Solving Integer Programming Problems
In practical terms, there are two methods that can be used to solve integer programming
problems. One is called the branch and bound method; which involves solving a sequence
of subproblems until the optimal solution has been identified. The other is enumeration; it
involves preparing a list of all possible alternatives, weeding out the ones that are infeasible,
then determining which of the feasible solutions is optimal. The enumeration approach is
particularly well suited for 0–1-type problems because the possible outcomes are easily
identified and fairly limited for small problems. The branch and bound approach is more
general; it lends itself to all types of integer programming problems. Consequently, the discussion
here will focus primarily on the branch and bound approach, although the enumeration
approach will be illustrated in the context of 0–1 problems.
The branch and bound method will be illustrated graphically for instructional
purposes. However, as a practical matter, branch and bound solutions rely on computer
support—each subproblem must be solved using the computer. Manual solution of subproblems
using simplex is theoretically possible, but the computational burden would be
enormous. Graphical method could be used, but only for two-variable problems.With that
in mind, let’s consider a graphical solution to a very simple problem in order to better understand
the nature of integer programming problems and solutions.
Figure 7S-1 shows the graph of a two-variable, pure-integer problem. The black dots
represent points at which both variables are integers. Note that the standard LP solution is
not an integer solution. The reason is that the optimal integer solution lies in the interior
problems. One is called the branch and bound method; which involves solving a sequence
of subproblems until the optimal solution has been identified. The other is enumeration; it
involves preparing a list of all possible alternatives, weeding out the ones that are infeasible,
then determining which of the feasible solutions is optimal. The enumeration approach is
particularly well suited for 0–1-type problems because the possible outcomes are easily
identified and fairly limited for small problems. The branch and bound approach is more
general; it lends itself to all types of integer programming problems. Consequently, the discussion
here will focus primarily on the branch and bound approach, although the enumeration
approach will be illustrated in the context of 0–1 problems.
The branch and bound method will be illustrated graphically for instructional
purposes. However, as a practical matter, branch and bound solutions rely on computer
support—each subproblem must be solved using the computer. Manual solution of subproblems
using simplex is theoretically possible, but the computational burden would be
enormous. Graphical method could be used, but only for two-variable problems.With that
in mind, let’s consider a graphical solution to a very simple problem in order to better understand
the nature of integer programming problems and solutions.
Figure 7S-1 shows the graph of a two-variable, pure-integer problem. The black dots
represent points at which both variables are integers. Note that the standard LP solution is
not an integer solution. The reason is that the optimal integer solution lies in the interior
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