Share on Facebook Share on Twitter Email
Answers.com

operations research

 
Dictionary: operations research
Home > Library > Literature & Language > Dictionary

n.
Mathematical or scientific analysis of a process or operation, used in making decisions.


Search unanswered questions...

Enter a question here...
Search: All sources Community Q&A Reference topics

Britannica Concise Encyclopedia: operations research
Top
Home > Library > Miscellaneous > Britannica Concise Encyclopedia

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.

For more information on operations research, visit Britannica.com.

Sci-Tech Encyclopedia: Operations research
Top
Home > Library > Science > Sci-Tech Encyclopedia

The application of scientific methods and techniques to decision-making problems. A decision-making problem occurs where there are two or more alternative courses of action, each of which leads to a different and sometimes unknown end result. Operations research is also used to maximize the utility of limited resources. The objective is to select the best alternative, that is, the one leading to the best result.

To put these definitions into perspective, the following analogy might be used. In mathematics, when solving a set of simultaneous linear equations, one states that if there are seven unknowns, there must be seven equations. If they are independent and consistent and if it exists, a unique solution to the problem is found. In operations research there are figuratively “seven unknowns and four equations.” There may exist a solution space with many feasible solutions which satisfy the equations. Operations research is concerned with establishing the best solution. To do so, some measure of merit, some objective function, must be prescribed.

In the current lexicon there are several terms associated with the subject matter of this program: operations research, management science, systems analysis, operations analysis, and so forth. While there are subtle differences and distinctions, the terms can be considered nearly synonymous. See also Systems engineering.

Methodology

The success of operations research, where there has been success, has been the result of the following six simply stated rules: (1) formulate the problem; (2) construct a model of the system; (3) select a solution technique; (4) obtain a solution to the problem; (5) establish controls over the system; and (6) implement the solution.

The first statement of the problem is usually vague and inaccurate. It may be a cataloging of observable effects. It is necessary to identify the decision maker, the alternatives, goals, and constraints, and the parameters of the system. A statement of the problem properly contains four basic elements that, if correctly identified and articulated, greatly eases the model formulation. These elements can be combined in the following general form: “Given (the system description), the problem is to optimize (the objective function), by choice of the (decision variable), subject to a set of (constraints and restrictions).”

In modeling the system, one usually relies on mathematics, although graphical and analog models are also useful. It is important, however, that the model suggest the solution technique, and not the other way around.

With the first solution obtained, it is often evident that the model and the problem statement must be modified, and the sequence of problem-model-technique-solution-problem may have to be repeated several times. The controls are established by performing sensitivity analysis on the parameters. This also indicates the areas in which the data-collecting effort should be made.

Implementation is perhaps of least interest to the theorists, but in reality it is the most important step. If direct action is not taken to implement the solution, the whole effort may end as a dust-collecting report on a shelf.

Mathematical programming

Probably the one technique most associated with operations research is linear programming. The basic problem that can be modeled by linear programming is the use of limited resources to meet demands for the output of these resources. This type of problem is found mainly in production systems, but is not limited to this area. See also Linear programming.

Stochastic processes

A large class of operations research methods and applications deals with stochastic processes. These can be defined as processes in which one or more of the variables take on values according to some, perhaps unknown, probability distribution. These are referred to as random variables, and it takes only one to make the process stochastic.

In contrast to the mathematical programming methods and applications, there are not many optimization techniques. The techniques used tend to be more diagnostic than prognostic; that is, they can be used to describe the “health”of a system, but not necessarily how to “cure” it. See also Probability; Queueing theory; Stochastic process.

