/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 AntiFam, a hunger-relief organiz... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 16

AntiFam, a hunger-relief organization, has earmarked between $$\$ 2$$ and $$\$ 2.5$$ million (inclusive) for aid to two African countries, country \(\mathrm{A}\) and country B. Country \(\mathrm{A}\) is to receive between $$\$ 1$$ million and $$\$ 1.5$$ million (inclusive), and country B is to receive at least $$\$ 0.75$$ million. It has been estimated that each dollar spent in country A will yield an effective retum of $$\$ .60$$, whereas a dollar spent in country B will yield an effective return of $$\$ .80 .$$ How should the aid be allocated if the money is to be utilized most effectively according to these criteria? Hint: If \(x\) and \(y\) denote the amount of money to be given to country A and country B, respectively, then the objective function to be maximized is \(P=0.6 x+0.8 y\).

Short Answer

Expert verified
To allocate the aid most effectively, AntiFam should give: - \(1.5\) million to country A - \(1\) million to country B This allocation results in a total effective return of \(P=1.7\) million.

Step by step solution

01

Write down the constraints for the problem

First, let's write down the given constraints for the problem in terms of \(x\) and \(y\). The constraints for the total aid allocated, country A's aid, and country B's aid are given as follows: 1. Total aid allocated: \(2 \le x+y \le 2.5\) (in millions) 2. Aid to country A: \(1 \le x \le 1.5\) (in millions) 3. Aid to country B: \(y \ge 0.75\) (in millions) But we also have one more constraint which is the non-negativity constraint, as the amount of aid cannot be negative: 4. Non-negativity: \(x \ge 0,\ y \ge 0\)
02

Graph the constraint inequalities

To find the solution, we need to graph the constraint inequalities on the \(x-y\) plane and find the feasible region, which represents the possible combinations of aid allocations that will meet all the constraints. 1. Plot the constraint \(x+y = 2\) and \(x+y = 2.5\). The feasible region will be between these two parallel lines. 2. Plot the constraint \(x=1\) and \(x=1.5\). The feasible region will be between these two vertical lines. 3. Plot the constraint \(y=0.75\). The feasible region will be above this horizontal line. 4. Plot the constraint \(x=0\) and \(y=0\). The feasible region will be in the first quadrant. You will find that the feasible region is bounded by the lines \(x=1\), \(x=1.5\), \(y=0.75\), and \(x+y=2.5\).
03

Calculate the vertices of the feasible region

To find the allocation that maximizes the objective function P, we will evaluate the objective function's value at the vertices of the feasible region. The four vertices of the feasible region are: 1. Intersection of \(x=1\) and \(x+y = 2\): \((1,1)\) 2. Intersection of \(x=1\) and \(y = 0.75\): \((1,0.75)\) 3. Intersection of \(x=1.5\) and \(y = 0.75\): \((1.5,0.75)\) 4. Intersection of \(x=1.5\) and \(x+y = 2.5\): \((1.5,1)\)
04

Determine the maximum value of the objective function

Calculate the objective function, \(P=0.6x+0.8y\), at each vertex and find the maximum value. Here are the values of \(P\) at each vertex: 1. At \((1,1)\): \(P = 0.6(1) + 0.8(1) = 1.4\) 2. At \((1,0.75)\): \(P = 0.6(1) + 0.8(0.75) = 1.2\) 3. At \((1.5,0.75)\): \(P = 0.6(1.5) + 0.8(0.75) = 1.65\) 4. At \((1.5,1)\): \(P = 0.6(1.5) + 0.8(1) = 1.7\) The maximum value of \(P\) occurs at the vertex \((1.5,1)\), with a value of \(1.7\).
05

Conclusion

The aid should be allocated as follows to be utilized most effectively according to the estimated effective returns: - Country A: \(x = 1.5\) million - Country B: \(y = 1\) million With this allocation, the total effective return is \(P=1.7\) million.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Linear Programming
Linear programming is a mathematical approach used for finding the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. It is extensively used in business, economics, engineering, and military applications, among other fields. At the heart of this technique lies the objective to maximize or minimize a linear objective function, which represents some quantity we wish to optimize, such as profit, cost, or efficiency.

