BREADTH OF MANAGEMENT SCIENCE TECHNIQUES
MARKOV PROCESS MODELS
QUEUING THEORY/WAITING-LINE THEORY
Management science generally refers to mathematical or quantitative methods for business decision making. The term “operations research” may be used interchangeably with management science.
Frederick Winslow Taylor is credited with the initial development of scientific management techniques in the early twentieth century. In addition, several management science techniques were further developed during World War II. Some even consider the World War II period as the beginning of management science, as this global conflict posed many military, strategic, logistic, and tactical problems. Operations research teams of engineers, mathematicians, and statisticians were developed to use the scientific method to find solutions for many of these problems.
Nonmilitary management science applications developed rapidly after World War II. Based on quantitative methods developed during World War II, several new applications emerged. The development of the simplex method by George Dantzig in the 1940s made application of linear programming practical. C. West Churchman, Russell Ackoff, and Leonard Arnoff made management science even more accessible by publishing the first operations research textbook in the 1950s.
Computer technology continues to play an integral role in management science. Practitioners and researchers are able to use ever-increasing computing power in conjunction with management science methods to solve larger and more complex problems. In addition, management scientists are constantly developing new algorithms and improving existing algorithms; these efforts also enable management scientists to solve larger and more complex problems.
The scope of management science techniques is broad. These techniques include:
- Mathematical programming
- Linear programming
- Simplex method
- Dynamic programming
- Goal programming
- Integer programming
- Nonlinear programming
- Stochastic programming
- Markov processes
- Queuing theory/waiting-line theory
- Transportation method
Management science techniques are used on a wide variety of problems from a vast array of applications. For example, integer programming has been used by baseball fans to allocate season tickets in a fair manner. When seven baseball fans purchased a pair of season tickets for the Seattle Mariners, the Mariners turned to management science and a computer program to assign games to each group member based on member priorities.
In marketing, optimal television scheduling has been determined using integer programming. Variables such as time slot, day of the week, show attributes, and competitive effects can be used to optimize the scheduling of programs. Optimal product designs based on consumer preferences have also been determined using integer programming.
Similarly, linear programming can be used in marketing research to help determine the timing of interviews. Such a model can determine the interviewing schedule that maximizes the overall response rate while providing appropriate representation across various demographics and household characteristics.
In the area of finance, management science can be employed to help determine optimal portfolio allocations, borrowing strategies, capital budgeting, asset allocations, and make-or-buy decisions. In portfolio allocations, for instance, linear programming can be used to help a financial manager select specific industries and investment vehicles (e.g., bonds versus stocks) in which to invest.
With regard to production scheduling, management science techniques can be applied to scheduling, inventory, and capacity problems. Production managers can deal with multi-period scheduling problems to develop low-cost production schedules. Production costs, inventory holding costs, and changes in production levels are among the types of variables that can be considered in such analyses.
Workforce assignment problems can also be solved with management science techniques. For example, when some personnel have been cross-trained and can work in more than one department, linear programming may be used to determine optimal staffing assignments.
Airports are frequently designed using queuing theory (to model the arrivals and departures of aircraft) and simulation (to simultaneously model the traffic on multiple runways). Such an analysis can yield information to be used in deciding how many runways to build and how many departing and arriving flights to allow by assessing the potential queues that can develop under various airport designs.
Mathematical programming deals with models comprised of an objective function and a set of constraints. Linear, integer, nonlinear, dynamic, goal, and stochastic programming are all types of mathematical programming.
An objective function is a mathematical expression of the quantity to be maximized or minimized. Manufacturers may wish to maximize production or minimize costs, advertisers may wish to maximize a product's exposure, and financial analysts may wish to maximize rate of return.
Constraints are mathematical expressions of restrictions that are placed on potential values of the objective function. Production may be constrained by the total amount of labor at hand and machine production capacity, an advertiser may be constrained by an advertising budget, and an investment portfolio may be restricted by the allowable risk.
Linear programming problems are a special class of mathematical programming problems for which the objective function and all constraints are linear. A classic example of the application of linear programming is the maximization of profits given various production or cost constraints.
Linear programming can be applied to a variety of business problems, such as marketing mix determination, financial decision making, production scheduling, work-force assignment, and resource blending. Such problems are generally solved using the “simplex method.”
Media Selection Problem. The local Chamber of Commerce periodically sponsors public service seminars and programs. Promotional plans are under way for this year's program. Advertising alternatives include television, radio, and newspaper. Audience estimates, costs, and maximum media usage limitations are shown in Exhibit 1.
If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? Linear programming can find the answer.
|Audience per ad||100,000||18,000||40,000|
|Cost per ad||2,000||300||600|
The simplex method is a specific algebraic procedure for solving linear programming problems. The simplex method begins with simultaneous linear equations and solves the equations by finding the best solution for the system of equations. This method first finds an initial basic feasible solution and then tries to find a better solution. A series of iterations results in an optimal solution.
Simplex Problem. Assume that Georgia Television buys components that are used to manufacture two television models. One model is called High Quality and the other is called Medium Quality. A weekly production schedule needs to be developed given the following production considerations.
The High Quality model produces a gross profit of $125 per unit, and the Medium Quality model has a $75 gross profit. Only 180 hours of production time are available for the next time period. High Quality models require a total production time of six hours and Medium Quality models require eight hours. In addition, there are only forty-five Medium Quality components on hand.
To complicate matters, only 250 square feet of warehouse space can be used for new production. The High Quality model requires 9 square feet of space while the Medium Quality model requires 7 square feet.
Given the above situation, the simplex method can provide a solution for the production allocation of High Quality models and Medium Quality models.
Dynamic programming is a process of segmenting a large problem into a several smaller problems. The approach is to solve all the smaller, easier problems individually in order to reach a solution to the original problem.
This technique is useful for making decisions that consist of several steps, each of which also requires a decision. In addition, it is assumed that the smaller problems are not independent of one another given they contribute to the larger question.
Dynamic programming can be utilized in the areas of capital budgeting, inventory control, resource allocation, production scheduling, and equipment replacement. These applications generally begin with a longer time horizon, such as a year, and then break down the problem into smaller time units such as months or weeks. For example, it may be necessary to determine an optimal production schedule for a twelve-month period.
Dynamic programming would first find a solution for smaller time periods, for example, monthly production schedules. By answering such questions, dynamic programming can identify solutions to a problem that are most efficient or that best serve other business needs given various constraints.
