Mathematical Programming:An Overview1Management science is characterized by a scientific approach to managerial decision making. It attemptsto apply mathematical methods and the capabilities of modern computers to the difficult and unstructuredproblems confronting modern managers. It is a young and novel discipline. Although its roots can betraced back to problems posed by early civilizations, it was not until World War II that it became identifiedas a respectable and well defined body of knowledge. Since then, it has grown at an impressive pace,unprecedented for most scientific accomplishments; it is changing our attitudes toward decision-making, andinfiltrating every conceivable area of application, covering a wide variety of business, industrial, military, andpublic-sector problems.Management science has been known by a variety of other names. In the United States, operationsresearch has served as a synonym and it is used widely today, while in Britain operational research seemsto be the more accepted name. Some people tend to identify the scientific approach to managerial problem-solving under such other names as systems analysis, cost–benefit analysis, and cost-effectiveness analysis.We will adhere to management science throughout this book.Mathematical programming, and especially linear programming, is one of the best developed and mostused branches of management science. It concerns the optimum allocation of limited resources amongcompeting activities, under a set of constraints imposed by the nature of the problem being studied. Theseconstraints could reflect financial, technological, marketing, organizational, or many other considerations. Inbroad terms, mathematical programming can be defined as a mathematical representation aimed at program-ming or planning the best possible allocation of scarce resources. When the mathematical representation useslinear functions exclusively, we have a linear-programming model.In 1947, George B. Dantzig, then part of a research group of the U.S. Air Force known as Project SCOOP(Scientific Computation Of Optimum Programs), developed the simplex method for solving the generallinear-programming problem. The extraordinary computational efficiency and robustness of the simplexmethod, together with the availability of high-speed digital computers, have made linear programming themost powerful optimization method ever designed and the most widely applied in the business environment.Since then, many additional techniques have been developed, which relax the assumptions of the linear-programming model and broaden the applications of the mathematical-programming approach. It is thisspectrum of techniques and their effective implementation in practice that are considered in this book.1.1 AN INTRODUCTION TO MANAGEMENT SCIENCESince mathematical programming is only a tool of the broad discipline known as management science,let us first attempt to understand the management-science approach and identify the role of mathematicalprogramming within that approach.12 Mathematical Programming: An Overview 1.2It is hard to give a noncontroversial definition of management science. As we have indicated before,this is a rather new field that is renewing itself and changing constantly. It has benefited from contributionsoriginating in the social and natural sciences, econometrics, and mathematics, much of which escape therigidity of a definition. Nonetheless it is possible to provide a general statement about the basic elements ofthe management-science approach.Management science is characterized by the use of mathematical models in providing guidelines tomanagers for making effective decisions within the state of the current information, or in seeking furtherinformation if current knowledge is insufficient to reach a proper decision.There are several elements of this statement that are deserving of emphasis. First, the essence of manage-ment scienceisthe model-building approach—thatis, an attempt to capture the most significant features of thedecision under consideration by means of a mathematical abstraction. Models are simplified representationsof the real world. In order for models to be useful in supporting management decisions, they have to be simpleto understand and easy to use. At the same time, they have to provide a complete and realistic representationof the decision environment by incorporating all the elements required to characterize the essence of theproblem under study. This is not an easy task but, if done properly, it will supply managers with a formidabletool to be used in complex decision situations.Second, through this model-design effort, management science tries to provide guidelines to managers or,in other words, to increase managers’ understanding of the consequences of their actions. There is never anattempt to replaceor substitutefor managers, but rather theaim isto supportmanagement actions. Itis critical,then, to recognize the strong interaction required between managers and models. Models can expedientlyand effectively account for the many interrelationships that might be present among the alternatives beingconsidered, and can explicitlyevaluatethe economicconsequences ofthe actionsavailable to managers withinthe constraints imposed by the existing resources and the demands placed upon the use of those resources.Managers, on the other hand, should formulate the basic questions to be addressed by the model, and theninterpret the model’s results in lightof theirownexperienceand intuition, recognizingthe model’slimitations.The complementarity between the superior computational capabilities provided by the model and the higherjudgmentalcapabilities ofthe human decision-makeris the key to asuccessful management-scienceapproach.Finally, it is the complexity of the decision under study, and not the tool being used to investigate thedecision-making process, that should determine the amount of information needed to handle that decisioneffectively. Models have been criticized for creating unreasonable requirements for information. In fact,this is not necessary. Quite to the contrary, models can be constructed within the current state of availableinformation and they can be used to evaluate whether or not it is economically desirable to gather additionalinformation.The subject of proper model design and implementation will be covered in detail in Chapter 5.1.2 MODEL CLASSIFICATIONThe management-science literature includes several