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 of the warehouses while minimizing the shipping costs for Acrosonic, we can follow these steps:
1. Determine the maximum number of loudspeaker systems that can be shipped from Plant I and Plant II.
- The output at Plant I is at most 800/month.
- The output at Plant II is at most 600/month.
2. Determine the minimum monthly requirements of each warehouse.
- Warehouse A requires a minimum of 500 systems.
- Warehouse B requires a minimum of 400 systems.
- Warehouse C requires a minimum of 400 systems.
3. Calculate the shipping costs from Plant I and Plant II to each warehouse.
- Shipping costs from Plant I to Warehouse A: $8 per loudspeaker system.
- Shipping costs from Plant I to Warehouse B: $10 per loudspeaker system.
- Shipping costs from Plant I to Warehouse C: $11 per loudspeaker system.
- Shipping costs from Plant II to Warehouse A: $9 per loudspeaker system.
- Shipping costs from Plant II to Warehouse B: $8 per loudspeaker system.
- Shipping costs from Plant II to Warehouse C: $7 per loudspeaker system.
Now, we can determine the shipping schedule and the minimum cost:
- Calculate the number of systems that need to be shipped from Plant I to each warehouse while minimizing costs:
- To Warehouse A: Ship as many systems as possible from Plant I (up to 800) since the shipping cost ($8) is the lowest. So, the number of systems from Plant I to Warehouse A is 500.
- To Warehouse B: Ship the remaining required systems, i.e., (400 - 500) = 0 systems since the requirement is already met.
- To Warehouse C: Ship the remaining required systems, i.e., (400 - 500) = 0 systems since the requirement is already met.
- Calculate the number of systems that need to be shipped from Plant II to each warehouse while minimizing costs:
- To Warehouse A: Ship the remaining required systems, i.e., (500 - 600) = 0 systems since the requirement is already met.
- To Warehouse B: Ship as many systems as possible from Plant II (up to 600) since the shipping cost ($8) is lower than from Plant I. So, the number of systems from Plant II to Warehouse B is 400.
- To Warehouse C: Ship the remaining required systems, i.e., (400 - 400) = 0 systems since the requirement is already met.
Therefore, the shipping schedule will be as follows:
- Plant I to Warehouse A: 500 systems
- Plant I to Warehouse B: 0 systems
- Plant I to Warehouse C: 0 systems
- Plant II to Warehouse A: 0 systems
- Plant II to Warehouse B: 400 systems
- Plant II to Warehouse C: 0 systems
The minimum cost can be calculated by multiplying the number of systems shipped with their respective shipping costs and adding them up:
Total cost = (500 * $8) + (400 * $8) = $4,400
So, the minimum cost is $4,400.