The binding constraints for this problem are the first and second. Min s.t. X1 +
ID: 357210 • Letter: T
Question
The binding constraints for this problem are the first and second. Min s.t. X1 + 2x2 X1t x2 2 300 2xi+ x2 2 400 2x 5x 750 a. Keeping c2 fixed at 2, over what range can ci vary before there is a change in the optimal solution point? Keeping ci fixed at 1, over what range can c2 vary before there is a change in the optimal solution point? c. If the objective function becomes Min 1.5x 2x2, what will be the optimal values ofx1, X2, and the objective function? If the objective function becomes Min 7x1+6x, what constraints will be binding?. b. C. e. Find the dual price for each constraint in the original problem.Explanation / Answer
(a) Binding constraint is only the first one and not the second constraint. Therefore slope of .
Slope of first constraint (ratio of coefficients of x1 to x2) is 1
So the slope of objective function must be greater than 0 and less than slope of first constraint (=1).
Therefore, value of c1 can vary between 0 and 2.
(b) Similarly as above, c2 can vary between 1 and infinity, such that slope of objective function remains between 0 and 1
(c) The objective coefficient of x1 is change, but it is still between the optimal range (0 to 2). Therefore, optimal solution remains unchanged (x1 = 300, x2 = 0)
Objective function value = 1.5*300+2*0 = 450
(d) Now the slope of objective function is greater than the slope of constraint 1 and 3, therefore these (first and third) are the binding constraints.
(e) In the original problem, only the first constraint is binding. If we increase the value of RHS of first constraint by 1, Then we need to increase the value of first variable by 1, this results in increase in objective function by 1 . Therefore, dual price of first constraint is 1. The other two constraints are non-binding. So their dual price is 0
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