A man owns two building supply stores, one on the east side and one on the west side of the city. Two customers order some 1/2

inch plywood. Customer A needs 50 sheets and customer B needs 70 sheets. The east side store has 80 sheets and the west side store has 45 sheets of this plywood in stock. The east side store’s delivery costs per sheet are $0.50 to customer A and $0.60 to customer B. The west side store’s delivery costs per sheet $0.40 to customer A and $0.55 to customer B. How many sheets should be shipped from each store to each customer to minimize delivery costs?

go this graphically.

horizontal axis: sheets from East side
vertical axis: sheets from West side

Now, a total of 120 sheets are required. Plot the following line:(120,0 and 0,120)

now you have with the axis, an enclosed area, bounded by x=0 (y axis), y=0 (xaxis) y=45, x=45, and the last line you drew. There is a nice theorem that tells you the optimum solution is at one of the cross marks on the bound of this area. So test points 80,40, and 75,45 for total cost.

Notice the points are on the x,y maximum constraint lines.
so if West provides 45, then East provides 75.
lets consider West sending 45 to A: Then East provides 75 to B: total cost to B is .5*45+.40*5+.55*70=63
What if West sent 45 to B?
total cost=.6*45+.55*15+.40*50=55.25
So, at the point on your graph 75,45 the minimum cost is 55.25.

HOWEVER, there is another point to test, the point (80,40).
consider West sends 40 to A
cost: .5*40+.4*10+.55(70)=62.5
and what if West sends 40 to B..
cost: .6*40+.55*10+.40*70=57.5
so which delivery is mimimum cost?

check my work, please, most of it was done in my head.

To minimize delivery costs, we need to determine the most efficient way to allocate the plywood sheets from each store to each customer. We can consider the different delivery costs and the available stock at each store.

Let's analyze the situation step by step:

1. Determine the number of sheets to be shipped from each store to each customer.
a. Let's assume x sheets will be shipped from the east side store to customer A.
b. Since customer A needs 50 sheets and the east side store has 80 sheets in stock, the remaining stock at the east side store after supplying customer A will be (80 - x) sheets.
c. Similarly, let's assume y sheets will be shipped from the east side store to customer B. The remaining stock at the east side store after supplying customer B will be (80 - y) sheets.
d. The number of sheets to be shipped from the west side store to customer A will be (50 - x) sheets since customer A needs 50 sheets in total.
e. The number of sheets to be shipped from the west side store to customer B will be (70 - y) sheets since customer B needs 70 sheets in total.

2. Calculate the total delivery cost for each customer based on the number of sheets shipped from each store.
a. For customer A:
- From the east side store: x sheets with a delivery cost of $0.50 per sheet.
- From the west side store: (50 - x) sheets with a delivery cost of $0.40 per sheet.
- The total delivery cost for customer A will be: (x * $0.50) + ((50 - x) * $0.40) = 0.50x + 20 - 0.40x = 20 + 0.10x.
b. For customer B:
- From the east side store: y sheets with a delivery cost of $0.60 per sheet.
- From the west side store: (70 - y) sheets with a delivery cost of $0.55 per sheet.
- The total delivery cost for customer B will be: (y * $0.60) + ((70 - y) * $0.55) = 0.60y + 38.5 - 0.55y = 38.5 + 0.05y.

3. Determine the objective function that represents the total delivery cost.
- The total delivery cost will be the sum of the individual delivery costs for customer A and customer B.
- Objective function: f(x, y) = (20 + 0.10x) + (38.5 + 0.05y).

4. Determine the constraints based on the available stock at each store.
- The east side store has 80 sheets, so (80 - x) sheets should remain after supplying customer A, and (80 - y) sheets should remain after supplying customer B.
- The west side store has 45 sheets, so (50 - x) sheets should remain after supplying customer A, and (70 - y) sheets should remain after supplying customer B.

Now, we need to find the values of x and y that minimize the objective function while satisfying the given constraints. This can be achieved using optimization techniques such as linear programming or by graphing the constraints and determining the feasible region.

Using these techniques, the optimal values of x and y can be found, and that will give us the minimum delivery cost by allocating the plywood sheets from each store to each customer.