Problem 41 - Algebra for College Students

Question

Products \(\mathrm{A}\) and \(\mathrm{B}\) are produced by a company according to the following production information. (a) To produce one unit of product A requires 1 hour of working time on machine I, 2 hours on machine II, and 1 hour on machine III. (b) To produce one unit of product B requires 1 hour of working time on machine I, 1 hour on machine II, and 3 hours on machine III. (c) Machine I is available for no more than 40 hours per week, machine II for no more than 40 hours per week, and machine III for no more than 60 hours per week. (d) Product \(A\) can be sold at a profit of \(\$ 2.75\) per unit and product B at a profit of \(\$ 3.50\) per unit. How many units each of product \(A\) and product \(B\) should be produced per week to maximize profit?

Step-by-Step Solution

Answer

Produce 10 units of product A and 30 units of product B per week to maximize profit.

Step 1: Define the Variables

Let \( x \) represent the number of units of product A produced per week and \( y \) the number of units of product B produced per week. These variables will help us formulate the objective function and constraints based on our production needs.

Step 2: Set Up the Objective Function

The objective is to maximize profit, given by the expression: \( 2.75x + 3.50y \), where \( 2.75 \) is the profit per unit of product A and \( 3.50 \) is the profit per unit of product B.

Step 3: Establish the Constraints

We need to convert the production limitations into inequalities with respect to the machines' working hours. The constraints are as follows: 1. Machine I: \( x + y \leq 40 \)2. Machine II: \( 2x + y \leq 40 \)3. Machine III: \( x + 3y \leq 60 \).4. Additionally, \( x \geq 0 \) and \( y \geq 0 \) due to non-negativity of production.

Step 4: Graph the Feasible Region

Draw the lines for each inequality on a graph. The feasible region is the area where all these inequalities overlap. Identify the coordinates of the vertices of the feasible region.

Step 5: Evaluate the Objective Function at Each Vertex

Substitute the coordinates of each vertex of the feasible region into the objective function \( 2.75x + 3.50y \) to calculate the profit. The vertex with the highest profit will indicate the optimal production quantities.

Step 6: Select the Optimal Solution

After evaluating the vertices, identify which combination of \( x \) and \( y \) gives the highest profit. That combination will be the optimal number of units for products A and B to produce.

Key Concepts

Objective Function

In linear programming, the **objective function** represents the goal of the problem, usually to maximize or minimize some quantity. In our exercise, the objective is to maximize profit. This is achieved by finding the best combination of products A and B that brings in the highest return.
The objective function, in this case, is expressed as: \( 2.75x + 3.50y \).

• Here, \( x \) stands for the units of product A, each unit providing a profit of \\(2.75.
• \( y \) represents the units of product B, which gives \\)3.50 of profit per unit.

The goal is to determine values of \( x \) and \( y \) that maximize this profit equation, considering the constraints set by the resources available.

Constraints

**Constraints** are restrictions or limitations on the decision variables in a linear programming problem. They define the conditions that must be met for a solution to be considered feasible.
In our scenario, constraints originate from the limited working hours of machines:

• Machine I: The sum of hours worked on both products should not exceed 40 hours. This can be expressed as: \( x + y \leq 40 \).
• Machine II: Similarly, the total hours must not be more than 40. The constraint is \( 2x + y \leq 40 \).
• Machine III: This has a higher limit of 60 hours, represented by \( x + 3y \leq 60 \).
• Another constraint stems from reality: you can't produce a negative quantity of products. Hence, \( x \geq 0 \) and \( y \geq 0 \).

The constraints limit the possible values of \( x \) and \( y \) that we can choose to ensure resources are used effectively.

Feasible Region

The **feasible region** in a linear programming problem is the set of all possible points that satisfy all the constraints. Think of it as a boundary within which any point represents a potential solution.
To determine this region, you'll need to graph the inequalities. Where the lines

• \( x + y = 40 \)
• \( 2x + y = 40 \)
• \( x + 3y = 60 \)

intersect or overlap on the graph, they form a polygon, which is your feasible region. This polygon contains all the points where both products can be produced without exceeding any machine's capacity.
In this region, any point on the graph represents a valid production level of products A and B. However, not every point will maximize profit.

Vertex Evaluation

**Vertex evaluation** is a crucial step in linear programming when trying to find the optimal solution. The optimal solution often occurs at a vertex of the feasible region, hence why it needs thorough evaluation.
Once your feasible region is graphed, identify the vertices—the corner points of this polygon. Let's say these are points A, B, C, and D. You then evaluate the objective function \( 2.75x + 3.50y \) at each of these vertices. It will show how much profit each vertex yields:

• Substitute each vertex into the profit formula.
• Calculate the profit for each vertex.

The vertex producing the highest profit indicates the most profitable production level of products A and B which satisfies all machine time constraints.