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Chapter 5: Problem 2

Find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) Objective function: $$ z=2 x+8 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 2 x+y & \leq 4 \end{aligned} $$

Short Answer

Expert verified
The minimum value of the objective function is 0, which occurs at the point (0,0), and the maximum value is 32, which occurs at the point (0,4).

Step by step solution

01

Identify the Feasible Region

The given constraints form a system of linear inequalities that defines a feasible region on the coordinate plane. To visualize this region, represent each inequality on a graph. The feasible region is the area where the three shaded regions overlap, it's the region above the x-axis, to the right of the y-axis, and below the line \(2x + y = 4\).
02

Identify the Corner Points

The corner points of the feasible region are the intersections of the lines defining the region. In this problem, the corner points are (0, 0), (0, 4) and (2, 0).
03

Evaluate the Objective Function at the Corner Points

Substitute the coordinates of the corner points into the objective function \(z = 2x + 8y\) and compute the values. For (0,0): \(z=2(0)+8(0)=0\). For (0, 4): \(z=2(0)+8(4)=32\). For (2, 0): \(z=2(2)+8(0)=4\).
04

Determine the Minimum and Maximum Values

The smallest value of z is 0 at the point (0,0) and the largest value is 32 at point (0,4). Therefore, the minimum and maximum values of the objective function are 0 and 32, respectively.

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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 the equation that you want to either maximize or minimize. It is typically expressed as a linear equation in terms of decision variables. In this specific exercise, the objective function is given as \( z = 2x + 8y \). Here,
  • \( z \) represents the value that needs to be maximized or minimized.
  • \( x \) and \( y \) are the decision variables you control.
Your task is to find the combination of \( x \) and \( y \) that produces the highest or lowest value of \( z \), within the specified constraints. The objective function serves as the ultimate goal of the optimization problem, guiding every decision made within the feasible region.
Feasible Region
The feasible region represents all the possible combinations of decision variables \( x \) and \( y \) that satisfy the given constraints. This region is found by graphing the system of linear inequalities. In this exercise, the constraints are:
  • \( x \geq 0 \)
  • \( y \geq 0 \)
  • \( 2x + y \leq 4 \)
These constraints form a shape on the graph that shows where the solutions are valid. The feasible region is the shaded area that lies:
  • Above the x-axis (\( x \geq 0 \))
  • To the right of the y-axis (\( y \geq 0 \))
  • Below the line \( 2x + y = 4 \)
Finding this region accurately is crucial because it contains all the potential solutions to the problem, within which we can evaluate the objective function.
Corner Points
Corner points, also known as vertices, occur where the boundaries of the feasible region intersect. These points are crucial in linear programming problems because, according to the theory of linear programming, the maximum and minimum values of the objective function will occur at one or more of these corner points. In the given exercise, the feasible region is bounded by the inequalities, and the corner points discovered are (0,0), (0,4), and (2,0). These are found by:
  • Identifying where the constraints intersect each other.
  • Solving the equations of the intersecting lines to find exact points.
Once these points are established, the objective function is evaluated at each to find potential maximum and minimum values. This process simplifies the search for optimal solutions.
Linear Inequalities
Linear inequalities form the backbone of constraints in linear programming exercises. They describe the permissible region through inequalities rather than equations and define space on the graph where solutions can exist.In this exercise, we deal with inequalities like \( x \geq 0 \), \( y \geq 0 \), and \( 2x + y \leq 4 \). Each inequality individually represents a half-plane on the graph:
  • \( x \geq 0 \) ensures solutions are in the non-negative x-direction.
  • \( y \geq 0 \) ensures solutions are in the non-negative y-direction.
  • \( 2x + y \leq 4 \) represents all points below the line \( 2x + y = 4 \).
Graphically, these inequalities are drawn as a series of half-planes, and the overlapping part of these planes constitutes the feasible region. Understanding these inequalities is key to solving linear programming problems as they define where a feasible solution can exist.

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Most popular questions from this chapter

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