Can s_j=max{ a_j, s_i+t_ij } be formulated as a linear constraint?

Question

My question is whether there is a trick to express the following as a linear constraint?

S_j=max{ a_j, S_i+t_ij }

where a_j and t_ij are constants

Many thanks in advance

Answer 1

Yes, you can do this by introducing a new binary variable y_j (0 or 1) and a couple of linear constraints.

We can rewrite your constraint as:

Sj = a      if a > Si + T
Sj = Si + T if a < Si + T


Sj = a y_j + (Si + T) (1 - yj)       .....(1)

Let's say that y_j = 0 if a is bigger

y_j = 1 if a is smaller than Si + T

M y_j + a - (Si + T) > 0   ....(2) 
where M is a big number, much bigger than a or Si or T.

If y_j is 1, the constraint above is trivially satisfied, and Sj will be equal to a. If y_j is 0, a has to be bigger for the constraints to be satisfied, and Sj = Si + T

In your formulation, just include (1) and (2) and you will have enforced the max constraint.