Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $
6
3
,
$
9
5
,
and $
1
3
5
,
respectively. Th

Question

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $
6
3
,
 ﻿$
9
5
,
 ﻿and $
1
3
5
,
 ﻿respectively. The production requirements per unit are as follows: Number of FansNumber of Cooling CoilsManufacturing Time 
(
hours
)
Economy
1
1
8
Standard
1
2
1
2
Deluxe
1
4
1
4
 ﻿For the coming production period, the company has 
2
2
0
 ﻿fan motors, 
3
4
0
 ﻿cooling coils, and 
2
,
6
0
0
 ﻿hours of manufacturing time available. How many economy models 
(
E
)
,
 ﻿standard models 
(
S
)
,
 ﻿and deluxe models 
(
D
)
 ﻿should the company produce in order to maximize profit? The linear programming model for the problem is as follows: Max

6
3
E
 
+
 
9
5
S
 
+
 
1
3
5
Ds
.
t
.
1
E
 
+
 
1
S
 
+
 
1
D
 
≤
 
2
2
0

Fan motors
1
E
 
+
 
2
S
 
+
 
4
D
 
≤
 
3
4
0

Cooling coils
8
E
 
+
 
1
2
S
 
+
 
1
4
D
 
≤
 
2
,
6
0
0

Manufacturing timeE, S
,
 ﻿D
 
≥
 
0
 ﻿The computer solution is shown below. Optimal Objective Value 
=
 
1
7
7
0
0
.
0
0
0
0
0
 ﻿VariableValueReduced CostE
1
0
0
.
0
0
0
0
0
0
.
0
0
0
0
0
S
1
2
0
.
0
0
0
0
0
0
.
0
0
0
0
0
D
0
.
0
0
0
0
0
−
2
4
.
0
0
0
0
0
 ﻿Con

VIPSolutions · Accepted Answer

Let's answer each part step by step, using the information from the computer solution (sensitivity analysis):

---

### (a) **Range of Optimality for Each Objective Function Coefficient**

From the sensitivity report, for each variable (E, S, D):

- **Economy (E)**:  
  - Objective Coefficient = 63  
  - Allowable Increase = 12  
  - Allowable Decrease = 15.5  
  - **Range of Optimality:**  
    $ 63 - 15.5 \leq \text{E coefficient} \leq 63 + 12 $  
    $ \boxed{47.5 \leq \text{E coefficient} \leq 75} $

- **Standard (S)**:  
  - Objective Coefficient = 95  
  - Allowable Increase = 31  
  - Allowable Decrease = 8  
  - **Range of Optimality:**  
    $ 95 - 8 \leq \text{S coefficient} \leq 95 + 31 $  
    $ \boxed{87 \leq \text{S coefficient} \leq 126} $

- **Deluxe (D)**:  
  - Objective Coefficient = 135  
  - Allowable Increase = 24  
  - Allowable Decrease = Infinite  
  - **Range of Optimality:**  
    $ 135 - \infty \leq \text{D coefficient} \leq 135 + 24 $  
    $ \boxed{-\infty < \text{D coefficient} \leq 159} $

---

### (b) **Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be?**

**New coefficients:**
- E: 63 + 6 = **69**
- S: 95 - 2 = **93**
- D: 135 + 4 = **139**

Now check if these new coefficients are **within the range of optimality** from (a):

- E: 69 (in [47.5, 75] ✔️)
- S: 93 (in [87, 126] ✔️)
- D: 139 (in (-∞, 159] ✔️)

**Since all new coefficients are within the range of optimality, the optimal solution (E = 100, S = 120, D = 0) does NOT change. Only the profit changes.**

**New profit calculation:**
$$
\text{Profit} = 69 \times 100 + 93 \times 120 + 139 \times 0 = 6900 + 11160 + 0 = \$18,060
$$

**Answers:**
- E units: **100**
- S units: **120**
- D units: **0**
- Profit: **\$18,060**

---

### (c) **Range of Feasibility for Right-hand-side (RHS) values**

From the sensitivity report for constraints (RHS Value, Allowable Increase, Allowable Decrease):

- **Constraint 1 (Fan motors):**  
  - RHS = 220  
  - Allowable Increase = 90  
  - Allowable Decrease = 50  
  - **Range:**  
    $ 220 - 50 \leq \text{RHS} \leq 220 + 90 $  
    $ \boxed{170 \leq \text{RHS} \leq 310} $

- **Constraint 2 (Cooling coils):**  
  - RHS = 340  
  - Allowable Increase = 90  
  - Allowable Decrease = 120  
  - **Range:**  
    $ 340 - 120 \leq \text{RHS} \leq 340 + 90 $  
    $ \boxed{220 \leq \text{RHS} \leq 430} $

- **Constraint 3 (Manufacturing time):**  
  - RHS = 2,600  
  - Allowable Increase = 360  
  - Allowable Decrease = Infinite  
  - **Range:**  
    $ 2600 - \infty < \text{RHS} \leq 2600 + 360 $  
    $ \boxed{-\infty < \text{RHS} \leq 2960} $

---

### (d) **If the number of fan motors available for production is increased by 110, will the dual value for that constraint change?**

- The current RHS value for fan motors is **220**.
- The allowable increase is **90**.
- Increasing by 110 would result in a new RHS of 330, which is **outside** the range of feasibility.
- **Therefore, the dual value will change because 110 is greater than the allowable increase of 90.**

**Correct answer:**
> Yes, the dual value will change because 110 is greater than the allowable increase of 90.

---

## **Summary Table**

| Part | Answer                                                                                           |
|------|--------------------------------------------------------------------------------------------------|
| (a)  | E: 47.5 to 75    S: 87 to 126    D: −∞ to 159                                |
| (b)  | E = 100, S = 120, D = 0, profit = $18,060                                                        |
| (c)  | Constraint 1: 170 to 310, Constraint 2: 220 to 430, Constraint 3: −∞ to 2960                     |
| (d)  | Yes, because 110 is greater than the allowable increase of 90.                                   |

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