Models
 Terminology
 Terminology
 Integer-Linear Programming Model (ILP) or Integer Programming Model (IP) Mixed-Integer-Linear Programming Model (MILP) or Mixed Integer Programming Model (MIP) IP Model with p strictly less than n. Pure Integer Programming Model IP with all variables required to be integer (p = n). Binary Programming Model IP with all integer variables restricted to 0 or 1. The model can be classified as a pure-binary programming model or mixed-binary programming model. Logical Constraint A linear constraint involving integer variables that models some logical condition on the decision variables. Concave Function of a Single Variable A function of a single variable with a decreasing derivative. The figure below shows a concave function of a decision variable. A piecewise linear approximation is shown by the dotted lines. A concave function in the objective function of a maximization problem can be represented by the sum of several linear expressions with a piecewise linear approximation (3 for the figure). No binary variables are required. For a minimization problem, a piecewise linear approximation requires binary variables to force the pieces to enter the solution in the proper order. The number of binary variables is one fewer than the number of pieces. Convex Function of a Single Variable A function of a single variable with a increasing derivative. The figure below shows a convex function of a decision variable. A piecewise linear approximation is shown by the dotted lines. A convex function in the objective function of a minimization problem can be represented by the sum of several linear expressions with a piecewise linear approximation (3 for the figure). No binary variables are required. For a maximization problem, a piecewise linear approximation requires binary variables to force the pieces to enter the solution in the proper order. The number of binary variables is one fewer than the number of pieces. Fixed Charge Function A nonlinear function that represents the cost of a decision. The cost is 0 if the variable is 0. The cost is a fixed value plus a cost which is linear with the amount of the variable. The fixed charge is modeled with a binary variable. The fixed charge multiplies the binary variable and the unit cost multiplies the decision variable. The model must include an implication constraint that forces the decision variable to 0 when the binary variable is zero. When the binary variable is 1, the constraint limits the decision variable to its upper bound. The model also includes nonnegativity restrictions and the upper bound on the binary variable.

Operations Research Models and Methods
Internet
by Paul A. Jensen