GATE 2025 MA – Question 24
Balanced transportation has cost rows S1:(2,6,20,11), S2:(12,7,4,10), S3:(8,14,16,11), supplies (α+10,α+λ+10,5) and demands (α+5,10,λ+5,α+λ), α,λ>0. Northwest-corner cost is 290. Which statements hold?
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Correct answer: (B) α²+αλ=150; (D) Optimal cost is 290
Explanation
Balance gives λ=5. Northwest allocations are x₁₁=α+5,x₁₂=5,x₂₂=5,x₂₃=10,x₂₄=α,x₃₄=5. Their cost is 12α+170=290, giving α=10. Thus α²+αλ=150, while α²+λ²=125. Potentials u=(0,1,2),v=(2,6,3,9) match all basic costs and every nonbasic reduced cost is nonnegative. The starting solution is already optimal at 290. **B,D**.