Williams, Paul

Professor Emeritus Paul Williams  

Department

Position held

Emeritus Professor of Operational Research

 

Experience keywords:

allocation of resources; combinatorial optimisation; decision theory; integer programming; linear integer programming; linear programming; mathematical models; operational research; travelling salesmen

Research summary > [Click to expand]

My main research interests are in the theory and application of optimisation to decision making with special reference to Linear Programming and its extensions. A major area of application is to Distribution (Facility Location, Vehicle Routing etc.). I am particularly interested in Discrete Optimisation (sometimes known as Integer Programming (IP). A lot of my work here has been on applications and modelling. This makes use of ideas from Logic. Some problems of Discrete Optimisation are notoriously difficult to solve (eg the famous Travelling Salesman problem) and I have done work on improved formulations and computational methods. Also I am doing work on Duality in LP and IP. This has economic applications (eg price allocation) as well as computational and mathematical importance. Professor Williams is examining the issue of 'fairness' and creating fair allocations by finding Minimax solutions to allocation problems.

Languages:

French [Spoken: Basic, Written: Basic]

Media experience:

Radio; TV

Contact Points

LSE phone number:

020 7955 7018

Publications

2017

Williams, H. Paul (2017) The dependency diagram of a mixed integer linear programme Journal of the Operational Research Society, 68 (7). 829-833. ISSN 0160-5682

2016

Williams, H. Paul (2016) The Red River Godrevy Press. ISBN 9781533177964

Williams, H. Paul and Hooker, J. N. (2016) Integer programming as projection Discrete Optimization, 22 (B). 291-311. ISSN 1572-5286

2015

Williams, H. Paul (2015) The dependency diagram of a linear programme Journal of the Operational Research Society, 67 (3). 450-456. ISSN 0160-5682

2014

Williams, H. Paul and Hooker, J. N. (2014) Integer programming as projection Working Paper, LSEOR 13.143. Management Science Group, Department of Management, The London School of Economics and Political Science, London, UK.

2013

Williams, H. Paul (2013) The dependency diagram of a linearprogramme Working paper, LSEOR 13.138. London School of Economics and Political Science, London, UK.

Williams, H. Paul (2013) Model building in mathematical programming 5th, Wiley. ISBN 9781118443330

Williams, H. P. (2013) The dependency diagram of a mixed integer linear programme Working paper, LSEOR 13.139. Management Science Group, Department of Management, The London School of Economics and Political Science, London, UK.

Williams, H. P. (2013) The general solution of a mixed integer linear programme over a cone Working paper, LSEOR 13.140. Management Science Group, Department of Management, The London School of Economics and Political Science, London, UK.

2012

Hooker, J. N. and Williams, H. Paul (2012) Combining equity and utilitarianism in a mathematical programming model Management Science, 58 (9). 1682-1693. ISSN 0025-1909

2011

Williams, H. Paul (2011) The problem with integer programming IMA Journal of Management Mathematics, 22 (3). 213-230. ISSN 1471-6798

2010

Williams, H. Paul (2010) What lies between + and x (and beyond)? Operational Research working papers, LSEOR 10.119. London School of Economics and Political Science, London, UK.

2009

Williams, H. Paul (2009) Logic and integer programming International series in operations research & management science. Springer, London, UK. ISBN 9780387922799

2007

Williams, H. Paul (2007) Stanley Skewes and the Skewes number Journal of the Royal Institution of Cornwall. 70-75. ISSN 0968-5396

Yan, Hong and Williams, H. Paul (2007) Convex hull representations of the at least predicate of constraint satisfaction Operational Research working papers, LSEOR 07.97. Operational Research Group, London School of Economics and Political Science, London, UK. ISBN 9780853280538

Williams, H. Paul (2007) A method for finding all solutions of a Linear Complementarity Problem Operational Research working papers, LSEOR 07.96. Operational Research Group, Department of Management, London School of Economics and Political Science, London, UK. ISBN 9780853280521

2006

Appa, Gautam and Pitsoulis, Leonidas and Williams, H. Paul, eds (2006) Handbook on modelling for discrete optimization International series in operations research & management science, 88. Springer, New York, USA. ISBN 9780387329413

Orman, A. J. and Williams, H. Paul (2006) A survey of different integer programming formulations of the travelling salesman problem In: Kontoghiorghes, Erricos John and Gatu, Cristian, (eds.) Optimisation, Econometric and Financial Analysis. Advances in computational management science, 9 (9). Springer, Berlin, Germany, 93-108. ISBN 9783540366256

Williams, H. Paul and Butler, Martin (2006) The allocation of shared fixed costs European Journal of Operational Research, 170 (2). 391-397. ISSN 0377-2217

