- operations research
the analysis, usually involving mathematical treatment, of a process, problem, or operation to determine its purpose and effectiveness and to gain maximum efficiency.[1940-45, Amer.]
* * *Application of scientific methods to management and administration of military, government, commercial, and industrial systems.It began during World War II in Britain when teams of scientists worked with the Royal Air Force to improve radar detection of enemy aircraft, leading to coordinated efforts to improve the entire system of early warning, defense, and supply. It is characterized by a systems orientation, or systems engineering, in which interdisciplinary research teams adapt scientific methods to large-scale problems that must be modeled, since laboratory testing is impossible. Examples include resource allocation and replacement, inventory control, and scheduling of large-scale construction projects.
* * *Introductionalso called operational research,application of scientific methods to the management and administration of organized military, governmental, commercial, and industrial processes.Basic aspectsOperations research attempts to provide those who manage organized systems with an objective and quantitative basis for decision; it is normally carried out by teams of scientists and engineers drawn from a variety of disciplines. Thus, operations research is not a science itself but rather the application of science to the solution of managerial and administrative problems, and it focuses on the performance of organized systems taken as a whole rather than on their parts taken separately. Usually concerned with systems in which human behaviour plays an important part, operations research differs in this respect from systems engineering, which, using a similar approach, tends to concentrate on systems in which human behaviour is not important. Operations research was originally concerned with improving the operations of existing systems rather than developing new ones; the converse was true of systems engineering. This difference, however, has been disappearing as both fields have matured.The subject matter of operations research consists of decisions that control the operations of systems. Hence, it is concerned with how managerial decisions are and should be made, how to acquire and process data and information required to make decisions effectively, how to monitor decisions once they are implemented, and how to organize the decision-making and decision-implementation process. Extensive use is made of older disciplines such as logic, mathematics, and statistics, as well as more recent scientific developments such as communications theory, decision theory, cybernetics, organization theory, the behavioral sciences, and general systems theory.In the 19th century the Industrial Revolution involved mechanization or replacement of human by machine as a source of physical work. Study and improvement of such work formed the basis of the field of industrial engineering. Many contemporary issues are concerned with automation or mechanization of mental work. The primary technologies involved are mechanization of symbol generation (observation by machines such as radar and sonar), mechanization of symbol transmission (communication by telephone, radio, and television), and mechanization of logical manipulation of symbols (data processing and decision making by computer). Operations research applies the scientific method to the study of mental work and provides the knowledge and understanding required to make effective use of personnel and machines to carry it out.HistoryIn a sense, every effort to apply science to management of organized systems, and to their understanding, was a predecessor of operations research. It began as a separate discipline, however, in 1937 in Britain as a result of the initiative of A.P. Rowe, superintendent of the Bawdsey Research Station, who led British scientists to teach military leaders how to use the then newly developed radar to locate enemy aircraft. By 1939 the Royal Air Force (Royal Air Force, The) formally commenced efforts to extend the range of radar equipment so as to increase the time between the first warning provided by radar and the attack by enemy aircraft. At first they analyzed physical equipment and communication networks, but later they examined behaviour of the operating personnel and relevant executives. Results of the studies revealed ways of improving the operators' techniques and also revealed unappreciated limitations in the network.Similar developments took place in the British Army and the Royal Navy, and in both cases radar again was the instigator. In the army, use of operations research had grown out of the initial inability to use radar effectively in controlling the fire of antiaircraft weapons. Since the traditional way of testing equipment did not seem to apply to radar gunsights, scientists found it necessary to test in the field under operating conditions, and the distinguished British physicist and future Nobel Laureate P.M.S. Blackett (Blackett, Patrick M.S., Baron Blackett of Chelsea) organized a team to solve the antiaircraft problem. Blackett's Antiaircraft Command Research Group included two physiologists, two mathematical physicists, an astrophysicist, an army officer, a former surveyor, and subsequently a third physiologist, a general physicist, and two mathematicians.By 1942 formal operations research groups had been established in all three of Britain's military services.Development of operations research paralleling that in Britain took place in Australia, Canada, France, and, most significantly for future developments, in the United States, which was the beneficiary of a number of contacts with British researchers. Sir Robert Watson-Watt (Watson-Watt, Sir Robert Alexander), who with A.P. Rowe launched the first two operational studies of radar in 1937 and who claims to have given the discipline its name, visited the United States in 1942 and urged that operations research be introduced into the War and Navy departments. Reports of the British work had already been sent from London by American observers, and James B. Conant, then chairman of the National Defense Research Committee, had become aware of operations research during a visit to England in the latter half of 1940. Another stimulant was Blackett's memorandum, “Scientists at the Operational Level,” of December 1941, which was widely circulated in the U.S. service departments.The first organized operations research activity in the United States began in 1942 in the Naval Ordnance Laboratory. This group, which dealt with mine warfare problems, was later transferred to the Navy Department, from which it designed the aircraft mining blockade of the Inland Sea of Japan.As in Britain, radar stimulated developments in the U.S. Air Force (United States Air Force, The). In October 1942, all Air Force commands were urged to include operations research groups in their staffs. By the end of World War II there were 26 such groups in the Air Force. In 1943 Gen. George Marshall suggested to all theatre commanders that they form teams to study amphibious and ground operations.At the end of World War II a number of British operations research workers moved to government and industry. Nationalization of several British industries was an important factor. One of the first industrial groups was established at the National Coal Board. Electricity and transport, both nationalized industries, began to use operations research shortly thereafter. Parts of the private sector began to follow suit, particularly in those industries with cooperative research associations; for example, in the British Iron and Steel Research Association.The early development of industrial operations research was cautious, and for some years most industrial groups were quite small. In the late 1950s, largely stimulated by developments in the United States, the development of industrial operations research in Britain was greatly accelerated.Although in the United States military research increased at the end of the war, and groups were expanded, it was not until the early 1950s that American industry began to take operations research seriously. The advent of the computer brought an awareness of a host of broad system problems and the potentiality for solving them, and within the decade about half the large corporations in the United States began to use operations research. Elsewhere the technique also spread through industry.Societies were organized, beginning with the Operational Research Club of Britain, formed in 1948, which in 1954 became the Operational Research Society. The Operations Research Society in America was formed in 1952. Many other national societies appeared; the first international conference on operations research was held at Oxford University in 1957. In 1959 an International Federation of Operational Research Societies was formed.The first appearance of operations research as an academic discipline came in 1948 when a course in nonmilitary techniques was