Define the concept of artificial variables in linear programming.

Artificial variables are introduced in linear programming to convert inequality constraints into equality constraints.

In linear programming, the objective is to maximize or minimize a linear function subject to a set of linear constraints. These constraints can be in the form of equalities or inequalities. However, the simplex method, which is commonly used to solve linear programming problems, requires all constraints to be in the form of equalities.

To convert inequality constraints into equality constraints, artificial variables are introduced. These variables are added to the problem as if they were decision variables, but they do not represent any physical quantity. They are used only to convert inequality constraints into equality constraints.

For example, consider the following linear programming problem:

Maximize 2x + 3y
Subject to:
x + y ≤ 5
2x + y ≥ 4
x, y ≥ 0

The second constraint is an inequality constraint. To convert it into an equality constraint, we introduce an artificial variable, say s, and rewrite the constraint as:

2x + y + s = 4

Now the problem becomes:

Maximize 2x + 3y
Subject to:
x + y ≤ 5
2x + y + s = 4
x, y, s ≥ 0

The artificial variable s is added to the problem, and it is set to zero in the optimal solution. Its purpose is to ensure that the constraint is satisfied as an equality constraint. Understanding the types of numbers involved in these constraints can further enhance comprehension of their application in linear programming.

A-Level Maths Tutor Summary: In linear programming, artificial variables are used to change inequality constraints (like "less than" or "more than") into equality constraints (exact equalities). They're added to problems when using the simplex method, which needs equalities to work properly. These variables don't represent anything real; they just help us meet the requirements of the method and are set to zero in the final solution.