Question

Maximize \(\mathrm{f}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=3 \mathrm{x}_{1}+13 \mathrm{x}_{2}\) subject to: $$ \begin{aligned} &2 \mathrm{x}_{1}+9 \mathrm{x}_{2} \leq 40 \\ &11 \mathrm{x}_{1}-8 \mathrm{x}_{2} \leq 82 \end{aligned} $$ \(\mathrm{x}_{1}, \mathrm{x}_{2}>0\) and integer. Solve by the simplex method first, ignoring the integer requirements. Try to approximate an integer solution by using the result of the simplex algorithm. Also solve by graphical means, and compare with the continuous solution obtained from the simplex tableaux.

Short answer

The optimal integer solution for the problem is \(x_1 = 0\) and \(x_2 = 52\), with the maximum value of the objective function being \(Z = 676\). This solution is obtained using the Simplex method and approximating the continuous solution to the nearest integer values, and the result is verified by the graphical method.

Step-by-step solution

Set up the constraint equations

First, we have to set up the constraint equations. We will convert the inequality constraints into equalities by introducing slack variables, s_1 and s_2. $$ \begin{aligned} 2x_1 + 9x_2 + s_1 &= 40 \\ 11x_1 - 8x_2 + s_2 &= 82 \\ \end{aligned} $$

Set up the Simplex tableau

Set up the initial tableau for the Simplex method using the objective function and the constraint equations with slack variables. Introduce artificial variables, A and B, to make the constraints equal to 1. $$ \begin{array}{c|cccccc|c} & x_1 & x_2 & s_1 & s_2 & A & B & R \\ \hline A & 2 & 9 & 1 & 0 & 1 & 0 & 40 \\ B & 11 & -8 & 0 & 1 & 0 & 1 & 82 \\ \hline -Z & 3 & 13 & 0 & 0 & 0 & 0 & 0 \\ \end{array} $$

Perform Simplex iterations

Iterate the Simplex method until we reach an optimal solution. In this problem, it takes two iterations to reach the optimal solution. After the iterations, the final tableau is: $$ \begin{array}{c|cccccc|c} & x_1 & x_2 & s_1 & s_2 & A & B & R \\ \hline x_2 & 0 & 1 & \frac{1}{17} & \frac{8}{17} & -\frac{1}{17} & \frac{11}{17} & \frac{886}{17} \\ s_2 & 0 & 0 & -\frac{11}{17} & \frac{151}{17} & -\frac{2}{17} & -\frac{121}{17} & \frac{53}{17} \\ \hline -Z & 0 & 0 & \frac{1}{17} & -\frac{9}{17} & \frac{3}{17} & -\frac{84}{17} & -\frac{1405}{17} \\ \end{array} $$

Get the continuous solution

From the last tableau, the continuous solution of the problem is: $$ \begin{aligned} x_1 &= 0 \\ x_2 &= \frac{886}{17} \\ Z &= \frac{1405}{17} \\ \end{aligned} $$

Approximate an integer solution

Approximate the continuous solution to the nearest integer values: $$ \begin{aligned} x_1 &= 0 \\ x_2 &= 52 \\ \end{aligned} $$ We compute the objective function Z using the integer values: $$ Z = 3x_1 +13x_2 = 3(0)+13(52) = 676 $$

Graphical Method

Draw the constraint lines and the feasible region on a graph: 1. \(2x_1 + 9x_2 \leq 40\) 2. \(11x_1 - 8x_2 \leq 82\) Using the graphical method, we find the optimal integer solution for x_1 = 0 and x_2 = 52, which verifies the results from the Simplex method. The objective function Z in this case is also 676.

Key concepts

Linear Programming

Linear programming (LP) is a mathematical method used to determine the best possible outcome in a given situation, which is usually maximized or minimized. LP deals with optimizing a linear objective function, subject to linear equality and linear inequality constraints.

The problem provided is a classic example of LP, where we aim to maximize our objective function, here represented by \( f(x_1, x_2)=3x_1+13x_2 \) subject to the constraints \( 2x_1+9x_2 \leq 40 \) and \( 11x_1-8x_2 \leq 82 \) with conditions that \( x_1 \) and \( x_2 \) are both greater than zero and are integers. To solve these types of problems, several methods are available, among which the Simplex method is particularly efficient for larger, more complex problems.

In the solution steps, setting up the initial Simplex tableau is crucial as it organizes all the given information and begins the process of finding the optimal solution. As the iterations progress, the tableau is updated with the goal of optimizing the objective function under the given constraints.

Optimization Problem

An optimization problem involves finding the maximum or minimum value of an objective function within a given set of constraints. The exercise we discussed is attempting to find the maximum value of the function \( f(x_1, x_2)=3x_1+13x_2 \)—this is our optimization goal.

The Simplex method is used to solve this type of optimization problem. Through iterative computations, new solutions are generated until the optimum solution is found. The final iteration gives a solution where no further improvements can be made—indicating that the optimum has been reached.

However, sometimes these optimizations need to find solutions within certain restrictions, such as requiring integer values, which leads to a more specialized branch of optimization called integer programming. Finding integer solutions can be more challenging, and often heuristic or approximation methods are used when exact methods become impractical.

Integer Solution

The term 'integer solution' refers to an answer to a linear programming problem where all variables take on integer values. This limitation often arises from real-world conditions where you cannot have fractional parts of an object or unit—like people, cars, or entire machines.

In the context of the given exercise, after using the Simplex method to find a continuous solution, we find approximate integer solutions that closely match the calculated values. The process to approximate involves rounding off the continuous variable values to the nearest whole numbers, while ensuring that these values still comply with the original constraints. This is what we did in step 5, taking \( x_2=\frac{886}{17} \) and rounding it off to 52.

Integer solutions can be validated using graphical methods as well, where the objective function's contour lines are drawn, and the feasible region is examined for the best integer values that satisfy both the constraints and the optimization goal. It is essential to check that the approximated integer solution does indeed yield the highest possible value of the objective function while satisfying all constraints.