Steinwelt Piano manufactures uprights and consoles in two plants, Plant 1 and plant 2. the output of plant 1 is at most 300/month, and the out put of plant 2 is at most 250/month. theses pianos are shipped to three warehouses that serve as distribution centers for Steinwelt. To fill current and projected future orders,Warehouse A requires a minimum of 200 piano/nibtm Warehouse B requires at least 150 pianos/month, and Wrehouse C requires at least 200 pianos/month. The shipping cost of each piano from Plant 1 to Warehouse A, Warehouse B, and Warehouse C is $60, $60, and $80, respectively, and the shipping cost of each piano from Plant 2 to Warehouse A, Warehouse B, and Warehouse C is $80, $70, and $50, repectively. What shipping schedule will enable Steinwelt to meet the requirements of the warehouses while keeping the shipping costs to a minimum? What is the minimum cost?
To determine the shipping schedule that minimizes the cost while meeting the requirements of the warehouses, we need to figure out how many pianos to ship from each plant to each warehouse.
Let's start by considering Plant 1. It has a maximum output of 300 pianos per month. Warehouse A requires a minimum of 200 pianos, Warehouse B requires at least 150 pianos, and Warehouse C requires at least 200 pianos per month. Therefore, we can ship 200 pianos from Plant 1 to Warehouse A, 150 pianos to Warehouse B, and the remaining (300 - 200 - 150 = 100) pianos to Warehouse C. This ensures that the minimum requirements of each warehouse are met using the output of Plant 1.
Next, let's consider Plant 2. It has a maximum output of 250 pianos per month. Since the requirements for Warehouse A and Warehouse B have already been met by Plant 1, we only need to ship pianos to Warehouse C from Plant 2. Warehouse C requires at least 200 pianos per month, so we will ship all 200 pianos from Plant 2 to Warehouse C.
Now, let's calculate the shipping costs. The shipping cost per piano from Plant 1 to Warehouse A is $60, and since we are shipping 200 pianos, the total cost is 200 * $60 = $12,000. Similarly, the shipping cost from Plant 1 to Warehouse B is also $12,000 (150 pianos * $60), and from Plant 1 to Warehouse C is 100 pianos * $60 = $6,000.
The shipping cost per piano from Plant 2 to Warehouse A is $80, and since we are not shipping any pianos from Plant 2 to Warehouse A, the cost is $0. The shipping cost from Plant 2 to Warehouse B is 0 pianos * $80 = $0, and from Plant 2 to Warehouse C is 200 pianos * $50 = $10,000.
Adding up all the shipping costs, we have a total cost of $12,000 + $12,000 + $6,000 + $0 + $0 + $10,000 = $40,000.
Therefore, the shipping schedule that enables Steinwelt to meet the requirements of the warehouses while keeping the shipping costs to a minimum is as follows:
- From Plant 1: Ship 200 pianos to Warehouse A, 150 pianos to Warehouse B, and 100 pianos to Warehouse C.
- From Plant 2: Ship 200 pianos to Warehouse C.
The minimum cost is $40,000.