Problem 51 - Finite Mathematics and Applied Calculus

Question

Create an interesting scenario leading to the following linear programming problem: Maximize \(\quad p=10 x+10 y\) \(\begin{aligned} \text { subject to } & 20 x+40 y \leq 1,000 \\ & 30 x+20 y \leq 1,200 \\ & x \geq 0, y \geq 0 \end{aligned}\)

Step-by-Step Solution

Answer

A factory produces two types of products: A and B, generating a profit of \(10 per unit for both products. The goal is to maximize the profit while considering resource constraints for production. Let x represent the number of units of product A and y represent the number of units of product B produced. Product A requires 20 units of resource 1 and 30 units of resource 2 per unit produced, whereas product B requires 40 units of resource 1 and 20 units of resource 2 per unit produced. The factory has 1000 units of resource 1 and 1200 units of resource 2 available. The linear programming problem is to find the number of units of products A and B (x and y) to produce in order to maximize the total profit (p) while adhering to the resource constraints and non-negativity constraints.

Step 1: Scenario Introduction

Let's imagine a factory that produces two types of products: A and B. The factory generates a profit of \(10 per unit of product A and \)10 per unit of product B. The factory's goal is to maximize its profit while taking into account resource constraints for production.

Step 2: Defining Variables

Let's define the variables: - x: The number of units of product A produced. - y: The number of units of product B produced. - p: The total profit generated from the production of both products. The given problem states Maximize \(p = 10x + 10y\), where x and y are the quantities of products A and B, respectively.

Step 3: Resource Constraints

The factory has limited resources available for the production of these products. These resources are represented by the constraints: 1. 20x + 40y ≤ 1000: Resource 1 constraint. 2. 30x + 20y ≤ 1200: Resource 2 constraint. The constraints state that product A requires 20 units of resource 1 and 30 units of resource 2 per unit produced, whereas product B requires 40 units of resource 1 and 20 units of resource 2 per unit produced. The factory has 1000 units of resource 1 and 1200 units of resource 2 available.

Step 4: Non-negativity Constraints

The given variables must be non-negative: - x ≥ 0: The number of units of product A produced must be non-negative. - y ≥ 0: The number of units of product B produced must be non-negative. This represents that the factory cannot produce negative quantities of products A and B. Now, the given linear programming problem can be interpreted as finding the number of units of products A and B (x and y) to produce in order to maximize the total profit (p) while adhering to the resource constraints and non-negativity constraints.

Key Concepts

Profit Maximization

Profit Maximization is the main objective in many linear programming problems, especially when we are considering the production of goods or services. Here, the factory's goal is to maximize profit from producing two products, A and B.
The profit function in this scenario is represented by the expression: \( p = 10x + 10y \). This means each unit of product A and product B contributes $10 to the total profit. Therefore, the total profit is directly proportional to the quantities of \( x \) and \( y \), which represent the units produced for products A and B, respectively.

• When formulating a profit maximization problem, define a clear profit function to be maximized.
• Ensure all factors contributing to profit are included in the mathematical expression.

This function becomes the keystone of the linear programming model and guides decision-making, ensuring that resources are allocated effectively to maximize returns.

Resource Constraints

Resource constraints are crucial as they dictate the boundaries within which the profit must be maximized. In our problem, we have two resource constraints that affect production:

• \(20x + 40y \leq 1000\): This represents the limitation posed by Resource 1.
• \(30x + 20y \leq 1200\): This is the constraint due to Resource 2.

Each constraint reflects the maximum availability of resources in the factory. For instance, product A uses 20 units of Resource 1 for each unit produced, while product B uses 40. Similarly, Resource 2 is consumed at 30 units per product A and 20 units per product B.
These constraints shape the feasible region—essentially the set of all possible production combinations of products A and B that can be produced without exceeding resource limitations.
Understanding these constraints is vital:

• They ensure the solutions are practical and economically feasible.
• They protect against over-utilization of resources, which could lead to increased costs or operational bottlenecks.

Non-Negativity Conditions

Non-negativity conditions are fundamental in linear programming. They ensure that solutions are realistic by restricting variables such as \( x \) and \( y \) to non-negative values. In our scenario, this means:

• \(x \geq 0\): You cannot produce a negative quantity of product A.
• \(y \geq 0\): Similarly, production of product B cannot be negative.

These conditions maintain the integrity of the model by ensuring that all quantities produced are valid in a real-world context.
When developing linear programming models, it's crucial to incorporate these conditions early on:

• They prevent illogical or impossible solutions (like negative production levels).
• They simplify the problem by focusing only on feasible production levels.

Non-negativity conditions are relatively simple yet profoundly significant in creating solutions that can be effectively implemented.