Question

Consider the following

LP: Max Z=X1+5X2+3X3

s.t. X1+2X2+X3=3

2X1-X2 =4 X1,X2,X3≥0

a.) Write the associated dual model

b.) Given the information that the optimal basic variables are X1 and X3, determine the associated optimal dual solution.

Answer 1

Answer:-

Given that:-

a)

Dual:

Min W = 3y1 + 4y2

1y1 + 2y2 = 1
2y1 - 1y2 = 5
1y1 + 0y3 = 3
y1, y2 = unrestricted

b)

X2 is non-basic. Therefore, X2=0

So, from the second constraint of the primal, 2X1 - 0 = 4 or X1 = 2

From the first constraint of the primal, 2 + 2*0 + X3 = 3 or, X3 = 1

So, the primal objective value at the optimality, Max Z = 1*2 + 5*0 + 3*1 = 5

Using the duality theorem, we can say that 'Min W' is also 5 and hence 3y1 + 4y2 = 5 ----(i)

Also, from the third constraint of the dual, y1 = 3 ----(ii),

So, from (i) and (ii), y2 = (5 - 3*3)/4 = -1

So, the solution to the dual is as follows:

y1 = 3
y2 = -1
Min W = 5