Problem 8 - Thinking Mathematically

Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=x+3 y $$ Constraints $$ \left\\{\begin{array}{l} x+y \geq 2 \\ x \leq 6 \\ y \leq 5 \\ x \geq 0 \\ y \geq 0 \end{array}\right\\} \text { Quadrant I and } $$

Step-by-Step Solution

Answer

The maximum value of the objective function is 21 at the point (6,5)

Step 1: Graph the Inequalities

Start by graphing each of the inequalities on the same system of axes. This can be done by first drawing the lines \(x+y = 2\), \(x = 6\), and \(y = 5\), then shading in the appropriate side of each line according to the inequality (greater than or less than). Also, the conditions \(x \geq 0\) and \(y \geq 0\) limit the solution to the first quadrant.

Step 2: Find the Intersection Points

Next, identify the corner points in the graph, which are the intersections of the lines. There should be four points: (0,2), (6,0), (6,5), and (2,5).

Step 3: Evaluate the Objective Function

Now, find the value of the objective function \(z = x + 3y\) at each of the corner points. The calculated values are: At (0,2), \(z = 6\); At (6,0), \(z = 6\); At (6,5), \(z = 21\); At (2,5), \(z = 17\).

Step 4: Determine the Maximal Value

Lastly, determine the maximum value of the objective function and the corresponding values of \(x\) and \(y\) by simply choosing the highest z-value found in step 3. In this case, the maximum value of the objective function is \(z = 21\) at the point (6,5).

Key Concepts

Objective Function

The objective function is a core concept in linear programming. It is what we aim to maximize or minimize. In the given exercise, the objective function is \(z = x + 3y\). This expression describes a line in a plane, and depending on different values of \(x\) and \(y\), we get different values of \(z\).
When working with an objective function:

• Understand what you want to achieve - either getting the highest or lowest value depending on the context (maximize or minimize).
• Use this function to determine your best possible solution within given constraints.

The constraints guide which combinations of \(x\) and \(y\) are possible, while the objective function guides us to the optimal outcome within those constraints.

Linear Inequalities

Linear inequalities are crucial in defining the restrictions or constraints in a linear programming problem. They set the boundaries within which the objective function can be examined. In this problem, we have several inequalities:

• \(x + y \geq 2\)
• \(x \leq 6\)
• \(y \leq 5\)
• \(x \geq 0\)
• \(y \geq 0\)

The graph of each inequality will divide the plane into two halves, and only one side of each line will contain the possible solutions. By shading the correct side for each inequality, you define the feasible region. This region contains all points that satisfy all inequalities simultaneously. In this case, the feasible region is confined to the first quadrant due to the inequalities \(x \geq 0\) and \(y \geq 0\).

Graphical Method

The graphical method is an intuitive and visual way to solve linear programming problems using graphs. The first step is to represent each inequality as a line on a coordinate plane to visualize the area where all conditions or constraints are satisfied together.
Here's how you can use the graphical method:

• Draw each inequality. Convert them temporarily to equalities to find their boundary lines, such as \(x + y = 2\).
• Determine which side of the line satisfies each inequality by testing a point not on the line (often, the origin is a convenient choice).
• Shade the area that satisfies the inequality.

When you have all constraints drawn, the overlap of all shaded regions forms the feasible region. This method helps to easily visualize potential solutions and the combinations of \(x\) and \(y\) that meet the constraints.

Corner Points

Corner points are the vertices of the feasible region formed by the overlapping of all inequalities in a linear programming problem. These points are critical because, according to the fundamental theorem of linear programming, if there is an optimal solution, it occurs at one of the corner points.
To identify corner points:

• Find where each pair of boundary lines intersect. These points lie on the edge of the feasible region.
• In this case, the corner points identified are (0,2), (6,0), (6,5), and (2,5).

Once you have the corner points, substitute these into the objective function to find which one gives the optimal value. For this exercise, the point (6,5) yields the maximum objective value, emphasizing the importance of corner points in finding solutions.