Question

Maximize $$ z=7 x_{1}+10 x_{2} $$ subject to: $$ \begin{aligned} 5 \mathrm{x}_{1}+4 \mathrm{x}_{2} & \leq 24 \\ 2 \mathrm{x}_{1}+5 \mathrm{x}_{2} & \leq 13 \\ \mathrm{x}_{1}, \mathrm{x}_{2} & \geq 0 \end{aligned} $$ Find an initial feasible solution.

Short answer

The initial feasible solution for the given linear programming problem is the origin \((0, 0)\), where both \(x_1\) and \(x_2\) are \(0\).

Step-by-step solution

Graph the constraints

First, we need to graph the constraint inequalities on a Cartesian coordinate system. To do this, we will treat the inequalities as equalities and find the boundary line for each constraint. For the constraint \(5x_1 + 4x_2 \leq 24\): The boundary line is \(5x_1 + 4x_2 = 24\), which can be rearranged to get \(x_2 = -\frac{5}{4}x_1 + 6 \). For the constraint \(2x_1 + 5x_2 \leq 13\): The boundary line is \(2x_1 + 5x_2 = 13\), which can be rearranged to get \(x_2 = -\frac{2}{5}x_1 + \frac{13}{5}\). Now, plot the boundary lines, shade the region that satisfies the constraints, and find the intersection points.

Identify the feasible region

Next, we need to identify the feasible region, which is the region in the Cartesian coordinate system that satisfies all the constraints. The feasible region will be the region where \(5x_1 + 4x_2 \leq 24\), \(2x_1 + 5x_2 \leq 13\), and \(x_1, x_2 \geq 0\) are all true. When examining the graph, the feasible region is the region which satisfies all the constraints, including the non-negativity constraints \(x_1, x_2 \geq 0\). This region should be a convex polygon.

Find the intersection points of the boundary lines

To identify the vertices of the feasible region, we need to find the points where the boundary lines intersect with each other and with the axes. There are 4 intersection points: 1. Intersection of \(5x_1 + 4x_2 = 24\) and \(2x_1 + 5x_2 = 13\). To find this point, solve the system of linear equations: \[\begin{cases} 5x_1 + 4x_2 = 24\\ 2x_1 + 5x_2 = 13 \end{cases}\] Using substitution or elimination method, we find the intersection point at \((1, 2)\). 2. Intersection of \(5x_1 + 4x_2 = 24\) with the \(x_1\) axis \((x_2=0)\). Plugging in \(x_2=0\), we get \(5x_1 = 24\), hence the intersection point is \((\frac{24}{5}, 0)\). 3. Intersection of \(2x_1 + 5x_2 = 13\) with the \(x_2\) axis \((x_1=0)\). Plugging in \(x_1=0\), we get \(5x_2 = 13\), hence the intersection point is \((0, \frac{13}{5})\). 4. The origin \((0,0)\), where both \(x_1\) and \(x_2\) are \(0\).

Identify the initial feasible solution

We have identified the vertices of the feasible region as \((1, 2)\), \((\frac{24}{5}, 0)\), \((0, \frac{13}{5})\), and \((0, 0)\). All of these points satisfy the constraints, so any of them can be considered as an initial feasible solution. For simplicity, the initial feasible solution can be chosen as the origin \((0, 0)\), where both \(x_1\) and \(x_2\) are \(0\). This means that the initial point is satisfying all the constraints, and we can start our optimization from this point.

Key concepts

Feasible Region

When we talk about linear programming, one of the most central concepts we encounter is the feasible region. This is essentially the area on a graph where all the constraints of a problem are met simultaneously. As seen in the exercise, after plotting the constraints, the feasible region is found where all lines come together to form a sort of shape, often a polygon, where all the given inequalities are true.

This region is significant because it contains all the possible solutions to the linear programming problem. The vertices of this polygonal region are of particular interest because, in a linear programming problem, the optimal solution will always be found at one of these vertices. That's why identifying the feasible region is a foundational step in solving these types of problems.

Constraint Inequalities

In linear programming, problems are defined by constraint inequalities. These are the mathematical expressions that set the limits within which solutions must fall. In the given exercise, we have two main constraints: \(5x_1 + 4x_2 \leq 24\) and \(2x_1 + 5x_2 \leq 13\), as well as the non-negativity constraints \(x_1, x_2 \geq 0\).

When graphing these constraints, we treat them as if they were equalities to find the boundary lines, which defines the edges of the feasible region. The area where these inequalities overlap, and taking into account the axes (since \(x_1, x_2 \geq 0\)), is the feasible region. Solving linear programming problems such as this involves finding the point or points within this region that optimize the objective function, here represented by the equation \(z = 7x_1 + 10x_2\).

Optimization

The ultimate goal of any linear programming problem, such as the one presented, is optimization — finding the 'best' solution according to a given objective function. In our scenario, the function to be maximized is \(z=7x_1+10x_2\).

Optimization in linear programming is a process by which we select the best possible outcome from the feasible region. The word 'best' will depend on whether we are trying to maximize or minimize our objective function. This solution is generally found at one of the corner points of the feasible region polygon because these points represent the extremes of the region given the constraints, and therefore, the potential for the optimum. In our exercise, these corner points were methodically evaluated to ensure the maximum value of the objective function.

Graphical Method

The graphical method is a way of solving linear programming problems by visually representing constraints and objective functions on a graph. As illustrated in the step-by-step solution of the exercise, the process involves drawing the constraints on a graph which intersect to outline the feasible region.

Each constraint's line is plotted by treating the inequality as an equality. The area that respects all constraints displays where potential solutions exist. By pinpointing where these lines cross, we find the coordinates of the feasible region's vertices. The graphical method makes it easy to visualize the problem at hand, making it particularly useful for linear programming problems with two variables, as we can clearly see which points lie within the feasible area and thus are potential candidates for optimization.