introduced at the Massachusetts Institute of Technology in Cambridge. In 1952 a curriculum leading to a master's and doctoral degree was established at the Case Institute of Technology (now Case Western Reserve University) in Cleveland. Since then many major academic institutions in the United States have introduced programs. In the United Kingdom courses were initiated at the University of Birmingham in the early 1950s. The first chair in operations research was created at the newly formed University of Lancaster in 1964. Similar developments have taken place in most countries in which a national operations research society exists.The first scholarly journal, the Operational Research Quarterly, published in the United Kingdom, was initiated in 1950; in 1978 its name was changed to the Journal of the Operational Research Society. It was followed in 1952 by the Journal of the Operations Research Society of America, which was renamed Operations Research in 1955. The International Federation of Operational Research Societies initiated the International Abstracts in Operations Research in 1961.Despite its rapid growth, operations research is still a relatively young scientific activity. Its techniques and methods, and the areas to which they are applied, can be expected to continue to expand rapidly. Most of its history lies in the future.Essential characteristicsThree essential characteristics of operations research are a systems orientation, the use of interdisciplinary teams, and the application of scientific method to the conditions under which the research is conducted.Systems orientationThe systems approach to problems recognizes that the behaviour of any part of a system has some effect on the behaviour of the system as a whole. Even if the individual components are performing well, however, the system as a whole is not necessarily performing equally well. For example, assembling the best of each type of automobile part, regardless of make, does not necessarily result in a good automobile or even one that will run, because the parts may not fit together. It is the interaction between parts, and not the actions of any single part, that determines how well a system performs.Thus, operations research attempts to evaluate the effect of changes in any part of a system on the performance of the system as a whole and to search for causes of a problem that arises in one part of a system in other parts or in the interrelationships between parts. In industry, a production problem may be approached by a change in marketing policy. For example, if a factory fabricates a few profitable products in large quantities and many less profitable items in small quantities, long efficient production runs of high-volume, high-profit items may have to be interrupted for short runs of low-volume, low-profit items. An operations researcher might propose reducing the sales of the less profitable items and increasing those of the profitable items by placing salesmen on an incentive system that especially compensates them for selling particular items.The interdisciplinary teamScientific and technological disciplines have proliferated rapidly in the last 100 years. The proliferation, resulting from the enormous increase in scientific knowledge, has provided science with a filing system that permits a systematic classification of knowledge. This classification system is helpful in solving many problems by identifying the proper discipline to appeal to for a solution. Difficulties arise when more complex problems, such as those arising in large organized systems, are encountered. It is then necessary to find a means of bringing together diverse disciplinary points of view. Furthermore, since methods differ among disciplines, the use of interdisciplinary teams makes available a much larger arsenal of research techniques and tools than would otherwise be available. Hence, operations research may be characterized by rather unusual combinations of disciplines on research teams and by the use of varied research procedures.MethodologyUntil the 20th century, laboratory experiments were the principal and almost the only method of conducting scientific research. But large systems such as are studied in operations research cannot be brought into laboratories. Furthermore, even if systems could be brought into the laboratory, what would be learned would not necessarily apply to their behaviour in their natural environment, as shown by early experience with radar. Experiments on systems and subsystems conducted in their natural environment (“operational experiments”) are possible as a result of the experimental methods developed by the British statistician R.A. Fisher (Fisher, Sir Ronald Aylmer) in 1923–24. For practical or even ethical reasons, however, it is seldom possible to experiment on large organized systems as a whole in their natural environments. This results in an apparent dilemma: to gain understanding of complex systems experimentation seems to be necessary, but it cannot usually be carried out. This difficulty is solved by the use of models (mathematical model), representations of the system under study. Provided the model is good, experiments (called “simulations”) can be conducted on it, or other methods can be used to obtain useful results.Phases of operations researchProblem formulationTo formulate an operations research problem, a suitable measure of performance must be devised, various possible courses of action defined (that is, controlled variables and the constraints upon them), and relevant uncontrolled variables identified. To devise a measure of performance, objectives are identified and defined, and then quantified. If objectives cannot be quantified or expressed in rigorous (usually mathematical) terms, most operations research techniques cannot be applied. For example, a business manager may have the acquisitive objective of introducing a new product and making it profitable within one year. The identified objective is profit in one year, which is defined as receipts less costs, and would probably be quantified in terms of sales. In the real world, conditions may change with time. Thus, though a given objective is identified at the beginning of the period, change and reformulation are frequently necessary.Detailed knowledge of how the system under study actually operates and of its environment is essential. Such knowledge is normally acquired through an analysis of the system, a four-step process that involves determining whose needs or desires the organization tries to satisfy; how these are communicated to the organization; how information on needs and desires penetrates the organization; and what action is taken, how it is controlled, and what the time and resource requirements of these actions are. This information can usually be represented graphically in a flowchart, which enables researchers to identify the variables that affect system performance.Once the objectives, the decision makers, their courses of action, and the uncontrolled variables have been identified and defined, a measure of performance can be developed and selection can be made of a quantitative function of this measure to be used as a criterion for the best solution.The type of decision criterion that is appropriate to a problem depends on the state of knowledge regarding possible outcomes. Certainty describes a situation in which each course of action is believed to result in one particular outcome. risk is a situation in which, for each course of action, alternative outcomes are possible, the probabilities of which are known or can be estimated. Uncertainty describes a situation in which, for each course of action, probabilities cannot be assigned to the possible outcomes.In risk situations, which are the most common in practice, the objective normally is to maximize expected (long-run average) net gain or gross gain for specified costs, or to minimize costs for specified benefits. A business, for example, seeks to maximize expected profits or minimize expected costs. Other objectives, not necessarily related, may be sought; for example, an economic planner may wish to maintain full employment without inflation; or different groups within an organization may have to compromise their differing objectives, as when an army and a navy, for example, must cooperate in matters of defense.In approaching uncertain situations one may attempt either to maximize the minimum gain or minimize the maximum loss that results from a choice; this is the “minimax” approach. Alternatively, one may weigh the possible outcomes to reflect one's optimism or pessimism and then apply the minimax principle. A third approach, “minimax regret,” attempts to minimize the maximum deviation from the outcome that would have been selected if a state of certainty had existed before the choice had been made.Each identified variable should be defined in terms of the conditions under which, and research operations by which, questions concerning its value ought to be answered; this includes identifying the scale used in measuring the