give full answer with final summary| Homework 6 - Question 3 - Build Plants - Transportation Model Objective: .
Net Cost Per Unit
Construction To Supply Customer
Plant Production Capacif Cost X Y 4

Question

give full answer with final summary| Homework 6 - Question 3 - Build Plants - Transportation Model Objective: .
Net Cost Per Unit
Construction To Supply Customer
Plant Production Capacif Cost X Y 4
1 40000 $1,325,000 $35 $30 $45
2 30000 $1,100,000 $45 $40 $50
3 50000 $1,500,000 $70 $65 $50
4 20000 $1,200,000 $20 $45 $25
5 40000 $1,400,000 $65 $45 $45

Suppose plants 1and 2 represent different building alternatives for the same site (that is, only one of these plants can be built).
Similarly, suppose plants 4 and 5 represent different building alternatives for another site.

The company wants to determine which plants to build in order to satisfy customer demand at minimum total cost.

Total cost=$[_]

A manufacturer is considering alternatives for building new plants in order to be located closer to three of its primary customers with
whom it intends to develop long-term relationships. The net cost of manufacturing and transporting each unit of the product to its
customers (X, Y, and Z) will vary depending on where the plant is built and the production capacity of the plant. These costs are
‘summarized in the following table:

[Pam [x [ v _ | 7

1 [ss | wo |

VIPSolutions · Accepted Answer

Let's analyze the information and calculate the minimum total cost by selecting appropriate plants based on their capacity construction cost, net cost per unit to supply customers.

### Step 1: Identify options for each plant
- **Plant 1:** Capacity = 40,000; Construction = $1,325,000
- **Plant 2:** Capacity = 30,000; Construction = $1,100,000
- **Plant 3:** Capacity = 50,000; Construction = $1,500,000
- **Plant 4:** Capacity = 20,000; Construction = $1,200,000
- **Plant 5:** Capacity = 40,000; Construction = $1,400,000

### Step 2: Customer demand
- **X:** 41,000 units
- **Y:** 26,000 units
- **Z:** 35,000 units

### Step 3: Determine optimal plant combinations

---

### **For Customer X (Demand = 41,000 units):**

- **Plant 1:** Capacity = 40,000 (short by 1,000 units)
- **Plant 2:** Capacity = 30,000 (short by 11,000 units)
- **Plant 3:** Capacity = 50,000 (sufficient)
- **Plant 4:** Capacity = 20,000 (short by 21,000 units)
- **Plant 5:** Capacity = 40,000 (short by 1,000 units)

**Best options:**

- **Plant 3:** 50,000 units (more than needed, but sufficient)
- **Plant 1 + Plant 2:** 40,000 + 30,000 = 70,000 units (more than needed, but possibly cost-effective)

Let's compare costs:

- **Plant 3 alone:**
  - Construction: $1,500,000
  - To supply 41,000 units with plant 3:
    - Use net cost $ \$70 $ per unit
    - Total variable cost: $ 41,000 \times \$70 = \$2,870,000 $

- **Total cost:** Construction + variable cost = $1,500,000 + $2,870,000 = **$4,370,000**

- **Plant 1 + Plant 2:**
  - Construction: $1,325,000 + $1,100,000 = $2,425,000
  - To supply 41,000 units:
    - Assign 40,000 units to Plant 1:
      - Cost: $ 40,000 \times \$35 = \$1,400,000 $
    - Remaining 1,000 units to Plant 2:
      - Cost: $ 1,000 \times \$45 = \$45,000 $
  - Total variable cost: $ \$1,400,000 + \$45,000 = \$1,445,000 $

- **Total cost:** Construction + variable cost = $2,425,000 + $1,445,000 = **$3,870,000**

*Conclusion:* The combination **Plant 1 + Plant 2** is cheaper for customer X.

---

### **For Customer Y (Demand = 26,000 units):**

- **Plant 1:** 40,000 capacity
- **Plant 2:** 30,000 capacity
- **Plant 4:** 20,000 capacity
- **Plant 5:** 40,000 capacity

Options:

- **Plant 2 alone:** 30,000 units
- **Plant 4 alone:** 20,000 units (short by 6,000 units)
- **Plant 1 or Plant 5:** Over capacity

Best options:

- **Plant 2:** 30,000 units with total cost:
  - Construction: $1,100,000
  - Variable: $ 26,000 \times \$40 = \$1,040,000 $
  - Total: $1,100,000 + $1,040,000 = **$2,140,000**

- **Plant 4 + Plant 4:** Not efficient (since only one plant can be built for a site)

Alternatively, using Plant 1:

- **Plant 1:** 40,000 units
  - Construction: $1,325,000
  - Variable: $ 26,000 \times \$30 = \$780,000 $
  - Total: $1,325,000 + $780,000 = **$2,105,000**

**Cheaper option:** Plant 1 alone

---

### **For Customer Z (Demand = 35,000 units):**

- **Plant 3:** 50,000 capacity
- **Plant 5:** 40,000 capacity
- **Plant 1:** 40,000 capacity
- **Plant 2:** 30,000 capacity
- **Plant 4:** 20,000 capacity

Options:

- **Plant 3:** 50,000 units
  - Construction: $1,500,000
  - Variable cost: $ 35,000 \times \$50 = \$1,750,000 $
  - Total: $3,250,000

- **Plant 5:** 40,000 units (short by 5,000 units)
- **Plant 1 + Plant 4:** 40,000 + 20,000 = 60,000 units (over capacity)

Choosing **Plant 3** seems most straightforward.

---

### **Final Selection**

| Customer | Selected Plant(s) | Total Cost Calculation                                             | Total Cost        |
|------------|-------------------|----------------------------------------------------------------------|------------------|
| X          | Plant 1 + Plant 2 | $1,325,000 + $1,100,000 + $40,000 + $5,000 = **$2,470,000** | **$2,470,000**  |
| Y          | Plant 1          | $1,325,000 + $780,000 = **$2,105,000**                          | **$2,105,000**  |
| Z          | Plant 3        | $1,500,000 + $1,750,000 = **$3,250,000**                          | **$3,250,000**  |

---

### **Final Total Cost:**

$$
\boxed{
\text{Total Cost} = \$2,470,000 + \$2,105,000 + \$3,250,000 = \mathbf{\$7,825,000}
}
$$

---

## **Summary:**
The company should build **Plant 1 and Plant 2** to supply customer X, **Plant 1** for customer Y, and **Plant 3** for customer Z to minimize total costs. The **total minimum cost** for building the plants and supplying the demands is **\$7,825,000**.

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