For example, in the exercise provided, AntiFam wants to maximize the return on their investment in two countries while adhering to certain financial constraints. By placing the aid amounts as variables and their respective returns as the coefficients in the objective function, linear programming can be used to determine the optimal allocation of funds to achieve the highest effectiveness.
Feasible Region
The feasible region in linear programming is the set of all possible points that satisfy the problem's constraints, including inequality constraints and non-negativity restrictions. It represents all the potential solutions before we apply the optimization criteria.

In the exercise, the feasible region is visually represented by a polygon on the Cartesian plane, bounded by the lines given by the inequality constraints of the aid to be given to country A and country B, and the total aid to be allocated. It is crucial to identify this region correctly since the optimum solution to the linear programming problem will always lie at one of the vertices of the feasible region, according to the fundamental theorem of linear programming.
Objective Function
The objective function in linear programming is a mathematical expression that defines what needs to be maximized or minimized in the problem. It is a linear equation formed by the decision variables and their respective coefficients that measure the performance of a solution.

In the case of AntiFam鈥檚 dilemma, the objective function is denoted as \(P=0.6x+0.8y\), where \(x\) and \(y\) are the amounts allocated to countries A and B, respectively. The coefficients 0.6 and 0.8 represent the estimated return per dollar for each country. To resolve the problem, AntiFam seeks to maximize \(P\), finding the values of \(x\) and \(y\) that yield the highest return under the given constraints.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Kane Manufacturing has a division that produces two models of hibachis, model A and model B. To produce each model-A hibachi requires \(3 \mathrm{lb}\) of cast iron and \(6 \mathrm{~min}\) of labor. To produce each model-B hibachi requires \(4 \mathrm{lb}\) of cast iron and \(3 \mathrm{~min}\) of labor. The profit for each modelA hibachi is $$\$ 2$$, and the profit for each model-B hibachi is $$\$ 1.50 .$$ If \(1000 \mathrm{lb}\) of cast iron and 20 labor-hours are available for the production of hibachis each day, how many hibachis of each model should the division produce in order to maximize Kane's profit? What is the largest profit the company can realize? Is there any raw material left over?

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 60 deluxe cabins and 160 standard cabins. whereas a type-B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15 -day cruise in May. It costs $$\$ 44,000$$ to operate a type-A vessel and $$\$ 54,000$$ to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

Everest Deluxe World Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Each advertisement in newspaper I is seen by 70,000 group-A customers, 40,000 group-B customers, and 20,000 group-C customers. Each advertisement in newspaper II is seen by 10,000 group-A, 20,000 group-B, and 40,000 group-C customers. Each advertisement in newspaper I costs $$\$ 1000$$, and each advertisement in newspaper II costs $$\$ 800$$. Everest would like their advertisements to be read by at least 2 million people from group A, \(1.4\) million people from group \(\mathrm{B}\), and 1 million people from group C. How many advertisements should Everest place in each newspaper to achieve its advertising goals at a minimum cost? What is the minimum cost?

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{cccrc|c} x & y & u & v & P & \text { Constant } \\ \hline 0 & 1 & \frac{5}{7} & -\frac{1}{7} & 0 & \frac{20}{7} \\ 1 & 0 & -\frac{3}{7} & \frac{2}{7} & 0 & \frac{30}{7} \\ \hline 0 & 0 & \frac{13}{7} & \frac{3}{7} & 1 & \frac{220}{7} \end{array} $$

An oil company operates two refineries in a certain city. Refinery I has an output of 200 , 100 , and 100 barrels of low-, medium-, and high-grade oil per day, respectively. Refinery II has an output of 100,200 , and 600 barrels of low-, medium-, and high-grade oil per day, respectively. The company wishes to produce at least 1000,1400 , and 3000 barrels of low-, medium-, and highgrade oil to fill an order. If it costs $$\$ 200 /$$ day to operate refinery I and $$\$ 300 /$$ day to operate refinery II, determine how many days each refinery should be operated in order to meet the production requirements at minimum cost to the company. What is the minimum cost?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.