Acrosonic manufactures a model G loudspeaker system in Plants I and II. The output at Plant I is at most 800/month, and the output at Plant II is at most 600/month. Model G loudspeaker systems are also shipped to the three warehouses—A, B, and C—whose minimum monthly requirements are 500, 400, and 400 systems, respectively. Shipping costs from Plant I to Warehouse A, Warehouse B, and Warehouse C are $8, $10, and $11 per loudspeaker system, respectively, and the shipping costs from Plant II to each of these warehouses are $9, $8, and $7, respectively. What shipping schedule will enable Acrosonic to meet the requirements of the warehouses while keeping its shipping costs to a minimum?

Plant I to Warehouse A ______ systems
Plant I to Warehouse B ______ systems
Plant I to Warehouse C ______ systems
Plant II to Warehouse A ______ systems
Plant II to Warehouse B ______ systems
Plant II to Warehouse C ______ systems

What is the minimum cost?
$ __________

To determine the shipping schedule that will meet the requirements and minimize shipping costs, we need to consider the maximum output of each plant and the minimum requirements of each warehouse.

Let's start by analyzing the shipping schedule from Plant I to the warehouses:

- Warehouse A requires a minimum of 500 systems, and the shipping cost from Plant I to Warehouse A is $8 per system. The output of Plant I is at most 800 systems. To minimize costs, we ship 500 systems from Plant I to Warehouse A, which incurs a cost of $8 * 500 = $4000.

- Warehouse B requires a minimum of 400 systems, and the shipping cost from Plant I to Warehouse B is $10 per system. Since the output of Plant I is at most 800 systems, we can ship 400 systems from Plant I to Warehouse B, incurring a cost of $10 * 400 = $4000.

- Warehouse C requires a minimum of 400 systems, and the shipping cost from Plant I to Warehouse C is $11 per system. Again, since the output of Plant I is at most 800 systems, we can ship 400 systems from Plant I to Warehouse C, resulting in a cost of $11 * 400 = $4400.

Now, let's analyze the shipping schedule from Plant II to the warehouses:

- Warehouse A requires a minimum of 500 systems, and the shipping cost from Plant II to Warehouse A is $9 per system. The output of Plant II is at most 600 systems. To meet the requirement, we can ship 600 systems from Plant II to Warehouse A, incurring a cost of $9 * 600 = $5400.

- Warehouse B requires a minimum of 400 systems, and the shipping cost from Plant II to Warehouse B is $8 per system. Since the output of Plant II is at most 600 systems, we can ship 400 systems from Plant II to Warehouse B, resulting in a cost of $8 * 400 = $3200.

- Warehouse C requires a minimum of 400 systems, and the shipping cost from Plant II to Warehouse C is $7 per system. Since the output of Plant II is at most 600 systems, we can ship 400 systems from Plant II to Warehouse C, incurring a cost of $7 * 400 = $2800.

Summing up the costs from each plant to each warehouse:

Plant I to Warehouse A: $4000
Plant I to Warehouse B: $4000
Plant I to Warehouse C: $4400
Plant II to Warehouse A: $5400
Plant II to Warehouse B: $3200
Plant II to Warehouse C: $2800

To find the minimum cost, let's add up all these costs:

Total cost = $4000 + $4000 + $4400 + $5400 + $3200 + $2800 = $23800

Hence, the minimum cost is $23800.

To determine the shipping schedule that will enable Acrosonic to meet the requirements of the warehouses while keeping its shipping costs to a minimum, we can use linear programming techniques.

Let's denote the number of loudspeaker systems shipped from Plant I to Warehouse A as x₁, from Plant II to Warehouse A as x₂, from Plant I to Warehouse B as y₁, from Plant II to Warehouse B as y₂, from Plant I to Warehouse C as z₁, and from Plant II to Warehouse C as z₂.

We can formulate the problem as follows:

Objective function:
Minimize the cost of shipping:
Cost = 8x₁ + 9x₂ + 10y₁ + 8y₂ + 11z₁ + 7z₂

Subject to the following constraints:
1. Output from Plant I cannot exceed 800: x₁ + y₁ + z₁ ≤ 800
2. Output from Plant II cannot exceed 600: x₂ + y₂ + z₂ ≤ 600
3. Warehouse A requires at least 500 systems: x₁ + x₂ ≥ 500
4. Warehouse B requires at least 400 systems: y₁ + y₂ ≥ 400
5. Warehouse C requires at least 400 systems: z₁ + z₂ ≥ 400

Solving this linear programming problem will give us the optimal shipping schedule and the corresponding minimum cost.