Appa, Gautam and Argyris, N. and Williams, H. Paul (2006) A methodology for cross-evaluation in DEA Operational Research working papers, LSEOR 06.81. Operational Research Group, Department of Management, London School of Economics and Political Science, London, UK. ISBN 0753017032

Appa, Gautam and Williams, H. Paul (2006) A new framework for the solution of DEA models European Journal of Operational Research, 172 (2). 604-615. ISSN 0377-2217

2005

Williams, H. Paul (2005) George B Dantzig, 1914–2005 Journal of the Operational Research Society, 56 (12). 1349-1350. ISSN 0160-5682

2002

Appa, Gautam and Williams, H. Paul (2002) A formula for the solution of DEA models Operational Research working papers, LSEOR 02.49. Department of Operational Research, London School of Economics and Political Science, London, UK.

Butler, Martin and Williams, H. Paul (2002) Fairness versus efficiency in charging for the use of common facilities Journal of the Operational Research Society, 53 (12). 1324-1329. ISSN 0160-5682

2001

Williams, H. Paul and Yan, Hong (2001) Representations of the 'all_different' predicate of constraint satisfaction in integer programming Informs Journal on Computing, 13 (2). 96-103. ISSN 1091-9856

2000

Williams, H. Paul (2000) The dual of a logical linear programme Journal of Global Optimization, 18 (2). 129-141. ISSN 0925-5001

1999

Williams, H. Paul (1999) Model building in mathematical programming 4th, John Wiley & Sons, Chichester. ISBN 9780471997887

Williams, H. Paul and Munford, A. G. (1999) Formulae for the L0, L1 and L∞ norms Journal of Statistical Computation and Simulation, 63 (2). 121-141. ISSN 0094-9655

1998

Williams, H. Paul (1998) Tutorials and recent reviews. In: 16th EURO Conference on Operational Analysis, July 1998, Brussels, Belgium

Williams, H. Paul (1998) Recent developments in combinatorial optimisation In: OR Keynote Papers 1998. The Operational Research Society, Birmingham, UK.

Williams, H. Paul and Wilson, John M. (1998) Connections between integer linear programming and constraint logic programming: an overview and introduction to the cluster of articles Informs Journal on Computing, 10 (3). 261-264. ISSN 1091-9856

1997

Williams, H. Paul and Kirby, D. (1997) Representing integral monoids by inequalities Journal of Combinatorial Mathematics and Combinatorial Computing, 23. 87-95. ISSN 0835-3026

Williams, H. Paul (1997) The splitting of variables and constraints in the formulation of integer programming models European Journal of Operational Research, 100 (3). 623-628. ISSN 0377-2217

Williams, H. Paul (1997) Integer programming and pricing revisited IMA Journal of Mathematics Applied in Business and Industry, 8 (3). 203-213. ISSN 0953-0061

Butler, Martin and Williams, H. Paul and Yarrow, Leslie-Ann (1997) The two-period travelling salesman problem applied to milk collection in Ireland Computational Optimization and Applications, 7 (3). 291-306. ISSN 0926-6003

1996

Williams, H. Paul (1996) Obituary of Professor Steven Vajda Times (3 Jan 1996) Website

Williams, H. Paul and Brailsford, S. C. (1996) Computational logic and integer programming In: Beasley, J. E., (ed.) Advances in Linear and Integer Programming. Oxford lecture series in mathematics and its applications (4). Oxford University Press, Oxford, UK, 249-281. ISBN 9780198538561

Williams, H. Paul (1996) The equivalence of two theorems of integer programming Bulletin of the London Mathematical Society, 28 (3). 311-316. ISSN 0024-6093

Williams, H. Paul (1996) Duality in mathematics and linear and integer programming Journal of Optimization Theory and Applications, 90 (2). 257-278. ISSN 0022-3239

Williams, H. Paul (1996) Constructing the value function for an Integer Linear Programme over a Cone Computational Optimization and Applications, 6 (1). 15-26. ISSN 0926-6003

Brailsford, S. C. and Hubbard, P. M. and Smith, B. M. and Williams, H. Paul (1996) Organizing a social event: a difficult problem of combinatorial optimization Computers and Operations Research, 23 (9). 845-856. ISSN 0305-0548

1995

Williams, H. Paul (1995) The Cornish caveman mathematician The Mathematical Intelligencer, 17 (1). 34, 64. ISSN 0343-6993

Rhodes, Frank and Williams, H. Paul (1995) Discrete subadditive functions as Gomory functions Mathematical Proceedings of the Cambridge Philosophical Society, 117 (03). 559-574. ISSN 0305-0041

