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Which of the following is not having feasible region given set of constraints for a linear programming problem? |
$x+2 y \leq 50,3 x+y \leq 80, x \geq 0, y \geq 0$ $x+2 y \leq 50,3 x+y \geq 80, x \geq 0, y \geq 0$ $x+y \leq 10,2 x+y \geq 40, x \geq 0, y \geq 0$ $2 x+y \leq 30,5 x+2 y \leq 100, x \geq 0, y \geq 0$ |
$x+y \leq 10,2 x+y \geq 40, x \geq 0, y \geq 0$ |
$\text{Check each set of constraints for feasibility.}$ $(A)\;x+2y\le50,\;3x+y\le80,\;x\ge0,\;y\ge0.$ $\text{Both inequalities allow a common region in first quadrant}.$ $(B)\;x+2y\le50,\;3x+y\ge80,\;x\ge0,\;y\ge0.$ $\text{Lines intersect in first quadrant, feasible region exists}.$ $(C)\;x+y\le10,\;2x+y\ge40,\;x\ge0,\;y\ge0.$ $\text{From }x+y\le10 \Rightarrow 2x+y\le20,$ $\text{but }2x+y\ge40\text{ cannot be satisfied simultaneously}.$ $\text{No feasible region}.$ $(D)\;2x+y\le30,\;5x+2y\le100,\;x\ge0,\;y\ge0.$ $\text{Feasible region exists}.$ $\text{Correct option: }(C).$ |
