In the linear programming models formulated and solved in the previous chapters, the implicit assumption was that solutions could be fractional or real numbers (i.e., non-integer). However, non-integer solutions are not always practical.
When only integer solutions are practical or logical, it is sometimes assumed that noninteger solution values can be "rounded off" to the nearest feasible integer values. This method would cause little concern if, for example, x 1 = 8,000.4 nails were rounded off to 8,000 nails because nails cost only a few cents apiece. However, if we are considering the production of jet aircraft and x 1 = 7.4 jet airliners, rounding off could affect profit (or cost) by millions of dollars. In this case we need to solve the problem so that an optimal integer solution is guaranteed . In this chapter the different forms of integer linear programming models are presented.