# Operations Research

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1370

## Abstract

Operations research (OR) is both a profession and an academic discipline. It involves the application of advanced analytical methods to improve executive and management decisions. This survey highlights the types of OR models and techniques in common use. It explores the roots of OR and its theoretical and professional evolution, and presents the current trends which shape its future.

## Keywords

Allocation problems Chance-constrained programming Computational methods Critical path method Dantzig, G. Data mining Dual method Dynamic pricing Dynamic programming E-commerce Financial engineering Game theory Globalization Graph theory Information technology Integer programming Inventory theory Kantorovich, L. V. Lattice programming Linear programming Marketing engineering Markov processes Multi-criteria programming Network flow optimization Operations research Polynomial algorithms Polynomial submodular set functions Probability theory Production control theory Program evaluation and review technique Quadratic programming Quesnay, F. Queuing theory Revenue management methods Simplex method for solving linear programs Simulation Stochastic programming Supermodularity Von Neumann, J. Walras, L## JEL Classifications

C44Operations research is commonly referred to as OR. In the United Kingdom, where the first formally recognized group of practitioners was formed, it is called ‘operational research’. Other names, such as ‘management science’, ‘operational analysis’ and ‘systems analysis’, are frequently used as synonyms.

Definitions of OR abound. The differences among these definitions reflect important dimensions of conflict in philosophy and perception of the field among the members of the various communities identifying themselves as operations researchers. It is instructive, therefore, to examine some of these definitions to identify areas of agreement about the distinctive characteristics of the field as well as those dimensions which are a cause of tension.

Operational Research is the application of the methods of science to the complex problems arising in the direction and management of large systems of men, machines, materials and money in industry, business, government and defense. The distinctive approach is to develop a scientific model of the system, incorporating measurements of factors such as chance and risk, with which to predict and compare outcomes of alternative decisions, strategies or controls. The purpose is to help management determine its policy and actions scientifically.

Its current website suggests that OR is the discipline of ‘applying advanced analytical methods to make better decisions’ (the OR Society). While these definitions see OR as an eclectic, problem-centred approach where scientific methods are employed to help management, definitions proposed in the United States view OR as a science or as a distinctive methodology providing scientific bases for decision-making. The constitution of the Operations Research Society of America (ORSA) referred to OR as ‘the science of operations research’ (House 1952, p. 28). This view was incorporated in 1982 in the Decision and Management Program of the U.S. National Science Foundation that referred to the emergence of a combined theoretical and empirical science of operational and managerial processes (Little 1986). The current website of the Institute for Operations Research and Management Science (INFORMS) which has succeeded ORSA, refers to OR as the ‘science of better’.

The scientific view of OR sees its goals as (*a*) the development of models of operations that represent the causal relationship between controlled variables, uncontrolled variables and system performance, and (*b*) the development of the computational means for identifying levels of controlled variables in ways that help managers of a system achieve systems outputs as close as possible to the ones they desire.

A broader and more proactive variant of the scientific view of OR was proposed in the first major textbook to be published on OR (Churchman et al. 1957). It suggested that the goal of OR is an overall understanding of optimal solutions to executive-type problems in organizations. This comprehensive goal of OR implies a normative prescriptive role with boundaries emancipated from mere reactive problem-solving.

Examination of the various definitions of OR establishes the following features upon which there is almost general agreement: (*a*) OR focuses upon executive and management-type decisions in organized systems; (*b*) a distinct feature of the methodologies used in OR is the development of quantitative models which relate controllable and uncontrollable variables to system performance measures; and (*c*) the outputs of OR models are solutions, that is, suggested levels of control variables that meet some prescribed restrictions. In addition, OR attempts to identify those solutions that are ‘better’ than others or are ‘best’ given an objective function and the validity of solutions ought to be tested empirically. OR, however, is concerned not only with the derivation of solutions but their relevance to management practice and their implementation.