Goal programming is a technique for solving multi-criteria rather than single-criteria decision problems, usually within the framework of linear programming. For example, in a location decision a bank would use not just one criterion, but several. The bank would consider cost of construction, land cost, and customer attractiveness, among other factors.
Goal programming establishes primary and secondary goals. The primary goal is generally referred to as a priority level 1 goal. Secondary goals are often labeled priority level 2, level 3, and so on. It should be noted that tradeoffs are not allowed between higher and lower level goals.
Assume a bank is searching for a site to locate a new branch. The primary goal is to be located in a five-mile proximity to a population of 40,000 consumers. A secondary goal might be to be located at least two miles from a competitor. Given the no trade-off rule, the bank would first search for a target solution of locating close to 40,000 consumers.
Blending Problem. The XYZ Company mixes three raw materials to produce two products: a fuel additive and a solvent. Each ton of fuel additive is a mixture of 2/5 ton of material A and 3/5 ton of material C. A ton of solvent base is a mixture of 1/2 ton of material A, 1/5 ton of material B, and 3/10 ton of material C. Production is constrained by a limited availability of the three raw materials. For the current production period XYZ has the following quantities of each raw material: 20 tons of material A, 5 tons of material B, and 21 tons of material C. Management would like to achieve the following priority level goals:
Goal 1. Produce at least 30 tons of fuel additive.
Goal 2. Produce at least 15 tons of solvent.
Goal programming would provide directions for production.
Integer programming is useful when values of one or more decision variables are limited to integer values. This
|Table 1 Integer Programming Example|
|Project||Estimated Net return||Year 1||Year 2||Year 3|
is particularly useful when modeling production processes for which fractional amounts of products cannot be produced. Integer variables are often limited to two values—zero or one. Such variables are particularly useful in modeling either/or decisions.
Areas of business that use integer linear programming include capital budgeting and physical distribution. For example, faced with limited capital, a firm needs to select capital projects in which to invest. This type of problem is represented in Table 1.
As can be seen in the table, capital requirements exceed the available funds for each year. Consequently, decisions to accept or reject regarding each of the projects must be made and integer programming would require the following integer definitions for each of the projects.
x1 = 1 if the new office project is accepted; 0 if rejected
x2 = 1 if the new warehouse project is accepted; 0 if rejected
x3 = 1 if the new branch project is accepted; 0 if rejected
A set of equations is developed from the definitions to provide an optimal solution.
Nonlinear programming is useful when the objective function or at least one of the constraints is not linear with respect to values of at least one decision variable. For example, the per-unit cost of a product may increase at a decreasing rate as the number of units produced increases because of economies of scale.
Stochastic programming is useful when the value of a coefficient in the objective function or one of the constraints is not known with certainty but has a known probability distribution. For instance, the exact demand for a product may not be known, but its probability distribution may be understood. For such a problem, random values from this distribution can be substituted into the problem formulation. The optimal objective function values associated with these formulations provide the basis of the probability distribution of the objective function.
Markov process models are used to predict the future of systems given repeated use. For example, Markov models are used to predict the probability that production machinery will function properly given its past performance in any one period. Markov process models are also used to predict future market share given any specific period's market share.
Computer Facility Problem . The computing center at a state university has been experiencing computer downtime. Assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data in Table 2 show the transition probabilities.
The Markov process would then solve for the following: if the system is running, what is the probability of the system being down in the next hour of operation?
Queuing theory is often referred to as waiting-line theory. Both terms refer to decision making regarding the management of waiting lines (or queues). This area of management science deals with operating characteristics of waiting lines, such as:
- The probability that there are no units in the system
- The mean number of units in the queue
- The mean number of units in the system (the number of units in the waiting line plus the number of units being served)
- The mean time a unit spends in the waiting line
- The mean time a unit spends in the system (the waiting time plus the service time)
- The probability that an arriving unit has to wait for service
- The probability of n units in the system
|Origin||Warehouse Location||3 month capacity|
Given the above information, programs are developed that balance costs and service delivery levels. Typical applications involve supermarket checkout lines and waiting times in banks, hospitals, and restaurants.
Bank Line Problem. XYZ State Bank operates a drive-in-teller window, which allows customers to complete bank transactions without getting out of their cars. On weekday mornings arrivals to the drive-in-teller window occur at random, with a mean arrival rate of twenty-four customers per hour or 0.4 customers per minute.
Delay problems are expected if more than three customers arrive during any five-minute period. Waiting line models can determine the probability that delay problems will occur.
The transportation method is a specific application of the simplex method that finds an initial solution and then uses iteration to develop an optimal solution. As the name implies, this method is utilized in transportation problems.
Transportation Problem. A company must plan its distribution of goods to several destinations from several warehouses. The quantity available at each warehouse is limited. The goal is to minimize the cost of shipping the goods. An example of production capacity can be found in Table 3. The forecast for demand is shown in Table 4.
|Destination||Location||3 month forecast|
The transportation method will determine the optimal amount to be shipped from each warehouse and determine the optimal destination.
Simulation is used to analyze complex systems by modeling complex relationships between variables with known probability distributions. Random values from these probability distributions are substituted into the model and the behavior of the system is observed. Repeated executions of the simulation model provide insight into the behavior of the system that is being modeled.
The different arithmetic models of management science can be incorporated into the strategic operational functions of organizations—such as integrated product management—to achieve greater efficiency in resource allocation and utilization. Integrated product management defines the end-to-end production activities such as operational costs, ethical issues, social factors, and environmental constraints. The incorporation of the tools, models, techniques, and concepts of management science into the overall product life-cycle processes optimizes the utility functions of all the elements in the chain of production to the advantage of the firm.
Management science keeps evolving because of the continued inventions of automated technologies and information management systems. In the book titled Programming Languages for Business Problem Solving, Wang and Wang reckon that computer literacy has become a prerequisite requirement for managers wishing to adopt the use of information technology in advancing business strategies in their organizations. The complex concepts and logical processes of management science can easily be conquered by the use of automated information technology systems tailored to suit organization-specific needs.
SEE ALSO Operations Management; Operations Scheduling; Operations Strategy; Production Planning and Scheduling
Anderson, David R., Dennis J. Sweeney, Thomas A. Williams, and R. Kipp Martin. An Introduction to Management Science: A Qualitative Approach to Decision Making. Cincinatti, OH: South-Western College Pub, 2007.