Scope of application

There are numerous areas where operations research has been applied. The following list is not intended to be all-inclusive, but is mainly to illustrate the scope of applications: optimal depreciation strategies; communication network design; computer network design; simulation of computer time-sharing systems; water resource project selection; demand forecasting; bidding models for offshore oil leases; production planning; classroom size mix to meet student demand; optimizing waste treatment plants; risk analysis in capital budgeting; electric utility fuel management; optimal staffing of medical facilities; feedlot optimization; minimizing waste in the steel industry; optimal design of natural-gas pipelines; economic inventory levels; optimal marketing-price strategies; project management with CPM/PERT/GERT; air-traffic-control simulations; optimal strategies in sports; optimal testing plans for reliability; optimal space trajectories. See also Decision theory; GERT; Inventory control; PERT.

Marketing Dictionary: operations research
Top
Home > Library > Business & Finance > Marketing Dictionary

Application of scientific and mathematical principles to business decisions; also called management science. Operations research uses mathematical models and computer simulations to better understand and predict market behaviors. It is also used by industrial buyers to determine the optimal time to order additional inventory. Manufacturers use operational research techniques to schedule production processes, to schedule tasks necessary to complete complex projects, and to optimize production work flows. See also heuristics; optimal order quantity.

Business Dictionary: Operations Research (OR)
Top
Home > Library > Business & Finance > Business Dictionary

Research concerned with the development of mathematical models of repetitive activities, using numerous variables, such as traffic flow, assembly lines, military campaigns, and optional production scheduling. Operations research makes extensive use of computer simulation.

US Military History Companion: Operations Research
Top
Home > Library > History, Politics & Society > US Military History Companion

In defense analysis, operations research uses statistics and mathematics to optimize the use of a weapon

To set up a radar/fighter defense against bomber aircraft, for example, the traditional method would be to draw advice from war‐experienced experts or apply current military doctrine. An operations researcher, however, starts with a numerical measure of effectiveness—perhaps the number of bombers expected to get through, or the warning time provided, or some combination. The next step is to analyze statistics from past systems, and feed these into a mathematical model combining scanning rates, the decision‐making speed of radar operators, and the average number of operational fighters. The best design might be found by pencil and paper analysis, or a computer program that simulates the system under various attacks.

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.]

Bibliography

  • Philip Morse and George Kimball, Methods of Operations Research, 1951.
  • Jerome Bracken, Moshe Kress, and Richard Rosenthal. Warfare Modeling, 1995
US Military Dictionary: operations research
Top
Home > Library > History, Politics & Society > US Military Dictionary

Also operational research or operations analysis the analytical study of military problems undertaken to provide responsible commanders and staff agencies with a scientific basis for decision on action to improve military operations.

See the Introduction, Abbreviations and Pronunciation for further details.

Veterinary Dictionary: operations research
Top
Home > Library > Animal Life > Veterinary Dictionary

Mathematical or scientific analysis of a process.

Military Dictionary: operations research
Top
Home > Library > History, Politics & Society > Military Dictionary

(DOD) The analytical study of military problems undertaken to provide responsible commanders and staff agencies with a scientific basis for decision on action to improve military operations. Also called operational research; operations analysis.

Wikipedia: Operations research
Top
Home > Library > Miscellaneous > Wikipedia

Operations research (N America) or Operational research (UK/Europe) "is a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control."[1] Other names for it include:

  • Operational analysis (UK Ministry of Defence from 1962)[2]
  • Quantitative management[3]

"The historical development of Operational Research (OR) is traditionally seen as the succession of several phases: the “heroic times” of the Second World War, the “Golden Age” between the fifties and the sixties during which major theoretical achievements were accompanied by a widespread diffusion of OR techniques in private and public organisations, a “crisis” followed by a “decline” starting with the late sixties, a phase during which OR groups in firms progressively disappeared while academia became less and less concerned with the applicability of the techniques developed."[4]. In the current phase, the increase in computing power coupled with the birth of related techniques like business intelligence (BI) and business analytics are leading a resurgence of OR.[5]

Model of Operation research study, based on Stafford Beer (1959).[6]

Contents

Overview

The terms operations research and management science are often used synonymously.[citation needed] When a distinction is drawn, management science generally implies a closer relationship to the problems of business management.[citation needed] The field of operations research is closely related to Industrial engineering.[citation needed] Industrial engineers typically consider Operations Research (OR) techniques to be a major part of their toolset.

