Problem 59 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Consider the linear programming problem $$ \begin{aligned} \text { Minimize } & C=-2 x+5 y \\ \text { subject to } & x+y \leq 3 \\ & 2 x+y \leq 4 \\ & 5 x+8 y \geq 40 \\ & x \geq 0, y \geq 0 \end{aligned} $$ a. Sketch the feasible set. b. Find the solution(s) of the linear programming problem, if it exists.

Step-by-Step Solution

Answer

In the linear programming problem, we identify the feasible region by graphing the constraints and find the corner points of the feasible set: \(A(1, 2)\), \(B(0, 3)\), and \(C(2, 0)\). Evaluating the objective function, \(C = -2x + 5y\), at these points, we find that the minimum value of \(C\) occurs at point \(A(1, 2)\) with a value of \(C = 1\). So, the optimal solution is \((x, y) = (1, 2)\) and the minimum value of the objective function is \(C = 1\).

Step 1: Identify and graph the feasible region

To begin, we must transform the inequality constraints to equations: 1. \(x + y = 3\) 2. \(2x + y = 4\) 3. \(5x + 8y = 40\) Now plot these lines on the \(xy\)-plane. This will divide the plane into regions. Determine which side of each line is part of the feasible region by plugging in any convenient point (e.g., the origin) into the inequalities and check if they hold. Shade the appropriate regions to identify the feasible region.

Step 2: Locate the corner points of the feasible set

Next, find the intersections of these lines, which represent the corner points of the feasible region. To do this, you can use different methods such as solving the equations simultaneously or by substitution: 1. Intersection of \(x + y = 3\) and \(2x + y = 4\): Solve the equations simultaneously to find the coordinates \((x,y)\). 2. Intersection of \(x + y = 3\) and \(5x + 8y = 40\): Solve the equations simultaneously to find the coordinates \((x,y)\). 3. Intersection of \(2x + y = 4\) and \(5x + 8y = 40\): Solve the equations simultaneously to find the coordinates \((x,y)\).

Step 3: Determine the objective function values at the corner points

Evaluate the objective function \(C = -2x + 5y\) at each of the corner points obtained in Step 2. Compare the values to determine which point results in the minimum value of \(C\). This point and the corresponding value of \(C\) will be the optimal solution to the linear programming problem. Note: If the feasible region is unbounded and there is no minimum or maximum value of the objective function, the problem will have no feasible solution. In this case, the objective function has a minimum value, and the problem has a feasible solution.

Key Concepts

Feasible Region

The feasible region is a fundamental concept in linear programming that represents all the possible points that satisfy the constraints of the problem. In simpler terms, it's the 'playing field' within which we search for the best solution.

Think of it as drawing a map based on given directions. For the linear programming exercise at hand, we have a set of inequalities that describe the constraints. Just as you would sketch out a treasure map's boundaries, we graph these inequalities on a coordinate plane. The feasible region is the shaded area where the 'treasure' (or in our case, the optimal solution) can be found.

The feasible region is vital for locating the best solution—it's where we apply the optimization method later on. To correctly sketch out the feasible set, remember to:

• Transform inequality constraints into equations.
• Graph the lines on an xy-plane.
• Use a test point like the origin to decide which side of the line to shade.

Only the common shaded area meeting all constraints is our feasible region. Here, we're looking for the corner points where the treasure might be buried!

Objective Function

Like the X that marks the spot on a treasure map, the objective function in linear programming points to our goal. The objective function is the formula we want to optimize—to either minimize costs or maximize profits, depending on the problem.

In our exercise, the objective function is given by the equation C = -2x + 5y. This represents what we are aiming to minimize. Think of each x and y as different paths to the treasure—the coefficients (-2 and 5, respectively) tell us how much each path costs or contributes to the goal.

How do we use it? Once we have located the feasible region and identified the corner points, we evaluate the objective function at each of these points. This is like checking each spot on the map where treasure could be hidden and seeing which one is the most valuable. The point that gives us the minimum (or maximum) value is where we dig for gold—the optimal solution. Remember to compare the objective function's values to identify the best solution!

Optimization Methods

Now that we know where to look for the treasure—the feasible region—and what we're looking for—the objective function value—we need a method for finding the ultimate treasure spot. Optimization methods in linear programming are the strategies used to find the best solution within the feasible region.

There are various methods for optimization, such as the Simplex method, Graphical method, and Interior Point methods, among others. For the exercise in question, we used the Graphical method. This involves plotting the feasible region, finding the corner points (aka potential treasure spots), and then evaluating the objective function at these points to determine the optimal solution.

The Graphical method works well for problems with two variables but becomes impractical with more variables. In such cases, more advanced methods like the Simplex algorithm, which can handle multiple dimensions, are used. Always remember to check if the problem has a bounded feasible region because some optimization problems might not have a solution if the objective function can be decreased indefinitely!