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Chapter 6: Problem 10

Solve each linear programming problem by the method of corners. $$ \begin{array}{l} \text { Maximize } P=4 x+2 y \\ \text { subject to } \quad x+y \leq 8 \\ \quad 2 x+y \leq 10 \\ x \geq 0, y \geq 0 \end{array} $$

Short Answer

Expert verified
By the method of corners, the maximum value of the objective function \(P = 4x + 2y\) occurs at the point (8, 0), with a value of P = 32, given the constraints \(x + y \leq 8\), \(2x + y \leq 10\), \(x \geq 0\), and \(y \geq 0\).

Step by step solution

01

Graph the Inequality Constraints

First, we need to graph the inequality constraints on the xy-plane. We will treat each inequality as an equation and find the lines that correspond to these equations. Then, shade the region that satisfies the inequality constraints. 1. For the inequality x + y 鈮� 8, the equation is x + y = 8. The line passes through points (0, 8) and (8, 0). The region below this line satisfies the inequality. 2. For the inequality 2x + y 鈮� 10, the equation is 2x + y = 10. The line passes through points (0, 10) and (5, 0). The region below this line satisfies the inequality. 3. Both x 鈮� 0 and y 鈮� 0 imply that we are only considering the first quadrant (x and y positive). Plot the lines and their respective regions, and find the intersection of all regions, which forms the feasible region of the problem.
02

Determine the Corner Points

The feasible region is a polygon, and we need to find its corner points. They are the vertices of the feasible region where the constraints intersect. In this case, we have 4 corner points: 1. Intersection of x-axis and the line x + y = 8: (8, 0). 2. Intersection of y-axis and the line 2x + y = 10: (0, 10). 3. Intersection of x-axis and y-axis: the origin (0, 0). 4. Intersection of the lines x + y = 8 and 2x + y = 10: Solve the equations simultaneously, which gives us (2, 6).
03

Evaluate the Objective Function at Corner Points

Substitute the x and y values of the corner points into the expression for P, which is P = 4x + 2y: 1. P at point (8, 0): \(P = 4(8) + 2(0) = 32\) 2. P at point (0, 10): \(P = 4(0) + 2(10) = 20\) 3. P at point (0, 0): \(P = 4(0) + 2(0) = 0\) 4. P at point (2, 6): \(P = 4(2) + 2(6) = 20\) Among these values, the maximum value of P is 32, which occurs at the point (8, 0).
04

Conclusion

Hence, by the method of corners, the maximum value of the objective function P is 32, and it occurs at the point (8, 0).

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Key Concepts

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

Linear Programming
Linear programming is a mathematical technique used for optimization, which involves finding the maximum or minimum value of a linear function, known as the objective function, subject to a set of linear inequality or equality constraints. Its applications are vast, ranging from resource allocation in business to diet planning in nutrition.

The cornerstone of solving a linear programming problem is to identify the objective function, constraints, and then determine the feasible region where all these constraints are satisfied. Once the feasible region is established, the values of the objective function at the corners or 'vertices' of this region are compared to find the maximum or minimum value required by the problem.
Inequality Constraints
Inequality constraints define the conditions that must be met for the variables within a linear programming problem. They are typically written as linear inequalities, restricting the values that the variables can assume. For example, the constraints \( x + y \leq 8 \) and \( 2x + y \leq 10 \) along with \( x \geq 0, y \geq 0 \) not only limit the range of possible values for x and y but also define the shape of the feasible region when graphed.

In plotting these constraints, we treat each inequality as if it were an equation to find its corresponding line, and then we identify which side of the line is included in the solution set by considering the inequality's direction.
Feasible Region
The feasible region is a graphical representation of all possible solutions that satisfy the inequality constraints of a linear programming problem. It's where all the conditions or limitations of the problem overlap, and it is often a polygon on the coordinate plane.

Identifying the feasible region is crucial because the optimal solution to the linear programming problem lies at one of its vertices, which are referred to as 'corner points.' For the case above, plotting the constraints on the graph provides a visual aid to establish this region. The intersection of the constraints delineates the region within which we need to search for our solution.
Objective Function
The objective function in linear programming is a formula representing the goal of the problem鈥攚hat you are trying to maximize or minimize. In the example \( P = 4x + 2y \), P represents the quantity that we want to optimize through the selection of x and y values within the feasible region.

Once the feasible region is determined, the next step is to evaluate the objective function at each corner point. The optimal solution is the value of the corner point that gives the highest or lowest result, depending on whether we are maximizing or minimizing, respectively. By analyzing the objective function at these key locations, we can conclude the best solution to the problem.

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Most popular questions from this chapter

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

National Business Machines manufactures two models of fax machines: A and B. Each model A costs $$\$ 100$$ to make, and each model \(\mathrm{B}\) costs $$\$ 150$$. The profits are $$\$ 30$$ for each model \(\mathrm{A}\) and $$\$ 40$$ for each model B fax machine. If the total number of fax machines demanded per month does not exceed 2500 and the company has earmarked no more than $$\$ 600,000 $$ month for manufacturing costs, how many units of each model should National make each month in order to maximize its monthly profit? What is the optimal profit?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A \subseteq B\), then \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & 2 x+y \leq 16 \\ & 2 x+3 y \leq 24 \\ y & \leq 6 \\ x & \geq 0, y \geq 0 \end{aligned} $$

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?
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