variable.Model constructionA model is a simplified representation of the real world and, as such, includes only those variables relevant to the problem at hand. A model of freely falling bodies, for example, does not refer to the colour, texture, or shape of the body involved. Furthermore, a model may not include all relevant variables because a small percentage of these may account for most of the phenomenon to be explained. Many of the simplifications used produce some error in predictions derived from the model, but these can often be kept small compared to the magnitude of the improvement in operations that can be extracted from them. Most operations research models are symbolic models because symbols represent properties of the system. The earliest models were physical representations such as model ships, airplanes, tow tanks, and wind tunnels. Physical models are usually fairly easy to construct, but only for relatively simple objects or systems, and are usually difficult to change.The next step beyond the physical model is the graph, easier to construct and manipulate but more abstract. Since graphic representation of more than three variables is difficult, symbolic (symbol) models came into use. There is no limit to the number of variables that can be included in a symbolic model, and such models are easier to construct and manipulate than physical models.Symbolic models are completely abstract. When the symbols in a model are defined, the model is given content or meaning. This has important consequences. Symbolic models of systems of very different content often reveal similar structure. Hence, most systems and problems arising in them can be fruitfully classified in terms of relatively few structures. Furthermore, since methods of extracting solutions from models depend only on their structure, some methods can be used to solve a wide variety of problems from a contextual point of view. Finally, a system that has the same structure as another, however different the two may be in content, can be used as a model of the other. Such a model is called an analogue. By use of such models much of what is known about the first system can be applied to the second.Despite the obvious advantages of symbolic models there are many cases in which physical models are still useful, as in testing physical structures and mechanisms; the same is true for graphic models. Physical and graphic models are frequently used in the preliminary phases of constructing symbolic models of systems.Operations research models represent the causal relationship between the controlled and uncontrolled variables and system performance; they must therefore be explanatory, not merely descriptive. Only explanatory models can provide the requisite means to manipulate the system to produce desired changes in performance.Operations research analysis is directed toward establishing cause-and-effect relations. Though experiments with actual operations of all or part of a system are often useful, these are not the only way to analyze cause and effect. There are four patterns of model construction, only two of which involve experimentation: inspection, use of analogues, operational analysis, and operational experiments. They are considered here in order of increasing complexity.In some cases the system and its problem are relatively simple and can be grasped either by inspection or from discussion with persons familiar with it. In general, only low-level and repetitive operating problems, those in which human behaviour plays a minor role, can be so treated.When the researcher finds it difficult to represent the structure of a system symbolically, it is sometimes possible to establish a similarity, if not an identity, with another system whose structure is better known and easier to manipulate. It may then be possible to use either the analogous system itself or a symbolic model of it as a model of the problem system. For example, an equation derived from the kinetic theory of gases has been used as a model of the movement of trains between two classification yards. Hydraulic analogues of economies and electronic analogues of automotive traffic have been constructed with which experimentation could be carried out to determine the effects of manipulation of controllable variables. Thus, analogues may be constructed as well as found in existing systems.In some cases analysis of actual operations of a system may reveal its causal structure. Data on operations are analyzed to yield an explanatory hypothesis, which is tested by analysis of operating data. Such testing may lead to revision of the hypothesis. The cycle is continued until a satisfactory explanatory model is developed.For example, an analysis of the cars stopping at urban automotive service stations located at intersections of two streets revealed that almost all came from four of the 16 possible routes through the intersection (four ways of entering times four ways of leaving). Examination of the percentage of cars in each route that stopped for service suggested that this percentage was related to the amount of time lost by stopping. Data were then collected on time lost by cars in each route. This revealed a close inverse relationship between the percentage stopping and time lost. But the relationship was not linear; that is, the increases in one were not proportional to increases in the other. It was then found that perceived lost time exceeded actual lost time, and the relationship between the percentage of cars stopping and perceived lost time was close and linear. The hypothesis was systematically tested and verified and a model constructed that related the number of cars stopping at service stations to the amount of traffic in each route through its intersection and to characteristics of the station that affect the time required to get service.In situations where it is not possible to isolate the effects of individual variables by analysis of operating data, it may be necessary to resort to operational experiments to determine which variables are relevant and how they affect system performance.Such is the case, for example, in attempts to quantify the effects of advertising (amount, timing, and media used) upon sales of a consumer product. Advertising by the producer is only one of many controlled and uncontrolled variables affecting sales. Hence, in many cases its effect can only be isolated and measured by controlled experiments in the field.The same is true in determining how the size, shape, weight, and price of a food product affect its sales. In this case laboratory experiments on samples of consumers can be used in preliminary stages, but field experiments are eventually necessary. Experiments do not yield explanatory theories, however. They can only be used to test explanatory hypotheses formulated before designing the experiment and to suggest additional hypotheses to be tested.It is sometimes necessary to modify an otherwise acceptable model because it is not possible or practical to find the numerical values of the variables that appear in it. For example, a model to be used in guiding the selection of research projects may contain such variables as “the probability of success of the project,” “expected cost of the project,” and its “expected yield.” But none of these may be calculable with any reliability.Models not only assist in solving problems but also are useful in formulating them; that is, models can be used as guides to explore the structure of a problem and to reveal possible courses of action that might otherwise be missed. In many cases the course of action revealed by such application of a model is so obviously superior to previously considered possibilities that justification of its choice is hardly required.In some cases the model of a problem may be either too complicated or too large to solve. It is frequently possible to divide the model into individually solvable parts and to take the output of one model as an input to another. Since the models are likely to be interdependent, several repetitions of this process may be necessary.Deriving solutions from modelsProcedures for deriving solutions from models are either deductive or inductive. With deduction one moves directly from the model to a solution in either symbolic or numerical form. Such procedures are supplied by mathematics; for example, the calculus. An explicit analytical procedure for finding the solution is called an algorithm.Even if a model cannot be solved, and many are too complex for solution, it can be used to compare alternative solutions. It is sometimes possible to conduct a sequence of comparisons, each suggested by the previous one and each likely to contain a better alternative than was contained in any previous comparison. Such a solution-seeking procedure is called heuristic.Inductive procedures involve trying and comparing different values of the controlled variables. Such procedures are said to be iterative (repetitive) if they proceed through successively improved solutions until either an optimal solution is reached or further calculation