## Logical and Integer Modeling

Tony Hürlimann

info@matmod.ch

November 12, 2023

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Abstract

Logical and Boolean conditions are often used in real-live problem solving. However,
its importance in practical modeling contrasts sharply with its treatment in modeling
books and courses. Many modeling techniques and applications to formulate logical
conditions are exposed in this paper.

It presents various methods of logical modeling, that is, mathematical modeling
containing logical propositions and Boolean mixed with mathematical expressions
within the constraints. Several modeling techniques are formulated in mathematical
notation and in the modeling system LPL (see [13]). Concrete and executable model
application examples are given for the different techniques.

A word of caution: It is not trivial in a practical context to use Boolean operators.
One needs to analyze carefully the context to use the right operators. “and”, “or”, and
other words in a a spoken text cannot be translated one-to-one to Boolean “and”,
“or”, for instance. Check multiple times before proceeding!

A second word of caution: The modeling formulation may be correct, the
implementation of the translation to a mathematical linear model might not always
correspond to this logical modeling. This is due to the fact that in most cases,
the automatically introduction of a new binary variable is realized in LPL as an
implication instead of an equivalence. In most cases, this is justifies but not always.
Hence, the resulting linear model in LPL should be checked anyway.

A ZIP file of all models can be found HERE