Anderson, David R., Dennis J. Sweeney, and Thomas A. Williams. Quantitative Methods for Business. Cincinnati, OH: South-Western Publishing, 2004.
Blumenfrucht, Israel, and Joel G. Segal. “Updating the Accountant on Financial Advisory Services: Financial Models, Latest Quantitative Techniques, and Other Recent Developments.” National Public Accountant, September 1998, 20–23.
Camm, D. Jeffrey, and James R. Evans. Management Science & Decision Technology. Cincinnati, OH: South-Western Publishing, 2000.
Celik, Sabri, and Costas Maglaras. “Dynamic Pricing and Lead-Time Quotation for a Multiclass Make-to-Order Queue.” Management Science. Vol. 54, No. 6, June 2008. Available from: http://mansci.journal.informs.org/cgi/reprint/54/6/iv.
The Institute For Operations Research and the Management Sciences. INFORMS Online. Available from: http://www.informs.org/.
McCulloch, C.E., B. Paal, and S.P. Ashdown. “An Optimisation Approach to Apparel Sizing.” Journal of the Operational Research Society, May 1998, 492—499.
Wang, S., and Wang, H. Programming Languages for Business Problem Solving. Auerbach Publications, 2007.
The roots of “operations research” (commonly referred to as OR) go back at least to the industrial revolution, which brought with it the mechanization of production, power generation, transportation, and communication. Machines replaced man as a source of power and made possible the development of the large industrial, military, and governmental complexes that we know today. These developments were accompanied by the continuous subdivision of industrial, commercial, military, and governmental management into more and more specialized functions and eventually resulted in the kind of multilevel structure that today characterizes most organizations in our culture.
As each new type of specialized manager appeared, a new specialized branch of applied science or engineering developed to provide him with assistance. For example, in industry this progression began with the emergence of mechanical and chemical engineering to serve production management and has continued into more recent times with the development of such specialties as industrial engineering, value analysis, statistical quality control, industrial psychology, and human engineering. Today, no matter how specialized the manager, at least one relevant type of applied science or engineering is available to him.
Whenever a new layer of management is created, a new managerial function, that of the executive, is also created at the next higher level. The executive function consists of coordinating and integrating the activities of diverse organizational units so that they serve the interests of the organization as a whole, or at least the interests of the unit that contains them. The importance of the executive function has grown steadily with the increase in the size and complexity of industrial, military, and governmental organizations.
The executive function in business and industry has developed gradually. The executive was not subjected to violent stimuli from new technology as was, for example, the manager of production. Consequently the executive “grew” into his problems, and these appeared to him to require for their solution nothing but good judgment based on relevant past experience. The executive, therefore, felt no need for a more rigorous scientific way of looking at his problems. However, the demands on his time grew, and he sought aid from those who had more time for, and more experience with, the problems that he faced. It was this need that gave rise to management consulting in the 1920s. Management consulting, however, was based on experience and qualitative judgment rather than on experimentation and quantitative analysis. The executive function was left without a scientific arm until World War II .
The major difference between the development of military executives and of their industrial counterparts is to be found in the twenty-year gap between the close of World War I and the opening of World War II. Because there was little opportunity to use military technology under combat conditions during this period, this technology developed too rapidly for effective absorption into military tactics and strategy. Thus, it is not surprising that British military executives turned to scientists for aid when the German air attack on Britain began. Initially they sought aid in incorporating the then new radar into the tactics and strategy of air defense. Small teams of scientists, drawn from any disciplines from which they could be obtained, worked on such problems with considerable success in 1939 and 1940. Their success bred further demand for such services, and their use spread to Britain’s allies—the United States, Canada, and France. These teams of scientists were usually assigned to the executive in charge of operations, and their work came to be known in the United Kingdom as “operational research” and in the United States by a variety of names: operations research, operations analysis, operations evaluation, systems analysis, systems evaluation, and management science. The name operations research was and is the most widely used in the United States.
At the end of the war very different things happened to OR in the United Kingdom and in the United States. In the United Kingdom expenditures on defense research were reduced. This led to the release of many OR workers from the military at a time when industrial managers were confronted with the need to reconstruct much of Britain’s manufacturing facilities that had been damaged during the war and to update obsolete equipment. In addition the British Labour party, which had come into power, began to nationalize several major and basic industries. Executives in these industries in particular sought and received assistance from the OR men coming out of the military. Coal, iron and steel, transport, and many other industries began to create industrial OR.
In contrast to the situation in Great Britain, defense research in the United States was increased at the end of the war. As a result military OR was expanded, and most of the war-experienced OR workers remained in the service of the military. Industrial executives did not ask for help because they were slipping back into a familiar peacetime pattern that did not involve either major reconstruction of plant or nationalization of industry.
During the late 1940s, however, the electronic computer became available and confronted the industrial manager with the possibility of automation —the replacement of man by machines as a source of control. The computer also made it possible for a man to control more effectively widely spread and large-scale activities because of its ability to process large amounts of data accurately and quickly. It provided the spark that set off what has sometimes been called the second industrial revolution. In order to exploit the new technology of control, industrial executives began to turn to scientists for aid as the military leaders had done before them. They absorbed the OR workers who trickled out of the military and encouraged academic institutions to educate additional men for work in this field.
Within a decade there were at least as many OR workers in academic, governmental, and industrial organizations as there were in the military. More than half of the largest companies in the United States have used or are using OR, and there are now about 4,000 OR workers in the country. A national society, the Operations Research Society of America, was formed in 1953. Other nations followed, and in 1957 the International Federation of Operational Research Societies was formed. Books and journals on the subject began to appear in a wide variety of languages. Graduate courses and curricula in OR began to proliferate in the United States and elsewhere.
In short, after vigorous growth in the military, OR entered its second decade with continued growth in the military and an even more rapid growth in industrial, academic, and governmental organizations.
Essential characteristics of OR . The essential characteristics of OR are its systems (or executive) orientation, its use of interdisciplinary teams, and its methodology.
Systems approach to problems. The systems approach to problems is based on the observation that in organized systems the behavior of any part ultimately has some effect on the performance of every other part. Not all these effects are significant or even capable of being detected. Therefore the essence of this orientation lies in the systematic search for significant interactions when evaluating actions or policies in any part of the organization. Use of such knowledge permits evaluation of actions and policies in terms of the organization as a whole, that is, in terms of their over-all effect.
