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Chapter 7: Problem 34

Describing an Unusual Characteristic, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=x+2 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+2 y \leq 4} \\ {2 x+y \leq 4}\end{array} $$

Short Answer

Expert verified
The solution region for this problem is a triangle formed with vertices at (0,0), (1,2), and (2,1). The maximum value of the objective function is 5, occurring at (1,2), while the minimum is 0 at (0,0). Unusual characteristic: the problem presents multiple optimal solutions for some specific objective function lines.

Step by step solution

01

Graph the Constraints

The constraints define the attainable region, which is the area of interest. The constraints \(x \geq 0\) and \(y \geq 0\) indicates that we are restricted to the first quadrant. The other two inequality constraints are \(x+2 y \leq 4\) and \(2 x+y \leq 4\), these will produce straight lines within the first quadrant. The intersection of all these lines and the axes will create a region in the first quadrant.
02

Identify the Solution Region

This is the region where all constraints are satisfied. From the graph, the point of intersection between the lines \(x+2 y = 4\) and \(2 x+y = 4\) gives the apex of the triangular solution region within the quadrant. The solution region is, therefore, the triangle formed with vertices at (0,0), (2,1), and (1,2).
03

Determine Maximum and Minimum values

Apply the objective function \(z=x+2 y\) at each of the vertices. For (0,0), z=0. For (1,2), z=5, and for (2,1), z=4. This implies the maximum of the objective function is 5 occurring at (1,2), and the minimum is 0 at (0,0).
04

Identify Unusual Characteristic

The unusual characteristic here is that a unique optimal solution may not always exist for some specific objective functions. In this exercise, objective function lines that are parallel to the line joining (2,1) and (1,2) will have all the points along this line as the solutions, which is somewhat unusual considering standard linear programming problem scenarios.

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Key Concepts

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

Constraints
Constraints in linear programming are conditions that any potential solution must satisfy. In this problem, the constraints are given by a set of inequalities:
  • \(x \geq 0\) ensures that we're only considering solutions in the positive range for x.
  • \(y \geq 0\) limits solutions to positive y-values.
  • \(x+2y \leq 4\) and \(2x+y \leq 4\) both form lines on our graph that determine additional regions of interest.
The importance of constraints is to form a boundary for our solutions. Without them, the objective function would have no specific range to evaluate.
In practice, constraints make sure we adhere to limitations such as resources, time, and other practical considerations.
Objective Function
The objective function in a linear programming problem is what you aim to optimize. It could be a goal such as maximizing profit or minimizing cost. In this exercise, the objective function is
\(z = x + 2y\)
This function tells us that each unit of y contributes twice as much to our objective as each unit of x.
Using this expression, once the solution region is identified, we can evaluate this function at different points to find the most desirable outcome, like a maximum or minimum value.
Here, the objective function helps in determining which points in the solution region give us our optimal results.
Solution Region
In linear programming, the solution region is the area on the graph where all constraints overlap and are satisfied. This is also known as the feasible region.
For our exercise, it forms a triangle bounded by the points (0,0), (2,1), and (1,2).
  • The intersection of the constraints creates this region.
  • Any point within or on the boundary of this triangle meets all the constraints.
Finding this region is crucial because it identifies where the optimal solutions could be located for the objective function. Only in this region can the objective function yield valid results under the given constraints.
Optimization
Optimization is the process of making something as effective or functional as possible. In linear programming, it involves finding the maximum or minimum value of the objective function within the solution region.
In this case, we calculate the value of the objective function \(z = x + 2y\) at key points: (0,0), (1,2), and (2,1).
  • For (0,0), z equals 0, which is the minimum in this scenario because it is the lowest value the function can achieve in our region.
  • For (1,2), z equals 5, which is the maximum value, indicating the best possible outcome given our constraints.

    • Optimization in this context is determining which point in the solution region gives the highest or lowest value for the objective function. It helps in decision-making and resource allocation.

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