Understanding Integer Linear Programming and Its Key Concepts in Optimization
Integer Linear Programming (ILP) is a specialized form of linear programming where some or all decision variables must be integers. LP relaxation involves removing the integer constraints, providing bounds on the objective value (OV) of the ILP. In maximization models, the relaxation offers an upper bound, while in minimization models, it provides a lower bound. Simply rounding LP solutions can lead to infeasibility or suboptimal solutions. ILP can be classified into All-Integer, Mixed Integer (MILP), and Binary Integer Programs, depending on the required integer status of decision variables.
Understanding Integer Linear Programming and Its Key Concepts in Optimization
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Presentation Transcript
Basic Concepts • Integer Linear Program(ILP): A linear program except that some or all of the decision variables must have integer values. • LP Relaxation: A LP model that is derived from an ILP by dropping the integer requirements for the variables. • In a Max model, the OV of the LP relaxation always provides an upper bound on the OV of ILP • In a Min model, the OV of the LP relaxation always provides a lower bound on the OV of ILP
Why not just use rounded solution of LP? • Rounded solution may be infeasible. • Even if one or more of the neighboring integer points is feasible: • Such a point need not be optimal for the ILP. • Such a point need not even be near the optimal ILP solution.(see figure 6.4 on p.293 of the text)
Types of Integer Program • All-Integer Linear Program: All of the decision variables are required to be integer. • Mixed Integer Linear Program(MILP): Some, but not all, of the decision variables in a LP are required to be integer. • Binary (0-1 )Integer Linear Program: The integer variables are only permitted to assume the values zero or one. It may be of either the all-integer or MILP type.