This way of approaching organizational problems is diametrically opposed to one based on “cutting a problem down to size.” OR workers almost always enlarge the scope of a problem that is given to them by taking into account interactions that were not incorporated in the initial formulation of the problem. New research methods had to be developed to deal with these enlarged and more complicated problems. These are discussed below.
As an illustration of the systems approach to organizational problems, consider the case of a company which has 5 plants that convert a natural material into a raw material and 15 finishing plants that use this raw material to manufacture the products sold by the company. The finishing plants are widely dispersed and have different capacities for manufacturing a wide range of finished products. No single finishing plant can manufacture all the products in the line, but any one product may be produced in more than one plant.
Many millions of dollars are spent each year in shipping the output of the first group of plants to the second group. The problem that management presented to an OR group, therefore, was how to allocate the output of the raw-material plants to the finishing plants so as to minimize total betweenplant transportation costs. So stated, this is a welldefined, self-contained problem for which a straightforward solution can be obtained by use of one of the techniques of OR, linear programming [seeProgramming].
In the initial phases of their work, the OR workers observed that whereas all the raw-material plants were operating at capacity, none of the finishing plants were. They inquired whether the unit-production costs at the finishing plants varied with the percentage of capacity in use. They found that this was the case and also that the costs varied in a different way in each plant. As a result of this inquiry the original problem was reformulated to include not only transportation costs, but also the increased costs of production resulting from shipping to a finishing plant less material than it required for capacity operation. In solving this enlarged problem it was found that increased costs of production outweighed transportation costs and that a solution to the original problem (as formulated by management) would have resulted in an increase in production costs that would have more than offset the saving in transportation costs.
The OR workers then asked whether the increased costs of production that resulted from unused capacity depended on how production was planned, and they discovered that it did. Consequently, another related study was initiated in an effort to determine how to plan production at each finishing plant so as to minimize the increase in unit-production costs that resulted from unused capacity. In the course of this study of production planning it also became apparent that production costs were dependent on what was held where in semifinished inventory. Therefore, another study was begun to determine at what processing stage semifinished inventories should be held and what they should contain. Eventually the cost of shipping finished products to customers also had to be considered.
In the sequence of studies briefly described, it was not necessary to wait until all were completed before the results of the first could be applied. Solutions to each part of the total problem were applied immediately because precautions had been taken not to harm other operations. With each successive finding previous solutions were suitably adjusted. Eventually some change was made in every aspect of the organization’s activities, but each with an eye on its over-all effect. This is the essence of the systems approach to organizational problems.
The interdisciplinary team. Although division of the domain of scientific knowledge into specific disciplines is a relatively recent phenomenon, we are now so accustomed to classifying scientific knowledge in a way that corresponds either to the departmental structure of universities or to the professional organization of scientists that we often act as though nature were structured in the same way. Yet we seldom find such things as pure physical problems, pure psychological problems, pure economic problems, and so on. There are only problems; the disciplines of science simply represent different ways of looking at them. Nearly every problem may be looked at through the eyes of every discipline, but, of course, it is not always fruitful to do so.
If we want to explain an automobile’s being struck by a locomotive at a grade crossing, for example, we could do so either in terms of the laws of motion, or the engineering failure of warning devices, or the state of physical or mental health of the driver, or the social use of automobiles as an instrument of suicide, and so on. The way in which we look at the event depends on our purposes in doing so. A highway engineer and a driving instructor would look at it quite differently.
Though experience indicates a fruitful way of looking at most familiar problems, we tend to deal with unfamiliar and complicated situations in the way that is most familiar to us. It is not surprising, therefore, that given the problem, for example, of increasing the productivity of a manufacturing facility, a personnel psychologist will try to select better workers or improve the training that workers are given. A mechanical engineer will try to improve the machines. An industrial engineer will try to improve the plant layout, simplify the operations performed by the workers, or offer them more attractive incentives. The systems and procedures analyst will try to improve the flow of information into and through the plant, and so on. All may produce improvements, but which is best? For complicated problems we seldom can know in advance. Hence it is desirable to consider and evaluate as wide a range of approaches to the problem as possible. OR has greatly enlarged our capacity to deal with all the complexities of and the approaches to a given problem and has therefore expanded our opportunities to benefit from the use of interdisciplinary teams in solving problems.
Since more than a hundred scientific disciplines, pure and applied, have been identified, it is clearly not possible to incorporate each in most research projects. But in OR as many diverse disciplines are used on a team as possible, and the team’s work is subjected to critical review by as many of the disciplines not represented on the team as possible.
Methodology. Experimentation lies at the heart of scientific method, but it is obvious that the kind of organized man-machine system with which industrial, military, and governmental managers are concerned can never be brought into the laboratory, and only infrequently can such systems be manipulated enough in their natural environment to experiment on them there. Consequently, the OR worker finds himself in much the same position as the astronomer, and he takes a way out of his difficulty much like that taken by the astronomer. If he cannot manipulate the system itself, he builds a representation of the system, a model of it, that he can manipulate. In OR such models are abstract(symbolic) representations that may be very complicated from a mathematical point of view. From a logical point of view, however, they are quite simple. In general they take the form of an equation in which the performance of the system, P, is expressed as a function, f, of a set of controlled variables, C, and a set of uncontrolled variables, U:
The controlled variables represent the aspects of the system that management can manipulate, for example, production quantities, prices, range of product line, and so on. Such variables are often called decision variables since managerial decision making may be thought of as assigning values to these variables. The uncontrolled variables represent aspects of the system and its environment that significantly affect the system’s performance but are not under the control of management, for example, product demand, competitors’ prices, cost of raw material, and location of customers.
The measure of performance of the system may be very difficult to construct since it must reflect the relative importance of each relevant objective of the organization. This measure is sometimes called the criterion or objective function since it provides the basis for selecting the “best” or “better “courses of action.
Limitations or restrictions may be imposed on the possible values of the controlled variables. For example, in preparing a budget a limitation is normally placed on the total amount that may be allocated to different departments, or there may be legal constraints on the decision-making activities of managers. Such restrictions can usually be expressed mathematically as equations or inequalities and can be incorporated in the model.
Once the decision maker’s choices and the system involved have been represented by a mathematical model, the researcher must find a set of values of the controlled variables that yields the best (or as close as possible to the best) performance of the system. These “optimizing” values may be found either by experimenting on the model(i.e., by simulation) or by mathematical analysis. In either case the result is a set of equations, one for each controlled variable, giving the value of that controlled variable relative to a particular set of values of the uncontrolled variables and other controlled variables that yields the best performance of the system as a whole [seeSimulation].
