Question

Consider the following problem: Maximize \(\mathrm{P}=5 \mathrm{x}_{1}+8 \mathrm{x}_{2}\) subject to $$ \begin{array}{r} 2 \mathrm{x}_{1}+\mathrm{x}_{2} \leq 14 \\ \mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 12 \\ \mathrm{x}_{2} \leq 3 \\ \mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0 \end{array} $$ Suppose that an additional constraint on \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) is imposed: $$ \mathrm{x}_{1}+\mathrm{x}_{2} \leq \mathrm{K} $$ where \(\mathrm{K}\) is some unspecified amount. How does the solution of \((1),(2)\) and (3) change as \(\mathrm{K}\) varies from zero to very large values?

Short answer

The solution to the given linear programming problem consists of finding the feasible region based on the given constraints, identifying the vertices of this region, and evaluating the objective function P at each vertex to find the optimal solution. The additional constraint \(x_1 + x_2 \leq K\) introduces variability in the feasible region, affecting the vertices and the optimal solution. As K varies from zero to very large values, the optimal vertex may change, affecting the overall solution of the problem. To analyze the impact of K on the solution, consider how the intersection with other constraints changes, and observe how the optimal solution depends on the value of K.

Step-by-step solution

Rewrite the constraints in slope-intercept form

To visualize the feasible region, we rewrite the inequalities as equations in slope-intercept form. For the constraints, we have: 1. \(2x_1 + x_2 = 14\) -> \(x_2 = -2x_1 + 14\) 2. \(x_1 + 3x_2 = 12\) -> \(x_2 = -\frac{1}{3}x_1 + 4\) 3. \(x_2 = 3\) 4. \(x_1 = 0\) (Boundary for \(x_1 \geq 0\)) 5. \(x_2 = 0\) (Boundary for \(x_2 \geq 0\)) 6. \(x_1 + x_2 = K\)

Find the feasible region

Now, we can plot the inequalities on a coordinate plane to visualize the feasible region: 1. For constraint \(2x_1 + x_2 \leq 14\), the feasible region is below the line \(x_2 = -2x_1 + 14\). 2. For constraint \(x_1 + 3x_2 \leq 12\), the feasible region is below the line \(x_2 = -\frac{1}{3}x_1 + 4\). 3. For constraint \(x_2 \leq 3\), the feasible region is below the line \(x_2 = 3\). 4. For constraint \(x_1 \geq 0\), the feasible region is right of the line \(x_1 = 0\). 5. For constraint \(x_2 \geq 0\), the feasible region is above the line \(x_2 = 0\). 6. For additional constraint \(x_1 + x_2 \leq K\), the feasible region is below the line \(x_2 = -x_1 + K\). By intersecting these regions, we find a feasible region, in the form of a polygon or an unbounded region if K varies from zero to very large values.

Find the vertices of the feasible region

We identify the vertices of the feasible region by considering the intersections among the lines. Since we have six lines and not all of them intersect at the same time, we will consider the intersection of pairs of lines. 1. \(x_2 = -2x_1 + 14\) and \(x_2 = -\frac{1}{3}x_1 + 4\) 2. \(x_2 = -2x_1 + 14\) and \(x_2 = 3\) 3. \(x_2 = -2x_1 + 14\) and \(x_1 = 0\) 4. \(x_2 = -\frac{1}{3}x_1 + 4\) and \(x_2 = 3\) 5. \(x_2 = -\frac{1}{3}x_1 + 4\) and \(x_1 = 0\) 6. \(x_1\) = 0 and \(x_2 = -x_1 + K\) Solve these systems of equations to find the coordinates of the vertices. Keep in mind that as K varies, the intersection of \(x_2 = -x_1 + K\) with other lines will change, creating different vertices, and determining the overall feasible region.

Find the optimal solution

The solution of the LP problem can be found at one of the vertices of the feasible region. Evaluate P at each vertex found in the previous step and pick the one(s) that gives the maximum value of P. Keep in mind that as K varies, the vertices and the optimal vertex may change, affecting the overall solution of the problem. Consider how the addition of the constraint \(x_1 + x_2 \leq K\) influences the feasible region as K goes from zero to very large values, and observe how the optimal solution depends on the value of K.

Key concepts

Feasible Region

The feasible region is a foundational concept in linear programming, representing all possible solutions that satisfy a given set of constraints. When plotting the constraints of a linear programming problem, the feasible region is depicted as the overlapping area where all the inequalities intersect.

For instance, consider a simple system involving two variables, \( x_1 \) and \( x_2 \). Each constraint bounds the values these variables can take. In a graphical sense, the feasible region is a shape, often a polygon, on the coordinate plane. Each edge of the polygon corresponds to a constraint, and any point within this region (including the edges) is a possible solution to the system of inequalities. In the exercise discussed, varying the value of \( K \) in the added constraint \( x_1 + x_2 \leq K \) will shift one of the boundaries, effectively reshaping the feasible region. The feasible region is particularly important because it contains the optimal solution to the linear programming problem.

Constraints in Slope-Intercept Form

Linear programming problems often involve constraints that limit the values of the decision variables. Expressing these constraints in slope-intercept form, \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept, greatly simplifies the process of graphing these constraints and identifying the feasible region.

For example, in the step-by-step solution provided, the constraint \( 2x_1 + x_2 \leq 14 \) can be rewritten in slope-intercept form as \( x_2 = -2x_1 + 14 \). This transformation from an inequality to an equation facilitates the plotting of a straight line, which represents the boundary of the feasible region for that particular constraint. By converting all constraints into this form and graphing them, one can easily visualize where the lines intersect to form the feasible region and thus lay the groundwork for identifying the optimal solution.

Optimization Problem

An optimization problem in the context of linear programming involves finding the maximum or minimum value of an objective function, subject to a set of constraints. The objective function, typically denoted as \( P \), \( Z \), or similar, is a linear equation in terms of the decision variables. It represents a quantity to be optimized, such as profit, cost, or time.

In the given exercise, the objective function is \( P = 5x_1 + 8x_2 \), and the goal is to maximize this function. To solve an optimization problem, one must find a point within the feasible region where \( P \) has the highest possible value (for maximization problems) or the lowest possible value (for minimization problems). This point is typically found at one of the vertices of the feasible region. It is important to note that altering the constraints, such as by changing the value of \( K \), can shift the feasible region and thus the location of the optimal solution.

System of Inequalities

A linear programming problem is characterized by a system of inequalities, which are used to describe constraints on the decision variables. Each inequality restricts the range of feasible values these variables can take. The system is composed of several linear inequalities that together define the feasible region where a solution must lie.

As seen in the exercise, we have inequalities such as \( 2x_1 + x_2 \leq 14 \) and \( x_1 + 3x_2 \leq 12 \), which constrain the values that \( x_1 \) and \( x_2 \) can take. Additionally, the non-negativity constraints \( x_1 \geq 0 \) and \( x_2 \geq 0 \) are typically present to exclude negative values. When an additional constraint is added to the system, such as \( x_1 + x_2 \leq K \) with a variable \( K \), it introduces a new boundary to the feasible region. The intersections of these inequalities create vertices on the graph, and the optimal solution of the linear programming problem is usually located at one of these vertices.