While OR definitions reflect the ideals and aspirations of many leaders of OR communities, a commonly held view is that OR is a collection of techniques (National Academy of Sciences 1976). In fact, the success in practice of some OR techniques, such as linear programming and the proliferation of accessible optimization packages, is responsible for this perception. In part, it is also a reflection of imbalances in the work of OR academics. Examination of the content of OR journals and textbooks, for example, would support such a proposition. Indeed, much of the academic effort since the mid-1980s has focused on articulating the supporting mathematical theories of OR models, the development of alternative models and computational methods with a glaring absence of empirical testing (Denizel et al. 2003). This, however, was more a result of a natural progression of the life cycle of the field than a paradigm shift.

Disagreements in the OR communities exist with regard to the following questions. First, what is the level of generality that OR models can attain (that is, what are the prospects of OR becoming a science of operations as opposed to an approach to problem-solving in specific organizational contexts)? Second, what is the degree of comprehensiveness of OR missions, in particular the degree to which a systems approach should characterize OR activities (that is, focus of OR methodologies upon overall effects of a proposed solution on an organization rather than a narrower problem-solving focus)? Third, what is the role of interdisciplinary teamwork in OR?

## OR Models and Techniques

Models and computational techniques are key elements in the OR methodology. Models in OR, as opposed to models developed by mathematicians, derive their legitimacy from the real world (as in other sciences) and from their potential uses. Thus one can classify OR models and techniques according to the type of management problems or decision areas they deal with.

Allocation problems

Inventory problems

Queuing problems

Scheduling problems

Competitive problems

Renewal and replacement problems

Search problems

Revenue management

Supply chain management

Financial and marketing engineering

Data mining

Each of these problem areas is characterized by some typical structures which have stimulated the development of certain classes of mathematical models as well as their supporting mathematical theories. Often, however, a type of mathematical model developed for a specific problem area can be used to model processes with similar structures in other problem areas.

Let us consider, for example, allocation problems. These are the typical economic problems of allocating scarce resources between competing demands so as to maximize net benefits. The allocation, however, must satisfy some prescribed constraints. The first primitive mathematical programs were formulated by economists late in the 18th century. The typical structure of mathematical programs is the maximization or minimization of an objective function subject to a set of constraints. The properties of the objective function and the special structures of the constraints determine the methods and difficulty of finding optimal values. For example, *linear programming* postulates a system with a linear objective function, linear constraints and non-negative control variables. Thus sub-areas of mathematical programming designate the mathematical structure of the optimization problem at hand: *integer programmin*g requires integer solutions; *quadratic programming* postulates a quadratic objective function; *stochastic programming* assumes that stochastic parameters describe the objective function; *chance-constrained programming* assumes that the restrictions on a problem are given as probabilities of satisfying each constraint, and so on.

An interesting allocation problem arises in situations with multiple decision units with separate conflicting objectives, when rules for trade-offs or reconciliation of conflict are not given. This problem led to the emergence of *multi-criteria programming*, a technique that postulates several objective functions subject to a set of joint constraints.

As we indicated, while mathematical programming emerged as a means of dealing with allocation problems, its applications cut across most areas of OR endeavour.

*Queuing theory* evolved primarily to help design service policies to deal with congestion and waiting lines. The theory has its roots in probability theory. The application of the theory demonstrates well a problem which characterizes many OR models – limited empirical validity. Indeed, in many practical situations, the probability distributions which characterize arrival and service time depart from those postulated by the basic theory. In such cases problems become analytically intractable and simulation techniques are used.

*Inventory and production control theory* can be divided into the tractable but unrealistic deterministic cases, and the more problematic stochastic cases. The theory has contributed important insights as to the shape of optimal policies, but specific solutions to problems arising in the real world are typically obtained by simulation.

*Simulation* is indeed the most prolific OR technique. It is used in practice especially to model stochastic processes and provide solutions to analytically difficult or intractable problems. A computer model representing the system provides the vehicle for low-cost, fast experimentation with alternative patterns of control variables.

Competitive problems have led to the emergence of *game theory*. While the theory has had some important applications (for example, designing optimal stable policies of inspections associated with international nuclear-testing restrictions), its restrictive assumptions with respect to the rationality of players have limited its usefulness for modelling many competitive business situations. *Gaming* and simulation techniques are often used to improve strategic decisions in competitive situations.

