91Ó°ÊÓ

Social ProGRAMS PLANNING AntiFam, a hunger-relief organization, has earmarked between \(\$ 2\) and \(\$ 2.5\) million (inclusive) for aid to two African countries, country 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 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 \(P=0.6 x+0.8 y\).

Step by step solution

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1. Define variables and objective function

Let's define the variables first: - \(x\) (in million dollars): amount of money given to country A - \(y\) (in million dollars): amount of money given to country B According to the given hint, the objective function that we need to maximize is the total return, which is given by: \(P= 0.6x + 0.8y\)

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2. Set up constraints

Now, we need to set up the constraints based on the given information: 1. Country A is to receive between \(\$ 1\) million and \(\$ 1.5\) million (inclusive): \(1 \le x \le 1.5\) 2. Country B is to receive at least \(\$ 0.75\) million: \(y \ge 0.75\) 3. Total funding is between \(\$ 2\) and \(\$ 2.5\) million (inclusive): \(2 \le x + y \le 2.5\)

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3. Analyze_constraints

Let's rewrite inequality constraints in the form of equations to analyze them: - \(x = 1\) - \(x = 1.5\) - \(y = 0.75\) - \(x + y = 2\) - \(x + y = 2.5\) Next, we'll draw graphs for these equations to find the feasible region for the given constraints.

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4. Find feasible_region

Looking at the graphs and considering all constraints, we have a quadrilateral as the feasible region with vertices at the points: - \((1,1)\) - \((1,1.5)\) - \((1.5,0.75)\) - \((1.5,1)\)

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5. Calculate_values

Now, we'll calculate the value of the objective function at each vertex of the feasible region: At \((1,1)\): \(P(1,1) = 0.6(1) + 0.8(1) = 1.4\) At \((1,1.5)\): \(P(1,1.5) = 0.6(1) + 0.8(1.5) = 1.8\) At \((1.5,0.75)\): \(P(1.5,0.75) = 0.6(1.5) + 0.8(0.75) = 1.65\) At \((1.5,1)\): \(P(1.5,1) = 0.6(1.5) + 0.8(1) = 1.7\)

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6. Find maximum value

Thus, among all the values of the objective function at the vertices of the feasible region, the maximum value of \(P\) is \(1.8\) (in million dollars), which is found at point \((1, 1.5)\). Therefore, the most effective allocation of funds according to the given criteria is to give \(\$ 1\) million to country A and \(\$ 1.5\) million to country B. This will yield an effective return of \(\$ 1.8\) million.

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Key Concepts

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

Objective Function

In linear programming, the objective function is what we aim to either maximize or minimize. It's the core goal of the optimization problem and often reflects real-world scenarios where we try to achieve the best outcome given certain conditions.
For AntiFam, the objective function is employed to determine how to allocate funds to two countries most effectively. Here, maximizing the return on investment is crucial. Given:

• Each dollar spent in country A yields a \(0.60 return.
• Each dollar spent in country B yields a \)0.80 return.

Thus, the objective function formulated as:\[ P = 0.6x + 0.8y \]represents the total expected return, where:

• \(x\) is the amount allocated to country A.
• \(y\) is the amount allocated to country B.

This equation is then used as the focal point around which the rest of the linear programming problem is established.

Feasible Region

The feasible region in linear programming is the set of all possible solutions that satisfy the given constraints. It is typically depicted graphically in the context of two variables, and its shape is influenced by the constraints in place.

In this scenario, the feasible region was identified as a quadrilateral formed by intersecting inequalities provided by budget allocations:

• Country A allocation constraints: \(1 \le x \le 1.5\)
• Country B allocation constraint: \(y \ge 0.75\)
• Total fund allocation constraint: \(2 \le x + y \le 2.5\)

Each of these inequalities is represented graphically, and the area they enclose is the feasible region. Each vertex of this enclosed area provides a potential solution to the problem—providing different combinations of fund distribution between the countries that meet all stipulated conditions.

Constraints

Constraints in linear programming are the conditions or limits within which the solution must be found. For any given problem, constraints define the boundaries of the feasible region and limit the choices available for the decision variables.

For the given problem of allocating funds to anti-famine efforts in two countries, the key constraints were:

• Country A to receive between \(1 million and \)1.5 million, creating the constraints \(1 \le x \le 1.5\).
• Country B to receive at least \(0.75 million, giving rise to the constraint \(y \ge 0.75\).
• Combined funds must be between \)2 million and $2.5 million, yielding the constraint \(2 \le x + y \le 2.5\).

These constraints ensure that the solution remains practical and within the available budget, while also capturing the intended distribution models for funds.

Optimization

Optimization in the context of linear programming refers to finding the best possible solution to a problem within the constraints provided. This generally involves maximizing or minimizing the objective function depending on the goals.
In this aid allocation problem, the aim is to maximize the return value captured by the objective function \(P = 0.6x + 0.8y\). By evaluating the objective function at each vertex of the feasible region, we identify:

• Highest yielding point: When \(x = 1\) and \(y = 1.5\), the yield is maximized at an effective return of \(1.8 million.

This output represents the optimal allocation of resources, directing \)1 million to country A and $1.5 million to country B. Through the process of optimization, the organization's intended investment is strategically leveraged to maximize benefits.