Traditional optimization techniques, such as those based on classical calculus, are useful and work well in decision environments that are not complex. When a decision environment becomes complex either due to resource limitations or the presence of interdependent goals and alternatives, traditional optimization techniques do not always produce the best results. Given that most decision environments are complex, one needs to use more efficient tools like mathematical programming that can deal with these complexities more effectively. Mathematical programming is a generic term used for a group of programming models where each model is based on a set of interrelated, but well-defined structures, equations, and assumptions. Models are abstract representations of the real world. As abstractions of the real world, they can only deal with the most important elements of the world. It is not possible for any model to capture every single element of a system in all its intricate details; it is costly, time consuming, and even difficult to analyze. Mathematical programming models are no exception. As quantitative tools, they attempt to capture only the most important elements and relationships that exist in a real system using mathematical relations. As a general rule, the more complex a system, the more complex the relations.
This chapter presents four of the most widely discussed programming models: linear programming, integer programming, dynamic programming, and heuristic programming. Because of its widespread use and importance, the chapter examines the linear programming model in much greater detail than the rest of the models.
Linear programming is the most well known among all mathematical programming models. Developed during World War II as a mathematical technique to deal with the problems of military logistics, 1 it has since been applied to a wide variety of problems and disciplines ranging from industrial technology to education to medicine to economics and business. It is the forerunner of all mathematical programming models