GATE 2021 MA – Question 65
The point x̄=(11/3,2/3,0)ᵀ is optimal for max 4x₁+x₂−3x₃ subject to 2x₁+4x₂+ax₃≤10,x₁−x₂+bx₃≤3,2x₁+3x₂+5x₃≤11,x≥0, where a,b∈ℝ. If (p,q,r)ᵀ is dual optimal, find p+q+r to two decimals.
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Correct answer: 3.14 to 3.18
Explanation
At x̄ the first two constraints are tight and the third has value 28/3<11. Complementary slackness forces r=0. Since x₁,x₂>0, their dual inequalities are tight: 2p+q+2r=4 and 4p−q+3r=1. Thus p=5/6,q=7/3, giving sum 19/6=**3.17**. The premise of optimality ensures the remaining dual inequality for x₃ is satisfied; a,b do not affect this sum.