amount.)
f. To determine if the changes are within the range for multiple changes, we first compute the ratio of the amount of each change to the end of the range in the same direction. For product 2, it is $1/$2.40 = .417; for product 3, it is −$.50/ − $1.50 = .333. Next, we compute the sum of these ratios: .417 + .333 = .750. Because this does not exceed 1.00, we conclude that these changes are within the range. This means that the optimal values of the decision variables will not change. We can compute the change to the value of the objective function by multiplying each product’s optimal quantity by its changed profit per unit: 5($1) + 32(−$.50) = −$11. Hence, with these changes, the value of the objective function would decrease by $11; its new value would be $548 − $11 = $537.