Max z = 30x+50y

Question

Max z = 30x+50y

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

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.

Answer 6

To determine the range of optimality for the coefficient of x, we need to find the range in which the optimal solution remains unchanged.

Let's start by calculating the objective function's value at the given optimal solution (x = 1.25, y = 0.625):

z = 30(1.25) + 50(0.625)
z = 37.5 + 31.25
z = 68.75

Since this is the maximum value of the objective function, the optimal solution is (1.25, 0.625) at z = 68.75.

To determine the range of optimality for the coefficient of x, we will increase the coefficient of x by a small amount (let's call it "ε") and calculate the new optimal solution.

Let the new coefficient of x be (30 + ε).

The new objective function becomes:
z = (30 + ε)x + 50y

Now, we need to solve the problem again with the updated objective function. Here are the constraints:

2x + 4y ≤ 5
9x + 8y ≥ 5
3x + 2y ≤ 5
x ≥ 0
y ≥ 0

By solving these equations, we will find the new optimal solution for the updated objective function.

Note: Calculating the new optimal solution involves solving a new linear programming problem, which might require a complex process. Due to the limitations of this text-based medium, I am unable to walk you through the entire calculation. However, I can provide you with this step-by-step explanation.

Once you calculate the new optimal solution, compare it to the original optimal solution (1.25, 0.625). If the new optimal solution remains the same, then the range of optimality for the coefficient of x is ε ≥ 0. If the optimal solution changes, then the range of optimality for the coefficient of x is ε > 0.

I suggest using software or a calculator specifically designed for solving linear programming problems to obtain the exact new optimal solution and determine the range of optimality for the coefficient of x accurately.

Answer 7

To determine the range of optimality for the coefficient of x, we need to analyze the sensitivity of the objective function to changes in the coefficient of x.

First, let's understand the current optimal solution at (1.25, 0.625). In this solution, the coefficient of x is 30. This means that the objective function value increases by 30 units for every unit increase in x.

To find the range of optimality for the coefficient of x, we can perform a sensitivity analysis by introducing a small change to the coefficient of x and observing how it affects the optimal solution.

Let's denote the coefficient of x as c. We can redefine the objective function as:

z' = 30c * x + 50y

Consider two cases:

1. Increasing the coefficient of x: Assume c = 30 + δ, where δ is a small positive value. By solving the linear programming problem again, we can find the new optimal solution. Let's denote it as (x', y').

Note the change in the objective function value:

z' = (30 + δ) * x' + 50y'

Compare the change in the objective function value with the original optimal solution. If the change is positive, it means the optimal solution has improved, indicating an increase in the range of optimality for the coefficient of x. If the change is negative, it means the optimal solution has worsened, suggesting a decrease in the range of optimality.

2. Decreasing the coefficient of x: Assume c = 30 - δ, where δ is a small positive value. Again, solve the linear programming problem to find the new optimal solution (x', y').

Compare the change in the objective function value:

z' = (30 - δ) * x' + 50y'

Analyze the change in the objective function value as we did in the previous case.

By observing the changes in the objective function value, we can determine the range of optimality for the coefficient of x. If the changes are consistently positive or negative across various values of δ, then the range of optimality can be determined accordingly.

It's important to note that these sensitivity analysis steps require solving the linear programming problem multiple times by adjusting the coefficient of x.