Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve each linear programming problem by the simplex method. $$ \begin{array}{rr} \text { Maximize } & P=15 x+12 y \\ \text { subject to } & x+y \leq 12 \\ & 3 x+y \leq 30 \\ & 10 x+7 y \leq 70 \\ x & \geq 0, y \geq 0 \end{array} $$

Step-by-Step Solution

Answer

The optimal solution for the given linear programming problem using the simplex method is \(x = 0.85714\), \(y = 12.85714\), and the maximum value for the objective function is \(P = 279.85714\).

Step 1: Convert inequalities to equalities by introducing slack variables

Add slack variables s₁, s₂, and s₃ to each inequality constraint to convert them to equalities: 1. \(x + y + s_1 = 12\) 2. \(3x + y + s_2 = 30\) 3. \(10x + 7y + s_3 = 70\) Now we have the following linear programming problem: Maximize \(P = 15x + 12y\) subject to 1. \(x + y + s_1 = 12\) 2. \(3x + y + s_2 = 30\) 3. \(10x + 7y + s_3 = 70\) 4. \(x, y, s_1, s_2, s_3 \geq 0\)

Step 2: Construct initial simplex tableau

Construct the initial simplex tableau for the given problem: $$ \begin{array}{c | c c c c c | c} & x & y & s_1 & s_2 & s_3 & RHS \\\hline s_1 & 1 & 1 & 1 & 0 & 0 & 12 \\\ s_2 & 3 & 1 & 0 & 1 & 0 & 30 \\\ s_3 & 10 & 7 & 0 & 0 & 1 & 70 \\\hline P & -15 & -12 & 0 & 0 & 0 & 0 \end{array} $$

Step 3: Perform simplex iterations

Perform the simplex iteration steps: Iteration 1: Choose the entering variable (the most negative value in the last row): In this case, it's x with a coefficient of -15. Choose the leaving variable by dividing the RHS by the corresponding value of the entering variable, taking the smallest non-negative ratio: \(12/1 = 12\), \(30/3 = 10\), and \(70/10 = 7\). The smallest ratio is 7, so s₃ will be the leaving variable. Now we perform pivot operation on the element in position (3,1) and update the tableau: $$ \begin{array}{c | c c c c c | c} & x & y & s_1 & s_2 & s_3 & RHS \\\hline s_1 & 0 & -0.3 & 1 & 0 & -0.1 & 5 \\\ s_2 & 0 & 0.7 & 0 & 1 & -0.3 & 9 \\\ x & 1 & 0.7 & 0 & 0 & 0.1 & 7 \\\hline P & 0 & 4.5 & 0 & 0 & 1.5 & 105 \end{array} $$ Iteration 2: Choose the entering variable: In this case, it's y with a coefficient of 4.5. Choose the leaving variable: \(5/(-0.3) = -16.67\) (discarded as it's negative), \(9/0.7 = 12.86\). So, s₂ will be the leaving variable. We perform pivot operation on the element in position (2,2) and update the tableau: $$ \begin{array}{c | c c c c c | c} & x & y & s_1 & s_2 & s_3 & RHS \\\hline s_1 & 0 & 0 & 1 & -0.42857 & -0.142857 & 3.85714 \\\ y & 0 & 1 & 0 & 1.42857 & -0.428571 & 12.85714 \\\ x & 1 & 0 & 0 & -1 & 0.428571 & 0.85714 \\\hline P & 0 & 0 & 0 & -3 & 3.428571 & 279.85714 \end{array} $$ There are no more negative values in the bottom row, so we have reached the optimal solution.

Step 4: Determine the optimal solution

From the last tableau, we could see that the optimal solution is: \(x = 0.85714\), \(y = 12.85714\), and the maximum value for the objective function is \(P = 279.85714\).

Key Concepts

Linear Programming

Linear programming is a mathematical method used to determine the best possible outcome or solution from a set of parameters or list of requirements that are presented in the form of linear relationships. It's a powerful technique used in a variety of fields such as economics, business, engineering, and military applications. It helps in maximizing or minimizing a linear objective function, subject to a set of linear inequality or equality constraints.

The primary elements of a linear programming problem include:

• An objective function, such as maximizing profits or minimizing costs.
• A series of linear inequalities or equations as constraints that model resource limits.
• Non-negative variables, expressed through the conditions that they are greater than or equal to zero.

In our exercise, the objective is to maximize the expression \(P = 15x + 12y\), subject to linear constraints. This signifies the goal to determine values for variables \(x\) and \(y\) that maximize \(P\) while staying within the defined limitations.

Slack Variables

In linear programming, constraints are often expressed as inequalities, which can pose a challenge for solution methods. Here is where slack variables come into play. They are introduced to transform inequalities into equalities, making the constraints easier to manage.

For instance, if you have a constraint like \(x + y \leq 12\), you add slack variable \(s_1\) to reformulate it as \(x + y + s_1 = 12\). This equation now accounts for any 'slack' or unused capacity in the system up to the constraint limit.

• Each inequality constraint needs its own slack variable.
• Slack variables represent how much of the available resources remain unused.
• In optimization, they contribute to forming a balanced and complete system for analysis.

In our problem, slack variables \(s_1, s_2,\) and \(s_3\) have been added to three inequalities, converting them into equations, setting the stage for the Simplex Method calculations.

Simplex Tableau

The Simplex Tableau is a tabular representation used in the Simplex Method to simplify and solve linear programming problems. It systematically helps in tracking the calculation progress towards finding the optimal solution.

Here’s how a Simplex Tableau works:

• It arranges the coefficients of the variables along with the slack variables and right-hand side constants.
• The tableau is updated through pivot operations, which involve a series of row transformations to move towards the optimal solution.
• The bottom row of the tableau represents the objective function equation.

The initial tableau begins with the system of equations derived from the constraints and slack variables. For each iteration, a variable enters and one leaves the basis, guided by the coefficients indicating the largest possible increase in the objective function. This structured approach facilitates the transition to a maximum or minimum value point indicated by the absence of negative values in the objective row.

Optimization Problems

Optimization problems involve finding the best solution from a set of feasible solutions. In linear programming, these are often related to maximizing or minimizing an objective function subject to several constraints.

Here's how this ties into our problem:

• The goal is to maximize the profit (objective function \(P = 15x + 12y\)) under certain conditions.
• These conditions are the constraints formed as inequalities in the beginning but resolved through the addition of slack variables.
• The Simplex Method allows us to find the values of \(x\) and \(y\) that provide the maximum value of \(P\) while satisfying all constraints.

In our problem, the solution \(x = 0.85714\) and \(y = 12.85714\) indicates the optimal point at which the objective function is maximized to \(P = 279.85714\), demonstrating how optimization can yield the most efficient and effective results within given limitations.