2. The figure shows the graphical model of a linear program. The large numbers o

Question

2. The figure shows the graphical model of a linear program. The large numbers on the right (1,2, and 3) indicate the constraints The feasible region is shown in white, and the infeasible region is shaded. The vari- ables are restricted to nonnegative values. The small numbers (0, 1, 2, and 3) indi- cate four feasible corner points: 0, 1,2,3 Three objective functions are under con- sideration, as indicated by the three lines labeled A, B, and C. The arrows represent the directions of increasing objective func- tion. Objective B is parallel to constraint 3. In each case, specify the location of the optimal solution. If there is more than one optimal solution, characterize all of them. (a) Maximize A (b) Maximize B (c) Maximize C (d) Minimize A (e) Minimize B (0) Minimize C (g) Drop x, 2 0 and minimize C

Explanation / Answer

a

To maximize A

We will keep moving in direction where A increases until we have at least one point in the feasible reason which is intersection of feasible point and this line. Therefore, in this way we will reach point 2 at which intersection of this line and feasible reason will became a single point. So we cannot move ahead so point 2 will be optimal

b

To maximize B

We will keep moving in direction where B increases until we have at least one point in the feasible reason and on the line. Therefore, in this way, we can maximize until we reach line 3 So whole line 3 will be its optimal solution which will be same as value at point 3

c

To maximize C

We will keep moving in direction where C increases until we have at least one point in the feasible reason and on the line. Therefore, in this way, we can maximize it as long as we want to infinity so no point is optimal for this.

d

To minimize A

We will keep moving in direction where A decreases until we have at least one point in the intersection of feasible reason and this line. Therefore, in this way, we can minimize A as much as we want so it cannot be optimize or no optimal point.

e

To minimize B

We will keep moving in direction where B decreases until we have at least one point in the intersection of feasible reason and this line. Therefore, in this way, we can minimize B as much as we want so it cannot be optimize or no optimal point.

f

To minimize C

We will keep moving in direction where C decreases until we have at least one point in the intersection of feasible reason and this line. Therefore, in this way, we can minimize C until we reach zero so 0 is feasible point.

g

if we drop x1>=0

then C can be minimize further to a point where constrraint 1 and x2 greater than equal to 0 intersect. In same way as all above.

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