Scheduling problems are typically modelled as *network flow optimization* problems. Two techniques have received great attention and have been employed widely in project planning: the program evaluation and review technique (PERT) and the critical path method (CPM). Network flow optimization is used extensively to deal with many transportation and communication problems.

An important area of OR modelling is the area of Markov and related processes. In a Markov process, knowledge of the present makes the future independent of the past. Markov chains have been used extensively in manpower planning.

*Dynamic programming* is a method of analysing multi-stage decision processes in which each decision in a sequence depends upon those preceding it as well as exogenous factors. The technique reduces significantly the computational effort by eliminating the need to enumerate and consider the consequences of all possible decision sequences. The method is used in a wide variety of problem areas.

In the 1980s *revenue management methods* combining accurate demand forecasts with intelligent dynamic pricing were developed for and adopted by airlines, and their use spread to other sectors. In the 1990s increasing globalization and the emergence of complex business networks of suppliers and producers created the need for better *supply chain management*. Advances in computing power, communications and operation research methods created new modelling opportunities responding to the challenge of finding best overall combinations of suppliers, transportation, production, warehousing and inventory. Recent modelling efforts in this domain incorporate ‘game like’ situations in cooperative networks where incentives of different participants may be misaligned. The late 1990s saw the emergence of e-commerce and powerful information technology applications in business generating high volumes of customer data. Large, high-quality data stimulated the development of *data mining* techniques to use the data to improve business strategies and operations. The proliferation of personal powerful computers created opportunities for the development of OR applications for a variety of business functions. *Financial and marketing engineering* are examples of OR applications to traditional functional fields of business.

The lack of definite boundaries as to what constitutes OR makes it difficult to determine whether some techniques originating in other fields, but frequently used by OR practitioners, should be designated as OR techniques. Statistical analysis, forecasting methodologies and evaluation techniques are good examples.

### The Roots of OR

The beginnings of OR can be traced to the emergence of the executive function and the complex organization brought about by the Industrial Revolution of the 19th century. The mathematical roots of OR can be traced earlier to the work of Quesnay (1759), who formulated primitive mathematical programming models. This fundamental work was followed by the work of Walras (1883), and by the work of von Neumann (1937) and Kantorovich (1939).

The roots of empirical OR can be traced to the scientific management movement. The work of Taylor, Gantt, Emerson and other pioneers of scientific management began around 1885. They proposed that scientific methods of analysis and measurement could and should be used in production management and business decisions. In 1909, Erlang, a Danish mathematician, published his study of traffic congestion in a telephone network, pioneering the modelling of queues. In 1916 Lanchester published his ‘N-square law’, assessing the fighting power of opposing forces. The theory was tested retrospectively against Admiral Nelson’s plan of the battle of Trafalgar.

The appearance of OR as an organized activity is associated with preparation in the UK for the Second World War. In 1936 the British government decided to set up radar stations. The need to study the operational use of radar chains in order to increase their ability to detect aircraft led to the establishment of a study group of scientists called ‘the operational research group’. Their success led to the adoption of OR by other branches of the military. In 1942 an OR section was established by the US Air Force. OR was soon adopted by other branches of the US military. Under the aegis of the US Air Force, a team of economists and mathematicians began in 1947 to model the military structure and the economy. During this period Dantzig (1963) developed the simplex method of solving linear programs.

## The Evolution of OR

The diffusion of OR to the industrial world was slow. Only in the early 1950s did the tools and methods of OR begin to be used outside the military. The first important industrial application of OR was the use of linear programming to schedule a petroleum refinery (Charnes and Cooper 1961).

The Operational Research Club of Britain was formed in 1948, and the Operations Research Society of America (ORSA) was established in 1951. Other national societies for OR soon followed, and in 1957 the International Federation of Operational Research Societies was formed. Books, journals and university programmes specializing in OR proliferated in the 1960s. A gradual process of change in the membership of most OR communities started, bringing a shift towards a higher proportion of university-based members. While the ORSA constitution saw as one of its major missions the establishment and maintenance of professional standards of competence in OR, the evolution of the field caused more emphasis to be placed upon the academic mission of the development of methods and techniques of OR. The tension between practice and theory of OR indeed originated in the 1950s and 1960s. It is interesting to note that this period is viewed by some as the best of times for OR (Miser 1978) and by others as the worst of times (Churchman 1979).

