Question

Given the following all-integer linear program: (COMPLETE YOUR SOLUTION IN EXCEL USING SOLVER AND UPLOAD YOUR FILE. BE SURE THAT EACH WORKSHEET IN THE EXCEL FILE CORRESPONDS TO EACH QUESTION BELOW )

Max 15x1 + 2x2

s. t. 7x1 + x2 < 23

3x1 - x2 < 5

x1, x2 > 0 and integer

a. Solve the problem (using SOLVER) as an LP, ignoring the integer constraints.

b. What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution?

c. What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain.

d. Show that the optimal objective function value for the ILP is lower than that for the optimal LP (Eg. Resolve original problem using SOLVER with the Integer requirement).

e. Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal?

Answer 1

a) Solution using Solver is following

Formula:

D2 =SUMPRODUCT(B2:C2,$B$6:$C$6) copy to D2:D4

solution: x1 = 2.8, x2 = 3.4

objective value = 48.8

b) solution obtained by rounding off

x1 = 3, x2 = 3

No, this is not optimal solution

c) Solution obtained by rounding down is

x1=2, x2 = 3

This is also not optimal

d) solution using integer requirement is following

Objective value is 348, which is lower than before.