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linear programming math

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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?

  • linear programming math -

    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.

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