GATE 2020 MA – Question 33
A balanced minimum-cost transportation problem has basic variables $\{x_{13},x_{22},x_{23}=10,x_{31},x_{32},x_{34}\}$. The cost rows for destinations D1,D2,D3,D4 are O1: (6,2,−1,0), supply 10; O2: (4,2,2,3), supply λ+5; O3: (3,1,2,1), supply 3λ. Demands are (10,μ−5,μ+5,15). Find x₃₂.
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Correct answer: 5 to 5
Explanation
All nonbasic variables are zero. Row O1 therefore gives x₁₃ = 10. Column D3 gives $x_{13}+x_{23}=20=\mu+5$, so μ = 15. Total demand is 55 and total supply is $10+(\lambda+5)+3\lambda=15+4\lambda$, yielding λ = 10. Row O2 gives x₂₂ = 15 − 10 = 5. Column D2 then gives $x_{32}=\mu-5-x_{22}=15-5-5=5$.