Question

Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and the number of support personnel required for each project, and the expected profits for each project are summarized in the following table: Project 1 2 3 4 5 6 Engineers Required 20 55 47 38 90 63 Support Personnel Required 15 45 50 40 70 70 Profit ($1,000,000s) 1.0 1.8 2.0 1.5 3.6 2.2 Formulate an integer program that maximizes Tower's profit subject to the following management constraints: 1) Use no more than 175 engineers 2) Use no more than 150 support personnel 3) If either project 6 or project 4 is done, both must be done 4) Project 2 can be done only if project 1 is done 5) If project 5 is done, project 3 must not be done and vice versa 6) No more than three projects are to be done.

Show Process.

Answer 1

Let Xj be the binary integer such that Xj=1 when project-j is selected and Xj=0 otherwise.

Maximize Z = total profit = 1.0X1 + 1.8X2 + 2.0X3 + 1.5X4 + 3.6X5 + 2.2X6

Subject to,

20X1 + 55X2 + 47X3 + 38X4 + 90X5 + 63X6 <= 175 ----(1)

15X1 + 45X2 + 50X3 + 40X4 + 70X5 + 70X6 <= 150 ----(2)

X6 - X4 = 0 ----(3)

X1 - X2 >= 0 ----(4)

X3 + X5 <= 1 ----(5)

X1 + X2 + X3 + X4 + X5 + X6 <= 3 ----(6)

Xj = {0,1}

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Truth table for constraint (3)

X6	X4	Remarks
1	1	Accept
0	0	Accept
1	0	Reject
0	1	Reject

Truth table for constriant (4)

X1	X2	Remarks
1	0	Accept
1	1	Accept
0	0	Accept
0	1	Reject

Truth table for constraint (5)

X3	X5	Remarks
1	0	Accept
0	1	Accept
0	0	Accept
1	1	Reject

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LINDO Code

Max 1.0X1 + 1.8X2 + 2.0X3 + 1.5X4 + 3.6X5 + 2.2X6
s.t.
20X1 + 55X2 + 47X3 + 38X4 + 90X5 + 63X6 <= 175
15X1 + 45X2 + 50X3 + 40X4 + 70X5 + 70X6 <= 150
X6 - X4 = 0
X1 - X2 >= 0
X3 + X5 <= 1
X1 + X2 + X3 + X4 + X5 + X6 <= 3
end
INT 6

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