Question

Ahmet is a farmer who wants to maximize his crop yield by choosing between two types He needs to produce at least 72 kg of com and 80 kg of wheat. Fertilizer A provides 4 kg of corn and 5 kg of wheat per bag, while Fertilizer B provides 6 kg of corn and 4 kg of wheat per bag. 1bag of fertilizer A costs $10, whereas one bag of fertilizer B costs $20. However, Ahmet needs to limit his use of Fertilizer B to no more than 10 bags

a. Formulate an LP which helps Ahmet to minimize the total cost of using fertilizers. due to environmental regulations. Let the first decision variable be related to fertilizer A and the second decision variable to fertilizer B
b. Solve the LP you build graphically.

Answer 1

To minimize the cost for Ahmet's fertilizer use, we set up an LP with two variables, x and y, representing the number of bags of Fertilizer A and B, respectively. The objective function is to minimize Z = 10x + 20y, subject to the constraints based on corn and wheat production and the limit on Fertilizer B. The problem can be solved graphically by finding the optimal corner point in the feasible region defined by the constraints.

Step-by-step explanation:

To help Ahmet minimize the total cost of using fertilizers while meeting his production targets, we can set up a Linear Programming (LP) problem with two decision variables: the number of bags of Fertilizer A (let's call this x) and the number of bags of Fertilizer B (let's call this y).

The objective function, which we want to minimize, represents the total cost of the fertilizers:

Minimize Z = 10x + 20y

The constraints based on the required production are:

• 
  
• 4x + 6y ≥ 72 (for corn)
• 
  
• 5x + 4y ≥ 80 (for wheat)
• 
  
• y ≤ 10 (limit on Fertilizer B due to environmental regulations)

We also have the non-negativity constraints:

• 
  
• x ≥ 0
• 
  
• y ≥ 0

To graphically solve the LP, we plot the constraints on a coordinate system where x is on the horizontal axis and y is on the vertical axis. The feasible region is determined by the area that satisfies all constraints, and the optimal solution lies at a corner point of this region. By examining the cost function at these corner points, we find the point that minimizes the total cost.

Answer 2

a. The linear programming (LP) formulation to help Ahmet minimize the total cost of using fertilizers is as follows:

`\[ \text{Minimize } Z = 10x + 20y \]`

subject to the constraints:

`\[ 4x + 6y \geq 72 \]`

`\[ 5x + 4y \geq 80 \]`

`\[ y \leq 10 \]`

`\[ x, y \geq 0 \]`

b. The graphical solution involves plotting the feasible region defined by the system of inequalities and finding the corner point that minimizes the total cost. Solving the equations, we find the corner point (x, y) = (10, 10), where the minimum cost Z is $300.

Step-by-step explanation:

a. The objective function Z represents the total cost of using fertilizers, with decision variables x and y corresponding to the number of bags of Fertilizer A and fertilizer B, respectively. The constraints ensure that Ahmet produces at least 72 kg of corn and 80 kg of wheat while adhering to environmental regulations limiting the use of Fertilizer B to no more than 10 bags.

b. Graphically, we plot the feasible region determined by the system of inequalities on a graph. The corner point (10, 10) satisfies all constraints and minimizes the total cost Z. Substituting these values back into the objective function, we find that the minimum cost is $300, achieved when Ahmet purchases 10 bags of Fertilizer A and 10 bags of Fertilizer B. The graphical solution provides a visual representation of the optimization problem, helping Ahmet make cost-effective decisions in fertilizer selection.