Let's analyze the data step-by-step to determine the optimal plants to build to satisfy demand at minimum total cost. --- ## ** 1: Summarize the Data** ### **Production Capacity & Cost for Each Plant** | Plant | Capacity | Construction Cost | Cost to Customer X | Cost to Customer Y | Cost to Customer Z | |---------|--------------|---------------------------|------------------------|------------------------|------------------------| | 1 | 40,000 | \$1,325,000 | \$35 | \$30 | \$45 | | 2 | 30,000 | \$1,100,000 | \$45 | \$40 | \$50 | | 3 | 50,000 | \$1,500,000 | \$70 | \$65 | \$50 | | 4 | 20,000 | \$1,200,000 | \$20 | \$45 | \$25 | | 5 | 40,000 | \$1,400,000 | \$65 | \$45 | \$45 | ### **Customer Demands** | Customer | Demand | |------------|---------| | X | 41,000 | | Y | 26,000 | | Z | 35,000 | --- ## **Step 2: Decide Which Plants to Build** ### **Step 2.1: Choose the least costly plant for each customer** | Customer | Least cost plant and unit cost | |--------------|----------------------------------------| | X | Plant 4 (\$20/unit) | | Y | Plant 1 (\$30/unit) | | Z | Plant 4 (\$25/unit) | ### **Step 2.2: Check if capacities of these plants are sufficient** - **Plant 4**: Capacity = 20,000 units - Customer Z demand = 35,000 units → Not enough - **Plant 1**: Capacity = 40,000 units - Customer X demand = 41,000 units → Not enough **Conclusion:** The cheapest options are insufficient to meet demands. We need to consider other plants or combinations. --- ## **Step 3: Consider alternative plants with low costs** ### **Customer X (demand 41,000):** - Plant 4 (cost \$20/unit): Capacity 20,000 - Plant 1 (cost \$35/unit): Capacity 40,000 - Plant 2 (cost \$45/unit) - Plant 3 (cost \$70/unit) - Plant 5 (cost \$65/unit) **Possible options:** - Use Plant 4 (20,000 units) + Plant 1 (remaining 21,000 units) ### **Customer Y (demand 26,000):** - Plant 1 (cost \$30/unit): Capacity 40,000 - Plant 2 (cost \$40/unit): Capacity 30,000 - Plant 3 (cost \$65/unit) - Plant 4 (cost \$45/unit) - Plant 5 (cost \$45/unit) Cheapest: Plant 1 (\$30/unit) ### **Customer Z (demand 35,000):** - Plant 4 (cost \$25/unit): Capacity 20,000 - Plant 1 (cost \$45/unit): Capacity 40,000 Cheapest: Plant 4, but capacity insufficient, so combine Plant 4 (20,000) + Plant 1 (15,000). --- ## **Step 4: Construct the Cost-Effective Plan** ### **Customer X:** - Use Plant 4 (20,000 units, \$20/unit) - Remaining demand: 21,000 units - Use Plant 1 (21,000 units, \$35/unit) ### **Customer Y:** - Use Plant 1 (26,000 units, \$30/unit) ### **Customer Z:** - Use Plant 4 (20,000 units, \$25/unit) - Remaining demand: 15,000 units - Use Plant 1 (15,000 units, \$45/unit) --- ## **Step 5: Check Capacity and Cost** ### **Plant 1:** - Used for Customer X (21,000), Y (26,000), Z (15,000) - Total: 21,000 + 26,000 + 15,000 = **62,000 units** **Capacity of Plant 1:** 40,000 units **Issue:** Exceeds capacity. So, need to adjust. --- ## **Step 6: Adjust for Capacity Constraints** ### **Option:** - Assign Plant 1 to Customer Y (26,000 units) (since it's cheapest for Y) - Assign Plant 4 to Customer X (20,000 units) - Remaining for Customer X: 41,000 - 20,000 = 21,000 units - Assign Plant 1 to Customer X **(for remaining 21,000 units)** **Total for Plant 1:** - Customer Y: 26,000 units - Customer X: 21,000 units Total = 47,000 units, which **exceeds capacity (40,000)**. So, split assignments: - Assign Plant 1 to Customer Y: 26,000 units - Remaining for Customer X: 41,000 units **Use Plant 2 or Plant 3 for Customer X?** **Check costs:** - Plant 2: \$45/unit - Plant 3: \$70/unit Cheapest for Customer X: Plant 2 (\$45/unit) **Assign Plant 2 to Customer X:** 30,000 units Remaining for Customer X: 41,000 - 30,000 = 11,000 units - Use Plant 1 for remaining 11,000 units Total for Plant 1: 11,000 units (cost \$35/unit) Total for Customer X: 30,000 + 11,000 units. **Total for Plant 1:** 26,000 (Y) + 11,000 (X) = **37,000 units** (within capacity). --- ## **Step 7: Final Assignment** | Plant | Customer | Units | Cost per Unit | Total Cost | |--------|------------|---------|--------------|--------------| | 1 | Y | 26,000 | \$30 | \$780,000 | | 1 | X | 11,000 | \$35 | \$385,000 | | 2 | X | 30,000 | \$45 | \$1,350,000 | | 4 | X | 20,000 | \$20 | \$400,000 | | 4 | Z | 20,000 | \$25 | \$500,000 | Total capacity used: - Plant 1: 26,000 + 11,000 = 37,000 (within 40,000) - Plant 2: 30,000 (within 30,000) - Plant 4: 20,000 + 20,000 = 40,000 (full capacity) --- ## **Step 8: Calculate Total Cost** ### **Construction Costs:** - Plant 1: \$1,325,000 - Plant 2: \$1,100,000 - Plant 4: \$1,200,000 Total construction cost: \[ \$1,325,000 + \$1,100,000 + \$1,200,000 = \boxed{\$3,625,000} \] ### **Transportation Costs:** - Plant 1 to Y: 26,000 units × \$30 = \$780,000 - Plant 1 to X: 11,000 units × \$35 = \$385,000 - Plant 2 to X: 30,000 units × \$45 = \$1,350,000 - Plant 4 to X: 20,000 units × \$20 = \$400,000 - Plant 4 to Z: 20,000 units × \$25 = \$500,000 Total transportation cost: \[ \$780,000 + \$385,000 + \$1,350,000 + \$400,000 + \$500,000 = \boxed{\$3,415,000} \] --- ## **## Final Total Cost** \[ \boxed{ \text{Total Cost} = \text{Construction} + \text{Transportation} = \$3,625,000 + \$3,415,000 = \boxed{\$7,040,000} } \] --- ## **Summary** - **Optimal plants to build:** Plants 1, 2, and 4 - **Total minimum cost:** **\$7,040,000** - **Note:** This solution minimizes total costs considering construction and transportation costs while satisfying demand and capacity constraints. --- If you need further optimization or alternative configurations, advanced linear programming methods can be applied, but this provides a practical, cost-effective plan based on the given data.