Williams, Paul

Professor Emeritus Paul Williams

Department	Position held
Department of Management
Management Science Group (Department of Management)	Emeritus Professor of Operational Research

Experience keywords:

linear programming; decision theory; integer programming; linear integer programming; combinatorial optimisation; operational research; travelling salesmen; mathematical models; allocation and fairness

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. Connected to this, 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	LSE email: h.p.williams@lse.ac.uk

Publications

The following references are sourced from LSE Research Online|. References that are linked lead to the full text.

2013

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

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

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). pp. 1682-1693. ISSN 0025-1909

2011

Williams, H. Paul (2011) The problem with integer programming. IMA journal of management mathematics, 22 (3). pp. 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. Springer, London, UK. ISBN 9780387922799

2007

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

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

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

2006

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

Appa, Gautam and Pitsoulis, Leonidas and Williams, H. Paul, (eds.) (2006) Handbook on modelling for discrete optimization. Springer, New York, USA. ISBN 9780387329413

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

Williams, H. Paul and Butler, Martin (2006) The allocation of shared fixed costs. European journal of operational research, 170 (2). pp. 391-397. ISSN 0377-2217

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. Springer, Berlin, Germany, pp. 93-108. ISBN 9783540366256

2005

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

2002

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). pp. 1324-1329. ISSN 0160-5682

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

2001

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

2000

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

1999

Williams, H. Paul (1999) Model building in mathematical programming. 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). pp. 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 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). pp. 261-264. ISSN 1091-9856

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

1997

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

Williams, H. Paul and Kirby, D. (1997) Representing integral monoids by inequalities. Journal of combinatorial mathematics and combinatorial computing, 23 pp. 87-95. ISSN 0835-3026

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). pp. 291-306. ISSN 0926-6003

Williams, H. Paul (1997) Integer programming and pricing revisited. IMA journal of mathematics applied in business & industry, 8 (3). pp. 203-213. ISSN 0953-0061

1996

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 & operations research, 23 (9). pp. 845-856. ISSN 0305-0548

Williams, H. Paul (1996) Duality in mathematics and linear and integer programming. Journal of optimization theory and applications, 90 (2). pp. 257-278. ISSN 0022-3239

Williams, H. Paul (1996) Obituary of Professor Steven Vajda. Times Newspapers Ltd.

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 University Press, Oxford, UK, pp. 249-281. ISBN 9780198538561

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

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

1995

Williams, H. Paul (1995) Logic applied to integer programming and integer programming applied to logic. European journal of operational research, 81 (3). pp. 605-616. ISSN 0377-2217

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

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, September 19 - 22, 1995.. Springer, Berlin, Germany, pp. 36-52. ISBN 9783540602996

Williams, H. Paul (1995) The Cornish caveman mathematician. The mathematical intelligencer, 17 (1). ISSN 0343-6993

1994

Williams, H. Paul (1994) An alternative explanation of disjunctive formulations. European journal of operational research, 72 (1). pp. 200-203. ISSN 0377-2217

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, pp. 25-32.

1993

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

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

1992

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

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

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). pp. 325-334. ISSN 0377-2217

1990

Williams, H. Paul (1990) How important are models to operational research? IMA journal of mathematics applied in business & industry, 2 pp. 189-195. ISSN 0953-0061

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

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

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

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, pp. 41-56.

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) Evolution, games theory and polyhedra. Journal of mathematical biology, 25 (4). pp. 393-409. ISSN 0303-6812

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) Recent developments in mathematical programming. In: Thomas, L. C., (ed.) Recent developments in operational research. Pergamon Press, Oxford, UK, pp. 33-44.

Williams, H. Paul (1987) Linear and integer programming applied to the propositional calculus. Systems research and information science, 2 pp. 81-100. ISSN 0882-3014

1986

Williams, H. Paul (1986) Fourier's method of linear programming and its dual. The American mathematical monthly, 93 (9). pp. 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). pp. 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, pp. 25-35.

1984

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

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). pp. 705-709. ISSN 0160-5682

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

1983

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

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. Springer-Verlag, Berlin, Germany, pp. 87-107. ISBN 0387115528

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

1982

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

1981

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

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

1980

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

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

1979

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 23-27, 1976. North Holland, Amsterdam, The Netherlands, pp. 567-586. ISBN 0444850333

Williams, H. Paul (1979) Four examples of artless modelling. Omega: the international journal of management science, 7 (2). pp. 163-165. ISSN 0305-0483

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). pp. 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 pp. 18-20. ISSN 0950-5628

1976

Williams, H. Paul (1976) Fourier-Motzkin elimination extension to integer programming problems. Journal of combinatorial theory, series A, 21 (1). pp. 118-123. ISSN 0097-3165

1975

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). pp. 54-83. ISSN 0025-5610

Williams, H. Paul (1975) The formulation of mathematical programming models. Omega: the international journal of management science, 3 (5). pp. 551-556. ISSN 0305-0483

1974

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

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

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

1973

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

Williams, H. Paul (1973) Simplifying linear programming problems. 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). pp. 77-89. ISSN 0029-4527

1968

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

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