Consider the following all-integer linear program: Max 5x1 +8x2 s.t. 6x1 + 5x2 <30,
9x1+4x2<36,1x1+2x2<10, x1,x2 > 0 and integer.
a. Graph the constraints for this problem, use dots to indicate all feasible integer solutions.
b. Find the optimal solution to the LP relaxation, round down to find a feasible integer solution.
c. Find the integer solution. Is it the same solution obtained in part (b) by rounding down?
(a) The constraint 6x1 + 5x2 < 30 corresponds to the origin side of the line
6x1 + 5x2 = 30
The constraint 9x1 + 4x2 < 36 corresponds to the origin side of the line
9x1 + 4x2 = 36
The constraint x1 + 2x2 < 10 corresponds to the origin side of the line
x1 + 2x2 = 10
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The dots in the feasible region indicate integer solutions.
(b) The relaxed LP solution to the problem lies at one of the vertices of the graph
To find the vertices, we read off their coordinates (Alternately, we solve the system of
equations in pairs and find the intersection points).
The possible LP relaxed solutions are at (x1, x2) =
(A) (10/7, 30/7) or (B)(20/7, 18/7)
The objective function is 5x1 + 8x2
At A(10/7, 30/7), the function value is 5(10/7) + 8(30/7) = 290/7
At B(20/7, 18/7), the function value is 5(20/7) + 8(18/7) = 244/7
The LP relaxed solution is (x1, x2) = (10/7, 30/7) = (1.4286, 4.286)
On rounding down the solution is (x1, x2) = (1, 4)
(c) The integer values of (x1, x2) which maximize the function 5x1 + 8x2 are
(x1, x2) = (1, 4), which is the same as that obtained in (b) above.
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