Integer Linear Programming

The integer linear programming problem has the form $$ \begin{gathered} {\text{maximize}}\quad \sum\limits_{{{\text{j = 1}}}}^{{\text{n}}} {{{{\text{c}}}_{{\text{j}}}}{{{\text{x}}}_{{\text{j}}}}} \hfill \\ {\text{subject}}\;{\text{to}}\quad \sum\limits_{{{\text{j = 1}}}}^{{\text{n}}} {{{{\text{a}}}_{{{\text{ij}}}}}{{{\text{x}}}_{{\text{j}}}}} \quad \leqslant \quad {{{\text{b}}}_{{\text{i}}}}\quad ({\text{i = 1,2,}} \ldots {\text{,m}}) \hfill \\ \quad \quad \quad {{{\text{x}}}_{{\text{j}}}}\quad \geqslant \quad 0\quad ({\text{j = 1,2,}} \ldots {\text{,n}}) \hfill \\ {{{\text{x}}}_{{\text{j}}}}\;{\text{is}}\;{\text{an}}\;{\text{integer}}\quad ({\text{j = 1,2,}} \ldots {\text{,m}}) \hfill \\ \end{gathered} $$ such that c_j, a_ij and b_i are real numbers.