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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 90: Operations research, mathematical programming

Introduction

Operations research may be loosely described as the study of optimal resource allocation. Mathematically, this is the study of optimization. Depending on the options and constraints in the setting, this may involve linear programming, or quadratic-, convex-, integer-, or boolean-programming.

History

Some links to the history of Operations Research can be found at the Military Operations Research Society and on J. E. Beasley's home page.

Applications and related fields

For numerical optimization techniques (conjugate gradient, simulated annealing, etc.) see 65, Numerical Analysis.

Discrete optimization problems (traveling salesman, etc.) are principally treated in Combinatorics.

The word "programming" in this context is essentially unrelated to computer programming; for that topic see Computer Science

Subfields

• 90B: Operations research and management science (for discrete assignment problems see also 05-XX.)
• 90C: Mathematical programming, see also 49MXX, 65KXX

This was among the larger of the areas of the Math Reviews database. 90C (Mathematical programming) is one of the largest 3-digit areas (and 90C30 (nonlinear programming) is one of the largest 5-digit areas!), but the other subfields were also fairly large.

Starting in the year 2000 sections A and D were removed from this heading; a new primary classification Game theory, economics, social and behavioral sciences will be added which will include most of what has been in those sections.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Some references for management and operations research:

• Bodin, Lawrence; Golden, Bruce; Assad, Arjang; Ball, Michael: "Routing and scheduling of vehicles and crews. The state of the art", Comput. Oper. Res. 10 (1983), no. 2, 63--211. MR85g:90062
• Grosh, Doris Lloyd: "A primer of reliability theory", John Wiley & Sons, Inc., New York, 1989. 373 pp. ISBN 0-471-63820-X MR91d:90048
• Alj, A.; Faure, R.: "Guide de la recherche opérationnelle" (French: Guide to operations research) in two volumes. Vol. I.: "Les fondements (Foundations)"; Masson, Paris, 1986. 265 pp. ISBN 2-903607-55-9 MR91g:90060. Vol. 2: "Les applications (Applications)"; Masson, Paris, 1990. 434 pp. ISBN 2-903607-61-3 MR91g:90061.

Some references for mathematical programming and optimization:

• "State of the art in global optimization", Computational methods and applications (Conf. Princeton University, Princeton, New Jersey, April 1995) Edited by C. A. Floudas and P. M. Pardalos. Nonconvex Optimization and its Applications, 7. Kluwer Academic Publishers, Dordrecht, 1996. 651 pp. ISBN 0-7923-3838-3 MR97a:90004 (For previous years' conference see MR96f:90013, MR96f:90005)
• "Mathematical programming: the state of the art", (Proc. 11th Intern. Symp. Mathematical Programming, University of Bonn, Bonn, August, 1982), Edited by Achim Bachem, Martin Grötschel and Bernhard Korte. Springer-Verlag, Berlin-New York, 1983. 655 pp. ISBN 3-540-12082-3 MR84j:90004
• V. Chvátal: "Linear Programming", W.H. Freeman 1983, ISBN 0-7167-1195-8.
• Gomory, R. E.: "Mathematical programming", Amer. Math. Monthly 72 1965 no. 2, part II 99--110. MR30#4595
• Geoffrion, Arthur M.: "A guided tour of recent practical advances in integer linear programming", ACM SIGMAP Newslett. No. 17 (1974), 22--32. MR54#14776
• Paris, Quirino: "A primer on Karmarkar's algorithm for linear programming", Stud. Develop. 12 (1985), no. 1-2, 131--155. MR87i:90148

Linear programming FAQ: World Wide Web version or Plain-text version

Nonlinear programming FAQ: World Wide Web version or Plain-text version

Newsgroups sci.op-research

Software and tables

Operations Research test data sets

Numerical optimization software is discussed as part of 65K: Mathematical programming, optimization and variational techniques.

Other web sites with this focus

• SIAM's Optimization Theory page.
• A compendium of NP optimization problems
• Here are the AMS and Goettingen resource pages for area 90.
• Pointer to comprehensive Operations Research page
• Overviews and pointers maintained by OpsResearch.com
• INFORMS (Institute for Operations Research and the Management Sciences) home page.
• glossary
• Michael Trick's Operations Research page
• WORMS (World-Wide-Web for Operations Research and Management Science)
• UTK archives page

Selected topics at this site

• Citations for implementation of simplex method
• The cutting stock problem: how to divide line segments (or rectangles, or...) into preassigned shapes with minimal loss? (also known as bin-packing, etc.)
• Using linear programming to answer questions with binary variables (an example of a transportation problem).
• What is the assignment problem?
• How to handle multiple-objective optimization problems?
• Optimization over R with general polynomial constraints is NP-hard.
• How do they schedule elevators?
• A question whose answer is "operations research"
• Minimizing "makespan" (the total time spent completing a multi-stage task).
• "Typical" mathematical modelling problem (operations research)
• Can you get the sofa into the elevator and close the door?
• Pointers to Bin-packing algorithms
• Matrix formulation of Hitchcock-Koopmans transportation problem
• Vehicle Routing problem (partitioning workload)
• Weiszfeld algorithm to solve Fermat-Weber (Steiner) facility location problem
• Assignment Problem: reordering matrix rows to minimize diagonals
• Pointer to a Linear Programming FAQ
• Adapting the simplex algorithm for optimization of nonlinear functions
• Using linear optimization theorems to test for intersections of half-spaces in R^n
• Survey of good algorithms to optimize quadratic function under linear constraints
• Example of Dynamic Programming: select a subset with one linear function constant to minimize another
• Duality gap in optimization of nonlinear programs