# Modelling set of binaries as SOS1

H.P. Williams states in his book "Model Building in Mathematical Programming:

"An SOS1 is a set of variables (continuous or integer) within which exactly one variable must be non-zero."

So, if I have two binary variables a, b which also form a SOS1 then the SOS1 property together with the constraint a + b = 1 impose integrality on a, b and therefore I don't need to require explicitely the integrality of a,b.

CPLEX defines a SOS1 as:

"SOS Type 1 is a set of variables where at most one variable may be nonzero."

Is there a way to avoid explicitely requiring the integrality of a,b as in the case of Williams?

Using the CPLEX definition, the constraint $$a+b=1$$ with $$a$$ and $$b$$ forming a type 1 SOS still implies integrality of $$a$$ and $$b.$$
• With $a+b\le 1$ I don't think there is a way to avoid declaring $a$ and $b$ to be integer.
• So $c$ would appear in just that one equation and the SOS1 set and nowhere else. That would seem to get the job done.