Question

Consider the problem: \(\operatorname{maximize} \quad \mathrm{x}_{1}+3 \mathrm{x}_{2}\) subject to: $$ \begin{aligned} 6 \mathrm{x}_{1}+19 \mathrm{x}_{2} & \leq 100 \\ 3 \mathrm{x}_{1}+5 \mathrm{x}_{2} & \leq 40 \\ \mathrm{x}_{1}-3 \mathrm{x}_{2} & \leq 33 \\ \mathrm{x}_{2} & \leq 25 \\ \mathrm{x}_{1} & \leq 42 \\ \mathrm{x}_{1}, \mathrm{x}_{2} & \geq 0 \end{aligned} $$ Find its dual problem.

Short answer

The dual problem of the given LP problem is: Minimize: \(100y_1 + 40y_2 + 33y_3 + 25y_4 + 42y_5\) Subject to: \[ \begin{aligned} 6 y_1 + 3 y_2 + y_3 + y_5 & \geq 1 \\ 19 y_1 + 5 y_2 -3 y_3 + y_4 & \geq 3 \\ y_1, y_2, y_3, y_4, y_5 & \geq 0 \end{aligned} \]

Step-by-step solution

Rewrite the given LP problem in standard form

We will first rewrite the given problem in the standard form by introducing slack variables \(s_1\), \(s_2\), \(s_3\), \(s_4\), and \(s_5\) to convert the inequality constraints into equality constraints. The problem becomes: Maximize: \(\ \mathrm{x}_{1}+3 \mathrm{x}_{2}\) Subject to: \[ \begin{aligned} 6 \mathrm{x}_{1}+19 \mathrm{x}_{2} + s_1 & = 100 \\ 3 \mathrm{x}_{1}+5 \mathrm{x}_{2} + s_2 & = 40 \\ \mathrm{x}_{1}-3 \mathrm{x}_{2}+s_3 & = 33 \\ \mathrm{x}_{2}+s_4 &= 25 \\ \mathrm{x}_{1}+s_5 &= 42 \\ \mathrm{x}_{1}, \mathrm{x}_{2}, s_1, s_2, s_3, s_4, s_5 & \geq 0 \end{aligned} \] Now we have our LP problem in standard form.

Construct the dual problem

Using the standard form of the original problem, we now construct the dual problem. It will be a minimization problem. Let the dual variables be \(y_1, y_2, y_3, y_4\), and \(y_5\). We build the dual LP problem: Minimize: \(100y_1 + 40y_2 + 33y_3 + 25y_4 + 42y_5\) Subject to: \[ \begin{aligned} 6 y_1 + 3 y_2 + y_3 + y_5 & \geq 1 \\ 19 y_1 + 5 y_2 -3 y_3 + y_4 & \geq 3 \\ y_1, y_2, y_3, y_4, y_5 & \geq 0 \end{aligned} \] This is the dual problem of the given LP problem.

Key concepts

Linear Programming

Linear programming (LP) is a mathematical method used to determine the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. More formally, it is a technique that optimizes a linear objective function, subject to linear equality and inequality constraints.

For instance, the LP problem in our exercise aims to maximize the function \( \mathrm{x}_{1}+3 \mathrm{x}_{2} \) subject to a set of constraints. These constraints represent limitations such as resource availability or certain conditions that must be met. The goal is to find the values of \( \mathrm{x}_{1} \) and \( \mathrm{x}_{2} \) that maximize the objective function without violating any constraints, which is a common task in fields such as economics, engineering, and military planning.

Standard Form Linear Programming

The standard form of an LP problem is a specific way to write it that simplifies the process of finding a solution. An LP problem is said to be in standard form if all the constraints are written as equalities with the addition of slack variables and all the decision variables are non-negative.

In our exercise, we converted the original inequalities into equations by introducing slack variables \(s_1, s_2, s_3, s_4, s_5\) to 'take up the slack' between the left and right sides of each inequality. This transformation is crucial because it sets up the problem for applying various algorithms, such as the Simplex method, which find solutions to linear programming problems efficiently. The standard form makes the next steps, such as constructing a dual problem, more straightforward.

Slack Variables

Slack variables are additional variables added to linear programming problems to transform inequality constraints into equality constraints. Essentially, they measure the 'unused' portion of the resources within the constraints. For every 'less than or equal to' inequality, a non-negative slack variable is added to convert it into an equation.

In our exercise example, slack variables \(s_1\) to \(s_5\) allow the inequalities to be converted into equations by accounting for the difference between the left-hand side and the right value within the constraints. This technique is widely used because it allows the application of the Simplex method, which requires constraints to be in equality form. It maintains the integrity of the original problem while allowing for a format that is conducive to optimization.

Optimization Constraints

Optimization constraints are the restrictions or limitations that bound the feasible region in a linear programming problem. These constraints must be satisfied for the solution to be valid. They are typically represented by linear inequalities or equations that define the conditions under which the optimization is performed.

In the exercise, the constraints given are inequalities that limit the values that \( \mathrm{x}_{1} \) and \( \mathrm{x}_{2} \) can take. These values are influenced by conditions such as resource capacity or other defined limits. As part of the process, establishing the dual to this problem involves constraints as well, but instead, these constraints limit the values of the dual variables \(y_1\) through \(y_5\) and seek a minimum rather than a maximum. The beauty of dual problems lies in their representation of the original problem from a different perspective, often shedding light on resource pricing or shadow prices.