If the problem is a recurrent one, then the values of the uncontrolled variables (for example, demand) may change from one decision-making period to another. In such cases a procedure must also be provided for determining when values of the uncontrolled variables have undergone significant change and for adjusting the solution appropriately. Such a procedure is called a solution-control system.
The output of an OR study, then, is usually a set of rules for determining the optimal values of the controlled variables together with a procedure for continuously checking the values of the uncontrolled variables. It must be borne in mind, however, that a single, unified, and comprehensive OR study is seldom possible in an organization of any appreciable size. Rather, what usually occurs is a sequence of interrelated studies, each of which is designed to be adjustable to the results of the others.
Ten years of constructing and working with models of managerial problems in industry have shown that, despite the fact that no two problems are ever exactly alike in content, most problems fall into one, or a combination, of a small number of basic types. These problem-types have now been studied extensively so that today we have considerable knowledge about how to construct and solve models that are relevant to them. Adequate definitions of these problem-types require more space than is available here, but the following brief characterizations indicate their nature.
Inventory problem—to determine the amount of a resource to be acquired or the frequency of acquisition when there is a penalty for having either too much or too little available.
Allocation problem—to determine the allocation of resources to a number of jobs where available resources do not permit each job to be done in the best possible way, so as to do all (or as many as possible) of the jobs in such a way as to achieve the best over-all performance, given criteria for measuring performance.
Queuing problem—to determine the amount of service facilities required or how to schedule arrival of tasks at service facilities so that losses associated with idle facilities, waiting, and turned-away tasks are minimized.
Sequencing problem—to determine the order in which a set of tasks should be performed in a multistage facility so as to minimize costs associated with the performance of the tasks and delays in completing them.
Routing problem—to determine which path or route through a network of points or locations is shortest (or longest), has maximum (or minimum) capacity, or is least (or most) costly to traverse subject to certain limitations on the paths or routes that are permissible.
Replacement problem—to determine when to replace instruments, tools, or facilities so that acquisition, maintenance, and operating and failure costs are minimized.
Competition problem—to determine the rule to be followed by a decision maker that yields the best results when the outcome of his decision depends in part on decisions made by others.
Search problem—to determine the amount of resources to employ and how to allocate them in seeking information to be used for a particular purpose so as to minimize the costs associated with the search and with the errors that can result from use of incorrect information.
The future of OR. OR has been primarily concerned with the executive’s decision-making or control process. There are, of course, other approaches to improving the performance of organizations, for example, selecting better personnel, providing better personnel training, better motivating personnel, accelerating their operations through work study, changing equipment and materials, modifying communications, changing organizational structure. This multiplicity of available approaches presents the executive with the additional problem of selecting which approaches to pursue. He seldom has an objective basis for doing so. Clearly it would be desirable to develop an integrated and comprehensive approach to organizations, one that rationally selects from or combines different points of view. OR and other systems-oriented interdisciplinary research are taking steps to develop such an overall approach to organizational problems. This is leading to mathematical descriptions of organizational structures and communications systems, thus providing the ultimate possibility of integrating studies of organizational structure, communication, and control.
Precise solutions of some limited problems of organizational structure have already been found. For example, given an organization’s over-all objective and a description of its task and environment, it is possible to determine the number and types of units into which the organization should be divided and the objectives to be assigned to these units so as to minimize inefficiency arising from the organization’s structure. This is a problem in structural design. Or, given an organization that has an inefficient structure, it is possible to determine the types of decentralized control to be applied to decentralized decision making so as to minimize inefficiency. This is a problem in structural control.
Such developments are leading to an integrated theory of, and generalized methodology for, research on organized systems. Since all of these systems are, in some sense, social systems, the participation of the social scientist in these interdisciplinary efforts is essential.
Russell L. Ackoff
[See alsoSystems Analysis.]
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Eddison, R. T.; Pennycuick, K.; and RIVETT, B. H. P.1962 Operational Research in Management. New York: Wiley.
Miller, David W.; and Starr, Martin K. 1960 Executive Decisions and Operations Research. Englewood Cliffs, N.J.: Prentice-Hall.
Progress in Operations Research. Volume 1. Edited by Russell L. Ackoff. Operations Research Society of America, Publications in Operations Research, No. 5. 1961 New York: Wiley.
Sasieni, Maurice; Yaspan, Arthur; and Friedman, Lawrence 1959 Operations Research: Methods and Problems. New York: Wiley.
Management science is the experimental study and systematic application of a coordinated process for reaching individual and collective goals by working with and through human and nonhuman resources to continually improve value added to the world.
In historical non-Western civilizations, management topics were addressed by Egyptian pharaohs, Chinese mandarins, pre-Hispanic American chiefs, and Arabic scholars like Ibn Khaldun (1332–1406). However, in Western civilization after the Protestant Reformation, worldly prosperity achieved through hard work in this life became a sign of divine approval, and individuals became motivated to exercise self-discipline at work to earn eternal salvation. Adam Smith (1723–1790) provided the economic rationale for free trade and capitalism to increase individual, group, and national wealth—while critiquing the effects of the division of labor. During the Industrial Revolution, the technological development of steam power, railroad transportation, enormous factories, and production changes required new management skills to rapidly handle large flows of material, people, and information over great distances.
Frederick W. Taylor (1856–1915) was the first to develop a science of the management of others’ work for controlled adaptation of labor to the needs of financial capital in his 1911 work, The Principles of Scientific Management. His model was the machine with its cheap, interchangeable parts, each of which does one specific function. Taylor attempted to do to complex organizations what engineers had done to machines, and this involved making individuals into the equivalent of machine parts. Just as machine parts were easily interchangeable, cheap, and passive, so too should the human parts be the same in the machine model of organizations.
This approach involved breaking down each task to its smallest unit to figure out the one best way to do each job. Then the engineer, after analyzing the job, would teach it to the worker and make sure the worker performed only those motions essential to the task. Taylor attempted to control each element of work and to limit human behavioral variability in order to increase productivity.
The results were profound. Productivity under Taylorism went up dramatically. New departments, such as industrial engineering, personnel, and quality control, arose. There was also growth in middle management as there evolved a separation of planning from operations. Rational rules replaced trial and error; management became formalized and effectiveness increased. Henry Ford (1863–1947) successfully applied scientific management processes in the mass production of inexpensive automobiles using the assembly line. He reduced car assembly time from 728 hours to 93 minutes and increased market share by 40 percent.
