Question

Maximize \(\mathrm{x}_{0}=4 \mathrm{x}_{1}+3 \mathrm{x}_{2}\) subject to: \(\quad 2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 6\) \(-3 \mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 3\) \(2 \mathrm{x}_{2} \leq 5\) \(2 \mathrm{x}_{1}+\quad \mathrm{x}_{2} \leq 4\) \(\mathrm{x}_{1}, \mathrm{x}_{2} \geq 0\) Find the standard form of this linear programming problem.

Short answer

The standard form of the given linear programming problem is: Minimize \(x_0 = -4x_1 - 3x_2\) Subject to: 1. \(2x_1 + 3x_2 + s_1 = 6\) 2. \(-3x_1 + 2x_2 + s_2 = 3\) 3. \(2x_2 + s_3 = 5\) 4. \(2x_1 + x_2 + s_4 = 4\) With non-negativity constraints: \(x_1, x_2, s_1, s_2, s_3, s_4 \geq 0\)

Step-by-step solution

change maximize to minimize

Since we are looking for a standard form, we need to change the objective of maximizing to minimizing. We can do that by introducing a new variable, \(x_0\), as follows: Minimize \(x_0 = -4x_1 - 3x_2\).

Introduce slack variables

We have four inequalities that need to be converted into equations: 1. \(2x_1 + 3x_2 \leq 6\) 2. \(-3x_1 + 2x_2 \leq 3\) 3. \(2x_2 \leq 5\) 4. \(2x_1 + x_2 \leq 4\) We'll now introduce a slack variable for each inequality, which we'll call \(s_i\), where \(i\) is the inequality number: 1. \(2x_1 + 3x_2 + s_1 = 6\) 2. \(-3x_1 + 2x_2 + s_2 = 3\) 3. \(2x_2 + s_3 = 5\) 4. \(2x_1 + x_2 + s_4 = 4\) The new problem also now has non-negativity constraints for both variables and slack variables: \(x_1, x_2, s_1, s_2, s_3, s_4 \geq 0\)

Rewrite the problem with changes

Now that we've introduced slack variables and changed the objective to minimize, we can rewrite the linear programming problem in standard form: Minimize \(x_0 = -4x_1 - 3x_2\) Subject to: 1. \(2x_1 + 3x_2 + s_1 = 6\) 2. \(-3x_1 + 2x_2 + s_2 = 3\) 3. \(2x_2 + s_3 = 5\) 4. \(2x_1 + x_2 + s_4 = 4\) With non-negativity constraints: \(x_1, x_2, s_1, s_2, s_3, s_4 \geq 0\)

Key concepts

Standard Form

Linear programming problems can be expressed in various forms, but the standard form is one of the most widely used. In standard form, a linear programming problem involves minimizing a linear objective function, subject to a set of equality constraints. The problem also includes non-negativity constraints for all variables involved. The transition from an inequality to equality is made by introducing slack variables. This simplification allows the application of many simplex algorithm approaches effectively.
When converting a problem to standard form from a maximization problem, as in our example, it's necessary to shift the objective to minimization by flipping the sign of the coefficients of the terms in the objective function. Hence, "Maximize \(4x_1+3x_2\)" becomes "Minimize \(-4x_1-3x_2\)."
This step is essential for professional and academic applications since most optimization tools and theoretical approaches assume standard form.

Slack Variables

To transform inequalities into equalities, you introduce slack variables. These are auxiliary variables added to each inequality constraint. The slack variables precisely account for the difference between the two sides of an inequality, effectively "slacking off" the constraint by adding unused capacity. For instance, if an inequality is expressed as \(2x_1 + 3x_2 \leq 6\), incorporating a slack variable \(s_1\) modifies it to \(2x_1 + 3x_2 + s_1 = 6\).
Slack variables must also satisfy non-negativity constraints. They are assumed to be non-negative since they represent additional positive capacity. It's important to remember:

• A distinct slack variable is added to each inequality.
• By introducing slack variables, the problem formulation maintains a structure suitable for solving using matrix methods.

This transformation is critical for implementing the Simplex method, a popular method for solving linear programming problems efficiently.

Objective Function

The objective function in linear programming defines the criterion you want to maximize or minimize. It represents the goal of the optimization task. In many real-world problems, businesses aim to maximize profit or minimize costs, which can be mathematically represented by an objective function.
In our exercise, the initial objective was to maximize the expression "\(4x_1 + 3x_2\)." For standard form, this is translated to "minimize \(-4x_1 - 3x_2\)."
Key points about objective functions include:

• It comprises decision variables multiplied by their respective coefficients, which reflect their contribution to the objective goal.
• The function can be linear, and in this format, it's suitable for applying linear programming techniques.

The formulation of an appropriate objective function is crucial as it directly influences the results derived from the optimization problem.

Non-Negativity Constraints

Non-negativity constraints are an essential aspect of linear programming. They ensure that all decision variables take on values that are zero or positive. Many real-life quantities, such as quantities produced, money amounts, and resource capacities, naturally require these constraints for a realistic and feasible solution.
In the context of the exercise, the non-negativity constraints are expressed as: \(x_1, x_2, s_1, s_2, s_3, s_4 \geq 0\).
These ensure:

• Both the original decision variables \(x_1\) and \(x_2\) must not be negative, aligning with physical or logical bounds in practical scenarios.
• Slack variables \(s_1, s_2, s_3, s_4\) hold non-negative conditions because they represent extra unused resources.

Adhering to non-negativity is critical for obtaining a valid solution in linear programming since negative values for certain types of quantities are typically non-sensible.