Consider the following linear programming problem : Maximize \(z = 5x_{1} + 2x_{2} + 3x_{3} - x_{4} + x_{5}\) Subject to : \(x_{1} + 2x_{2} + 2x_{3} + x_{4} = 8\), \(3x_{1} + 4x_{2} + x_{3} + x_{5} = 7\), \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \geq 0\) What will be the basic feasible solution after iteration 1 ?
\(x_{1} = 0, x_{2} = 0, x_{3} = 4, x_{4} = 0, x_{5} = 3, z = 15\)
x1 = 0, x2 = 0, x3 = 4, x4 = 0, x5 = 3, z = 15 option is correct.
The simplex method starts from an initial basic feasible solution and, at each iteration, brings the non-basic variable with the most positive reduced cost into the basis. Given the two equality constraints in the problem, the natural starting basis is \(\{x_{4}, x_{5}\}\) because they appear alone in the two rows, yielding the initial values \(x_{4} = 8, x_{5} = 7\) and objective \(z = -x_{4} + x_{5} = -8 + 7 = -1\).
Step 1 — compute the reduced costs \(\bar{c}_{j} = c_{j} - c_{B}^{T} B^{-1} A_{j}\) for each non-basic variable. With \(c_{B} = (-1, 1)^{T}\) these evaluate to \(\bar{c}_{1} = 3, \bar{c}_{2} = 0, \bar{c}_{3} = 4, \bar{c}_{4} = 0, \bar{c}_{5} = 0\). Step 2 — the entering variable is \(x_{3}\) because \(\bar{c}_{3} = 4\) is largest. Step 3 — the minimum-ratio test gives \(8/2 = 4\) in row 1 versus \(7/1 = 7\) in row 2, so \(x_{4}\) leaves. Step 4 — after pivoting, the new basis is \(\{x_{3}, x_{5}\}\) with \(x_{3} = 4, x_{5} = 3\) and objective \(z = 3 \cdot 4 + 1 \cdot 3 = 15\), all other decision variables equal to zero.
The other choices either use a wrong entering variable (leading to smaller objective 12 or 10) or claim infeasibility, which is inconsistent with the presence of a basic feasible starting point.
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