Modeling number of machine start-ups

I came a across an extension of a well-known problem: counting the number of start-ups of a machine.

In the power generation models we often see constraints related to the number of start-ups of a generator: because of wear and tear we may like to minimize or restrict the number of times a generator is turned on. In the picture below the x(i) vector is the state of a generator (on or off) and the s(i) vector indicates when a startup occurs (i is time period). Both x and s are binary variables.

If we are minimizing the number of startups or have an upper bound, this can be easily modeled with a single inequality:

Here s(i) is forced to be one if there is a start-up, otherwise it is unrestricted. I.e. as long as the bound U is not binding we may have a few elements in s being 1 while there is no real start-up there. This is a structure we often see in models.

A lower bound on the number of startups is somewhat unusual, but lets see what happens. If we have a lower bound on the number of start-ups we need to be more precise and force s(i)=0 if there is no start-up. This can be modeled as:

This nonlinear expression can be linearized easily and leads to the three inequalities shown above. This formulation is so tight we can even relax the s variables to continuous variables between zero and one (they will be automatically integer for any feasible solution).

The state in period zero is assumed to be known. This matters if we have x(i)=1 in the first period. This is only counted as a startup in case the initial state is 0. In GAMS we can model this compactly as:

The trick with the initial state resembles how we can deal with iniitial inventory in a supply chain model. See: http://yetanothermathprogrammingconsultant.blogspot.com/2014/10/handling-of-inventory.html.