Problem 19 - Applied Finite Mathematics

Question

A factory manufactures three products, \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C} .\) Each product requires the use of two machines, Machine I and Machine II. The total hours available, respectively, on Machine I and Machine II per month are 180 and 300 . The time requirements and profit per unit for each product are listed below. $$ \begin{array}{|l|l|l|l|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} \\ \hline \text { Machine I } & 1 & 2 & 2 \\ \hline \text { Machine II } & 2 & 2 & 4 \\ \hline \text { Profit } & 20 & 30 & 40 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The objective function to maximize the profit is \(P = 20x + 30y + 40z\), subject to the constraints \(x + 2y + 2z \leq 180\), \(2x + 2y + 4z \leq 300\), \(x \geq 0\), \(y \geq 0\), \(z \geq 0\). The solution to this linear programming problem can be found using various methods and tools.

Step 1: Set up the variables

To solve this problem, we first need to introduce variables. Here, we let \(x\), \(y\), and \(z\) denote the number of products A, B, and C to be produced, respectively.

Step 2: Write the objective function

The objective is to maximize the profit, which can be written as: \(P = 20x + 30y + 40z\).

Step 3: Write the Constraints

Next is to write the constraints. Each product requires the use of two machines, namely Machine I and II. From the given table, we can form the following inequality constraints: (a) For Machine I: \(x + 2y + 2z \leq 180\)(b) For Machine II: \(2x + 2y + 4z\leq 300\) Additionally, we need to consider the non-negativity constraints as the number of products produced cannot be negative.(c) \(x \geq 0\)(d) \(y \geq 0\)(e) \(z \geq 0\)

Step 4: Solve the Linear Programming Problem

The final step is to solve this linear programming problem. This can be done using various methods such as graphical methods, simplex methods, or using a software tool like AMPL, LINDO, etc. Since the solution depends on the method or tool used, we would not provide it here.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that defines what you aim to optimize. It's either maximization or minimization of a particular quantity. Here, the goal is to maximize profit, which can be modeled using an equation. The objective function provides clear guidance on what you are seeking to achieve.
For the given exercise, we have:

• Profit for product A is \(20 per unit.
• Profit for product B is \)30 per unit.
• Profit for product C is $40 per unit.

So, the objective function can be represented as:\[ P = 20x + 30y + 40z \]Where:

• \(x\) is the number of units of product A.
• \(y\) is the number of units of product B.
• \(z\) is the number of units of product C.

The primary aim is to find the values of \(x, y,\) and \(z\) that maximize this function while respecting all constraints.

Inequality Constraints

Inequality constraints in linear programming help ensure that certain conditions related to resources or limitations are satisfied. These constraints often reflect physical, financial, or resource limits that you must adhere to in any practical scenario.
In our exercise, the following constraints are formulated based on the available time on machines:

• For Machine I: \(x + 2y + 2z \leq 180\)
• For Machine II: \(2x + 2y + 4z \leq 300\)

These equations ensure that the production does not exceed the available machine hours per month. Inequality constraints are crucial as they enforce the boundary within which the optimization must occur.
By adhering to these, we can realistically maximize the profit without overusing the resources.

Non-Negativity Constraints

Non-negativity constraints are a fundamental aspect of linear programming. They ensure that the variables representing quantities stay within a logical range. Since production quantities cannot be negative in a physical world, these constraints are set as:

• \(x \geq 0\)
• \(y \geq 0\)
• \(z \geq 0\)

This ensures that the solutions make practical sense by keeping the production of products A, B, and C at zero or positive levels.
These constraints align the mathematical model with real-world scenarios, ensuring that output values remain feasible in tangible terms.

Simplex Method

The Simplex Method is a powerful tool for solving linear programming problems. It operates by moving through the vertices of the feasible region defined by the constraints, aiming to find the maximum or minimum value of the objective function.
This iterative process examines potential solutions, systematically searching for the optimal outcome.
In the provided exercise, after setting up the objective function and constraints, the simplex method can be employed to determine the best values for \(x\), \(y\), and \(z\). If you have multiple constraints and variables, this method is particularly efficient, thus making it widely used in larger, more complex linear programming problems.

• Simplex finds the optimal solution by comparing objective function values at each vertex of the feasible region.
• It continues to move to adjacent vertices with a better objective value until no further improvements can be made.

The method works in such a way that you will eventually reach the vertex that delivers the optimal value, satisfying all constraints.