Problem 9 - Thinking Mathematically

Question

Use the directions for Exercises 5-8 to solve Exercises 9-12. Objective Function $$ z=5 x-2 y $$ Constraints $$ \left\\{\begin{array}{l} 0 \leq x \leq 5 \\ 0 \leq y \leq 3 \\ x+y \geq 2 \end{array}\right. $$

Step-by-Step Solution

Answer

The maximum value of the objective function is 25 at point (5,0) and the minimum value is -4 at point (0,2).

Step 1: Identify the Feasible Region

The feasible region is determined by the constraints. It is the area enclosed by \(0 \leq x \leq 5\), \(0 \leq y \leq 3\), \(x + y \geq 2\). From the constraint \(x + y \geq 2\), we can infer that the feasible region lies above the line \(x + y = 2\).

Step 2: Determine the Corner Points

The corner points of the feasible region are the points of intersection of the constraints. There are three constraints so by solving them pair-wise will give the corner points. The corner points are found to be (5,3), (2,1), (5,0), (0,2).

Step 3: Evaluate the Objective Function

Next, substitute the corner points into the objective function. For (5,3), \(z = 5*5 - 2*3 = 19\). For (2,1), \(z = 5*2 - 2*1 = 8\). For (5,0), \(z = 5*5 - 2*0 = 25\). For (0,2), \(z = 5*0 - 2*2 = -4\).

Step 4: Determine the Maximum and Minimum Values

The maximum and minimum values of the objective function are the maximum and minimum z-values obtained. Hence, the maximum value is 25 at point (5,0) and the minimum value is -4 at point (0,2).

Key Concepts

Objective Function

The objective function in linear programming serves as the mathematical representation of the goal you want to achieve. In this specific exercise, the objective function is given by:\[ z = 5x - 2y \] Here, **z** represents the value of the objective function, which needs to be maximized or minimized. The terms **5x** and **-2y** assign respective weights to the variables **x** and **y**. This function essentially tells you how each unit change in **x** or **y** affects the value of **z**. In the context of this exercise, increasing **x** increases **z**, while increasing **y** decreases **z**. The ultimate aim in such problems is to find the values of **x** and **y** such that **z** is maximized or minimized, within given limitations or constraints.

Constraints

Constraints are the conditions that define the limitations within which the objective function must operate. These are essentially the rules that the solution must obey. In this exercise, the constraints are:

• \(0 \leq x \leq 5\)
• \(0 \leq y \leq 3\)
• \(x + y \geq 2\)

Constraints are critical as they define the boundaries of possible solutions. They are given as inequalities or equations and can be visualized graphically as lines on a plane. The role of constraints is to carve out a feasible area on a graph, where any point within this area can be a potential solution to the problem. It is crucial to respect these constraints; ignoring them can lead to solutions that are not feasible in real-world applications. Each constraint mathematically limits the possibilities for the values of **x** and **y**.

Feasible Region

The feasible region is the graphical representation of all possible solutions that satisfy all the constraints simultaneously. It is typically a polygon on the graph where each side is defined by a constraint line.In this exercise, the feasible region is determined by the area that is:

• Between \(x = 0\) and \(x = 5\)
• Between \(y = 0\) and \(y = 3\)
• Above the line \(x + y = 2\)

This region is not just a random area but an essential part of linear programming problems. The solution to your problem must lie within this feasible region. Finding this region involves graphing all the constraint lines and identifying the intersection area where they all overlap. This region could either be bounded, forming a polygon, or unbounded extending infinitely in one direction. However, your solution, especially optimal solution, needs to find a spot within this feasible region.

Corner Points

Corner points, also known as vertices, are crucial in linear programming because they represent potential optimal solutions. Each corner point is where two or more constraint lines intersect in the feasible region. In this exercise, by examining the intersection of constraints, the corner points discovered are:

• (5, 3)
• (2, 1)
• (5, 0)
• (0, 2)

These points are important because, according to the nature of linear functions, the optimal solution (maximum or minimum value of the objective function) will occur at one of these corner points. In practical terms, these are the extreme points that need to be tested in the objective function to confirm which provides the best outcome. By calculating the objective function at these points, we establish which one gives the highest or lowest value, depending on the requirement of the problem.