cannot be justified. A rational basis for terminating such a process—known as “stopping rules”—involves the determination of the point at which the expected improvement of the solution on the next trial is less than the cost of the trial.Such well-known algorithms as linear, nonlinear, and dynamic programming are iterative procedures based on mathematical theory. Simulation and experimental optimization are iterative procedures based primarily on statistics.Testing the model and the solutionA model may be deficient because it includes irrelevant variables, excludes relevant variables, contains inaccurately evaluated variables, is incorrectly structured, or contains incorrectly formulated constraints. Tests for deficiencies of a model are statistical in nature; their use requires knowledge of sampling and estimation theory, experimental designs, and the theory of hypothesis testing (see also statistics).Sampling-estimation theory is concerned with selecting a sample of items from a large group and using their observed properties to characterize the group as a whole. To save time and money, the sample taken is as small as possible. Several theories of sampling design and estimation are available, each yielding estimates with different properties.The structure of a model consists of a function relating the measure of performance to the controlled and uncontrolled variables; for example, a business may attempt to show the functional relationship between profit levels (the measure of performance) and controlled variables (prices, amount spent on advertising) and uncontrolled variables (economic conditions, competition). In order to test the model, values of the measure of performance computed from the model are compared with actual values under different sets of conditions. If there is a significant difference between these values, or if the variability of these differences is large, the model requires repair. Such tests do not use data that have been used in constructing the model, because to do so would determine how well the model fits performance data from which it has been derived, not how well it predicts performance.The solution derived from a model is tested to find whether it yields better performance than some alternative, usually the one in current use. The test may be prospective, against future performance, or retrospective, comparing solutions that would have been obtained had the model been used in the past with what actually did happen. If neither prospective nor retrospective testing is feasible, it may be possible to evaluate the solution by “sensitivity analysis,” a measurement of the extent to which estimates used in the solution would have to be in error before the proposed solution performs less satisfactorily than the alternative decision procedure.The cost of implementing a solution should be subtracted from the gain expected from applying it, thus obtaining an estimate of net improvement. Where errors or inefficiencies in applying the solution are possible, these should also be taken into account in estimating the net improvement.Implementing and controlling the solutionThe acceptance of a recommended solution by the responsible manager depends on the extent to which he believes the solution to be superior to alternatives. This in turn depends on his faith in the researchers involved and their methods. Hence, participation by managers in the research process is essential for success.Operations researchers are normally expected to oversee implementation of an accepted solution. This provides them with an ultimate test of their work and an opportunity to make adjustments if any deficiencies should appear in application. The operations research team prepares detailed instructions for those who will carry out the solution and trains them in following these instructions. The cooperation of those who carry out the solution and those who will be affected by it should be sought in the course of the research process, not after everything is done. Implementation plans and schedules are pretested and deficiencies corrected. Actual performance of the solution is compared with expectations and, where divergence is significant, the reasons for it are determined and appropriate adjustments made.The solution may fail to yield expected performance for one or a combination of reasons: the model may be wrongly constructed or used; the data used in making the model may be incorrect; the solution may be incorrectly carried out; the system or its environment may have changed in unexpected ways after the solution was applied. Corrective action is required in each case.Controlling a solution requires deciding what constitutes a significant deviation in performance from expectations; determining the frequency of control checks, the size and type of sample of observations to be made, and the types of analyses of the resulting data that should be carried out; and taking appropriate corrective action. The second step should be designed to minimize the sum of the costs of carrying out the control procedures and the errors that might be involved.Since most models involve a variety of assumptions, these are checked systematically. Such checking requires explicit formulation of the assumptions made during construction of the model.Effective controls not only make possible but often lead to better understanding of the dynamics of the system involved. Through controls the problem-solving system of which operations research is a part learns from its own experience and adapts more effectively to changing conditions.Computers (computer) and operations researchComputers have had a dramatic impact on the management of industrial production systems and the fields of operations research and industrial engineering. The speed and data-handling capabilities of computers allow engineers and scientists to build larger, more realistic models of organized systems and to get meaningful solutions to those models through the use of simulation techniques.Simulation consists of calculating the performance of a system by evaluating a model of it for randomly selected values of variables contained within it. Most simulation in operations research is concerned with “stochastic” variables; that is, variables whose values change randomly within some probability distribution over time. The random sampling employed in simulation requires either a supply of random numbers or a procedure for generating them. It also requires a way of converting these numbers into the distribution of the relevant variable, a way of sampling these values, and a way of evaluating the resulting performance.A simulation in which decision making is performed by one or more real decision makers is called “operational gaming.” Such simulations are commonly used in the study of interactions of decision makers as in competitive situations. Military gaming has long been used as a training device, but only relatively recently has it been used for research purposes. There is still considerable difficulty, however, in drawing inferences from operational games to the real world.Experimental optimization is a means of experimenting on a system so as to find the best solution to a problem within it. Such experiments, conducted either simultaneously or sequentially, may be designed in various ways, no one of which is best in all situations.Decision analysis and supportSince their widespread introduction in business and government organizations in the 1950s, the primary applications of computers have been in the areas of record keeping, bookkeeping, and transaction processing. These applications, commonly called data processing, automate the flow of paperwork, account for business transactions (such as order processing and inventory and shipping activities), and maintain orderly and accurate records. Although data processing is vital to most organizations, most of the work involved in the design of such systems does not require the methods of operations research.In the 1960s, when computers were applied to the routine decision-making problems of managers, management information systems (MIS) emerged. These systems use the raw (usually historical) data from data-processing systems to prepare management summaries, to chart information on trends and cycles, and to monitor actual performance against plans or budgets.More recently, decision support systems (DSS) have been developed to project and predict the results of decisions before they are made. These projections permit managers and analysts to evaluate the possible consequences of decisions and to try several alternatives on paper before committing valuable resources to actual programs.The development of management information systems and decision support systems brought operations researchers and industrial engineers to the forefront of business planning. These computer-based systems require knowledge of an organization and its activities in addition to technical skills in computer programming and data handling. The key issues in MIS or DSS include how a system will be modeled, how the