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  • The feasible region for a LPP is shown in Fig. 12.10. Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.

The feasible region for a LPP is shown in Fig. 12.10. Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.

 

Answers (1)

It is subject to constraints

x+2 y \geq 4, x+y \geq 3, x \geq 0, y \geq 0

Now let us convert the given inequalities into equation
We obtain the following equation

\\ x+2 y \geq 4 \\ \Rightarrow x+2 y=4 \\ x+y \geq 3 \\ \Rightarrow x+y=3 \\ y \geq 0 \\ \Rightarrow x=0 \\ y \geq 0\\ \Rightarrow y=0

The region representing x + 2y $ \geq $ 4: \\

The line that is x + 2y=4 meets the other coordinate axes (4,0) and (0,2) respectively. Once we join these points we get the result x + 2y=4. After joining the lines, it is then clarified that it does not satisfy the inequation x + 2y $ \geq $ 4: \\. So, then the region that does not contain the origin then represents the set of solutions.

The region that is represented by x +y $ \geq $ 3:\\

The other line that is x +y=3  meets the coordinate axes (3,0) and (0,3) respectively. When we join these points we obtain the line x +y=3. It is then cleared that it does not satisfy the inequation. Therefore, the region that does not contain the origin further represents the solution set.

Region represented by $x \geq 0$\ and\ $y \geq 0$ is first quadrant, since every point in the first quadrant satisfies these inequations
The graph of these equations is given.

The shaded region ABC is the feasible region is bounded, so, minimum value will occur at a corner point of the feasible region.
Corner Points are A(0,3), B(2,1), and C(4,0)  .
When we see the region, we can see that it is unbounded, and 3 is not the minimum value of Z.

When we decide this issue, we can graph the inequality 4x + y < 3 and then further check that whether the resulting does has the open half no point in common, otherwise Z has no minimum value.

The graph that is shown above, we can see that it is clear that there is no point in common with the region and therefore, Z has a minimum value of 3 at (0, 3).

Henceforth, the minimum value of Z is 3 at the point (0, 3).

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