This chapter addressed the solution of mixed-integer linear (MILP) problems for which first simple methods are introduced, namely exhaustive enumeration of all 0-1 combinations, and solution through relaxation and rounding. The branch and bound method is then formally introduced, identifying major properties such as lower bounding in order to use these tofathom nodes, or to obtain feasibleinteger solutions which yield upper bounds. The concept of cutting planes is also introduced with Gomory's cutting plane. Finally, the combination of branch and bound and cutting planes is discussed. Finally, a brief discussion on software such as CPLEX, GUROBI, and XPRESS is presented.

In this chapter, we return to linear programming. We begin by covering Phase I pivoting, so that now readers can solve any linear programming problem, whether it is a standard form problem or not. Then we return to sensitivity analysis, but now with the benefit of the simplex algortihm, we show how it becomes much easier to calculate revised optimal solutions and stable ranges of parameters by using some simple matrix algebra.Finally, we consider integer programming, where we require that the solution values of some or all of the decision variables to be integers. While there are several possible approaches, we stick to the branch and bound method in this text. We also observe that integer constraints can be easily handled when solving integer programming problems using Microsoft Excel.