Some of the primary tools used by operations researchers are statistics, optimization, probability theory, queuing theory, game theory, graph theory, decision analysis, mathematical modeling and simulation. Because of the computational nature of these fields, OR also has ties to computer science, and operations researchers use both custom-written and off-the-shelf software.

Operations research is distinguished by its frequent use to examine an entire management information system, rather than concentrating only on specific elements (though this is often done as well). An operations researcher faced with a new problem is expected to determine which techniques are most appropriate given the nature of the system, the goals for improvement, and constraints on time and computing power. For this and other reasons, the human element of OR is vital. Like any other tools, OR techniques cannot solve problems by themselves.

History

Some say that Charles Babbage (1791-1871) is the "father of operations research" because his research into the cost of transportation and sorting of mail led to England's universal "Penny Post" in 1840, and studies into the dynamical behaviour of railway vehicles in defence of the GWR's broad gauge.[7] The modern field of operations research arose during World War II.

Modern operations research originated at the Bawdsey Research Station in the UK in 1937 and was the result of an initiative of the station's superintendent, A. P. Rowe. Rowe conceived the idea as a means to analyse and improve the working of the UK's early warning radar system, Chain Home (CH). Initially, he analyzed the operating of the radar equipment and its communication networks, expanding later to include the operating personnel's behaviour. This revealed unappreciated limitations of the CH network and allowed remedial action to be taken. [3]

Patrick Blackett.

Scientists in the United Kingdom including Patrick Blackett, Cecil Gordon, C. H. Waddington, Owen Wansbrough-Jones, Frank Yates, Jacob Bronowski and Freeman Dyson, and in the United States with George Dantzig looked for ways to make better decisions in such areas as logistics and training schedules. After the war it began to be applied to similar problems in industry.

Second World War

During the Second World War close to 1,000 men and women in Britain were engaged in operational research. About 200 operational research boffins worked for the British Army.[8]

Patrick Blackett worked for several different organizations during the war. Early in the war while working for the Royal Aircraft Establishment (RAE) he set up a team known as the "Circus" which helped to reduce the number of anti-aircraft artillery rounds needed to shoot down an enemy aircraft from an average of over 20,000 at the start of the Battle of Britain to 4,000 in 1941[9]

In 1941 Blackett moved from the RAE to the Navy, first to the Royal Navy's Coastal Command, in 1941 and then early in 1942 to the Admiralty.[10] Blackett's team at Coastal Command's Operational Research Section (CC-ORS), included, two future Nobel prize winners, and many other people who went on to be preeminent in their fields,[11] undertook a number of crucial analyses that aided the war effort. Britain introduced the convoy system to reduce shipping losses, but while the principle of using warships to accompany merchant ships was generally accepted, it was unclear whether it was better for convoys to be small or large. Convoys travel at the speed of the slowest member, so small convoys can travel faster. It was also argued that small convoys would be harder for German U-boats to detect. On the other hand, large convoys could deploy more warships against an attacker. Blackett's staff showed that the losses suffered by convoys depended largely on the number of escort vessels present, rather than on the overall size of the convoy. Their conclusion, therefore, was that a few large convoys are more defensible than many small ones.[12]

While performing an analysis of the methods used by RAF Coastal Command to hunt and destroy submarines, one of the analysts asked what colour the aircraft were. As most of them were from Bomber Command they were painted black for nighttime operations. At the suggestion of CC-ORS a test was run to see if that was the best colour to camouflage the aircraft for daytime operations in the grey North Atlantic skies. Tests showed that aircraft painted white were on average not spotted until they were 20% closer than those painted black. This change indicated that 30% more submarines would be attacked and sunk for the same number of sightings.[13]