Williams, H. Paul (1995) Logic applied to integer programming and integer programming applied to logic European Journal of Operational Research, 81 (3). 605-616. ISSN 0377-2217

Smith, Barbara M. and Brailsford, Sally C. and Hubbard, Peter M. and Williams, H. Paul (1995) The progressive party problem: integer linear programming and constraint programming compared In: Montanari, Ugo and Rossi, Francesca, (eds.) Principles and Practice of Constraint Programming: Proceedings of First International Conference, Cp '95, Cassis, France, Septem. Lecture notes in computer science (976). Springer, Berlin, Germany, 36-52. ISBN 9783540602996

1994

Williams, H. Paul (1994) The contractor allocation problem: an integer programming problem that wasn't In: Ciriani, T. and Leachman, R., (eds.) Optimisation in Industry. Wiley, Chichester, UK, 25-32.

Williams, H. Paul (1994) An alternative explanation of disjunctive formulations European Journal of Operational Research, 72 (1). 200-203. ISSN 0377-2217

1993

Williams, H. Paul (1993) Model solving in mathematical programming Wiley, Chichester, UK. ISBN 9780471935810

Williams, H. Paul (1993) How long is the coast of Cornwall? Mathematics Review, 3. 2-5. ISSN 0957-1280

1992

Williams, H. Paul (1992) Solving systems of linear inequalities Mathematical Medley, 20 (1). 59-67. ISSN 0217-2976

Williams, H. Paul (1992) The elimination of integer variables Journal of the Operational Research Society, 43 (5). 387-393. ISSN 0160-5682

1991

Baston, V. J. D. and Rahmouni, M. K. and Williams, H. Paul (1991) The practical conversion of linear programmes to network flow models European Journal of Operational Research, 50 (3). 325-334. ISSN 0377-2217

1990

Williams, H. Paul (1990) Optimisation and operational research Bulletin of the Institute of Mathematics and Its Applications, 26. 76-85. ISSN 0950-5628

Williams, H. Paul (1990) How important are models to operational research? IMA Journal of Mathematics Applied in Business and Industry, 2. 189-195. ISSN 0953-0061

1989

Williams, H. Paul (1989) A method of finding all equilibrium solutions of a 2-person matrix game Faculty of Mathematical Studies working papers, OR22. Southampton University, Southampton, UK.

Williams, H. Paul (1989) Mathematical programming modelling In: Shahani, Arjan K. and Stainton, Roy, (eds.) Tutorial Papers in Operational Research. Operational Research Society, Birmingham, UK, 41-56.

McKinnon, K. I. M. and Williams, H. Paul (1989) Constructing integer programming models by the predicate calculus Annals of Operations Research, 21. 227-245. ISSN 0254-5330

1988

Williams, H. Paul (1988) An alternative form of the value function of an integer programme Faculty of Mathematical Studies working papers, OR16. Faculty of Mathematical Studies, University of Southampton, Southampton, UK.

1987

Williams, H. Paul (1987) Recent developments in mathematical programming In: Thomas, L. C., (ed.) Recent Developments in Operational Research. Pergamon Press, Oxford, UK, 33-44.

Williams, H. Paul (1987) Linear and integer programming applied to the propositional calculus Systems Research and Information Science, 2. 81-100. ISSN 0882-3014

Williams, H. Paul (1987) A note: orthogonality in linear congruence duality Faculty of Mathematical Studies working papers, OR9. University of Southampton, Southampton, UK.

Williams, H. Paul (1987) Evolution, games theory and polyhedra Journal of Mathematical Biology, 25 (4). 393-409. ISSN 0303-6812

1986

Williams, H. Paul (1986) Fourier's method of linear programming and its dual The American Mathematical Monthly, 93 (9). 681-695. ISSN 0002-9890

Williams, H. Paul (1986) Optimisation is best. In: Inaugural lecture as Professor of Operational Research, Nov 1986, University of Southampton

Day, R. E. and Williams, H. Paul (1986) MAGIC: The design and use of an interactive modelling language for mathematical programming IMA Journal of Management Mathematics, 1 (1). 53-65. ISSN 1471-678X

1985

Williams, H. Paul (1985) Model building in linear and integer programming In: Schittkowski, K., (ed.) Proceedings of Nato Advanced Study Institute on Mathematical Programming. Springer, Berlin, Germany, 25-35.

