Discussion
For the linear programming problem:
\[ \begin{aligned} \text{Maximize } Z &= 2x_{1} + 4x_{2} + 4x_{3} - 3x_{4} \\ \text{Subject to } \alpha x_{1} + x_{2} + x_{3} &= 4, \\ x_{1} + \beta x_{2} + x_{4} &= 8, \\ x_{1}, x_{2}, x_{3}, x_{4} &\ge 0, \end{aligned} \]
consider the following two statements:
S1: If \(\alpha = 2\) and \(\beta = 1\), then \((x_{1}, x_{2})^{T}\) forms an optimal basis.
S2: If \(\alpha = 1\) and \(\beta = 4\), then \((x_{3}, x_{2})^{T}\) forms an optimal basis.
Then, which one of the following is correct?
Answer: B
S2 is TRUE and S1 is FALSE.
S1: with \(\alpha = 2\), \(\beta = 1\), setting \(x_{3} = x_{4} = 0\) gives \(2x_{1} + x_{2} = 4\) and \(x_{1} + x_{2} = 8\), so \(x_{1} = -4\). The basic solution is infeasible, so \((x_{1}, x_{2})\) cannot be an optimal basis.
S2: with \(\alpha = 1\), \(\beta = 4\), setting \(x_{1} = x_{4} = 0\) gives \(4x_{2} = 8\), so \(x_{2} = 2\) and \(x_{3} = 2\): feasible. With \(c_{B} = (4, 4)\) the simplex multipliers are \(y = (4, 0)\), and the reduced costs of the non-basic variables are
\[ \begin{aligned} x_{1}: &\ 2 - (4 \cdot 1 + 0 \cdot 1) = -2 \\ x_{4}: &\ -3 - (4 \cdot 0 + 0 \cdot 1) = -3 \end{aligned} \]
Both are negative, so no entering variable can increase Z; the basis is optimal with Z = 16.
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