Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve each linear programming problem by the method of corners. Find the maximum and minimum of \(P=4 x+3 y\) subject to $$ \begin{aligned} 3 x+5 y & \geq 20 \\ 3 x+y & \leq 16 \\ -2 x+y & \leq 1 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

The minimum value of the objective function \(P = 4x + 3y\) is 3 at the point (0,1), and the maximum value is 40 at the point (6,4).

Step 1: Identify and graph the inequalities

Let's first plot the given constraints on a graph: 1. \(3x + 5y \geq 20\) 2. \(3x + y \leq 16\) 3. \(-2x + y \leq 1\) 4. \(x \geq 0\) 5. \(y \geq 0\) First, we need to plot the x-axis (x ≥ 0) and y-axis (y ≥ 0), making sure we are working in the first quadrant as both x and y are non-negative. Next, we are going to plot each inequality as an equation, and then shade the appropriate half-planes: 1. Plot \(3x + 5y = 20\), shade the region where \(3x + 5y \geq 20\) 2. Plot \(3x + y = 16\), shade the region where \(3x + y \leq 16\) 3. Plot \(-2x + y = 1\), shade the region where \(-2x + y \leq 1\) Now we have a graph for each constraint, with shaded regions.

Step 2: Find the corner points of the feasible region

From the graph, we should be able to find the corner points of our feasible region. The feasible region is defined as the intersection of all the constraints. Remember that we are working in the first quadrant where x and y are non-negative. Corner points (the points where the shaded regions of the inequalities intersect): 1. (0, 4) - Intersection of x-axis and line 1 2. (0, 1) - Intersection of x-axis and line 3. 3. (5, 0) - Intersection of y-axis and line 2 4. (6, 4) - Intersection of lines 1 and 2 5. (2, 3) - Intersection of lines 2 and 3

Step 3: Calculate the objective function value for each corner point

Next, we will substitute the coordinates of the corner points into the objective function P = 4x + 3y and calculate the P values for each point: 1. P(0, 4) = 4(0) + 3(4) = 12 2. P(0, 1) = 4(0) + 3(1) = 3 3. P(5, 0) = 4(5) + 3(0) = 20 4. P(6, 4) = 4(6) + 3(4) = 40 5. P(2, 3) = 4(2) + 3(3) = 17

Step 4: Determine the maximum and minimum values for the objective function

Now that we have P evaluated at each corner point, we can find the maximum and minimum values for the objective function P = 4x + 3y: Minimum value of P: From the calculated P values, the minimum value of P is 3, which occurs at the point (0,1). Maximum value of P: From the calculated P values, the maximum value of P is 40, which occurs at the point (6,4). Hence, the minimum value of the objective function P is 3 at the point (0,1), and the maximum value is 40 at the point (6,4).

Key Concepts

Method of Corners

The Method of Corners is a straightforward technique used in linear programming to find the optimal value of an objective function. It involves evaluating the objective function at each corner (vertex) of the feasible region. Why corners? Because, in linear programming, the maximum or minimum value of the objective function occurs at one of these points.
To apply the method, follow these steps:

• Graph the constraints to identify the feasible region.
• Find all corner points where the constraint lines intersect.
• Calculate the objective function at each corner point.

This method ensures that you find the highest or lowest value of the objective function depending on whether you are maximizing or minimizing.

Objective Function

In linear programming, the objective function is the formula you want to maximize or minimize. It's a linear equation typically represented as something like \(P = ax + by\). Here, \(P\) is the objective value to optimize.
The objective function gives direction to the problem by indicating what needs to be achieved, such as maximizing profit or minimizing costs. In our example, the given function is \(P = 4x + 3y\). The coefficients 4 and 3 represent the contribution of variables \(x\) and \(y\) to the objective value.
Ultimately, you substitute each corner point into this function to determine which one gives you the desired optimal value.

Feasible Region

The feasible region is the area on the graph where all constraints are satisfied simultaneously. It's the set of possible solutions that meet the requirements given by the inequalities.
To identify this region, you need to:

• Graph each constraint line.
• Shade the region that satisfies each inequality.
• Locate the overlap, which represents the feasible solutions.

The feasible region is bounded by the lines from the constraints and is often a polygon. The corner points of this region are crucial as they are used to determine the optimal solution. Only solutions within this area are valid for the problem.

Constraints in Linear Programming

Constraints are conditions that limit the possible solutions to a linear programming problem. They are represented as linear inequalities that define the bounds of your problem.
In the example, the constraints include:

• \(3x + 5y \geq 20\)
• \(3x + y \leq 16\)
• \(-2x + y \leq 1\)
• \(x \geq 0\)
• \(y \geq 0\)

These constraints form the edges of the feasible region on the graph. They dictate the possible combinations of \(x\) and \(y\) that meet all the defined requirements. Understanding constraints is key to knowing how the feasible region is shaped and where the solution may lie.