A manufacturer has a maximum of 240, 360, and 180 kilograms of wood, plastic and steel available. The company produces two products, A and B. Each unit of A requires 1, 3 and 2 kilograms of wood, plastic and steel respectively; each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively, and each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively. The profit per unit of A and B is $4.00 and $6.00 respectively.
Identify all constraints.
Identify all applicable corner points of the feasibility region.
How many units of A and B should be manufactured in order to maximize profits? What would the maximum profit be?
Constraints:
Wood: x + 3y ≤ 240
Plastic: 3x + 4y ≤ 360
Steel: 2x + y ≤ 180
Corner Points:
(0, 0), (80, 0), (0, 90), (60, 45)
The maximum profit would be achieved by producing 60 units of A and 45 units of B, for a total profit of $420.
To solve this problem, let's follow these steps:
Step 1: Identify all constraints:
The constraints can be determined by the availability of wood, plastic, and steel.
From the given information, we have:
Wood constraint: 1A + 3B ≤ 240 kg
Plastic constraint: 3A + 4B ≤ 360 kg
Steel constraint: 2A + B ≤ 180 kg
Step 2: Identify all applicable corner points of the feasibility region:
To find the corner points, we need to find the intersections of the lines formed by the constraints. We can solve the equations pairwise to find the corner points:
1A + 3B = 240 kg, 3A + 4B = 360 kg
Wood and plastic intersection: A = 120, B = 40
3A + 4B = 360 kg, 2A + B = 180 kg
Plastic and steel intersection: A = 90, B = 60
2A + B = 180 kg, 1A + 3B = 240 kg
Steel and wood intersection: A = 60, B = 60
So, the corner points are: (120, 40), (90, 60), and (60, 60).
Step 3: Determine the units of A and B to maximize profits:
To maximize profits, we need to find the point where the profit function reaches its highest value. The profit function is given by:
Profit = 4A + 6B
Now, we can substitute the values of A and B from the corner points to calculate the profits for each point:
For (120, 40):
Profit = 4(120) + 6(40) = $760
For (90, 60):
Profit = 4(90) + 6(60) = $660
For (60, 60):
Profit = 4(60) + 6(60) = $600
Hence, the maximum profit is $760, and it can be achieved by manufacturing 120 units of A and 40 units of B.
To solve this problem, we need to identify the constraints and find the feasible region before determining the maximum profit and the corresponding units of products A and B.
1. Identify all constraints:
From the given information, we can identify the following constraints:
- Wood constraint: The maximum amount of wood available is 240 kilograms.
- Plastic constraint: The maximum amount of plastic available is 360 kilograms.
- Steel constraint: The maximum amount of steel available is 180 kilograms.
- Unit production constraints: The number of units of product A and B must be greater than or equal to zero.
2. Identify all applicable corner points of the feasibility region:
To find the corner points of the feasibility region, we need to solve the equations formed by the constraints. We can use a graph or linear programming techniques like the simplex method to solve this. However, since there are only three constraints, we can use a graphical method.
By graphing the feasible region determined by the constraints, we can identify the corner points where the constraints intersect. Each corner point represents a combination of units of A and B that satisfies all constraints simultaneously.
3. Determine the maximum profit and units of A and B:
Once we have identified the corner points, we need to evaluate the profit at each point to determine the maximum profit.
For each corner point, calculate the profit obtained by multiplying the number of units of A and B with their respective profit per unit. Compare the profits at each corner point and select the combination that yields the maximum profit.
The maximum profit will be the highest profit obtained among all the corner points.
Note: Without specific values for the profit per unit and the amount of wood, plastic, and steel required for products A and B, I cannot provide the exact numerical solution. However, by following the steps outlined above, you should be able to determine the maximum profit and the corresponding units of products A and B given the specific numerical values provided in the problem.