Of course, scientific management did not come about without criticism. First, the old-line managers resisted the notion that management was a science to be studied, not a status that was conferred at birth or by inheritance or acquired through an intimidating physical presence or force of personality. Second, many workers resisted the dehumanization of their work due to managerial overcontrol of the labor process in a way that ignored workers’ conceptual and craft skills and deprived them of discretionary time-motion options and work design input. The industrial engineer with a clipboard and a stopwatch, standing over workers to control their task motions and work pace, became a hated figure and led to group resistance.
To counter and complement Taylor’s emphasis on external control of labor, three major approaches have developed over the years, focusing on the following managerial emphases: internal flexibility, internal control, and external flexibility.
First, with respect to emphasizing the internal flexibility of managers, the human relations approach, based on the experiments of Fritz J. Roethlisberger (1898–1974) and William J. Dickson (1904–1989) described in Management and the Worker (1939), demonstrated that work is not merely a mechanical, physical act but an expression of multiple psychosocial needs requiring managerial flexibility. When managers give special attention to employees, productivity is likely to improve regardless of whether working conditions improve—the Hawthorne effect. In The Functions of the Executive (1938), Chester I. Barnard (1886–1961) reinforced this view of the organization as a social system requiring employee cooperation and acceptance of workplace authority. Kurt Lewin (1890–1947), in Frontiers in Group Dynamics (1946), further demonstrated that human work systems could not be understood—much less improved—without factoring in the social-psychological impact of group dynamics in decision making and work performance.
Second, with respect to emphasizing the internal control of managers, Max Weber (1864–1920) maintained that efficient, impersonal coordination of standardized procedures by means of a hierarchy of formal authorities to ensure bureaucratic internal control of labor was the prototypical new role for managers in any organization. Herbert A. Simon (1916–2001) further demonstrated, in The New Science of Management Decision (1960), that the internal organization of firms was the result of decisions made by managers facing uncertainty about the future and costs in acquiring information in the present. Under decision theory, Simon claimed that managers have only “bounded rationality” and are forced to make decisions not by “maximizing” but by “satisficing,” that is, setting an aspiration level that, if achieved, they will be happy enough with. If the aspiration level is not achieved, managers will try to change either their aspiration level or their decision to internally control labor.
Third, with respect to emphasizing external flexibility, Paul Lawrence and Jay Lorsch, in Organization and Environment (1968), maintained that the context of fastpaced uncertainty and global risks required modern managers to creatively adapt and acquire external resources to thrive. They argued that a rapidly changing external environment demanded a dynamic, opensystems approach to management with an organizational feedback structure more like an “adhocracy” than a bureaucracy. One of the first contingency theorists, Fred E. Fiedler, in Leader Attitudes and Group Effectiveness (1958), argued that managerial performance depended (was contingent) upon the manager’s match with three situational factors: leader member relations, task structure, and position power.
In summary, the four major approaches to management compete and complement each other, requiring both control and flexibility to demonstrate managerial excellence.
SEE ALSO Labor; Management; Taylorism
Barnard, Chester I. 1938. The Functions of the Executive. Cambridge, MA: Harvard University Press.
Fayol, Henri.  1949. General and Industrial Management. Trans. Constance Storrs. London: Pittman.
Fiedler, Fred E. 1958. Leader Attitudes and Group Effectiveness. Urbana: University of Illinois Press.
Lawrence, Paul, and Jay Lorsch. 1967. Organization and Environment: Managing Differentiation and Integration. Cambridge, MA: Harvard University Press.
Mayo, Elton. 1933. The Human Problems of an Industrial Civilization. New York: Macmillan.
Roethlisberger, Fritz J., and William J. Dickson. 1939. Management and the Worker: An Account of a Research Program Conducted by the Western Electric Company, Hawthorne Works, Chicago. Cambridge, MA: Harvard University Press.
Simon, Herbert Alexander. 1960. The New Science of Management Decision. New York: Harper.
Taylor, Frederick. 1911. The Principles of Scientific Management. New York: Harper.
Joseph A. Petrick
Operations research is, by its very nature, a multidisciplinary field in which a group of experts attempts to manage and optimize complex undertakings. These undertakings can be military operations, industrial systems management, governmental operations, or commercial ventures. The heart of operations research is the mathematical modeling of part or all of these ventures, which can provide more insight into the interactions of certain variables of interest. The technique of linear programming was developed, in part, to help solve some of these problems. Operations research proved invaluable in solving some military problems in World War II and in subsequent wars and has had a significant impact on the way complex operations are planned and managed.
With the Industrial Revolution in the nineteenth century, production systems became more automated and mechanized, many activities became more complex, and people began to supervise and manage machines that produced goods. This led to the field of industrial engineering, in which designers tried to make machines and production lines as efficient as possible to maximize overall efficiency and corporate profitability.
During World War II, problems arose that required new methods to reach a satisfactory solution. One of these problems was finding German submarines in the open ocean; another was maximizing the utility of Britain's early radar warning networks; yet another was planning complex operations such as the invasion of Normandy. Each of these was exceedingly complex and required the integration of several specialties. In addition, it turned out that each of these problems could be solved, in part, using mathematical analysis to arrive at an optimum solution. Put more succinctly, operations research involves the scientific and mathematical examination of systems and processes to make the most effective use possible of personnel and machines.
Please note that, in this context, "optimum" does not necessarily mean "ideal"; it simply means the best solution possible given the constraints under which the problem is solved. For example, when all is taken into account, the optimum amount of time in which to move three divisions of soldiers and all of their equipment from the United States to the Middle East might be three weeks or to move only the soldiers in two days. If the soldiers and their equipment are required in three days, the "optimum" solutions to this problem are hardly what the planners would desire, but they are the best ones possible, given certain factors such as the number of transport planes available, the number of soldiers and equipment that can fit into a single transport plane, the number and length of runways for both takeoff and landing, the loading and unloading capacity at all airports involved in the mobilization, the amount of fuel available on short notice, the airspeed of the aircraft, the availability of aircrews to load and fly the planes, and many other factors.