Other work by the CC-ORS indicated that on average if the depth at which aerial delivered depth charges (DC's) was changed from 100 feet to 25 feet, the kill ratios would go up. This was because if a U-boat saw an aircraft only shortly before it arrived over the target then at 100 feet the charges would do no damage, and if it saw the aircraft a long way from the target it had time to alter course under water so the chances of it being within the 20 feet kill zone of the charges was small. It was more efficient to attack those submarines close to the surface who's location was known than those at a greater depth who's position could only be guessed. Before the change from 100 feet to 25 feet 1% of submerged U-boats were sunk and 14% damaged, after the change 7% were sunk and 11% damaged (if caught on the surface the numbers were 11% sunk and 15% damaged). Blackett observed "there can be few cases where such a great operational gain had been obtained by such a small and simple change of tactics[14]

Bomber Command's Operational Research Section (BC-ORS), analysed a report of a survey carried out by RAF Bomber Command.[citation needed] For the survey, Bomber Command inspected all bombers returning from bombing raids over Germany over a particular period. All damage inflicted by German air defenses was noted and the recommendation was given that armour be added in the most heavily damaged areas. Their suggestion to remove some of the crew so that an aircraft loss would result in fewer personnel loss was rejected by RAF command. Blackett's team instead made the surprising and counter-intuitive recommendation that the armour be placed in the areas which were completely untouched by damage in the bombers which returned. They reasoned that the survey was biased, since it only included aircraft that returned to Britain. The untouched areas of returning aircraft were probably vital areas, which, if hit, would result in the loss of the aircraft.[citation needed]

Map of Kammhuber Line

When Germany organised its air defences into the Kammhuber Line, it was realised that if the RAF bombers were to fly in a bomber stream they could overwhelm the night fighters who flew in individual cells directed to their targets by ground controllers. It was then a matter of calculating the statistical loss from collisions against the statistical loss from night fighters to calculate how close the bombers should fly to minimise RAF losses.[15]

The "exchange rate" ratio of output to input was a characteristic feature of operations research. By comparing the number of flying hours put in by Allied aircraft to the number of U-boat sightings in a given area, it was possible to redistribute aircraft to more productive patrol areas. Comparison of exchange rates established "effectiveness ratios" useful in planning. The ratio of 60 mines laid per ship sunk was common to several campaigns: German mines in British ports, British mines on German routes, and United States mines in Japanese routes.[16]

Operations research doubled the on-target bomb rate of B-29s bombing Japan from the Marianas Islands by increasing the training ratio from 4 to 10 percent of flying hours; revealed that wolf-packs of three United States submarines were the most effective number to enable all members of the pack to engage targets discovered on their individual patrol stations; revealed that glossy enamel paint was more effective camouflage for night fighters than traditional dull camouflage paint finish, and the smooth paint finish increased airspeed by reducing skin friction.[16]

On land, the operational research sections of the Army Operational Research Group (AORG) of the Ministry of Supply (MoS) were landed in Normandy in 1944, and they followed British forces in the advance across Europe. They analysed, among other topics, the effectiveness of artillery, aerial bombing, and anti-tank shooting.

After World War II

From 1962, military operational research in the United Kingdom became known as "operational analysis" (OA) within the UK Ministry of Defence,[2] where OR stands for "Operational Requirement". With expanded techniques and growing awareness, military OR or OA was no longer limited to only operations, but was extended to encompass equipment procurement, training, logistics and infrastructure.

Scope of operations research

Examples of applications in which operations research is currently used include:

  • critical path analysis or project planning: identifying those processes in a complex project which affect the overall duration of the project
  • designing the layout of a factory for efficient flow of materials
  • constructing a telecommunications network at low cost while still guaranteeing quality of service (QoS) or Quality of Experience (QoE) if particular connections become very busy or get damaged
  • road traffic management and 'one way' street allocations i.e. allocation problems.
  • determining the routes of school buses (or city buses) so that as few buses are needed as possible
  • designing the layout of a computer chip to reduce manufacturing time (therefore reducing cost)
  • managing the flow of raw materials and products in a supply chain based on uncertain demand for the finished products
  • efficient messaging and customer response tactics
  • robotizing or automating human-driven operations processes
  • globalizing operations processes in order to take advantage of cheaper materials, labor, land or other productivity inputs
  • managing freight transportation and delivery systems (Examples: LTL Shipping, intermodal freight transport)
  • scheduling:
    • personnel staffing
    • manufacturing steps
    • project tasks
    • network data traffic: these are known as queueing models or queueing systems.
    • sports events and their television coverage
  • blending of raw materials in oil refineries
  • determining optimal prices, in many retail and B2B settings, within the disciplines of pricing science

