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Chapter 6: Problem 44

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 return of $$\$ .60$$, whereas a dollar spent in country \(\mathrm{B}\) 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 utilize the money most effectively, AntiFam should allocate \(1.5\) million dollars to country A and \(1.25\) million dollars to country B. This will yield a return of \(1.9\) million dollars.

Step by step solution

01

Define the decision variables

Let's define x as the amount of money (in million dollars) allocated to country A and y as the amount allocated to country B.
02

Define the objective function

We want to maximize the returns, which are given by the return rates for each country. So, the objective function can be defined as: \( P = 0.6x + 0.8y\)
03

Define the constraints

We need to consider the constraints given in the problem statement: 1. Total funds allocated: \(2 \le x + y \le 2.5\) 2. Funds allocated to country A: \(1 \le x \le 1.5\) 3. Funds allocated to country B: \(y \ge 0.75\)
04

Solve the linear programming problem

Now that we have the objective function and the constraints, we can solve the linear programming problem graphically. We would plot the feasible region based on the constraints and then analyze the objective function along the boundary of the feasible region to find the optimal allocation. The feasible region is determined by the constraints: 1. The total funds constraint forms two parallel lines, with x+y=2 and x+y=2.5, which represent the budget limit. 2. The funds allocated to country A create vertical lines, x=1 and x=1.5, which are the minimum and maximum allocations for Country A. 3. The funds allocated to country B constraint creates a horizontal line, y=0.75, which represents the minimum allocation for Country B. The feasible region is a rectangle with vertices at (1, 1), (1, 1.5), (1.5, 1), and (1.5, 1.25). The objective function can be rewritten as \(y = \frac{1}{0.8}(P - 0.6x)\). To maximize the returns, we need to analyze the objective function along the boundary of the feasible region.
05

Check for optimality at vertices

We can evaluate the objective function at each of the vertices: 1. At (1, 1): \(P = 0.6(1) + 0.8(1) = 1.4\) 2. At (1, 1.5): \(P = 0.6(1) + 0.8(1.5) = 1.8\) 3. At (1.5, 1): \(P = 0.6(1.5) + 0.8(1) = 1.7\) 4. At (1.5, 1.25): \(P = 0.6(1.5) + 0.8(1.25) = 1.9\) The maximum return occurs at the vertex (1.5, 1.25), where the return is 1.9 million dollars.
06

Optimal allocation

Therefore, to utilize the money most effectively, AntiFam should allocate 1.5 million dollars to country A and 1.25 million dollars to country B. This will yield a return of 1.9 million dollars.

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Key Concepts

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

Optimization
In the context of linear programming, optimization refers to the process of finding the most advantageous solution from a set of possible choices. Here, the goal is to allocate funds in such a way that the total effective return is maximized. This means looking for a combination of allocations to Country A and Country B that results in the highest possible return. The optimization process considers all given conditions, or constraints, and attempts to achieve the best objective value, often using methods like the graphical method or simplex algorithm in more complex scenarios. Understanding optimization is crucial for determining efficient resource allocation in various real-world situations.
Constraints
Constraints are the rules or limits that define the possible solutions to a linear programming problem. In our problem, constraints restrict how the funds can be distributed among the countries. These are defined by:
  • The total budget must be between 2 and 2.5 million dollars, so the combined funds for A and B should satisfy \(2 \leq x + y \leq 2.5\).
  • Country A should receive between 1 and 1.5 million dollars, which means \(1 \leq x \leq 1.5\).
  • Country B must receive no less than 0.75 million dollars, represented by \(y \geq 0.75\).
Constraints help in mapping out the feasible region and play a crucial role in identifying the set of all possible allocations that satisfy every listed condition.
Objective Function
The objective function is the mathematical expression that needs to be optimized鈥攅ither maximized or minimized鈥攊n a linear programming problem. For this exercise, the objective function is defined as \(P = 0.6x + 0.8y\). This equation represents the total return based on the allocation of funds between Country A and Country B.
  • The term \(0.6x\) indicates that every dollar allocated to Country A yields a return of 0.6 dollars.
  • Similarly, the \(0.8y\) reflects a return of 0.8 dollars per dollar allocated to Country B.
The objective function guides the linear programming process by providing the mathematical basis for measuring the effectiveness of different solutions, helping to select the option that achieves the highest return.
Feasible Region
The feasible region is a crucial component in linear programming鈥攊t represents all possible solutions that satisfy each of the problem's constraints. Geometrically, it is the intersection area of all inequalities on the graph. For the given problem, the constraints form a rectangular feasible region, confined by a combination of horizontal, vertical, and diagonal boundaries.
  • The vertical boundaries are determined by the allocation limits for Country A \(x = 1\) and \(x = 1.5\).
  • The horizontal boundary arises from Country B's minimum allocation \(y = 0.75\).
  • The diagonal limits are determined by the total budget constraints \(x + y = 2\) and \(x + y = 2.5\).
Finding the optimal solution involves evaluating the objective function at the vertices of this feasible region, as these points represent potential solutions where maximum or minimum values of the objective function are commonly found.

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Most popular questions from this chapter

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $$\$ 45$$, and the profit for each chair is $$\$ 20 .$$ In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture in order to maximize its profit? What is the maximum profit?

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?

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=4 x+6 y+5 z \\ \text { subject to } & x+y+z \leq 20 \\ & 2 x+4 y+3 z \leq 42 \\ & 2 x+3 z \leq 30 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=x+2 y-z \\ \text { subject to } & 2 x+y+z \leq 14 \\ & 4 x+2 y+3 z \leq 28 \\ & 2 x+5 y+5 z \leq 30 \\ & x \geq 0, y \geq 0, z \geq 0 \end{array} $$

Consider the linear programming problem $$ \begin{aligned} \text { Minimize } & C=-2 x+5 y \\ \text { subject to } & x+y \leq 3 \\ & 2 x+y \leq 4 \\ & 5 x+8 y \geq 40 \\ & x \geq 0, y \geq 0 \end{aligned} $$ a. Sketch the feasible set. b. Find the solution(s) of the linear programming problem, if it exists.
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