It was quickly noted that many operations research problems are very complex. In theory, a large number of these problems cannot be solved easily because the number of possibilities increases exponentially with the number of factors to consider. In actuality, however, many possibilities can be disregarded because, although mathematically valid, they are not considered in real life. For example, in trying to find the optimum routing for supplies from a variety of production locations to a central shipping location, it is mathematically valid to consider shipping supplies from one location to that same location, although in real life this would not be done. Similarly, it is not reasonable to assume that an attacking airplane will originate its flight from over friendly territory, so attack vectors that originate behind friendly lines may be disregarded in planning an air defense system. These and other similar possibilities may be disregarded when setting up such a problem, simplifying it to the point of being solvable.
The inaugural use of operations research came in the late 1930s when future Nobel laureate P.M.S. Blackett (1897-1974) was asked to develop a way to make best use of new radar gunsights for anti-aircraft guns being introduced to the British air defense system. Blackett assembled a team that included physiologists, physicists, mathematicians, an astronomer, an army officer, and a surveyor. Each person brought unique insights and job skills to the task, collectively reaching a solution in a relatively short period of time. This success was noticed quickly by other branches of the British military and by the Americans, all of whom adopted operations research by the early 1940s. Other problems addressed in this manner included hunting submarines in the Atlantic Ocean, designing the mining blockade in Japan's Inland Sea, and helping to plan many major amphibious operations during the war.
After the end of the war, operations research was also adopted by many American businesses, correctly seeing this as a powerful tool that could be adapted to nonmilitary use. In the business world, operations research helped design efficient flow paths for parts being incorporated into a final product, designed transportation systems, and optimized electrical distribution systems. At this time, too, the first mainframe computers were being built, giving researchers new capabilities. More sophisticated models were made possible by high-speed computers and the use of linear programming (a mathematical technique in which operations are described in equations in which no variables are raised to exponential powers or are multiplied together). Provided a system could be described in mathematically simple terms without losing touch with reality, linear programming made it possible to solve problems exactly, rather than by guesswork and experimentation.
Two milestones in the field of linear programming were the development of the technique by Soviet mathematician Leonid Kantorovich and the discovery of "polynomial-time" algorithms for many problems by another Soviet mathematician, Leonid Khachian. Both of these breakthroughs made the solution to many previously intractable problems possible. Linear programming is used to construct an equation that describes the interactions of various aspects of a problem, such as the military mobilization problem mentioned above. For example, if the number of men (nm) that can be transported in a single plane is 200 and five planes (np) are available, then no more than 1,000 men can be transported at a single time. If the time (tl) to load a plane, fly to the destination, unload it, refuel, and return to the point of origin is 18 hours, then in three days (tt), the number of men that could be moved is equal to:
In addition, certain inequalities will be set up as constraints on the problem. For example, a plane cannot hold more than 200 people, so we can say that nm = 200. By programming a computer to solve all of these equations simultaneously while recognizing the inequalities, such problems can be "solved" by coming up with approaches that make the best possible use of available resources to reach a desired goal. Similar, albeit more complex analyses can be used to determine the best trade-off in this example, suggesting how to get the highest number of fully armed soldiers on the ground in the shortest period of time.
Operations research had an immdediate effect on the military success of Allied forces in the Second World War. While the Allies are likely to have won the war in any case, these techniques probably helped shorten the war and allowed it to be won on more favorable terms with fewer casualties. The same is true for subsequent wars, including the Persian Gulf War. In fact, operations research is increasingly important to the U.S. military in an era of declining enlistment, decreasing numbers of planes, ships, and tanks, and more complex military problems in a wider variety of scenarios. This combination requires the most careful possible planning to meet both national priorities and military obligations around the world.
Operations research has also influenced business and industry, giving companies the ability to analyze their operations for the most efficient use of personnel, transportation systems, and production systems. This, in turn, has increased productivity, reduced the use of resources, and streamlined operations, adding to profitability and corporate efficiency. Thes techniques have also been used by government organizations to increase efficiency in many areas.
Finally, in the area of science, operations research was perhaps the first truly interdisciplinary field of inquiry. As such, it not only pioneered many of mathematical techniques now used in other fields, but also helped pave the way for the current increase in interdisciplinary research. Assembling scientists from different specialties offers many different views of a problem and provides a wider variety of intellectual tools that can be brought to a solution. In a way, it is the equivalent of parallax: looking at an object from different vantage points to fix its position in space. This "intellectual parallax" not only gives scientists a better overall view of a problem, but allows the team a much better chance of reaching a solution because they can more quickly and more accurately outline the problem's boundaries.
P. ANDREW KARAM
Operations Research (OR) is defined, according to the International Federation of Operational Research Societies, as a scientific approach to the solution of problems in the management of complex systems. Unlike the natural sciences, OR is a science of the artificial in that its object is not natural reality but rather human-made reality, the reality of complex human-machine systems. OR involves not just theoretical study but also practical application. Its purpose is not only to understand the world as it is, but also to develop guidelines about how to change it in order to achieve aims or to solve certain problems. Ethical considerations are thus crucial to almost all aspects of OR research and practice.
OR as a specific scientific discipline dates back to the years immediately preceding World War II. First in the United Kingdom and later in the United States, interdisciplinary groups were constituted with the objective of improving military operations through a scientific approach. A typical example is the British Anti-Aircraft Command Research Group, better known as Blackett's Circus, which consisted of three physiologists, four physicists, two mathematicians, one army officer, and one surveyor.
Experience with OR in the military context during the war was the basis for new applications in industry afterward. The development of complex, large, and decentralized industrial organizations together with the introduction of computers and the mechanization of many functions required novel scientific approaches to decision making and management. This need led to the establishment, not only in industry but also in academia, of what formally became known as operational research in the United Kingdom, and operations research or management science in the United States (these last two terms are often used synonymously).
The first national OR scientific society was founded in 1948 in the United Kingdom. The U.S. societies, Operations Research Society of America (ORSA) and the Institute of Management Science (TIMS), which later merged as the Institute for Operational Research and the Management Sciences (INFORMS), followed a few years later. In 1959 the International Federations of Operational Research Societies was established.
Optimization plays a major role in OR methodologies: Problems are formulated by means of a set of constraints (equalities or inequalities) and an objective function. The maximization or minimization of the objective function subject to the constraints provides the problem's solution.
Codes versus Principles
Ethics in any applied science develop along two complementary lines. First scientific or professional codes of ethics can be created. These are typically sets of rules, sometimes well-defined, sometimes generic. Useful as they are, ethics codes are external directives not evolved from any individual's ethical beliefs and may lead to double standards. Some evidence suggests that people apply ethical standards at work that are often different and significantly lower than those they follow in their private lives. Although no major national OR society has a formal ethics code, the codes of related scientific disciplines may be applied to OR.
