# Unrestricted Variable Problems

**Introduction:**

The non–negativity constraint of a linear programming problem restricts that the values of all variables in the problem say for example *x*, *y *and *z *or *a*, *b*, *c *and d etc. must be = 0. Sometimes we may come across a situation that the values of the variables are unrestricted, *i.e*., may assume any value (0 or > 1 or < 1) *i.e., *to say that the ≥ sign is not required. In such cases to maintain non-negativity restriction for all variables, each variable is replaced by two non-negative variables, say for example: *x*, *y *and *z *are replaced by *x' *and *x"*, *y' *and *y"*, *z' *and *z" *respectively. If *x' *> *x"*, **then x is positive, while**

*x*' <*x"*then*x*is negative.

**Example:**Maximize *Z *= 2*x * 3*y *s.t.

–1*x * 2*y *≤ 4

1*x * 1*y *≤ 6

1*x * 3*y *≤ 9 and x and y are unrestricted.

As it is given that both *x *and *y *are unrestricted in sign they are replaced by non–negative variables, *x' *and *x"*, *y' *and *y" *respectively. This is subjected to *x *= *x' *– *x" *and *y *= *y' *– *y"*. By introducing slack variables and non–negative variables the simplex format is: Maximize *Z *= 2 (*x' *– *x"*) 3 (*y*' – *y"*) 0*S*_{1} 0*S*_{2} 0*S*_{3} s.t.

– 1 (*x' *– *x"*) 2 (*y' *– *y"*) 1*S*_{1} 0*S*_{2} 0*S*_{3} = 4

1 (*x' *– *x"*) 1 (*y' *– *y"*) 0*S*_{1} 1*S*_{2} 0*S *= 6

1 (*x' *– *x"*) 3 (*y' *– *y"*) 0*S*_{1} 0*S*_{2} 1*S*_{3} = 9

*x'*, *x"*, *y'*, *y" *and *S*_{1}, *S*_{2} and *S*_{3} all ≥ 0.