model of the system will be handled by the computer, what data will be used, how far into the future trends will be extrapolated, and so on. In much of this work, as well as in more traditional operations research modeling, simulation techniques have proved invaluable.New software tools for decision makingThe explosive growth of personal computers (computer program) in business organizations in the early 1980s spawned a parallel growth in software to assist in decision making. These tools include spreadsheet programs for analyzing complex problems with trails that have different sets of data, data base management programs that permit the orderly maintenance and manipulation of vast amounts of information, and graphics programs that quickly and easily prepare professional-looking displays of data. Business programs (software) like these once cost tens of thousands of dollars; now they are widely available, may be used on relatively inexpensive hardware, are easy to use without learning a programming language, and are powerful enough to handle sophisticated, practical business problems.The availability of spreadsheet, data base, and graphics programs on personal computers has also greatly aided industrial engineers and operations researchers whose work involves the construction, solution, and testing of models. Easy-to-use software that does not require extensive programming knowledge permits faster, more cost-effective model building and is also helpful in communicating the results of analysis to management. Indeed, many managers now have a computer on their desk and work with spreadsheets and other programs as a routine part of their managerial duties.Examples of operations research models and applicationsAs previously mentioned, many operational problems of organized systems have common structures. The most common types of structure have been identified as prototype problems, and extensive work has been done on modeling and solving them.Though all the problems with similar structures do not have the same model, those that apply to them may have a common mathematical structure and hence may be solvable by one procedure. Some real problems consist of combinations of smaller problems, some or all of which fall into different prototypes. In general, prototype models are the largest that can be solved in one step. Hence, large problems that consist of combinations of prototype problems usually must be broken down into solvable units; the overall model used is an aggregation of prototype and possibly other models.Resource allocationAllocation problems involve the distribution of resources (allocation of resources) among competing alternatives in order to minimize total costs or maximize total return. Such problems have the following components: a set of resources available in given amounts; a set of jobs to be done, each consuming a specified amount of resources; and a set of costs or returns for each job and resource. The problem is to determine how much of each resource to allocate to each job.If more resources are available than needed, the solution should indicate which resources are not to be used, taking associated costs into account. Similarly, if there are more jobs than can be done with available resources, the solution should indicate which jobs are not to be done, again taking into account the associated costs.If each job requires exactly one resource (e.g., one person) and each resource can be used on only one job, the resulting problem is one of assignment. If resources are divisible, and if both jobs and resources are expressed in units on the same scale, it is termed a transportation or distribution problem. If jobs and resources are not expressed in the same units, it is a general allocation problem.An assignment problem may consist of assigning workers to offices or jobs, trucks to delivery routes, drivers to trucks, or classes to rooms. A typical transportation problem involves distribution of empty railroad freight cars where needed or the assignment of orders to factories for production. The general allocation problem may consist of determining which machines should be employed to make a given product or what set of products should be manufactured in a plant during a particular period.In allocation problems the unit costs or returns may be either independent or interdependent; for example, the return from investing a dollar in selling effort may depend on the amount spent on advertising. If the allocations made in one period affect those in subsequent periods, the problem is said to be dynamic, and time must be considered in its solution.Russell L. AckoffLinear programming (LP) refers to a family of mathematical optimization techniques that have proved effective in solving resource allocation problems, particularly those found in industrial production systems. Linear programming methods are algebraic techniques based on a series of equations or inequalities that limit a problem and are used to optimize a mathematical expression called an objective function. The objective function and the constraints placed upon the problem must be deterministic and able to be expressed in linear form. These restrictions limit the number of problems that can be handled directly, but since the introduction of linear programming in the late 1940s, much progress has been made to adapt the method to more complex problems.Since linear programming is probably the most widely used mathematical optimization technique, numerous computer programs are available for solving LP problems. For example, LP techniques are now used routinely for such problems as oil and chemical refinery blending, choosing vendors or suppliers for large, multiplant manufacturing corporations, determining shipping routes and schedules, and managing and maintaining truck fleets.Inventory controlInventories (inventory) include raw materials, component parts, work in process, finished goods, packing and packaging materials, and general supplies. The control of inventories, vital to the financial strength of a firm, in general involves deciding at what points in the production system stocks shall be held and what their form and size are to be. As some unit costs increase with inventory size—including storage, obsolescence, deterioration, insurance, investment—and other unit costs decrease with inventory size—including setup or preparation costs, delays because of shortages, and so forth—a good part of inventory management consists of determining optimal purchase or production lot sizes and base stock levels that will balance the opposing cost influences. Another part of the general inventory problem is deciding the levels (reorder points) at which orders for replenishment of inventories are to be initiated.Inventory control is concerned with two questions: when to replenish the store and by how much. There are two main control systems. The two-bin system (sometimes called the min-max system) involves the use of two bins, either physically or on paper. The first bin is intended for supplying current demand and the second for satisfying demand during the replenishment period. When the stock in the first bin is depleted, an order for a given quantity is generated. The reorder-cycle system, or cyclical-review system, consists of ordering at fixed regular intervals. Various combinations of these systems can be used in the construction of an inventory-control procedure. A pure two-bin system, for example, can be modified to require cyclical instead of continuous review of stock, with orders being generated only when the stock falls below a specific level. Similarly, a pure reorder-cycle system can be modified to allow orders to be generated if the stock falls below the reorder level between the cyclical reviews. In yet another variation, the reorder quantity in the reorder-cycle system is made to depend on the stock level at the review period or the need to order other products or materials at the same time or both.Samuel EilonThe classic inventory problem involves determining how much of a resource to acquire, either by purchasing or producing it, and whether or when to acquire it to minimize the sum of the costs that increase with the size of inventory and those that decrease with increases in inventory. Costs of the first type include the cost of the capital invested in inventory, handling, storage, insurance, taxes, depreciation, deterioration, and obsolescence. Costs that decrease as inventory increases include shortage costs (arising from lost sales), production setup costs, and the purchase price or direct production costs. Setup costs include the cost of placing a purchase order or starting a production run. If large quantities are ordered, inventories increase but the frequency of ordering decreases, hence setup costs decrease. In general, the larger the quantity ordered the lower the unit purchase price because of quantity discounts and the lower production cost per unit resulting from the greater efficiency of long production runs. Other relevant variables include demand for the resource and the time