A second way to develop a particular ethics is through an individual approach based upon principles and values instead of rules that govern behavior. According to philosopher Hans Jonas (1903–1993), the following principle can be the basis of an ethical discourse: People have a responsibility toward others, be it humankind (past, present, and future generations) or nature. Another principle of responsibility complements this general rule: Knowledge in all forms must be shared and made available to everyone; cooperation rather than competition should be at the basis of research activity. The latter is called the sharing and cooperation principle (Gallo 2003). These principles are basic to confronting two issues that are crucial to the survival of society: increasing societal inequalities and sustainability.
Models and Methods
If the above are accepted as appropriate principles of responsibility, they can be applied to OR, and, in particular, to model building, which is the fundamental OR activity. The first issue in this regard is determining whether ethics has anything to say about model construction. In his excellent book on ethics and models, William A. Wallace (1994) reports a consensus in the OR research community to the effect that "one of the ethical responsibilities [of modelers] is that the goal of any model building process is objectivity with clear assumptions, reproducible results, and no advocacy" (Wallace 1994, p. 6), and on the "need for model builders to be honest, to represent reality as faithfully as possible in their models, to use accurate data, to represent the results of the models as clearly as possible, and to make clear to the model user what the model can do and what its limitations are" (Wallace 1994, p. 8).
But might responsibility also arise at an earlier stage, when choosing the methodology to create the model? In other words, are methodologies (and hence models) value neutral? This is a controversial issue. It can be argued that behind the role of optimization in OR and the parallel development of optimality as a fundamental principle in the analysis of economic activities and in decision making related to such activities, there are assumptions with ethical implications. Among these is whether self-interest is the only motivation for individual economic choices; whether maximization of the utility function is the best formal way to model individual behavior; and whether, by applying the proper rate of substitution, anything can be traded for anything else, with the consequence that everything can be assigned a monetary value.
These considerations have led some, including J. Pierre Brans (2002), to advocate the use of multicriteria approaches in order to balance objective, subjective, and ethical concerns in model building and problem solving. Such approaches do not reduce, by weighting, different, often noncommensurable, criteria (including those derived from ethical considerations) to one single criterion. Instead each criterion maintains its individuality, leading to a solution that is acceptable to or appropriate for the parties, rather than one that is objectively optimal.
Another issue in the application of principles of responsibility is that optimization-based models are often solution oriented: The final goal of the model is the solution, for instance, the recommendation of action to be made to the client. Some argue that the process is more important than the solution: creating a learning process in which all parties involved acquire a better understanding of the problem and of the system in which the problem arises, with its structure and its dynamics, and have a say in the final decision. These concerns, which call for a broader sense of responsibility not only with respect to the client but to all stakeholders, have led to divisions in the OR community. The development of alternative approaches such as systems thinking and soft operational research are some results.
Clients and Society
Another important question concerns the kind of clients served. As pointed out by Jonathan Rosenhead (1994), OR practitioners "have worked almost exclusively for one type of client: the management of large, hierarchically structured work organizations in which employees are constrained to pursue interests external to their own" (Rosenhead 1994, p. 195). Yet these are not the only possible clients. Other types of organizations exist, operating by consensus rather than chain of command, and representing various interests in society (health, education, housing, employment, environment). Such organizations usually have limited resources though the problems they face are no less challenging for the OR profession.
This fact has ethical relevance. Because the use of models constitutes a source of power, the OR profession runs the risk of aiding the powerful and neglecting the weak, thus contributing to the imbalance of power in society. A positive but rather isolated example of OR assistance outside the sphere of big business is community operational research in the United Kingdom. This initiative has allowed many OR researchers and practitioners to work with community groups, such as associations, cooperatives and trades unions.
Another way OR may contribute to power imbalances at the international level is in the strict enforcement of patents and intellectual property rights. Wide dissemination of methodologies and software, in accordance with the sharing and cooperation principle mentioned above, might reduce the technology divide between rich and poor countries.
Brans, J. Pierre. (2002). "Ethics and Decisions." European Journal of Operational Research 136: 340–352.
Gallo, Giorgio. (2003). "Operations Research and Ethics: Responsibility, Sharing and Cooperation." European Journal of Operational Research 153: 468–476.
Rosenhead, Jonathan. (1994). "One-Sided Practice—Can We Do Better?" In Ethics in Modeling, ed. William A. Wallace. Tarrytown, NY: Pergamon.
Wallace, William A., ed. (1994). Ethics in Modeling. Tarrytown, NY: Pergamon.
Operations research was first used widely in Britain in 1939–40 at the start of World War II. It spread to the United States, where it solved problems such as the placement of bomber‐dropped naval mines to destroy Japanese shipping. Another question involved a patrol plane coming upon a submarine on the surface—the submarine dives and the patrol plane must set an optimal detonation depth for its depth charge. Operations researchers also improved the likelihood that bombers would destroy an industrial target. They recommended reducing the size of a flight to about a dozen planes, assigning the best bombardier to the lead plane and have the rest follow his cue, and minimizing the time between successive bombs released from each plane. Photo reconnaissance showed an approximately fourfold improvement.
Sometimes operations research has exposed an important simple truth, but sometimes it has oversimplified an essentially complex situation. Starting in the late 1960s, it figured in the public debate over antimissile defenses and the survivability of the Minuteman intercontinental ballistic missile. The problem was construed as Soviet missiles destroying American missiles in their silos, but it became clear that the adversary would attack communications and control centers, and that U.S. policy was not to wait and “ride out” such an attack. The scenario of missiles attacking silos received attention partly because it was simple enough to solve.
Historically, there has been tension between the mathematical/scientific training of operations researchers and the military background of those implementing their ideas. In the early 1960s, officers generally resented Department of Defense secretary Robert S. McNamara's civilian whiz kids. Organizational savvy and the proven worth of the method have bridged this gap, and today any major campaign, such as Desert Storm, in the Persian Gulf War, would be preceded by extensive computer simulations.
[See also Disciplinary Views of War: History of Science and Technology; Disciplinary Views of War: Peace History; Game Theory; Neumann, John von; Science, Technology, War, and the Military; World War II: Military and Diplomatic Course.]
Philip Morse and and George Kimball , Methods of Operations Research, 1951.
Jerome Bracken,, Moshe Kress,, and and Richard Rosenthal . Warfare Modeling, 1995.