Operations research is also used extensively in government where evidence-based policy is used.

Societies and journals

Societies

The International Federation of Operational Research Societies[17] is an umbrella organization for operations research societies worldwide. Significant among these are:

  • Institute for Operations Research and the Management Sciences (INFORMS)[18]
  • The Operational Research Society.[19]
  • EURO is the association of European Operational Research Societies.[20]
  • CORS is the Canadian Operational Research Society.[21]
  • ASOR is the Australian Society for Operations Research.[22]
  • MORS is the Military Operations Research Society[23]: based in the United States since 1966 with the objective of enhancing the quality and usefulness of military operations research analysis in support of defense decisions. (MORS)[24]
  • ORSNZ is the Operations Research Society of New Zealand.[25]
  • ORSP is the Operations Research Society of the Philippines[26]
  • ORSI the Operational Research Society of India,[27] and
  • ORSSA the Operations Research Society of South Africa.[28]

Other important Operations Research organizations are:

In 2004 INFORMS began an initiative to market the OR profession better, including a website entitled The Science of Better[31] which provides an introduction to OR and examples of successful applications of OR to industrial problems.

Journals

INFORMS publishes twelve scholarly journals about operations research, including the top two journals in their class, according to 2005 Journal Citation Reports.[32] They are:

Other journals
  • European Journal of Operational Research (EJOR): Founded in 1975 and is presently by far the largest operational research journal in the world, with its around 9,000 pages of published papers per year. In 2004, its total number of citations was the second largest amongst Operational Research and Management Science journals;
  • INFOR Journal: published and sponsored by the Canadian Operational Research Society;
  • Journal of Defense Modeling and Simulation (JDMS): Applications, Methodology, Technology: a quarterly journal devoted to advancing the science of modeling and simulation as it relates to the military and defense.[35]
  • Journal of the Operational Research Society (JORS): an official journal of The OR Society;[36]
  • Journal of Simulation (JOS): an official journal of The OR Society;[37]
  • Military Operations Research (MOR): published by the Military Operations Research Society;
  • Opsearch: official journal of the Operational Research Society of India;
  • OR Insight: a quarterly journal of The OR Society;[38]
  • TOP: the official journal of the Spanish Society of Statistics and Operations Research.[39]

