So far all our functions were linear. In many situations, it might not be the case.

1. So far all our functions were linear. • In many situations, it might not be the case. • Example: Production cost. • c1= $11/unit for first 5 items • c2=$8/unit for next 4 items • c3=$5/unit for next 7 items • c4=$7/unit for next 10 items • The cost of producing x items is an example of so-called piecewise linear function:

2. How to include piecewise linear cost functions in an objective function of LP? • Idea: Introduce a new variable for each cost segment. For i=1,2,3,4 yi = number of items produced at cost ci Then the total number of items is x =y1+y2+y3+y4 . We need constraints 0  y1  5, 0  y2  4, 0  y3 7, 0  y4 10 , (*) and the production cost in the objective function is 11y1 + 8y2 + 5y3 + 7y4 • What is the shortcoming of this model?

3. We should require that • y2>0 implies that y1=5(1) • y3>0 implies that y2=4(2) • y4>0 implies that y3=7(3) • Introduce new variables to translate these requirements into linear constraints. For i=1,2,3,4, • Proper constraints relating wi and yi will provide that requirements (1)-(3) are satisfied. y2  4w1and 5w1  y1provide (1) y3  7w2 and 4w2  y2provide (2) y4  10w3and 7w3  y3provide (3)

4. Summarizing, the bound constraints in (*) should be substituted with 5w1  y1 5, 4w2  y2 4w1 , 7w3  y3 7w2 , 0  y4  10w3 . • Generalizing, suppose we have k segments with lengths L1, L2, …, Lk . Then the necessary constraints: L1w1  y1 L1 , Liwi yi  Liwi-1for i = 2, …, k-1 0  yk  Lkwk-1