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Question 1176370: Electrocomp's management realizes that it forgot to include two critical constraints. In particular, management decides that there should be a minimum number of air conditioners produced in order to fulfill a contract. Also, due to an oversupply of fans in the preceding period, a limit should be placed on the total number of fans produced. (a) If Electrocomp decides that at least 20 air conditioners should be produced but no more than 80 fans should be produced, what would be the optimal solution? How much slack is there for each of the four constraints? (b) If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, what would be the optimal solution? How much slack is there for each of the four constraints at the optimal solution?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To answer this question, we need the original Electrocomp linear programming problem. I'll assume it's a typical production problem with:
* **Variables:**
* x = number of air conditioners
* y = number of fans
* **Objective Function (Maximize Profit):**
* Let's assume it's something like P = ax + by (where a and b are profit per air conditioner and fan, respectively).
* **Constraints (Example):**
* Resource 1: c1x + d1y ≤ e1
* Resource 2: c2x + d2y ≤ e2
* x ≥ 0, y ≥ 0
**Important:** You must replace these general constraints and objective function with the actual Electrocomp problem constraints and objective function.
**Now, let's address the new constraints:**
**(a) At least 20 air conditioners (x ≥ 20) and no more than 80 fans (y ≤ 80)**
1. **Add the new constraints:**
* x ≥ 20
* y ≤ 80
2. **Graph the feasible region:**
* Graph all the original constraints and the two new ones.
* Identify the corner points of the new feasible region.
3. **Evaluate the objective function:**
* Calculate the profit (P) at each corner point.
* Determine the corner point that yields the maximum profit.
4. **Calculate slack:**
* For each constraint, plug the optimal (x, y) values.
* If the constraint is "≤", slack is the difference between the right-hand side and the left-hand side.
* If the constraint is "≥", slack is the difference between the left-hand side and the right-hand side.
**(b) At least 30 air conditioners (x ≥ 30) and no more than 50 fans (y ≤ 50)**
1. **Add the new constraints:**
* x ≥ 30
* y ≤ 50
2. **Graph the feasible region:**
* Graph all the original constraints and the two new ones.
* Identify the corner points of the new feasible region.
3. **Evaluate the objective function:**
* Calculate the profit (P) at each corner point.
* Determine the corner point that yields the maximum profit.
4. **Calculate slack:**
* For each constraint, plug the optimal (x, y) values.
* If the constraint is "≤", slack is the difference between the right-hand side and the left-hand side.
* If the constraint is "≥", slack is the difference between the left-hand side and the right-hand side.
**Example (Illustrative - You must replace with actual constraints):**
Let's assume the example Electrocomp problem is:
* Maximize P = 20x + 15y
* Constraints:
* x + y ≤ 100 (Resource 1)
* 2x + y ≤ 150 (Resource 2)
* x ≥ 0, y ≥ 0
**(a) x ≥ 20, y ≤ 80**
1. **Feasible region:** The feasible region is bounded by the original constraints and the new ones.
2. **Corner points (after graphing):** (20, 80), (20, 60), (35, 65), (75, 0)
3. **P values:**
* (20, 80): P = 20(20) + 15(80) = 1600
* (20, 60): P = 20(20) + 15(60) = 1300
* (35, 65): P = 20(35) + 15(65) = 1675
* (75, 0): P = 20(75) + 15(0) = 1500
4. **Optimal solution:** (35, 65), P = 1675
5. **Slack:**
* x + y ≤ 100: 35 + 65 = 100 (slack = 0)
* 2x + y ≤ 150: 2(35) + 65 = 135 (slack = 15)
* x ≥ 20: 35 - 20 = 15 (slack = 15)
* y ≤ 80: 80 - 65 = 15 (slack = 15)
**(b) x ≥ 30, y ≤ 50**
Repeat the process with the new constraints.
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