Integer Programming

In all the optimization techniques considered so far, the design variables are assumed to be continuous, which can take any real value. In many situations it is entirely appropriate and possible to have fractional solutions. Among the several techniques available for solving the all-integer and mixed-integer linear programming problems, the cutting plane algorithm of Gomory and the branch-and-bound algorithm of Land and Doig have been quite popular. Although the zero–one linear programming problems can be solved by the general cutting plane or the branch-and-bound algorithms, Balas developed an efficient enumerative algorithm for solving those problems. Very little work has been done in the field of integer nonlinear programming. The generalized penalty function method and the sequential linear integer (discrete) programming method can be used to solve all integer and mixed-integer nonlinear programming problems. The chapter also summarizes the various solution techniques of solving integer programming problems.