Problem 12 - 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.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=40 x+45 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0}\end{array} $$ $$ \begin{array}{l}{8 x+9 y \leq 7200} \\ {8 x+9 y \geq 3600}\end{array} $$

Step-by-Step Solution

Answer

The minimum and maximum values of the objective function 'z' is 18000. These values occur at the intersection points (0,400) and (450,0) respectively.

Step 1: Rewrite Constraints in Y =

Rewriting the constraints in terms of 'y' (equate y in terms of x and other constants), we get \(y \leq 800 - \frac{8}{9}x\) and \(y \geq 400 - \frac{8}{9}x\)

Step 2: Find Intersection Points

We know that the feasible region lies between the constraints \(y \geq 400 - \frac{8}{9}x\) and \(y \leq 800 - \frac{8}{9}x\). In addition, we plot the points for \(x \geq 0\) and \(y \geq 0\). To find where these constraints intersect, we solve \(400 - \frac{8}{9}x = 800 - \frac{8}{9}x\), which gives one point of intersection as \((0,400)\). Another point can be found where \(y = 0\), which gives \(x = 450\) and thus another point is \((450,0)\).

Step 3: Evaluate Objective Function at Intersection Points

Substituting these intersection points into the objective function will yield its minimum and maximum values. Thus, \(z(0,400) = 40 * 0 + 45 * 400 = 18000\) and \(z(450,0) = 40 * 450+ 45 * 0 = 18000\). Because these two values are equal, the minimum and maximum values of 'z' occur at these points and are both '18000'.

Key Concepts

Objective Function

A core aspect of linear programming is the objective function. This is the mathematical expression that needs optimization—either maximization or minimization. In our problem, the objective function is given by \( z = 40x + 45y \). This equation represents a linear relationship. Here, \( z \) is what we want to optimize based on the values of \( x \) and \( y \). The coefficients 40 and 45 signify the contribution of variables \( x \) and \( y \) to the value of \( z \), respectively. It's crucial in such problems to clearly define your objective function, as this is where your answer will hinge on determining optimal points within the feasible region. Once the feasible region is identified, we evaluate \( z \) at each corner point to identify possible extremes of \( z \).

Let's break it down to basic understanding:

• The objective function aims to find optimal values.
• It must maintain a linear relationship within the problem scope.
• Values of the variables that satisfy the constraints are plugged into this function.
• The goal is to determine where \( z \) is as small or large as possible based on these values.

Constraints

Constraints are conditions that limit the solutions of the problem. They form the backbone of any linear programming task as they define the feasible region where a solution might lie. In this particular exercise, constraints are inequalities involving \( x \) and \( y \). They include:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( 8x + 9y \leq 7200 \)
• \( 8x + 9y \geq 3600 \)

These constraints essentially shape a region on a graph known as the feasible region. By converting these constraints into equations, we are able to clearly visualize feasible solutions. Each constraint restricts the environment, playing an essential role in determining the region wherein optimal points for the objective function must lie. Only combinations of \( x \) and \( y \) that satisfy all these inequalities exhibit potential solutions.

Feasible Region

In linear programming, the feasible region is the area on a graph where all the constraints overlap. It represents all possible combinations of \( x \) and \( y \) that satisfy every constraint in the system. Feasibility is key as it represents where potential solutions, if any, may be found.

In the given exercise, the constraints define a bounded area which can be visualized by plotting the lines:

• \( y \leq 800 - \frac{8}{9}x \)
• \( y \geq 400 - \frac{8}{9}x \)
• \( x \geq 0 \)
• \( y \geq 0 \)

These lines intersect to form a polygon on the graph, the interior of which is the feasible region. Solutions to the objective function can only occur here.

To ensure clarity:

• The feasible region is essential for determining potential solutions.
• All solutions must lie within this bounded area.
• It is represented as the overlap of all constraints on a graph.

Intersection Points

Intersection points are where two or more constraint lines meet. These points are significant in linear programming problems, as they are often where optimal solutions reside. In this exercise, intersection points are computed by solving the equality of constraints. For instance, where \( y = 800 - \frac{8}{9}x \) intersects with \( y = 400 - \frac{8}{9}x \). Solving such equations gives us critical points.

From our example:

• The intersection at \((0, 400)\) occurs when \( x=0 \) and translating \( y \) accordingly.
• The intersection at \((450, 0)\) is reached when \( y=0 \), indicating how far \( x \) stretches.

Such points are fundamental because the objective function values tested at these points yield the potential minimum or maximum values within the feasible region.

Key points to remember:

• Intersection points are derived from solving equalities between constraints.
• Optimal solutions often occur at these critical points.
• They are vital for evaluating the objective function effectively.