Problem 52 - College Algebra and Calculus: An Applied Approach

Question

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=x+y\) Constraints: \(x \geq 0\) \(y \geq 0\) \(-x+y \leq 1\) \(-x+2 y \leq 4\)

Step-by-Step Solution

Answer

The unusual characteristic of the problem lies in the feasibility area defined by the constraints and the fact that the region is unclosed. The maximum value occurs at the point (0,1) with a value of 1.

Step 1: Set up the Constraints Graph

First, it is necessary to sketch the region defined by the constraints. Start with the individual inequalities, plotting them based on a y-intercept and slope form. For \( -x+y \leq 1 \), when \( x=0 \), \( y=1 \), and when \( y=0 \), \( x=-1 \) which is outside our region as we only consider \( x \geq 0 \). So, plot the y-intercept and sketch the line, and then shade below it because it's a 'less than or equal to' inequality. Similarly for \( -x+2y \leq 4 \), when \( x=0, y=2 \), and when \( y=0, x=-4 \). Again shade the region below it.

Step 2: Intersect Constraint Regions

Our feasible region is the intersection of the areas defined by each inequality. This intersection also includes the area where \( x \geq 0 \) and \( y \geq 0 \), meaning the first quadrant. Therefore, the feasible (solution) region that satisfies all conditions is the region in the first quadrant below both lines.

Step 3: Identify Maximum Objective Function Value

The objective is to maximize the function \( z=x+y \). Visually, this can be interpreted as finding the highest intersection point of the function \( z=x+y \) and the solution region. One point of the region is (0,1) which lies on \(y \leq 1\), and doesn't intersect with the other line. The other point is obtained by solving the two lines: \( -x+y=1 \) and \( -x+2y=4 \). Solving them gives \( x=2 \) and \( y=1 \). This verifies that (0,1) is the only point to check as (2,1) lies outside the solution region. Thus the maximum value occurs at \( x=0 \) and \( y=1 \). Substituting these values into the objective function, gives the maximum value as \(0+1=1\).

Key Concepts

Objective Function Maximization

Linear programming is a method used to determine the best outcome, such as maximum profit or lowest cost, within a given mathematical model represented by linear relationships. Maximizing the objective function is the process of finding the highest value of a linear equation subject to a set of constraints.

The objective function in our example is given by the equation:

\( z = x + y \).

To maximize this objective function, we look for the highest value it can reach within the feasible region. The feasible region is determined by the system of constraints provided. Each constraint reduces the area where the maximization can occur. Once we have the feasible region, we evaluate the objective function at each vertex of this region, or by finding its highest point along the boundary if a vertex doesn't exist. The maximum value of the objective function is found at these critical points. In this exercise, the maximum value of the objective function occurs at the point where \( y = 1 \) and \( x = 0 \), resulting in a maximum value of \( z = 1 \).

Graphical Method Constraint Solving

The graphical method is a visual approach to solving linear programming problems. It involves plotting each inequality constraint on a graph, then identifying the common area that satisfies all constraints, known as the feasible region.

In our example, the constraints are plotted on the Cartesian plane. For instance, the constraint \( -x + y \leq 1 \) is a line, which we graph based on its intercepts. The inequality indicates that we need to consider the area below this line. We also plot \( -x + 2y \leq 4 \), shading the area below it too. The constraints \( x \geq 0 \) and \( y \geq 0 \) restrict our feasible region to the first quadrant of the coordinate plane.

To solve these graphically, we look for where these shaded areas overlap, which will be the feasible region. In this case, our feasible region is bounded by the two lines and the coordinate axes. In more complex problems, there may be several lines intersecting, and it's important to be precise in identifying the correct area of overlap.

Feasible Region Identification

Identifying the feasible region is a critical step in solving linear programming problems. The feasible region represents all possible solutions that satisfy the problem's constraints. It is always a convex polygon, which means a straight line between any two points within the region will not cross its boundary.

In the given exercise, the feasible region is confined to the first quadrant due to the non-negativity constraints \( x \geq 0 \) and \( y \geq 0 \). The other constraints \( -x + y \leq 1 \) and \( -x + 2y \leq 4 \) further restrict this region to a subset of the first quadrant. To accurately identify this area, the lines created by each inequality are drawn, and the region that falls within all the constraints is shaded in.

The unusual characteristic in this problem is that the feasible region has a vertex at the intersection of the y-axis and one of the constraints, meaning that the maximum value of the objective function is found on this boundary, rather than at the intersection of two lines representing constraints. Identifying such characteristics is essential for correctly applying the graphical method and accurately interpreting the results.