between placing and filling orders.Inventory problems arise in a wide variety of contexts; for example, determining quantities of goods to be purchased or produced, how many people to hire or train, how large a new production or retailing facility should be or how many should be provided, and how much fluid (operating) capital to keep available. Inventory models for single items are well developed and are normally solved with calculus. When the order quantities for many items are interdependent (as, for example, when there is limited storage space or production time) the problem is more difficult. Some of the larger problems can be solved by breaking them into interacting inventory and allocation problems. In very large problems simulation can be used to test various relevant decision rules.Japanese (Japan) approachesIn the 1970s several Japanese firms, led by the Toyota Motor Corporation, developed radically different approaches to the management of inventories. Coined the “just-in-time” approach, the basic element of the new systems was the dramatic reduction of inventories throughout the total production system. By relying on careful scheduling and the coordination of supplies, the Japanese ensured that parts and supplies were available in the right quantity, with proper quality, at the exact time they were needed in the manufacturing or assembly process.Two things made just-in-time work—a dogged attention to quality at all levels of the total system obviated the need for parts inventories to cover defectives found in the manufacturing process, and a close coordination of information and plans with suppliers and vendors permitted them to align their schedules and shipments with the last-minute needs of the manufacturer. Elements of the just-in-time approach now have been adopted by numerous companies in the United States and Europe, although many cannot use the system to its fullest extent because their supplier networks are larger and more widely dispersed than in Japan.A second Japanese technique, called kanban (“card”), also permits Japanese firms to schedule production and manage inventories more effectively. In the kanban system, cards or tickets are attached to batches, racks, or pallet loads of parts in the manufacturing process. When a batch is depleted in the assembly process, its kanban is returned to the manufacturing department and another batch is shipped immediately. Since the total number of parts or batches in the system is held constant, the coordination, scheduling, and control of the inventory is greatly simplified.Replacement and maintenanceReplacement problems involve items that degenerate with use or with the passage of time and those that fail after a certain amount of use or time. Items that deteriorate are likely to be large and costly (e.g., machine tools, trucks, ships, and home appliances). Nondeteriorating items tend to be small and relatively inexpensive (e.g., light bulbs, vacuum tubes, ink cartridges). The longer a deteriorating item is operated the more maintenance it requires to maintain efficiency. Furthermore, the longer such an item is kept the less is its resale value and the more likely it is to be made obsolete by new equipment. If the item is replaced frequently, however, investment costs increase. Thus the problem is to determine when to replace such items and how much maintenance (particularly preventive) to perform so that the sum of the operating, maintenance, and investment costs is minimized.In the case of nondeteriorating items the problem involves determining whether to replace them as a group or to replace individuals as they fail. Though group replacement is wasteful, labour cost of replacements is greater when done singly; for example, the light bulbs in a large subway system may be replaced in groups to save labour. Replacement problems that involve minimizing the costs of items, failures, and the replacement labour are solvable either by numerical analysis or simulation.The “items” involved in replacement problems may be people. If so, maintenance can be interpreted as training or improvements in salary, status, or fringe benefits. Failure can be interpreted as departure, and investment as recruiting, hiring, and initial training costs. There are many additional complexities in such cases; for example, the effect of one person's resigning or being promoted on the behaviour of others. Such controllable aspects of the environment as location of work and working hours can have a considerable effect on productivity and failure rates. In problems of this type, the inputs of the behavioral sciences are particularly useful.Queuing (queuing theory)A queue is a waiting line, and queuing involves dealing with items or people in sequence. Thus, a queuing problem consists either of determining what facilities to provide or scheduling the use of them. The cost of providing service and the waiting time of users are minimized. Examples of such problems include determining the number of checkout counters to provide at a supermarket, runways at an airport, parking spaces at a shopping centre, or tellers in a bank. Many maintenance problems can be treated as queuing problems; items requiring repair are like users of a service. Some inventory problems may also be formulated as queuing problems in which orders are like users and stocks are like service facilities.Job shop sequencingIn queuing problems, the order in which users waiting for service are served is always specified. Selection of that order so as to minimize some function of the time to perform all the tasks is a sequencing problem. The performance measure may account for total elapsed time, total tardiness in meeting deadlines or due dates, and the cost of in-process inventories.The most common context for sequencing problems is a batch, or job shop, production facility that processes many different products with many combinations of machines. In this context account may have to be taken of such factors as overlapping service (that is, if a customer consists of a number of items to be taken through several steps of a process, the first items completing the initial step may start on the second step before the last one finishes the first), transportation time between service facilities, correction of service breakdowns, facility breakdowns, and material shortages.Russell L. Ackoff A simplified job shop sequencing problem, with two jobs and four machines, is shown in the figure—>. At the top of the figure is the operations sequence of the two jobs. Job A must go first to machine 1, then to 2, then to 3, and finally to 4, and the order of processing on the four machines cannot be changed. The processing time for the job is one hour at each machine, for a total of four hours of machining time. In this example, the job can only be on one machine at a time, as if the job consisted of a single product being processed through four machine tools.Job B must follow a different sequence. It also starts on machine 1, but then it goes to machine 4, then to 2, and finally back to machine 4. Each machining operation on Job B also requires one hour.Underneath the charts showing the required sequence of operations, two alternative schedules are shown for the two jobs. (In a bar chart, time is shown on the horizontal line, and the bars or blocks represent the time that each operation is scheduled on each of the four machines.) The first schedule assumes that Job A is run first. Once Job A is laid out on the schedule, Job B's operations are placed on the chart as far to the left as possible, without violating the sequence constraints. In this case, the chart shows that both jobs (eight hours of work) can be completed in five hours. This is made possible by running both jobs at the same time (on separate machines) during the second, third, and fourth hours. The second schedule assumes that Job B is run first. This schedule requires a total of six hours, one more than the previous schedule. If the total elapsed time for completion of the two jobs is an important criterion, the first schedule would be superior to the second.Although this problem is easily solved, solutions to actual job shop sequencing problems require the use of sophisticated models and the calculating power of computers. It is not unusual for job shops to have 5,000 customer orders in process at any given time, with each order requiring 50 or 60 distinct processing or machine operations. The number of combinations of feasible sequences is astronomical in such problems, and they provide many problems in modeling and systems development for operations researchers and industrial engineers.Manufacturing progress functionBecause of the enormous complexity of a typical mass production line and the almost infinite number of changes that can be made and alternatives that can be pursued, a body of quantitative theory of mass production manufacturing systems has not yet been developed. The volume of available observational data is, however, growing, and qualitative facts are emerging that may eventually serve as a basis for quantitative