What is the current solution (x1, x2, x3, x4, x5, x6)T ? What is the current objective function value?

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

3.     30 points) Following is an initial tableau of a linear programming problem (LP) in standard form.

 

x1	x2	x3	x4	x5    x6	RHS
2 a	b c	3 −2	−1 e	1      0 0      1	8 20
3	d	−11	7	0      0	0

Using a simplex method to solve it, after some iterations, we get the following tableau (T ∗).(T ∗) 

(a)     Find the values of a ∼ m.

[Use the values you obtained in (a) for the following questions.]

(b)     What is the current solution (x1, x2, x3, x4, x5, x6)T ? What is the current objective function value?

(c)     Is (T ∗) an optimal tableau? If not, find an optimal solution starting from (T ∗).

(d)     Find the dual of this LP and its optimal solution (the objective value and the value of the dual variables including structural, surplus/slack variables, if any). [Use the tableau - do not solve from scratch!]

 

[Problem (e)-(h) should be done independently.]

(e)    Suppose b, the RHS coefficient vector of these two constraints, is changed to (−9, 15)T . What is the effect of this change on the solution?

(f)     Would the optimal solution be altered, if a2, the second column of the coefficient matrix, is changed to (4, 2)T ? If yes, find the new optimal solution.

(g)     What effect will the addition of the following constraint have on the solution?x1 + x2 ≥ 2

 

(h)     Find the largest and smallest value of λ such that the given solution is still optimal if the cost vector c in the objective function is replaced by c + λ1, where 1 = (1, 1, 1, 1, 1, 1)

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