BUS 203 Quantitative Applications

Solutions to Assignment 10

Problem 7-2
Problem 7-3 thru 6
Problem 7-12

7-2.

a) x1=3.5; x2=4.5; OV= 12.5
b) There are 29 of them. Blue dots in the green area.
c) There are two optimal solutions x1=0; x2=6 and x1=2; x2=5, both have OV=12
d) We get x1=3; x2=4; OV=11. clearly not optimal.

7-3.
For a Min problem the LP relaxation would provide a Lower Bound . Since the LP relaxation is obtained by ignoring some constraints(integrality) the solution value would be better (lower) than if we had enforced those constraints.

7-4.
For a Min problem a feasible solution (obtained by rounding the LP solution or any other means) gives an Upper Bound . Since we are not interested in any solution whose value exceeds the value of this known solution.

7-5.
For a Max problem the LP provides an Upper Bound .

7-6.
For a Max problem rounding (to get a feasible solution would give a Lower Bound .

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7-12.
Let Aj, Bj and Cj the shipments from warehouse A, B and C to customer j and YA, YB, and YC zero/one switch variables denoting whether warehouse A, B, and C will be used. Since warehouses can be assumed to have unlimited capacity, for modeling purposes we might consider the capcities (if operated) to be equal to the total demand, the most that might ever be shipped from a single warehouse.

  MIN 15A1 + 32A2 + 21A3 + 9B1 + 7B2 + 6B3 + 11C1 + 18C2
    + 5C3 + 5000YA + 750YB + 600YC
  SUBJECT TO
            A1 + B1 + C1 >= 200
            A2 + B2 + C2 >= 150
            A3 + B3 + C3 >= 175
            A1 + A2 + A3 <= 525YA 
            B1 + B2 + B3 <= 525YB 
            C1 + C2 + C3 <= 525YC 
            YA + YB + YC >= 2
            Aj, Bj, Cj>=0, YA, YB, YC =0,1

Excel solution
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