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How to use system of linear equation in solving production planning in a manufacturing company?

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Question: How to use system of linear equation in solving production planning in a manufacturing company?
Solution:

Imagine you work for a manufacturing company that produces two types of products: Product A and Product B. The company has limited resources for production, including labor hours and raw materials. Your task is to determine how many units of each product should be produced to maximize profit while staying within the resource constraints.

Here are the relevant details:

- Product A requires 2 hours of labor and 3 units of raw material per unit produced.
- Product B requires 1 hour of labor and 4 units of raw material per unit produced.
- The company has a total of 1000 labor hours and 2000 units of raw material available for production.
- The profit per unit of Product A is $10, and the profit per unit of Product B is $15.

To solve this problem, we can set up a system of equations:

Let (x) be the number of units of Product A produced, and (y) be the number of units of Product B produced.

The constraints are:
1. Labor hours constraint: (2x + 1y <= 1000)
2. Raw material constraint: (3x + 4y <= 2000)

The objective is to maximize profit: (Z = 10x + 15y)

Now, we have the following system of equations:

2x + y <= 1000
3x + 4y <= 2000
Z = 10x + 15y

We'll solve this system using linear programming techniques.

The solution involves finding the corner points of the feasible region formed by the constraints and evaluating the objective function at each corner point to determine the optimal solution.

To find the solution, let's solve the system of equations:

{2x+y≤10003x+4y≤2000Z=10x+15yegin{cases} 2x + y leq 1000 \ 3x + 4y leq 2000 \ Z = 10x + 15y end{cases}

We'll solve this system using linear programming techniques. The solution involves finding the corner points of the feasible region formed by the constraints and evaluating the objective function at each corner point to determine the optimal solution.

First, let's find the corner points:

  1. Set 2x+y=10002x + y = 1000, which gives us y=1000−2xy = 1000 - 2x. We'll substitute this into the second constraint. 3x+4(1000−2x)≤20003x + 4(1000 - 2x) leq 2000
    Solving for xx, we get x=250x = 250.
    Substituting x=250x = 250 into the equation for yy, we get y=500y = 500.
    So, the first corner point is (250, 500).

  2. Set 3x+4y=20003x + 4y = 2000, which gives us y=2000−3x4y = rac{2000 - 3x}{4}. We'll substitute this into the first constraint. 2x+2000−3x4≤10002x + rac{2000 - 3x}{4} leq 1000
    Solving for xx, we get x=400x = 400.
    Substituting x=400x = 400 into the equation for yy, we get y=200y = 200.
    So, the second corner point is (400, 200).

Now, we'll evaluate the objective function Z=10x+15yZ = 10x + 15y at each corner point:

  1. At (250, 500): Z=10(250)+15(500)=2500+7500=10000Z = 10(250) + 15(500) = 2500 + 7500 = 10000
  2. At (400, 200): Z=10(400)+15(200)=4000+3000=7000Z = 10(400) + 15(200) = 4000 + 3000 = 7000

Since we want to maximize profit, the optimal solution occurs at (250, 500) with a profit of $10,000. Therefore, the manufacturing company should produce 250 units of Product A and 500 units of Product B to maximize profit while satisfying the resource constraints.

Calculate Box

Result: The optimal solution occurs at (250, 500) with a profit of $10000Notes: For Form - Javascript solving function applied: (eq) => eq.px * eq.x + eq.py * eq.y
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