Integer Programming formulation described as follows:
Assume a set of variable $V$ = ${v_1,...,v_m}$.
The set of total $S$ constraints is of the form:
$$v_1 + \overline{v_2} + v_3 \leq 1 \\ ... \\ \overline{v_2} + v_4 + v_6 \leq 1 $$
each called a clause, $C$.
Problem formulation (Objective function):
Find an assignment that satisfies maximum constraints (out of S constraints).
Formally: $$r(C) = max \sum_{C \in S} z_{C}$$
The variable $z_C$ will be 1 if the each corresponding constraint is true; for example in case $v_1 + \overline{v_2} + v_3 \leq 1$ is true and 0 otherwise.
I'm new to Integer programming and first tool that I tried MIP solver I can write all $S$ constraints easily. But I have no idea how to encode the objective function.
The objective function to be maximized is $\sum_{C\in S} z_C$. You can define it here.