theory. An example is the “manufacturing progress function.” This was first recognized in the airframe industry. Early manufacturers of aircraft observed that as they produced increasing numbers of a given model of airplane, their manufacturing costs decreased in a predictable fashion, declining steeply at first, then continuing to decline at a lower rate. When an actual cost graph is drawn on double logarithmic paper plotting the logarithm of the cost per unit as a function of the logarithm of the total number of units produced results in data points that almost form a straight line. Over the years similar relationships have been found for many products manufactured by mass production techniques. The slope of the straight line varies from product to product. For a given class of products and a given type of production technology, however, the slope appears remarkably constant.Manufacturing progress functions can be of great value to the manufacturer, serving as a useful tool in estimating future costs. Furthermore, the failure of costs to follow a well-established progress function may be a sign that more attention should be given to the operation in order to bring its cost performance in line with expectation.Though manufacturing progress functions are sometimes called “learning curves,” they reflect much more than the improved training of the manufacturing operators. Improved operator skill is important in the start-up of production, but the major portion of the long-term cost improvement is contributed by improvements in product design, machinery, and the overall engineering planning of the production sequence.Morris Tanenbaum William K. HolsteinNetwork routingA network (computer network) may be defined by a set of points, or “nodes,” that are connected by lines, or “links.” A way of going from one node (the “origin”) to another (the “destination”) is called a “route” or “path.” Links, which may be one-way or two-way, are usually characterized by the time, cost, or distance required to traverse them. The time or cost of traveling in different directions on the same link may differ.A network routing problem consists of finding an optimum route between two or more nodes in relation to total time, cost, or distance. Various constraints may exist, such as a prohibition on returning to a node already visited or a stipulation of passing through every node only once.Network routing problems commonly arise in communication and transportation systems. Delays that occur at the nodes (e.g., railroad classification yards or telephone switchboards) may be a function of the loads placed on them and their capacities. Breakdowns may occur in either links or nodes. Much studied is the “traveling salesman problem,” which consists of starting a route from a designated node that goes through each node (e.g., city) only once and returns to the origin in the least time, cost, or distance. This problem arises in selecting an order for processing a set of production jobs when the cost of setting up each job depends on which job has preceded it. In this case the jobs can be thought of as nodes, each of which is connected to all of the others, with setup costs as the analogue of distances between them. The order that yields the least total setup cost is therefore equivalent to a solution to the traveling salesman problem. The complexity of the calculations is such that even with the use of computers it is very costly to handle more than 20 nodes. Less costly approximating procedures are available, however. More typical routing problems involve getting from one place to another in the least time, cost, or distance. Both graphic and analytic procedures are available for finding such routes.Competitive problemsCompetitive problems deal with choice in interactive situations where the outcome of one decision maker's choice depends on the choice, either helpful or harmful, of one or more others. Examples of these are war, marketing, and bidding for contracts. Competitive problems are classifiable as certain, risky, or uncertain, depending on the state of a decision maker's knowledge of his opponent's choices. Under conditions of certainty, it is easy to maximize gain or minimize loss. Competitive problems of the risk type require the use of statistical analysis for their solution; the most difficult aspect of solving such problems usually lies in estimating the probabilities of the competitor's choices; for example, in bidding for a contract on which competitors and their bids are unknown.The theory of games was developed to deal with a large class of competitive situations of the uncertainty type in which each participant knows what choices he and each other participant has. There is a well-defined “end state” that terminates the interaction (e.g., win, lose, or draw), and the payoffs associated with each end state are specified in advance and are known to each participant. In situations in which all the alternatives are open to competition, or some of their outcomes are not known in advance, operational gaming can sometimes be used. The military have long constructed operational games; their use by business is more recent.Search problemsSearch problems involve finding the best way to obtain information needed for a decision. Though every problem contains a search problem in one sense, situations exist in which search itself is the essential process; for example, in auditing accounts, inspection and quality control procedures, in exploration for minerals, in the design of information systems, and in military problems involving the location of such threats as enemy ships, aircraft, mines, and missiles.Two kinds of error are involved in search: those of observation and those of sampling. Observational errors, in turn, are of two general types: commission, seeing something that is not there; and omission, not seeing something that is there. In general, as the chance of making one of these errors is decreased, the chance of making the other is increased. Furthermore, if fixed resources are available for search, the larger the sample (and hence the smaller the sampling error), the less resources available per observation (and hence the larger the observational error).The cost of search is composed of setup or design cost, cost of observations, cost of analyzing the data obtained, and cost of error. The objective is to minimize these costs by manipulating the sample size (amount of observation), the sample design (how the things or places to be observed are selected), and the way of analyzing the data (the inferential procedure).Almost all branches of statistics provide useful techniques for solving search problems. In search problems that involve the location of physical objects, particularly those that move, physics and some fields of mathematics (e.g., geometry and trigonometry) are also applicable.A “reversed-search” problem arises when the search procedure is not under control but the object of the search is. Most retailers, for example, cannot control the manner in which customers search for goods in their stores, but they can control the location of the goods. This type of problem also arises in the design of libraries and information systems, and in laying land and sea mines. These, too, are search problems, and solution techniques described above are applicable to them.Frontiers of operations researchOperations research is a rapidly developing application of the scientific method to organizational problems. Its growth has consisted of both technical development and enlargement of the class of organized systems and the class of problems to which it is applied.Strategic problemsTactics and strategy are relative concepts. The distinction between them depends on three considerations: (1) the longer the effect of a decision and the less reversible it is, the more strategic it is; (2) the larger the portion of a system that is affected by a decision, the more strategic it is; and (3) the more concerned a decision is with the selection of goals and objectives, as well as the means by which they are to be obtained, the more strategic it is.Strategy and tactics are separable only in thought, not in action. Every tactical decision involves a strategic choice, no matter how implicit and unconscious it may be. Since the strategic aspects of decisions are usually suppressed, an organization's strategy often emerges as an accidental consequence of its tactical decisions.Operations research is becoming increasingly concerned with strategic decisions and the development of explicit strategies for organizations so as to improve the quality of their tactical decisions and make even the most immediate and urgent of these contribute to its long-run goals.The system design problemOperations research has traditionally been concerned with finding effective solutions to specific operational problems. It has developed better methods, techniques, and tools for doing so. But operations researchers have found that too many of their solutions are not implemented and, of those that are, too few survive the inclination