See also

Notes

Text document with red question mark.svg
 This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (June 2009)
  1. ^ "Operational Research in the British Army 1939-1945, October 1947, Report C67/3/4/48, UK National Archives file WO291/1301
    Quoted on the dust-jacket of: Morse, Philip M, and Kimball, George E, Methods of Operations Research, 1st Edition Revised, pub MIT Press & J Wiley, 5th printing, 1954.
  2. ^ a b UK National Archives Catalogue for WO291 lists a War Office organisation called Army Operational Research Group (AORG) that existed from 1946 to 1962. "In January 1962 the name was changed to Army Operational Research Establishment (AORE). Following the creation of a unified Ministry of Defence, a tri-service operational research organisation was established: the Defence Operational Research Establishment (DOAE) which was formed in 1965, and it absorbed the Army Operational Research Establishment based at West Byfleet."
  3. ^ http://brochure.unisa.ac.za/myunisa/data/subjects/Quantitative%20Management.pdf
  4. ^ Boussou, Denis, Questioning the history of operational in order to prepare its future http://hal.ccsd.cnrs.fr/docs/00/02/86/41/PDF/cahierLamsade196.pdf
  5. ^ http://analyticsmagazine.com/
  6. ^ Stafford Beer (1959). Cybernetic and Management. English Universities Press.
  7. ^ M.S. Sodhi, "What about the 'O' in O.R.?" OR/MS Today, December, 2007, p. 12, http://www.lionhrtpub.com/orms/orms-12-07/frqed.html
  8. ^ Kirby, [http://books.google.co.uk/books?id=DWITTpkFPEAC&lpg=PA141&pg=PA117 p. 117
  9. ^ Kirby, [http://books.google.co.uk/books?id=DWITTpkFPEAC&lpg=PA141&pg=PA94 pp. 91-94
  10. ^ Kirby, p. 96,109
  11. ^ Kirby, p. 96
  12. ^ "Numbers are Essential": Victory in the North Atlantic Reconsidered, March-May 1943
  13. ^ Kirby, p. 101
  14. ^ (Kirby, pp. 102,103)
  15. ^ [1]
  16. ^ a b Milkman, Raymond H. (May 1968). Operations Research in World War II. United States Naval Institute Proceedings. 
  17. ^ IFORS
  18. ^ INFORMS
  19. ^ The OR Society
  20. ^ EURO
  21. ^ CORS
  22. ^ ASOR
  23. ^ MORS
  24. ^ [2]
  25. ^ ORSNZ
  26. ^ ORSP
  27. ^ ORSI
  28. ^ ORSSA
  29. ^ SISO
  30. ^ I/ITSEC
  31. ^ The Science of Better
  32. ^ INFORMS Journals
  33. ^ Decision Analysis
  34. ^ INFORMS Transactions on Education
  35. ^ JDMS
  36. ^ The OR Society;
  37. ^ The OR Society;
  38. ^ The OR Society;
  39. ^ TOP

References

  • Kirby, M. W. (Operational Research Society (Great Britain)). Operational Research in War and Peace: The British Experience from the 1930s to 1970, Imperial College Press, 2003. ISBN 1860943667, 9781860943669

Further reading

  • C. West Churchman, Russell L. Ackoff & E. L. Arnoff, Introduction to Operations Research, New York: J. Wiley and Sons, 1957
  • Joseph G. Ecker & Michael Kupferschmid, Introduction to Operations Research, Krieger Publishing Co.
  • Frederick S. Hillier & Gerald J. Lieberman, Introduction to Operations Research, McGraw-Hill: Boston MA; 8th. (International) Edition, 2005
  • Maurice W. Kirby, Operational Research in War and Peace, Imperial College Press, London, 2003
  • Michael Pidd, Tools for Thinking: Modelling in Management Science, J. Wiley & Sons Ltd., Chichester; 2nd. Edition, 2003
  • Hamdy A. Taha, Operations Research: An Introduction, Prentice Hall; 8th. Edition, 2006
  • Wayne Winston, Operations Research: Applications and Algorithms, Duxbury Press; 4th. Edition, 2003

External links

v • d • e
Systems and systems science
Systems categories
Systems
Theoretical fields
Systems scientists

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Best of the Web: operations research
Top

Some good "operations research" pages on the web:


Math
mathworld.wolfram.com
 
 
 

 

Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.  Read more
Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved.  Read more
Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.  Read more
Marketing Dictionary. Dictionary of Marketing Terms. Copyright © 2000 by Barron's Educational Series, Inc. All rights reserved.  Read more
Business Dictionary. Dictionary of Business Terms. Copyright © 2000 by Barron's Educational Series, Inc. All rights reserved.  Read more
US Military History Companion. The Oxford Companion to American Military History. Copyright © 2000 by Oxford University Press, Inc. All rights reserved.  Read more
US Military Dictionary. The Oxford Essential Dictionary of the U.S. Military. Copyright © 2001, 2002 by Oxford University Press, Inc. All rights reserved.  Read more
Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved.  Read more
Military Dictionary. US Department of Defense Dictionary of Military and Associated Words, 2003.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Operations research".  Read more