Operations research tools are directed toward the solution of resource management and planning problems. Models in operations research are representations of the structure of a physical object or a conceptual or business process. Using the tools of operations research involves the following:
defining a structural model of the system under investigation
collecting the data for the model
solving the model
interpreting the results
SAS/OR software is a set of procedures for exploring models of distribution networks, production systems, resource allocation problems, and scheduling problems using the tools of operations research.
The following list suggests some of the application areas in which optimization-based decision support systems have been used. In practice, models often contain elements of several applications listed here.
Product-mix problems find the mix of products that generates the largest return when several products compete for limited resources.
Blending problems find the mix of ingredients to be used in a product so that it meets minimum standards at minimum cost.
Time-staged problems are models whose structure repeats as a function of time. Production and inventory models are classic examples of time-staged problems. In each period, production plus inventory minus current demand equals inventory carried to the next period.
Scheduling problems assign people to times, places, or tasks so as to optimize people’s preferences or performance while satisfying the demands of the schedule.
Multiple objective problems have multiple, possibly conflicting, objectives. Typically, the objectives are prioritized, and the problems are solved sequentially in a priority order.
Capital budgeting and project selection problems ask for the project or set of projects that yield the greatest return.
Location problems seek the set of locations that meets the distribution needs at minimum cost.
Cutting stock problems find the partition of raw material that minimizes waste and fulfills demand.
The basic optimization problem is that of minimizing or maximizing an objective function subject to constraints imposed on the variables of that function. The objective function and constraints can be linear or nonlinear; the constraints can be bound constraints, equality or inequality constraints, or integer constraints. Traditionally, optimization problems are divided into various types depending on the sets of values that the variables are restricted to (real, integer, or binary, or a combination) and the nature of functional form of the constraints and objectives (linear, quadratic, or general nonlinear). An expression of an optimization problem in mathematical form is called a mathematical program.
When the complete description of a mathematical program is supplied to an appropriate algorithm (such as one of the solvers described in this book), the algorithm determines the optimal values for the decision variables so the objective is either maximized or minimized, the optimal values that are assigned to decision variables are on or between allowable bounds, and the constraints are obeyed. This process of solving mathematical programs is called mathematical programming, mathematical optimization, or just optimization.
When the constraints in an optimization problem are linear and the objective is either linear or quadratic, the optimization problem can be encapsulated in SAS data sets and then solved using the appropriate SAS/OR procedure: the OPTLP, OPTMILP, or OPTQP procedure.
Often optimization problems, and especially those with nonlinear elements, are formalized in an algebraic model that represents the problem. When formulated in its most abstract form, such an algebraic model is independent of problem data. A specific optimization problem instance (including the original problem) is then just an instantiation of the algebraic model with the specific data associated with that instance. An optimization modeling language (also called an algebraic modeling language) is a programming environment that has syntax, structures, and operations that enable you to express a mathematical program in a form that corresponds in a natural and transparent way to its algebraic model. The syntax, structures, and operations also enable you to populate an algebraic model with a specific data instance and then solve the resulting optimization problem instance with an appropriate solver. The OPTMODEL procedure is such an algebraic modeling language in SAS/OR software and can be viewed as a single, unified environment to formulate and solve mathematical programming problems of many different types.
Whether mathematical programs are represented in SAS data sets or in an algebraic model in PROC OPTMODEL, they can be saved, easily changed, and solved again. The SAS/OR procedures also output SAS data sets that contain the solutions. These data sets can then be used to produce customized reports or as input to other SAS procedures. This structure enables you to use the tools of operations research and other SAS tools as building blocks to build decision support systems.
This chapter describes how to use SAS/OR software to solve a wide variety of optimization problems. It describes various types of optimization problems, indicates which SAS/OR procedures you can use, and shows how you provide data, run the procedure, and obtain optimal solutions. For additional examples that demonstrate the features of the OPTMODEL procedure, see SAS/OR User's Guide: Mathematical Programming Examples.
The next section broadly classifies the SAS/OR procedures based on the types of mathematical programming problems they can solve.