5.3 Q-1

Question Statement

Maximize the function $f(x, y) = 2x + 5y$ subject to the following constraints:

2y - x \leq 8, \quad x - y \leq 4, \quad x \geq 0, \quad y \geq 0

Background and Explanation

This is a linear programming problem, where we are asked to maximize a linear objective function subject to certain constraints. The constraints define a feasible region, and the solution will be found at one of the corner points of this region. The method involves plotting the constraints, identifying the feasible region, and then evaluating the objective function at the corner points.

Solution

Step 1: Graph the constraints

We start by solving each inequality to find the boundary lines and determine the feasible region.

Constraint 1: $2y - x \leq 8$

Rearrange it to get:

x \geq 2y - 8 \tag{1}

Substitute $x = 0$ into equation (1):

2y - 0 = 8 \quad \Rightarrow \quad y = 4

So, the point $(0, 4)$ is on the line.

Now substitute $y = 0$ :

2(0) - x = 8 \quad \Rightarrow \quad x = -8

But since $x \geq 0$ in our constraints, this point is not valid.

Next, check when $x = 0, y = 0$ in the inequality:

0 - 0 < 8

This is true, so the inequality is satisfied.

Constraint 2: $x - y \leq 4$

Rearrange this to get:

x \leq y + 4 \tag{2}

Substitute $x = 0$ into equation (2):

0 - y = 4 \quad \Rightarrow \quad y = -4

But $y \geq 0$ , so this point is not valid.

Substitute $y = 0$ :

x - 0 = 4 \quad \Rightarrow \quad x = 4

So, the point $(4, 0)$ is on the line.

Now check for $x = 0, y = 0$ :

0 - 0 < 4

This is true, so the inequality is satisfied.

Step 2: Find the feasible region

The feasible region is the area where all constraints are satisfied simultaneously. We now identify the corner points of the feasible region by solving the intersection of the lines:

From the equation $x = 4 + 12 = 16$ and the previous equations, we calculate $y = 12$ .
The intersection points are $(16, 12)$ , $(0, 4)$ , $(0, 0)$ , and $(4, 0)$ .

Step 3: Evaluate the objective function at the corner points

The objective function is $f(x, y) = 2x + 5y$ . We now calculate the value of $f(x, y)$ at each corner point:

At $(16, 12)$ :

f(16, 12) = 2(16) + 5(12) = 32 + 60 = 92

At $(0, 4)$ :

f(0, 4) = 2(0) + 5(4) = 0 + 20 = 20

At $(0, 0)$ :

f(0, 0) = 2(0) + 5(0) = 0 + 0 = 0

At $(4, 0)$ :

f(4, 0) = 2(4) + 5(0) = 8 + 0 = 8

Key Formulas or Methods Used

Objective Function: $f(x, y) = 2x + 5y$
Linear Constraints: $2y - x \leq 8$ , $x - y \leq 4$
Corner Point Evaluation: The maximum value of the objective function occurs at a corner point of the feasible region.

Summary of Steps

Write the constraints in terms of equality to find boundary lines.
Graph the inequalities to identify the feasible region.
Find the corner points by solving the intersections of the constraints.
Evaluate the objective function at each corner point.
The maximum value of $f(x, y)$ is found at the corner point $(16, 12)$ , giving $f(16, 12) = 92$ .