1984

Williams, H. Paul (1984) A duality theorem for linear congruences Discrete Applied Mathematics, 7 (1). 93-103. ISSN 0166-218X

Williams, H. Paul (1984) Restricted vertex generation applied as a CRASHing procedure for linear programming Computers and Operations Research, 11 (4). 401-407. ISSN 0305-0548

Cheshire, M. and McKinnon, K. I. M. and Williams, H. Paul (1984) The efficient allocation of private contractors to public works Journal of the Operational Research Society, 35 (8). 705-709. ISSN 0160-5682

1983

Williams, H. Paul (1983) A duality relationship for integer programmes Department of Business Studies working papers, 1/83. Department of Business Studies, University of Edinburgh, Edinburgh, UK.

Williams, H. Paul (1983) A reduction procedure for linear and integer programming models In: Karwan, M. H. and Lotfi, V. and Telgen, J. and Zionts, S., (eds.) Redundancy in Mathematical Programming: a State-Of-The-Art Survey. Lecture notes in economics and mathematical systems (206). Springer-Verlag, Berlin, Germany, 87-107. ISBN 0387115528

Williams, H. Paul (1983) A characterisation of all feasible solutions to an integer program Discrete Applied Mathematics, 5 (1). 147-155. ISSN 0166-218X

1982

Williams, H. Paul (1982) Models with network duals Journal of the Operational Research Society, 33 (2). 161-169. ISSN 0160-5682

1981

Williams, H. Paul (1981) Mathematical modelling in management Bulletin of the Institute of Mathematics and Its Applications, 17. 51-52. ISSN 0950-5628

Williams, H. Paul (1981) Reallocating the cost of dependent decisions Applied Economics, 13 (1). 89-98. ISSN 0003-6846

1980

Williams, H. Paul and Thomas, P. G. (1980) The Open University interactive system for integer programming Iucc Bulletin, 2. ISSN 0142-2464

Williams, H. Paul (1980) Integer programming In: Lederman, Walter and Vajda, Steven, (eds.) Handbook of Applicable Mathematics: Vol. 1, Algebra. Wiley, Chichester, UK, 339-390. ISBN 0471277045

1979

Williams, H. Paul (1979) Four examples of artless modelling Omega, 7 (2). 163-165. ISSN 0305-0483

Williams, H. Paul (1979) The economic interpretation of duality for practical mixed integer programming models In: Prékopa, A., (ed.) Survey of Mathematical Programming : Proceedings of the 9th International Mathematical Programming Symposium, Budapest, August 2. North Holland, Amsterdam, The Netherlands, 567-586. ISBN 0444850333

1978

Williams, H. Paul (1978) Model building in management science: is the model more important than the mathematics?. In: Inaugural lecture as Professor of Management Science, 1978, University of Edinburgh

Williams, H. Paul (1978) The reformulation of two mixed integer programming problems Mathematical Programming, 14 (1). 325-331. ISSN 0025-5610

1977

Williams, H. Paul (1977) Logical problems and integer programming Bulletin of the Institute of Mathematics and Its Applications, 13. 18-20. ISSN 0950-5628

Williams, H. Paul (1976) Fourier-Motzkin elimination extension to integer programming problems Journal of Combinatorial Theory, Series A, 21 (1). 118-123. ISSN 0097-3165

1975

Williams, H. Paul (1975) The formulation of mathematical programming models Omega, 3 (5). 551-556. ISSN 0305-0483

Brearley, A. L. and Mitra, G. and Williams, H. Paul (1975) Analysis of mathematical programming problems prior to applying the simplex algorithm Mathematical Programming, 8 (1). 54-83. ISSN 0025-5610

1974

Williams, H. Paul (1974) Experiments in the formulation of integer programming problems Mathematical Programming Studies, 2. 180-197. ISSN 0303-3929

Williams, H. Paul (1974) Three integer programming models ACM Sigmap Bulletin, 16. 35-41. ISSN 1931-1184

Williams, H. Paul and Redwood, A. C. (1974) A structured linear programming model in the food industry Operational Research Quarterly, 25 (4). 517-527. ISSN 0030-3623

1973

Williams, H. Paul (1973) Simplifying linear programming problems Research report, 73-2. Operational Research Group, University of Sussex, Brighton, UK.

Williams, H. Paul (1973) Decision procedures in formal logic and mathematical programming algorithms Research report, 73-5. Operational Research Group, University of Sussex, Brighton, UK.

1970

Williams, H. Paul (1970) An algorithm for the solution of linear programming problems IBM working papers, IBM, UK.

1969

Williams, H. Paul (1969) A formalisation of the arithmetic of the ordinals less than Wω Notre Dame Journal of Formal Logic, 10 (1). 77-89. ISSN 0029-4527

1968

Williams, H. Paul (1968) The synthesis of logical nets consisting of NOR units The Computer Journal, 11 (2). 173-176. ISSN 0010-4620

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