Discussion

Consider the following Linear Programming Problem P

minimize \(5x_{1} + 2x_{2}\)
subject to \(2x_{1} + x_{2} \le 2,\)
\(x_{1} + x_{2} \ge 1,\)
\(x_{1}, x_{2} \ge 0.\)

The optimal value of the problem P is equal to

Answer: D

The optimal value is 2.

The feasible region is bounded by \(2x_{1} + x_{2} \le 2\), \(x_{1} + x_{2} \ge 1\) and the axes. Its corners are \((0, 1)\), \((0, 2)\), \((1, 0)\).

\[ \begin{aligned} (0, 1) &: 5(0) + 2(1) = 2 \\ (0, 2) &: 4 \\ (1, 0) &: 5 \end{aligned} \]

The minimum is 2 at \((0, 1)\). 4 and 5 are the values at the other corners, and 0 would need \((0, 0)\), which violates \(x_{1} + x_{2} \ge 1\).

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