Define the concept of surplus variables in linear programming.

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

In linear programming, constraints can be either equality or inequality. Equality constraints are of the form Ax = b, where A is a matrix of coefficients, x is the vector of decision variables, and b is a vector of constants. Inequality constraints are of the form Ax ≤ b or Ax ≥ b. However, linear programming problems are typically solved using only equality constraints.

To convert inequality constraints into equality constraints, surplus variables are introduced. For example, consider the inequality constraint 2x + 3y ≤ 5. We can introduce a surplus variable s such that 2x + 3y + s = 5. The surplus variable represents the amount by which the left-hand side of the inequality exceeds the right-hand side. This adjustment is essential in optimisation problems where the goal is to find the best solution within given constraints.

Similarly, we can introduce a slack variable to convert a greater-than or equal-to inequality into an equality constraint. For example, consider the inequality constraint 2x + 3y ≥ 5. We can introduce a slack variable t such that 2x + 3y - t = 5. The slack variable represents the amount by which the left-hand side of the inequality falls short of the right-hand side.

Surplus and slack variables do not affect the objective function or the feasible region of the linear programming problem. They simply allow us to convert inequality constraints into equality constraints, which makes the problem easier to solve using standard linear algebra techniques. Understanding these variables can also provide insight into other mathematical concepts, such as types of numbers used in different constraints.

A-Level Maths Tutor Summary: In simple terms, surplus variables help change 'less than or equal to' inequalities into straight equals in linear programming, making them easier to solve. They show how much we exceed a limit. Similarly, slack variables do the opposite for 'greater than or equal to' inequalities, showing how much we're under a limit. Neither changes the main goal or possible solutions of the problem. The application of these variables is crucial in real-world scenarios, which can be explored further in real-world applications of linear programming.