A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 960, 1020, and 840, respectively. The time requirements (in hours per unit) and the profit per unit for each product are as follows.
Product
A	Product
B	Product
C
Dept. I	2	1	2
Dept. II	3	1	2
Dept. II	2	2	1
Profit	$18	$12	$15
If management decides that the number of units of Product B manufactured must equal or exceed the number of units of products A and C manufactured, how many units of each product should the company produce to maximize its profit?
product A

product B

product C

What is the maximum profit (in dollars)?
$

To solve this problem, let's define the variables and set up a linear programming model:

### Variables
Let:
- $ x $: Number of units of Product A to produce.
- $ y $: Number of units of Product B to produce.
- $ z $: Number of units of Product C to produce.

### Objective Function
The company wants to maximize profit. The profit per unit for each product is:
- Product A: $18
- Product B: $12
- Product C: $15

The objective function, therefore, is:
\[
\text{Maximize } P = 18x + 12y + 15z
\]

### Constraints
The time requirements in each department set constraints on the number of products that can be produced:

1. **Department I (960 hours available):**
   \[
   2x + y + 2z \leq 960
   \]

2. **Department II (1020 hours available):**
   \[
   3x + y + 2z \leq 1020
   \]

3. **Department III (840 hours available):**
   \[
   2x + 2y + z \leq 840
   \]

4. **Production Constraint for Product B:**
   Management requires that the number of units of Product B produced must be greater than or equal to the number of units of Products A and C combined:
   \[
   y \geq x + z
   \]

5. **Non-negativity Constraints:**
   \[
   x, y, z \geq 0
   \]

### Solution Method
This problem can be solved using linear programming techniques, such as the Simplex method or a computational tool to find the optimal values of $ x $, $ y $, and $ z $ that maximize $ P $.

I'll compute the optimal solution using these constraints to determine the values of $ x $, $ y $, and $ z $, along with the maximum profit.The optimal solution for maximizing profit, given the constraints, is as follows:

- **Product A**: Produce 60 units
- **Product B**: Produce 200 units
- **Product C**: Produce 320 units

The maximum profit achievable under these conditions is **$8,280**.

Would you like more details on the solution steps or have any questions?

Here are some related questions to deepen understanding:
1. What changes in constraints would affect the maximum profit?
2. How would the result differ if the profit per unit for Product B increased?
3. What if there was an additional constraint on the maximum units of Product C?
4. How would a decrease in available hours for Department II impact production?
5. Could this method work for cases with more than three products or departments?

**Tip:** In linear programming, always double-check constraint inequalities to ensure they accurately reflect real-world limitations.

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 960, 1020, and 840, respectively. The time requirements (in hours per unit) and the profit per unit for each product are as follows. Product A: Department I 2 hours, Department II 3 hours, Department III 2 hours, Profit $18. Product B: Department I 1 hour, Department II 1 hour, Department III 2 hours, Profit $12. Product C: Department I 2 hours, Department II 2 hours, Department III 1 hour, Profit $15. If management decides that the number of units of Product B manufactured must equal or exceed the number of units of Products A and C manufactured, how many units of each product should the company produce to maximize its profit? What is the maximum profit (in dollars)?

Learn how to use linear programming to maximize profit in product manufacturing. This problem involves optimizing the production of Products A, B, and C across three departments with time constraints and profit maximization. The solution calculates the optimal number of units for each product to achieve a maximum profit of $8,280.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation