Problem 7 - Precalculus with Limits

Question

Solving a Linear Programming Problem, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=4 x+3 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+y \leq 5}\end{array} $$

Step-by-Step Solution

Answer

The maximum value of the objective function is 20 at the point (5,0) and the minimum value of the function is 0 at the point (0,0).

Step 1: Identifying the Constraints

There are three constraints in this problem, which are \( x \geq 0 \), \( y \geq 0 \), and \( x+y \leq 5 \). These constraints define a region in the xy-plane, known as the feasible region, which contains all possible solutions to the problem.

Step 2: Identifying the Region Determined by the Constraints

With \( x \geq 0 \) and \( y \geq 0 \), we are restricted to the first quadrant. Additionally, \( x+y \leq 5 \) confines the region by the line \( y=5-x \). Therefore, the feasible region is a right triangle in the first quadrant bounded by the axes and the line \( y=5-x \). The vertex coordinates of the triangle are (0,0), (5,0), and (0,5).

Step 3: Evaluating the Objective Function at the Corner Points

To find the minimum and maximum values of the objective function \( z=4x+3y \), we substitute the corner points into the objective function:For (0,0), \( z=4(0)+3(0)=0 \)For (5,0), \( z=4(5)+3(0)=20 \)For (0,5), \( z=4(0)+3(5)=15 \)So, the minimum value of the function is 0 at (0,0) and the maximum value of the function is 20 at (5,0).

Key Concepts

Objective Function

The objective function in a Linear Programming Problem (LPP) is a mathematical expression that we are seeking to maximize or minimize. In our exercise, the objective function is given by the formula:
\[ z = 4x + 3y \].
This function represents the goal of the LPP, where 'z' is the objective value we want to optimize, and 'x' and 'y' are the variables controlled within the constraints of the problem. To solve for the minimum and maximum values of 'z', we evaluate the objective function at the corner points of the feasible region, which are derived from the constraints. The process of evaluating the objective function helps us identify where our maximum or minimum values occur, essential in business applications like cost minimization or profit maximization. It's crucial to understand that the objective function embodies the very purpose of the LPP—without it, there's no criterion for decision-making.

Constraints

Constraints in a Linear Programming Problem are the conditions or restrictions that limit the feasible solutions to the problem. The constraints for our exercise are:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( x + y \leq 5 \)

These constraints are represented by linear inequalities that define the range of possible values for the variables 'x' and 'y'. The first two constraints signify that 'x' and 'y' cannot be negative, meaning that our solutions must lie in the first quadrant of the Cartesian plane. The third constraint indicates a boundary for the combined value of 'x' and 'y', so that their sum does not exceed 5. In essence, constraints are the rules of the game—they carve out the 'playing field' for the LPP, known as the feasible region.

Feasible Region

The feasible region in Linear Programming is a graphical representation of all potential solutions that satisfy the problem's constraints. In the provided exercise, the feasible region is a right triangle in the first quadrant of the xy-plane. It's illustrated by:

• The x-axis (since \( x \geq 0 \))
• The y-axis (since \( y \geq 0 \))
• The line \( y = 5 - x \) (from the constraint \( x + y \leq 5 \))

This region contains all the pairs (x, y) that fulfill the given limitations. To find the optimum value of the objective function within the feasible region, we focus on the corner points of the region. These points are where the maximum or minimum values of the objective function will reside because of the linear nature of both the objective function and the constraints. This is known as the 'corner-point principle.' In our exercise, we've identified the corner points as (0,0), (5,0), and (0,5). The feasibility of the solutions is critical, highlighting that even the best objective function value is useless if it doesn't fall within this allowed region.