Forget specific applications, and just think about what a linear program is. You're optimising (either maximising or minimising) some linear combination of variables, subject to certain linear constraints.
So embedding boolean logic in the linear program should allow you to extend this to boolean constraints as well. Rather than just finding any old assignment of truth values which satisfies the boolean expression, this gives you a natural way to find an optimal assignment, for some reasonable definition of "optimal".
Now can you think of an application?
This doesn't use the exact embedding, but Appel and George, Optimal Spilling for CISC Machines with Few Registers (2000) is one good example of a real-world optimisation problem based around boolean constraints and a linear objective function.