Question #203325

Write the dual of the following LPP after reducing it to canonical form.

Min Z = 3x1 + 4x2 + 3x3

Subject to

2x1+4x2 =12

5x1+3x3 ≥11

6x1+ x2 ≥ 8

x1,x2,x3≥0


1
Expert's answer
2021-06-17T11:44:05-0400

The LPP is given as;

3x1 + 4x2 + 3x3 = Min Z

2x1 + 4x2 + 0x3 =12

5x1 + 0x2 + 3x3 ≥ 11

6x1+ x2 + 0x3 ≥ 8

x1 ≥ 0

x2 ≥ 0

x3 ≥ 0


We construct the matrix of coefficients for the free problem, direct problem, and forward problem and name them A, B, C respectively, where;

A = (343)\begin{pmatrix} 3 & 4 &3\\ \end{pmatrix}

B= (12118)\begin{pmatrix} 12 \\ 11\\ 8 \end{pmatrix}

D =(240503610)\begin{pmatrix} 2& 4&0\\ 5 & 0&3\\ 6&1&0 \end{pmatrix}


To find the dual of A, we compute the transpose matrix of B and to compute the dual of B, we find the tranpose matrix of A. The dual of C is the transpose matrix of C

A dual = (12118)\begin{pmatrix} 12 & 11 &8\\ \end{pmatrix}

B dual = (343)\begin{pmatrix} 3 \\ 4\\ 3 \end{pmatrix}

C dual = (256401030)\begin{pmatrix} 2& 5&6\\ 4& 0&1\\ 0&3&0 \end{pmatrix}


To find dual of the problem, we maximize the objective function. Also, we have the same number of constraints in the dual problem as that of the direct problem.

A maximized objective function will contain either the inequality sign "\le " or " ="


Therefore, the dual LPP is given as follows;

12y1 +11y2 + 8y3 = Max Z

2y1 +5y2 + 6y3 \le 3

4y1 + y3 \le 4

3y2 \le 3


y1, y2, y3 ≥ 0



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