The unrestricted variable Sj is now the difference between two nonnegative variables 5j and Sf.
We can think of 5i as the number of temps hired and 5; as the number of temps fired. For example, if Sj = 5 and 5t = 0 then 5i
= S - 0 = +S, which represents hiring. If 5i = 0 and
5t = 7 then Sj =0 - 7 = -7, which represents firing. In the first case, the corresponding cost
of hiring is 2005i = 200 X S = $1000 and in the second case the corresponding cost offiring is
4005; =400 X 7 = $2800. Thisidea is the basisfor the development of the objective function.
First we need to address an important point: What if both 5j and 5t are positive? TIle answer is that this cannot happen because it impliesthat the solution callsfor both hiring and firing
in the same month. Interestingly, the theory oflinear programming (see Chapter 7) tells us that
5i and 5t can never be positive simultaneously, a result that confirms intuition.
We can now write the cost of hiring and firing asfollows: