Operations Research

A Practical Introduction

Michael W. Carter, Camille C. Price

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351990639
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Technology & Engineering Research & Skills
• Index: Technology & Engineering

Students with diverse backgrounds will face a multitude of decisions in a variety of engineering, scientific, industrial, and financial settings. They will need to know how to identify problems that the methods of operations research (OR) can solve, how to structure the problems into standard mathematical models, and finally how to apply or develop computational tools to solve the problems.

Perfect for any one-semester course in OR, Operations Research: A Practical Introduction answers all of these needs. In addition to providing a practical introduction and guide to using OR techniques, it includes a timely examination of innovative methods and practical issues related to the development and use of computer implementations. It provides a sound introduction to the mathematical models relevant to OR and illustrates the effective use of OR techniques with examples drawn from industrial, computing, engineering, and business applications.

Many students will take only one course in the techniques of Operations Research. Operations Research: A Practical Introduction offers them the greatest benefit from that course through a broad survey of the techniques and tools available for quantitative decision making. It will also encourage other students to pursue more advanced studies and provides you a concise, well-structured, vehicle for delivering the best possible overview of the discipline.

1 Introduction to Operations Research

CONTENTS

1.1 The Origins and Applications of Operations Research

1.2 System Modeling Principles

1.3 Algorithm Efficiency and Problem Complexity

1.4 Optimality and Practicality

1.5 Guide to Software Tools

1.6 Summary

Key Terms

References

1.1 THE ORIGINS AND APPLICATIONS OF OPERATIONS RESEARCH

Operations Research can be defined as the use of quantitative methods to assist analysts and decision-makers in designing, analyzing, and improving the performance or operation of systems. The systems being studied may be any kind of financial systems, scientific or engineering systems, or industrial systems; but regardless of the context, practically all such systems lend themselves to scrutiny within the systematic framework of the scientific method.

The field of Operations Research incorporates analytical tools from many different disciplines, so that they can be applied in a rational way to help decision-makers solve problems and control the operations of systems and organizations in the most practical or advantageous way. The tools of Operations Research can be used to optimize the performance of systems that are already well-understood or to investigate the performance of systems that are ill-defined or poorly understood, perhaps to identify which aspects of the system are controllable (and to what extent) and which are not. In any case, mathematical, computational, and analytical tools and devices are employed merely to provide information and insight; and ultimately, it is the human decision-makers who will utilize and implement what has been learned through the analysis process to achieve the most favorable performance of the system.

The ideas and methodologies of Operations Research have been taking shape throughout the history of science and mathematics, but most notably since the industrial revolution. In various ways, all of human knowledge seems to play a role in determining the goals and limitations underlying the decisions people make. Physical laws (such as gravity and the properties of material substances), human motivations (such as greed, compassion, and philanthropy), economic concepts (supply and demand, resource scarcity, division of labor, skill levels, and wage differentials), the apparent fragility of the environment (erosion, species decline), and political issues (territorial aggression, democratic ideals) all eventually are evident, at least indirectly, in the many types of systems that are studied with the techniques of Operations Research. Some of these are the natural, physical, and mathematical laws that are inherent and that have been discovered through observation, while others have emerged as a result of the development of our society and civilization. Within the context of these grand themes, decision-makers are called upon to make specific decisions — whether to launch a missile, introduce a new commercial product, build a factory, drill a well, or plant a crop.

Operations Research (also called Management Science) became an identifiable discipline during the days leading up to World War II. In the 1930s, the British military buildup centered around the development of weapons, devices, and other support equipment. The buildup was, however, of an unprecedented magnitude, and it became clear that there was also an urgent need to devise systems to ensure the most advantageous deployment and management of material and manpower.

Some of the earliest investigations led to the development and use of radar for detecting and tracking aircraft. This project required the cooperative efforts of the British military and scientific communities. In 1938, the scientific experts named their component of this project “operational research.” The term “operations analysis” was soon used in the U.S. military to refer to the work done by teams of analysts from various traditional disciplines who cooperated during the war.

Wartime military operations and supporting activities included contributions from many scientific fields. Chemists were at work developing processes for producing high-octane fuels; physicists were developing systems for detection of submarines and aircraft; and statisticians were making contributions in the area of utility theory, game theory, and models for various strategic and tactical problems. To coordinate the effectiveness of these diverse scientific endeavors, mathematicians and planners developed quantitative management techniques for allocating scarce resources (raw materials, parts, time, and manpower) among all the critical activities in order to achieve military and industrial goals. Informative overviews of the history of Operations Research in military operations are to be found in [White, 1985] and [McArthur, 1990].

The new analytical research on how best to conduct military operations had been remarkably successful, and after the conclusion of World War II, the skill and talent of the scientists that had been focused on military applications were immediately available for redirection to industrial, financial, and government applications. At nearly the same time, the advent of high-speed electronic computers made feasible the complex and time-consuming calculations required for many operations research techniques. Thus, the methodologies developed earlier for other purposes now became practical and profitable in business and industry.

