Question

राशि z का न्यूनतम मान ज्ञात कीजिए।

Z = X1+2X2

निम्न प्रतिबन्धो के अन्तर्गत

X1+X2 ≥1

X120, X2 20

Answer 1

SOLUTION

TO SOLVE

Find the minimum value of the quantity

$\sf{z = x_1 + 2x_2 }$

According to the following conditions

$\sf{x_1 + x_2 \geqslant 1 }$

$\sf{ x_1 \geqslant 0 \: \: , \: x_2 \geqslant 0}$

EVALUATION

The given linear programming problem is

$\sf{Minimize \: \: z = x_1 + 2x_2 }$

Subject to the constraints

$\sf{x_1 + x_2 \geqslant 1 }$

With non negativity conditions

$\sf{ x_1 \geqslant 0 \: \: , \: x_2 \geqslant 0}$

First we consider $\sf{ (x_1 \: , \: x_2 )}$ coordinate system

Any point $\sf{ (x_1 \: , \: x_2 )}$ satisfying the restrictions

$\sf{ x_1 \geqslant 0 \: \: , \: x_2 \geqslant 0}$ lies in the first quadrant only

The solution space satisfying the constraints is shown in the figure attached .

As seen from the direction of the arrows , the solution space is unbounded

The corner points are A(1,0) , B(1,0)

At A(1,0) we have z = 1 + ( 2 × 0 ) = 1

At B(0,1) we have z = 0 + ( 2 × 1 ) = 2

$\sf{Min \: z = 1 \: \: when \: \: x_1 = 1 \: , \: x_2 = 0 }$

FINAL ANSWER

$\sf{Min \: z = 1 \: \: when \: \: x_1 = 1 \: , \: x_2 = 0 }$

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