What is a pivot/basic variable? Free variable?
A pivot/basic variable corresponds to a pivot column. A free variable corresponds to a solution of the system for any value.
In linear programming, a pivot or basic variable is a decision variable that has a non-zero coefficient in the equation representing a basic feasible solution. In other words, a pivot variable is a variable that is chosen to represent a specific value of an inequality in the constraints of a linear programming problem.
On the other hand, a free variable is a decision variable in a linear programming problem that does not represent a basic feasible solution and can take on any value. Free variables have coefficients of zero in the equation representing the basic feasible solution.
To illustrate the difference between pivot and free variables, let’s consider the following linear programming problem:
Maximize: 2x + 3y
Subject to:
3x + 2y ≤ 18
x + y ≤ 8
2x – y ≤ 2
The basic feasible solution for this problem can be represented as follows:
x = 2
y = 6
s1 = 0
s2 = 0
s3 = 2
Here, x and y are the pivot variables, as they represent specific values that satisfy the constraints. The variables s1, s2, and s3 are the slack variables, which are used to convert the inequalities to equations.
Meanwhile, z, w, u, and v are the free variables, and they can take on any value without violating the constraints. In a linear programming problem, the number of free variables is equal to the number of variables in the problem minus the number of constraints.
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