Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the maximum and/or minimum value(s) of the objective function on the feasible set \(S .\) $$Z=3 x-y$$

Step-by-Step Solution

Answer

The maximum value of the objective function \(Z = 3x - y\) on the feasible set S is 30, which occurs at the vertex (10, 0). The minimum value of the objective function on the feasible set S is -10, which occurs at the vertex (0, 10).

Step 1: Identify the Constraints and Feasible Set

Begin by identifying the constraints that define the feasible set S. In this case, they are: 1. \(x \ge 0\) 2. \(y \ge 0\) 3. \(x + y \le 10\) Together, these constraints form a set of points in the \(xy\)-plane. Let's now find all intersections of these constraints.

Step 2: Find All Vertices of the Feasible Set

To find the vertices of the feasible set, we need to find all intersections of the constraints. We can do this by solving each pair of constraints as a system of linear equations: - Intersection of \(x \ge 0\) and \(y \ge 0\): \(x = 0, y = 0\). The point is (0, 0). - Intersection of \(x \ge 0\) and \(x + y \le 10\): \(x = 0, y + 0 \le 10\). The point is (0, 10). - Intersection of \(y \ge 0\) and \(x + y \le 10\): \(x + 0 \le 10, y = 0\). The point is (10, 0). The feasible set's vertices are (0, 0), (0, 10), and (10, 0).

Step 3: Evaluate the Objective Function at Each Vertex

Now, substitute the coordinates of each vertex into the objective function \(Z=3x-y\) and find its value at each vertex: - For vertex (0, 0): \(Z = 3(0) - 0 = 0\) - For vertex (0, 10): \(Z = 3(0) - 10 = -10\) - For vertex (10, 0): \(Z = 3(10) - 0 = 30\)

Step 4: Find the Maximum and/or Minimum Value(s)

After evaluating the objective function at each vertex, we can determine its maximum and minimum value(s): - Maximum value of Z: 30 (at vertex (10, 0)) - Minimum value of Z: -10 (at vertex (0, 10)) In conclusion, the maximum value of the objective function on the feasible set S is 30, which occurs at the vertex (10, 0). The minimum value of the objective function on the feasible set S is -10, which occurs at the vertex (0, 10).

Key Concepts

Objective Function

An objective function is a mathematical expression that one seeks to optimize—either maximize or minimize—during an optimization process. In this context, we focus on the function \( Z = 3x - y \), where \(Z\) represents the value we want to either maximize or minimize. Objective functions are central in optimization problems because they provide a way to measure how well a particular solution achieves the desired outcome.

• The goal is to find values of \(x\) and \(y\) within given constraints that result in the highest or lowest possible value of \(Z\).
• In many real-world problems, the objective function might represent profits, costs or other quantities one wants to optimize.

Understanding the role of the objective function is key because it helps us set our priorities in any optimization problem.

Linear Programming

Linear programming is a method used to achieve the best outcome in a mathematical model. Its functions are expressed as linear relationships. This approach is widely applicable in industries where decision-making is required.

• Linear programming models consist of an objective function, a set of constraints, and decision variables (like \(x\) and \(y\) in our specific problem).
• Linear equations form both the objective function and the constraints.

In our exercise, linear programming is used to determine the maximum and minimum values of the function \(Z = 3x - y\) considering the constraints imposed on \(x\) and \(y\). Solving linear programming problems generally involves analyzing feasible solutions, making it essential to understand the feasible set before proceeding.

Feasible Set

The feasible set is the collection of all possible points that satisfy the constraints of an optimization problem. These constraints often include inequalities or equalities that limit the values variables like \(x\) or \(y\) can take. In our problem, the constraints are:

• \(x \ge 0\)
• \(y \ge 0\)
• \(x + y \le 10\)

The feasible set is graphically represented by the region which satisfies all these constraints simultaneously. It assists in identifying the potential solutions for the objective function. By finding the vertices of the feasible set, we can insert these values into the objective function to evaluate and identify maximum or minimum outcomes.
Understanding the feasible set is vital in linear programming because it defines the boundary of possible solutions from which you can select the optimal one.