of organizations to return to familiar ways of doing things. Therefore, operations researchers have gradually come to realize that their task should not only include solving specific problems but also designing problem-solving and implementation systems that predict and prevent future problems, identify and solve current ones, and implement and maintain these solutions under changing conditions.The planning problemOperations researchers have come to realize that most problems do not arise in isolation but are part of an interacting system. The process of seeking simultaneous interrelated solutions to a set of interdependent problems is planning. More and more operations research effort is being devoted to developing a rational methodology of such planning, particularly strategic planning.Most organizations resist changes in their operations or management. The organizational need to find better ways of doing things is often not nearly as great as is the need to maximize use of what it already knows or has. This is apparent in many underdeveloped countries that, while complaining about the lack of required resources, use what resources they have with considerably less efficiency than do most developed countries. Operations research, therefore, has been addressing itself more and more to determining how to produce the willingness to change.Types of organizationOperations researchers have become increasingly aware of the need to distinguish between different types of organization because their distinguishing features affect how one must go about solving their problems. Two important classifications exist, the first of which is homogeneous–heterogeneous. Homogeneous organizations are those in which membership involves serving the objectives of the whole (e.g., a corporation or military unit), while heterogeneous organizations are those whose principal objective it is to serve the objectives of its members (e.g., a university or city). The second classification is unimodal–multimodal. Unimodal organizations are hierarchical organizations with a single decision-making authority that can resolve differences between any lower level decision makers. Multimodal organizations have no such authority but have diffused decision making and hence require agreement among the several decision makers in order to reach conclusions.Since current skills in operations research are largely restricted to homogeneous unimodal organizations, attempts are under way to develop methodologies adequate for improving the other three types of organization.In order to solve any of the preceding problems more effectively, operations research requires a better understanding of human behaviour, individual and collective, than is currently available. Furthermore, what understanding the behavioral sciences claim to provide is seldom available in a form that lends itself to symbolic representation and hence to operations research methodology. Operations researchers, therefore, are increasingly working with behavioral scientists to develop behavioral theories that are expressible in a more usable form.As the scope of problems to which operations research addresses itself increases, it becomes more apparent that the number of disciplines and interdisciplines that have an important contribution to make to their solution also increases. An attempt to provide such a higher order integration of scientific activity is being made in the management sciences.Russell L. AckoffAdditional ReadingHistoryJ.G. Crowther and R. Whiddington, Science at War (1948); Florence N. Trefethen, “A History of Operations Research,” in Joseph F. McCloskey and Florence N. Trefethen (eds.), Operations Research for Management, vol. 1, pp. 3–35 (1954); and Great Britain, Air Ministry, The Origins and Development of Operational Research in the Royal Air Force (1963).General textsRussell L. Ackoff and Maurice W. Sasieni, Fundamentals of Operations Research (1968); Stafford Beer, Decision and Control: The Meaning of Operational Research and Management Cybernetics (1966); David W. Miller and Martin K. Starr, Executive Decisions and Operations Research, 2nd ed. (1969); Harvey M. Wagner , Principles of Operations Research (1969); David R. Anderson, Dennis J. Sweeney, and Thomas A. Williams, An Introduction to Management Science: Quantitative Approaches to Decision Making, 4th ed. (1985); Elwood S. Buffa and James S. Dyer, Management Science/Operations Research: Model Formulation and Solution Methods, 2nd ed. (1981); Gilbert Gordon and Israel Pressman, Quantitative Decision Making for Business, 2nd ed. (1983); and G.D. Eppen and F.J. Gould, Quantitative Concepts for Management: Decision Making Without Algorithms, 2nd ed. (1985).MethodologyRussell L. Ackoff, Shiv K. Gupta, and J. Sayer Minas, Scientific Method: Optimizing Applied Research Decisions (1962, reprinted 1984); C. West Churchman, Prediction and Optimal Decision: Philosophical Issues of a Science of Values (1961, reprinted 1982); Peter C. Fishburn, Decision and Value Theory (1964), and Utility Theory for Decision Making (1970, reprinted 1979); and D.J. White, Decision Theory (1969). Marilyn Taylor Thompson, Management Information, Where to Find It (1981), is an annotated bibliography of resources.General techniquesBurton V. Dean, Maurice W. Sasieni, and Shiv K. Gupta, Mathematics for Modern Management (1963, reprinted 1978); James R. Emshoff and Roger L. Sisson, Design and Use of Computer Simulation Models (1970); Howard Raiffa and Robert Schlaifer, Applied Statistical Decision Theory (1961, reprinted 1970); Thomas L. Saaty, Mathematical Methods of Operations Research (1959); and Clayton J. Thomas and Walter L. Deemer, Jr., “The Role of Operational Gaming in Operations Research,” Operations Research, 5:1–27 (1957).Examples of operations research models and applications:Specific techniques are addressed in the following: on programming, Richard E. Bellman, Dynamic Programming (1957, reprinted 1972); Richard E. Bellman and Stuart E. Dreyfus, Applied Dynamic Programming (1962); Saul I. Gass, Linear Programming: Methods and Applications, 5th ed. (1985); Ronald A. Howard, Dynamic Programming and Markov Processes (1960, reprinted 1964); and G. Hadley, Linear Programming (1962), Nonlinear and Dynamic Programming (1964); on inventory control, Kenneth J. Arrow, Samuel Karlin, and Herbert Scarf, Studies in the Mathematical Theory of Inventory and Production (1958); G. Hadley and T.M. Whitin, Analysis of Inventory Systems (1963); and Fred Hanssmann, Operations Research in Production and Inventory Control (1962); on replacement and maintenance, Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability (1965); D.R. Cox, Renewal Theory (1962, reissued 1967); and John J. McCall, “Maintenance Policies for Stochastically Failing Equipment: A Survey,” Management Science, 11(5):493–524 (March 1965); on queuing, D.R. Cox and Walter L. Smith, Queues (1961, reprinted 1979); Thomas L. Saaty, Elements of Queueing Theory, with Applications (1961, reprinted 1983); and Lajos Takacs, Introduction to the Theory of Queues (1962, reprinted 1982); on sequencing and coordination, James E. Kelley, Jr., “Critical-Path Planning and Scheduling: Mathematical Basis,” Operations Research, 9(3):296–320 (May–June 1961); Robert W. Miller, Schedule, Cost, and Profit Control with PERT: A Comprehensive Guide for Program Management (1963); and Roger L. Sisson, “Sequencing Theory,” in Russell L. Ackoff (ed.), Progress in Operations Research, vol. 1, pp. 293–326 (1961); on network routing, L.R. Ford, Jr., and D.R. Fulkerson, Flows in Networks (1962); E.L. Lawler and D.E. Wood, “Branch-and-Bound Methods: A Survey,” Operations Research, 14(4):699–719 (July–August 1966); and John D.C. Little et al., “An Algorithm for the Traveling Salesman Problem,” Operations Research, 11(6):972–989 (November–December 1963); on competitive problems, Nigel Howard, “The Theory of Meta-Games,” General Systems, 11:167–186 (1966), and “The Mathematics of Meta-Games,” General Systems, 11:187–200 (1966); J.C.C. McKinsey, Introduction to the Theory of Games (1952); A. Rapoport, N-Person Game Theory: Concepts and Applications (1970); and John Von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, 3rd. ed. (1944, reissued 1980); and, on search problems, Russell L. Ackoff and Maurice W. Sasieni, Fundamentals of Operations Research, especially ch. 14 (1968); B.O. Koopman, “The Theory of Search,” Operations Research, 4(3):324–346 (June 1956), 4(5):503–531 (October 1956), and 5(5):613–626 (October 1957); and James B. Macqueen, “Optimal Policies for a Class of Search and Evaluation Problems,” Management Science, 10(4):746–759 (July 1964).Planning and implementationRussell L. Ackoff, A Concept of Corporate Planning (1970); H. Igor Ansoff, Corporate Strategy: An Analytic Approach to Business Policy for Growth and Expansion (1965, reissued 1968); James C. Emery, Organizational Planning and Control Systems: Theory and Technology (1969); and Jan H.B.M. Huysmans, The Implementation of Operations Research (1970).
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