Problem 8 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve each linear programming problem by the method of corners. $$ \begin{array}{lr} \text { Maximize } & P=x+2 y \\ \text { subject to } & x+y \leq 4 \\ & 2 x+y \leq 5 \\ & x \geq 0, y \geq 0 \end{array} $$

Step-by-Step Solution

Answer

The short answer is: The solution to the linear programming problem is x=0 and y=5, which gives the maximum value of the objective function \(P = 10\).

Step 1: Graph the Constraints

Plot the inequality constraints and identify the feasible region. The feasible region is the region that satisfies all the constraints. The constraints are: 1. x + y ≤ 4 2. 2x + y ≤ 5 3. x ≥ 0 4. y ≥ 0 Since x and y are both non-negative, the feasible region will be in the first quadrant of the xy-plane.

Step 2: Find the vertices of the feasible region

The vertices of the feasible region are the points where the constraints intersect. We need to find these intersection points: 1. Intersection of x + y = 4 and 2x + y = 5 2. Intersection of x + y = 4 and x = 0 3. Intersection of 2x + y = 5 and x = 0 4. Intersection of x = 0 and y = 0 (origin)

Step 3: Solve the system of linear equations

To find the intersection points from Step 2, we'll need to solve the following systems of linear equations: 1. x + y = 4 and 2x + y = 5: Subtract the first equation from the second to eliminate y: x = 1 Now, plug the value of x = 1 into the first equation to solve for y: 1 + y = 4 ⟹ y = 3 Vertex A: (1, 3) 2. x + y = 4 and x = 0: Plug in x = 0 into the first equation to solve for y: y = 4 Vertex B: (0, 4) 3. 2x + y = 5 and x = 0: Plug in x = 0 into the second equation to solve for y: y = 5 Vertex C: (0, 5) 4. x = 0 and y = 0: Vertex D: (0, 0) Now we have four vertices of the feasible region: A(1, 3), B(0, 4), C(0, 5), and D(0, 0).

Step 4: Evaluate the objective function at each vertex

The objective function is P(x, y) = x + 2y. Let's compute its value at each vertex: 1. P(A) = P(1, 3) = 1 + 2(3) = 7 2. P(B) = P(0, 4) = 0 + 2(4) = 8 3. P(C) = P(0, 5) = 0 + 2(5) = 10 4. P(D) = P(0, 0) = 0

Step 5: Determine the maximum value

Compare the values of the objective function at each vertex to find the maximum value: The maximum value of P(x, y) is 10 and occurs at vertex C(0, 5). Therefore, the solution to this linear programming problem is x=0 and y=5, and the maximum value of the objective function is P = 10.

Key Concepts

Method of Corners

The method of corners is a graphical solution technique for solving linear programming problems. It is especially useful for problems with two variables, where visualization on a graph is manageable. First, you must sketch the constraints and identify the feasible region. This region is bounded by the intersecting lines that represent the equations of the constraints.

Then, you locate the 'corners' or vertices of this region—points where these lines intersect. The optimal solution of a linear programming problem lies at one of these vertices. Calculate the value of the objective function at each vertex; the highest (or lowest, depending on the problem) value indicates the optimal solution. This method transforms a seemingly complex problem into a visual and more easily understood format.

Feasible Region

The feasible region in a linear programming problem is a graphical representation of all possible solutions that satisfy the constraints of the problem. It's depicted on a graph where each constraint is a line or curve, and their collective intersection forms an area.

In our exercise, the feasible region is in the first quadrant because both variables are non-negative. This region is crucial because it contains all potential solutions to the problem, and any solution outside this region either violates one of the constraints or is not practically viable. To enhance understanding, visualize each constraint as a 'fence', and the feasible region is the 'enclosed yard' where possible solutions 'live'.

System of Linear Equations

A system of linear equations consists of two or more linear equations that we aim to solve simultaneously. To find a common solution, typically where these lines intersect, we use algebraic methods like substitution or elimination.

In the context of linear programming, the system of linear equations arises from the intersection points of the constraints. These intersections determine the vertices of the feasible region. As an example, to solve for the intersection of the equations given in our problem, we manipulate the equations by substituting values or adding and subtracting whole equations until the values of the unknowns emerge.

Objective Function

The objective function in linear programming is a mathematical expression that defines what we are trying to maximize or minimize—for instance, profit, cost, distance, etc. It is always expressed in terms of the decision variables of the problem.

In the provided exercise, the objective function is to maximize the value of P(x, y) = x + 2y. To find the maximum, we calculate the value of P for each vertex in the feasible region. We then compare these values; the coordinates that give us the highest value of P represent our optimal solution. Understanding the objective function is key because it gives us the 'goal' of the entire problem-solving process.