The period saw some of the most exciting mathematical developments since the simplex algorithm. Examples are the important paper by Kuhn and Tucker (1951) laying the foundations of nonlinear programming; the paper by Gomory (1958) presenting a systematic computational technique for integer programming; the works of Bellman (1957) developing dynamic programming; the seminal book by Ford Jr. and Fulkerson (1962) articulating network flow optimization; and the volume edited by Arrow et al. (1958) on the mathematical theory of inventory and production processes. Other important developments during the period were the articulation of decision analysis (see, for example, Raiffa 1968), the development of stochastic programming and chance-constrained programming (see, for example, Charnes and Cooper 1959) and the development of the dual method (Lemke 1954) and the linear complementarity algorithm (Lemke 1965).

Yet, despite these developments, Churchman (1979, p. 13) called the period ‘dreary’, lamenting the separation of theoretical developments from application, describing OR modelling as a ‘study of the delights of algorithms; nuances of game theory; fascinating but irrelevant things that can happen in queues’.

The 1970s presented OR with an important mathematical theory – a theory focusing on its bounds rather than promises: the theory of NP-completeness. The theory presents a framework for the identification of bounds on computational efficiencies (Cook 1971; Karp 1972). Important breakthroughs in the early 1980s were associated with possible improvements on the simplex algorithm in solving linear programs – the development of polynomial algorithms by Khachian (1979, 1980) and Karmarkar (1984).

The 1980s also saw a breakthrough development in the inventory management field. Roundy (1985, 1986) found a simple heuristic and proved that it yields schedules within two per cent of the optimal solution; this work anticipated also the coordination problems characterizing supply chain management. The 1990s saw articulation of the general theory of supermodularity and lattice programming pioneered by Veinott, Edmonds and Topkis (see Topkis 1998). The theory provides fundamental insights to certain classes of optimization problems and issues related to monotone comparative statics, fundamental in economic analysis. The development of polynomial submodular set functions – an unresolved problem remaining – was solved simultaneously by Iwata, Fleischer and Fujishige and Schrijver (see Fleischer 2000).

The new millennium also saw a breakthrough in graph theory – the characterization of the strong perfect graphs by Chudnovsky, Robertson, Seymour and Thomas (see Cornuéjols 2003).

Perhaps more important to the future of operations research has been the great progress achieved since the mid-1990s in computational methods. Advances in computing machinery, software improvements and development combined to increase the practical significance of the various OR methods. The increased speed of computation and the huge increases in computer memory capacity have made it possible to solve much larger problems and use entirely different solution strategies (Bixby 2002). Improved software also allowed also better interface with users, increasing the accessibility of OR methods to a wider population of users.

The scope of OR was enlarged while the cohesiveness of its communities reduced. Fragmentation was identified by many as an explanation of the declining memberships of many OR and management science professional societies. OR appeared to some observers to be ‘in danger of losing its identity as a recognized activity and being assimilated into other fields of endeavor’ (Bonder 1979, p. 218). Thus, while the power of OR methods and their use increased, the period since the mid-1980s has witnessed some trends which are threatening the identity of OR as a distinct profession.

## The Future of OR

The apparent divorce of OR theory from practice and empirical testing led some leaders in the OR community to wonder whether ‘the future of OR is past’. The microcomputer revolution has increased the benefit–costs ratios of OR methods and increased the direct access of general business users to OR. OR groups and practitioners, however, have lost some of their unique advantages as gatekeepers to the application of OR methods. Much of the diffusion of OR methods to the industry is now accomplished through the sales of packaged programs, and is marketed through demonstration CDs. Many users of OR methods in business do not consider themselves OR practitioners. Thus, the dispersion of OR practice in business has resulted in a loss of professional identity (Geoffrion 1992).

Loss of professional identity reduces the flow of new recruits to the profession and limits the career opportunities of OR professionals. The success of OR methods may, therefore, entail the decline of the profession. The sustainability and health of the profession depends on its ability to adopt new business models that fit the new environment, turning threats to opportunities for growth.

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