In the early 1950s, interest in the subject was so widespread, both in academia and in industry, that professional societies sprang up to foster and promote the development and exchange of new ideas. The first was the Operational Research Society in Britain. In the U.S., the Operations Research Society of America (ORSA) and The Institute of Management Science (TIMS) were formed, and operated more or less as separate societies until the 1990s. These two organizations, however, had a large and overlapping membership and served somewhat similar purposes, and recently merged into a single organization known as INFORMS (Institute for Operations Research and the Management Sciences). National societies in many other countries are active and are related through IFORS (the International Federation of Operational Research Societies). Within INFORMS, there are numerous special-interest groups, and some specialized groups of researchers and practitioners have created separate societies to promote professional and scholarly endeavors in such areas as simulation, the decision sciences, and artificial intelligence. Furthermore, many mathematicians and computer scientists have interests that overlap those of operations researchers. Thus, the field of Operations Research is large and diverse. Some of the many activities and areas of research sponsored by INFORMS can be found at the Web site http://www.informs.org or in the journals associated with that organization. As will be apparent from the many illustrative applications presented throughout this book, the quantitative analysis techniques that found their first application over 60 years ago, are now used in many ways to - influence our quality of life today.

1.2 SYSTEM MODELING PRINCIPLES

Central to the practice of Operations Research is the process of building mathematical models. A model is a simplified, idealized representation of a real object, a real process, or a rea¹ system. The models used here are called mathematical models because the building blocks of the models are mathematical structures such as equations, inequalities, matrices, functions, and operators. In developing a model, these mathematical structures are used to capture and describe the most salient features of the entity that is being modeled. For example, a financial balance sheet may model the financial position of a corporation; mathematical formulae may serve as models of market activity or trends; and a probability distribution can be used to describe the frequency with which certain asynchronous events occur in a multiprocessor computer. Mathematical models may look very different, depending on the structure of the system or problem being modeled and the application area. In studying the various topics in this book, we will see that models do indeed take on various forms. Each chapter provides the opportunity to build different kinds of models. This chapter merely makes a few general observations pertinent to all modeling.

The first step in building a model often lies in discovering an area that is in need of study or improvement. Having established a need and a target for investigation, the analyst must determine which aspects of the system are controllable and which are not, and identify the goals or purpose of the system, and the constraints or limitations that govern the operation of the system. These limitations may result from physical, financial, political, or human factors. The next step is to create a model that implicitly or explicitly embodies alternative courses of action, and to collect data that characterize the particular system being modeled.

The process of “solving” the model or the problem depends entirely on the type of problem. It may involve applying a mathematical process to obtain a “best” answer, or it may necessitate the use of some other quantitative tools to determine, estimate, or project the behavior of the system being modeled. Realizing that the data may have been only approximate, and that the model may have been an imperfect representation of the real system, a successful analyst ultimately has the obligation to assess the practical applicability and flexibility of the solution suggested by the foregoing analysis. Merely finding an “optimal” solution to a model may be just the beginning of a manager’s job; a good manager must constantly reevaluate current practices, seek better ways to operate a system or organization, and notice trends in problem data that may not explicitly appear as part of a mathematical solution, such as excess production capacity, under-utilized labor, or a decreasing product demand over time. The entire modeling process is likely to require the skill and knowledge of a variety of individuals who are able to work effectively in teams and communicate clearly and convincingly among themselves, and then to explain and “sell” their recommendations to management.

Considerable skill is required in determining just how much detail to incorporate into a mathematical model. A very accurate representation of a system can be obtained with a large and sophisticated mathematical model. But if too many details are included, the model may be so complex and unwieldy that it becomes impossible to analyze or solve the system being modeled. Therefore, we do not even try to make a model as realistic as possible. On the other hand, a very simplistic model may not carry enough detail to provide an accurate representation of the real object or system; and in that case, any analysis that is performed on the model may not apply to the reality.

It is tempting to confuse detail (or precision) with accuracy. They are not the same, although many people are under the impression that the more detailed or complex a model, the more accurately it reflects reality. Not all details are correct, and not all details are relevant. The availability of powerful computing hardware and user-friendly software for building computer models almost seem to encourage runaway complexity and detail, as there seems to be no limit to what can be included almost effortlessly in a model. Nevertheless, it is possible to build models that are both realistic and simple, and doing so may spare an analyst from losing sight of the purpose of building the model in the first place.

The best model is one that strikes a practical compromise in representing a system as realistically as possible, while still being understandable and computationally tractable. It is, therefore, not surprising that developing a mathematical model is itself an iterative process, and a model can assume numerous forms during its development before an acceptable model emerges. An analyst might in fact need to see some numerical results of a solution to a problem in order to begin to recognize that the underlying model is incomplete or inaccurate.

The purpose of building models of systems is to develop an understanding of the real system; to predict its behavior; to learn the limiting capabilities of a system; and eventually to make decisions about the design, development, fabrication, modification, or operation of the real system being modeled. A thorough understanding of a model may make it unnecessary to build and experiment with the real system, and thus may avoid expense or alleviate exposure to dangerous situations.

Operations Research deals with decision-making. Decision-making is a human process that is often aided by intuition as well as facts. Intuition may serve well in personal decisions, but decisions made in political, government, commercial, and institutional settings that will affect large numbers of people require something more than intuition. A more systematic methodology is needed. Mathematical models that can be analyzed by well-understood methods and algorithms inspire more confid...