# If This and That Then Whatever

Written by: Paul Rubin

Primary Source: OR in an OB World

I was asked a question that reduced to the following: if $$x$$, $$y$$ and $$z$$ are all binary variables, how do we handle (in an integer programming model) the requirement “if $$x=1$$ and $$y=1$$ then $$z=1$$”? In the absence of any constraints on $$z$$ when the antecedent is not true, this is very easy: add the constraint

$$\displaystyle z \ge x + y – 1.$$

Verification (by substituting all possible combinations of 0 and 1 for the variables) is left to the reader as an exercise.

I thought I had covered this in a previous post, but looking back it appears that it never came up (at least in this form). This might be my shortest post ever. :-)

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#### Paul Rubin

Professor Emeritus at Michigan State University
I'm an apostate mathematician, retired from a business school after 33 years of teaching mostly (but not exclusively) quantitative methods courses. My academic interests lie in operations research. I also study Tae Kwon Do a bit on the side.