Max z = 30x+50y

Question

Subject to

2x+4y \le 5
9x + 8y \ge 5
3x+2y \le 5
x \ge 0
y \ge 0

The optimal solution is at (1.25,0.625) Determine the range of optimality for coefficient of x

Answer 1

To determine the range of optimality for the coefficient of x, we perform a sensitivity analysis. This involves calculating the reduced costs for the optimal solution and then changing the coefficient of x in the objective function to see how much it can change before affecting the optimal solution.

The reduced costs for the optimal solution are:

rc_x = 30 - 2/3y
rc_y = 50 - 5/3x

To determine the range of optimality for coefficient of x, we start by setting rc_x = 0 and solving for y:

0 = 30 - 2/3y
y = 45

This means that if the coefficient of x increases beyond its current value, the objective function will remain optimal as long as y stays below 45. To determine the upper limit on the coefficient of x, we set y = 45 in the constraint 2x+4y = 5 and solve for x:

2x + 4(45) = 5
x = -80

Since x cannot be negative, this means that the upper limit on the coefficient of x is 0, and any increase beyond its current value will result in an infeasible solution. Therefore, the range of optimality for the coefficient of x is:

0 <= x <= 1.25

Answer 2

Max z = 30x+50y
Subject to

2x+4y \le 5
3x+2y \le 5
x \ge 0
y \ge 0

The optimal solution is at (1.25,0.625)

Determine the range of optimality for coefficient of y :

[Final range ]

Answer 3

Similar to the previous question, to determine the range of optimality for the coefficient of y, we perform a sensitivity analysis. This involves calculating the reduced costs for the optimal solution and then changing the coefficient of y in the objective function to see how much it can change before affecting the optimal solution.

The reduced costs for the optimal solution are:

rc_x = 30 - 2/3y
rc_y = 50 - 5/3x

To determine the range of optimality for coefficient of y, we start by setting rc_y = 0 and solving for x:

0 = 50 - 5/3x
x = 30

This means that if the coefficient of y increases beyond its current value, the objective function will remain optimal as long as x stays below 30. To determine the upper limit on the coefficient of y, we set x = 30 in the constraint 2x+4y = 5 and solve for y:

2(30) + 4y = 5
y = -55/4

Since y cannot be negative, this means that the upper limit on the coefficient of y is 0, and any increase beyond its current value will result in an infeasible solution. Therefore, the range of optimality for the coefficient of y is:

0 <= y <= 0.625.

Answer 4

Aesop was a writer who was well known for a type of story writing in which his characters often learned a lesson or moral. These stories were known as fables. In addition to a lesson or moral, these fables usually provided a message to the audience about things that can be observed about human natura and life. These observations are known as theme. Analyze the main theme of the story, "The Shepherd's Boy and the Wolf."

Answer 5

The main theme of "The Shepherd's Boy and the Wolf" is the importance of honesty and the consequences of lying. The story follows a young shepherd who repeatedly tricks his fellow villagers by pretending that a wolf is attacking his flock. After he does this several times, the villagers stop believing him and refuse to help when a real wolf later attacks his flock. The shepherd learns that lying and deceiving others can have serious consequences, as it erodes trust and credibility. The fable is often interpreted as a warning against the dangers of making false accusations and spreading rumors, highlighting the importance of honesty and integrity in all human relationships.