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A factory manufactures two products, \(A\) and \(B\). Each product requires the use of three machines, Machine I, Machine II, and Machine III. The time requirements and total hours available on each machine are listed below. $$ \begin{array}{|l|l|l|l|} \hline & \text { Machine I } & \text { Machine II } & \text { Machine III } \\\ \hline \text { Product A } & 1 & 2 & 4 \\ \hline \text { Product B } & 2 & 2 & 2 \\ \hline \text { Total hours } & 70 & 90 & 160 \\ \hline \end{array} $$ If product A generates a profit of $$\$ 60$$ per unit and product \(B\) a profit of $$\$ 50$$ per unit, how many units of each product should be manufactured to maximize profit, and what is the maximum profit?

Step by step solution

01

Define the Variables

Let's define two variables for the number of units of products A and B to be manufactured, namely \( x \) for product A and \( y \) for product B.

02

Formulate the Constraint Equations

Next, we need to formulate the constraint equations based on the information given in the table. There are three machines with limited weekly working hours which should not be exceeded during the production of products. This gives us three inequalities: \n1. Machine I: \( x + 2y \leq 70 \) \n2. Machine II: \( 2x + 2y \leq 90 \) \n3. Machine III: \( 4x + 2y \leq 160 \)

03

Formulate the Objective Function

Given that product \(A\) generates a profit of \$60 per unit and product \(B\) generates a profit of \$50 per unit, the total profit, \(P\), can be written as: \(P = 60x + 50y\). Our goal is to maximize this profit \(P\).

04

Find the Feasible Region

The feasible region is the area of the graph where all constraints are satisfied simultaneously. In this case, we plot the inequalities on a graph (preferably done graphically if possible) and identify the region that satisfies all three inequalities.

05

Determine the Optimal Solution

The optimal solution lies at the vertices of the feasible region. Evaluate the objective function for each of these vertices to find the one that produces the maximum value, and this will give us our maximum profit and the number of units to be produced for each product to achieve this profit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Objective Function

When tackling any linear programming problem, one of the first tasks is to clearly define the objective function. This is a mathematical expression that symbolizes what we're trying to achieve. In our exercise, the objective is to maximize profit for a factory producing two products, A and B. The profit per unit for these products is given, with product A yielding a profit of \(60 and product B yielding \)50.

Therefore, we express our total profit, which we seek to maximize, as the equation:

• \( P = 60x + 50y \)

Where:

• \( x \) is the quantity of product A manufactured.
• \( y \) is the quantity of product B manufactured.

This equation will guide us through identifying how many units of each product will result in the maximum possible profit.

Constraint Equations

In any real-world scenario, resources are limited, which translates into constraints on how many products can be produced. For our problem, these constraints are represented by the constraints on the available hours of the three machines that are used.

• Machine I: \( x + 2y \leq 70 \)
• Machine II: \( 2x + 2y \leq 90 \)
• Machine III: \( 4x + 2y \leq 160 \)

These equations capture the fact that each machine has a maximum number of hours it can operate. Here, each inequality signifies a limitation that the production must respect.

These constraints are crucial as they help shape the production capabilities within which the factory must stay.

Feasible Region

The feasible region in linear programming represents all possible solutions that satisfy all the constraints simultaneously. To visualize this for our problem, one would graph each of the constraint inequalities:

• \( x + 2y \leq 70 \)
• \( 2x + 2y \leq 90 \)
• \( 4x + 2y \leq 160 \)

The intersection of these constraints is plotted on a graph, and the region where they overlap indicates feasible solutions — combinations of \( x \) and \( y \) — which are possible under the given restrictions.

The feasible region is essential because it visually shows all potential production combinations that comply with the machine restrictions, ensuring productive yet efficient manufacturing.

Optimal Solution

Finding the optimal solution in linear programming involves evaluating the objective function at each vertex (corner point) of the feasible region. In linear programming, it's known that the maximum or minimum value of the objective function will occur at one of these vertices.

For this exercise, once the feasible region is defined through graphing, each vertex is checked:

• Calculate \( P = 60x + 50y \) at each vertex.
• Identify which vertex provides the highest value for \( P \).

This vertex corresponds to the optimal number of units of products A and B that should be manufactured to achieve the maximum profit.

Understanding the optimal solution is vital as it ensures the factory achieves the highest possible profit within the constraints of machine hours.