Discussion
Consider the LPP :
Minimize \(Z = 4x_{1} + 6x_{2} + 18x_{3}\)
Subject to : \(x_{1} + 3x_{2} \ge 3\),
\(x_{2} + 2x_{3} \ge 5\)
and \(x_{1}, x_{2}, x_{3} \ge 0\).
Then, which of the following is not a constraint in the corresponding dual problem ? (Assume that \(w_{1}\) and \(w_{2}\) are the dual variables.)
Answer: C
The statement that is not a dual constraint is \(3w_{1} + w_{2} \ge 6\).
For a minimisation primal with \(\ge\) constraints and non-negative variables, the dual is to maximise \(3w_{1} + 5w_{2}\) subject to one \(\le\) constraint per primal variable:
- \(x_{1}\): \(w_{1} \le 4\)
- \(x_{2}\): \(3w_{1} + w_{2} \le 6\)
- \(x_{3}\): \(2w_{2} \le 18\), i.e. \(w_{2} \le 9\)
- \(w_{1}, w_{2} \ge 0\)
The constraint for \(x_{2}\) is \(3w_{1} + w_{2} \le 6\); option C has the inequality reversed, so it is not a constraint of the dual. The others are.
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