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?

cheapest is plant 2 to C,

so send 200 2-C $10,000
then 50 2-B $3,500
C has enough
Plant 1 maxed out

200 1-A $12,000
A has enough
100 1-B $6,000
B has enough (50 from 1, a00 from 2)
add up
$ 31,500

To find the shipping schedule that will fulfill the requirements of the warehouses while minimizing the shipping costs, we can use linear programming.

Let's define the decision variables:
x1: Number of pianos shipped from Plant 1 to Warehouse A
x2: Number of pianos shipped from Plant 1 to Warehouse B
x3: Number of pianos shipped from Plant 1 to Warehouse C
y1: Number of pianos shipped from Plant 2 to Warehouse A
y2: Number of pianos shipped from Plant 2 to Warehouse B
y3: Number of pianos shipped from Plant 2 to Warehouse C

We want to minimize the total shipping cost while ensuring that the requirements of the warehouses are met. Hence, the objective function is:
Minimize Cost = 60x1 + 60x2 + 80x3 + 80y1 + 70y2 + 50y3

Subject to the following constraints:
x1 + y1 ≤ 300 (Plant 1 capacity constraint)
x2 + y2 ≤ 250 (Plant 2 capacity constraint)
x1 + x2 + x3 ≥ 200 (Warehouse A minimum requirement)
x2 + y2 ≥ 150 (Warehouse B minimum requirement)
x3 + y3 ≥ 200 (Warehouse C minimum requirement)

Since we want to minimize the shipping costs, we need to solve this linear programming problem using a technique like the Simplex method or graphical method. The solution to this problem will provide the shipping schedule and the minimum cost.

Please note that in this formulation, we assume that pianos can be split between plants for shipment to different warehouses.