Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The problem Maximize \(\quad P=x y\) $$ \text { subject to } \begin{aligned} & 2 x+3 y \leq 12 \\ & 2 x+y \leq 8 \\ & x \geq 0, y \geq 0 \end{aligned} $$ is a linear programming problem.

Step-by-Step Solution

Answer

The statement is false because the objective function, P = xy, is not linear. While the constraints are linear inequalities and there are non-negativity constraints on the variables, a linear programming problem requires a linear objective function.

Step 1: Objective Function Linearity

In the given problem, the objective function is P = xy. This is not a linear function, as it is a product of x and y.

Step 2: Constraints Linearity

The constraints in the problem are: 1. \(2x + 3y \leq 12\) 2. \(2x + y \leq 8\) These constraints are both linear inequalities.

Step 3: Non-negativity Constraints

The variables x and y both have a non-negativity constraint. Since the objective function is not linear, the given problem is not a linear programming problem. The statement is #tag_underline#false.

Key Concepts

Objective Function

In linear programming, the objective function represents the goal that needs to be either maximized or minimized, such as profit, cost, or distance. It's essential for this function to be linear, which means it should consist of the sum of the decision variables, each multiplied by a constant. For example, an objective function could be something like \( c_1x_1 + c_2x_2 + \ldots + c_nx_n \), where \( c_i \) are constants and \( x_i \) are the variables.

In the discussed exercise, the objective function is \( P = xy \), which is a product of the two variables involved. As multiplication of variables introduces a nonlinearity (since the rate of change is not constant), this means the function is not linear, and therefore, the problem does not qualify as a linear programming problem. Our improvement advice is centered on recognizing the linear forms of objective functions to correctly categorize optimization problems as linear programming problems.

Non-negativity Constraints

The non-negativity constraints in linear programming are conditions that require all variables to be greater than or equal to zero, meaning that negative values are not permissible. These constraints typically represent real-world conditions where you can't have negative quantities, such as the amount of goods produced or time spent on a task.

These constraints are crucial for ensuring that the solutions to linear programming problems remain practical and applicable. In the provided exercise, we have \( x \geq 0 \) and \( y \geq 0 \), affirming that both \( x \) and \( y \) must be at least zero or positive. This is common practice in linear programming as it reflects reality where negative responses often do not make sense.

Linear Inequalities

Within the spectrum of math optimization, linear inequalities play a pivotal role. These inequalities involve linear expressions set within greater than, less than, greater than or equal to, or less than or equal to conditions. For instance, \( 2x + 3y \leq 12 \) and \( 2x + y \leq 8 \) are both linear inequalities.

These expressions offer boundary conditions that form a feasible region within which the solution must lie. If the inequalities were not linear, this would complicate the region or potentially make it impossible to define. That is why, when you are solving or setting up a linear programming problem, it is critical to ensure that all constraints are linear to allow for the solvable convex region to form. This is rightly identified in the step-by-step solution provided for the exercise.

Mathematical Optimization

The concept of mathematical optimization encapsulates the pursuit of the best solution from a set of available alternatives, guided by a precise set of criteria. In linear programming, this pertains to finding the maximum or minimum value of a linear objective function within a region determined by a set of linear inequalities.

Optimization can be applied to a plethora of fields such as economics, engineering, logistics, and management, making it an interdisciplinary tool. The exercise given provides a scenario where optimization techniques are required, but as noted, it doesn't qualify as linear programming due to the non-linear objective function. Emphasizing this distinction is a crucial learning point, which underscores the importance of identifying the right mathematical optimization technique based on the problem's characteristics.