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Chapter 4: Problem 51

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

Short Answer

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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 by step solution

01

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.
02

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.
03

Resource Constraints

The factory has limited resources available for the production of these products. These resources are represented by the constraints: 1. 20x + 40y 鈮� 1000: Resource 1 constraint. 2. 30x + 20y 鈮� 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.
04

Non-negativity Constraints

The given variables must be non-negative: - x 鈮� 0: The number of units of product A produced must be non-negative. - y 鈮� 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.

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Key Concepts

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

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鈥攅ssentially 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.

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Most popular questions from this chapter

Investments Your friend's portfolio manager has suggested two energy stocks: Exxon Mobil (XOM) and British Petroleum (BP). XOM shares cost \(\$ 80\), yield \(2 \%\) in dividends, and have a risk index of \(2.5\). BP shares cost \(\$ 50\), yield \(7 \%\) in dividends, and have a risk index of \(4.5 .{ }^{19}\) Your friend has up to \(\$ 40,000\) to invest, and would like to earn at least \(\$ 1,400\) in dividends. How many shares (to the nearest tenth of a unit) of each stock should she purchase to meet your requirements and minimize the total risk index for your portfolio? What is the minimum total risk index?

Enormous State University's Business School is buying computers. The school has two models from which to choose, the Pomegranate and the iZac. Each Pomegranate comes with \(400 \mathrm{MB}\) of memory and \(80 \mathrm{~GB}\) of disk space; each iZac has \(300 \mathrm{MB}\) of memory and \(100 \mathrm{~GB}\) of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least \(48,000 \mathrm{MB}\) of memory and at least \(12,800 \mathrm{~GB}\) of disk space. If the Pomegranate and the iZac cost \(\$ 2,000\) each, how many of each should the school buy to keep the cost as low as possible? HINT [See Example 4.]

f Purchasing Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for \(\$ 500,000\). Admiral Motors is offering 5 small, 10 medium, and 5 large cars for \(\$ 400,000\). Chrysalis is offering 10 small, 5 medium, and 5 large cars for \(\$ 300,000\). Federal would like to buy at least 550 small cars, at least 500 medium cars, and at least 550 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible? What will be the total cost?

Your company鈥檚 new portable phone/music player/PDA/bottle washer, the RunMan, will compete against the established market leader, the iNod, in a saturated market. (Thus, for each device you sell, one fewer iNod is sold.) You are planning to launch the RunMan with a traveling road show, concentrating on two cities, New York and Boston. The makers of the iNod will do the same to try to maintain their sales. If, on a given day, you both go to New York, you will lose 1,000 units in sales to the iNod. If you both go to Boston, you will lose 750 units in sales. On the other hand, if you go to New York and your competitor to Boston, you will gain 1,500 units in sales from them. If you

Give an example of a standard minimization problem whose dual is not a standard maximization problem. How would you go about solving your problem?
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