Professor Paul Williams was born in Cornwall, and still spends a lot of time there. After attending the local Grammar school he studied Mathematics at Cambridge followed by a PhD in Mathematical Logic at Leicester University under the late Professor R.L.Goodstein.
His first job was with IBM where he worked on Linear and Integer Programming and gained a lasting interest in these subjects. He realised that a lot about them, particularly methods of modelling, were not properly understood and he did a lot of the early work on developing modelling as a subject, as well as making contributions to the algorithmic and computational side. He then held a number of academic jobs, as a lecturer at Sussex University, Professor at Edinburgh, Southampton (Dean and Head of Department) Universities before joining LSE where he has served as Head of the Department of Operational Research (currently the Management Science Group).
Most of his research has been in Linear and Integer Programming, where he has further developed modelling methods, new algorithms and economic interpretations as well as many practical implementations. He has paid extended academic visits to many universities in the USA, Australia , Europe, Hong Kong and Singapore. For many years he also held a visiting Chair at Universiy College Dublin.
In retirement he continues to be very active in research, in particular working on the development of an economically meaningful dual of an Integer Programme and using Mathematical Programming to clarify the dichotomy between equity and utilitarianism in resource allocation. He also consults in helping organisations to formulate and solve their problems as mathematical models.
For a full list of Professor Williams' publications on LSE Research Online, please click here, otherwise see below.
Publications:
Books
Williams, H. Paul (2016) The Red River. Godrevy Press. ISBN 9781533177964
Williams, H. Paul (2013) Model building in mathematical programming. John Wiley & Sons. ISBN 9781118443330
Williams, H. Paul (2009) Logic and integer programming. International series in operations research & management science. Springer Berlin / Heidelberg, London, UK. ISBN 9780387922799
Williams, H. Paul (1999) Model building in mathematical programming. John Wiley & Sons, Chichester. ISBN 9780471997887
Williams, H. Paul (1993) Model solving in mathematical programming. John Wiley & Sons, Chichester, UK. ISBN 9780471935810
Articles
Williams, H. Paul (2017) The dependency diagram of a mixed integer linear programme. Journal of the Operational Research Society, 68 (7). pp. 829-833. ISSN 0160-5682
Williams, H. Paul and Hooker, J. N. (2016) Integer programming as projection. Discrete Optimization, 22 (B). pp. 291-311. ISSN 1572-5286
Williams, H. Paul (2015) The dependency diagram of a linear programme. Journal of the Operational Research Society, 67 (3). pp. 450-456. ISSN 0160-5682
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
Williams, H. Paul (2011) The problem with integer programming. IMA Journal of Management Mathematics, 22 (3). pp. 213-230. ISSN 1471-6798
Williams, H. Paul (2007) Stanley Skewes and the Skewes number. Journal of the Royal Institution of Cornwall. pp. 70-75. ISSN 0968-5396
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
Williams, H. Paul (2005) George B Dantzig, 1914–2005. Journal of the Operational Research Society, 56 (12). pp. 1349-1350. ISSN 0160-5682
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
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). pp. 96-103. ISSN 1091-9856
Williams, H. Paul (2000) The dual of a logical linear programme. Journal of Global Optimization, 18 (2). pp. 129-141. ISSN 0925-5001
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
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 (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 (1997) Integer programming and pricing revisited. IMA Journal of Mathematics Applied in Business and Industry, 8 (3). pp. 203-213. ISSN 0953-0061
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, 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
Brailsford, S. C., Hubbard, P. M., Smith, B. M. and Williams, H. Paul (1996) Organizing a social event: a difficult problem of combinatorial optimization. Computers and 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) 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
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
Williams, H. Paul (1995) The Cornish caveman mathematician. 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). pp. 559-574. ISSN 0305-0041
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 (1993) How long is the coast of Cornwall? Mathematics Review, 3. pp. 2-5. ISSN 0957-1280
Williams, H. Paul (1992) Solving systems of linear inequalities. Mathematical Medley, 20 (1). pp. 59-67. ISSN 0217-2976
Williams, H. Paul (1992) The elimination of integer variables. Journal of the Operational Research Society, 43 (5). pp. 387-393. ISSN 0160-5682
Baston, V. J. D., 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
Williams, H. Paul (1990) How important are models to operational research? IMA Journal of Mathematics Applied in Business and 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
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 (1987) Evolution, games theory and polyhedra. Journal of Mathematical Biology, 25 (4). pp. 393-409. ISSN 0303-6812
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
Williams, H. Paul (1986) Fourier's method of linear programming and its dual. American Mathematical Monthly, 93 (9). pp. 681-695. ISSN 0002-9890
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
Williams, H. Paul (1984) A duality theorem for linear congruences. Discrete Applied Mathematics, 7 (1). pp. 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). pp. 401-407. ISSN 0305-0548
Cheshire, M., 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 (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 (1982) Models with network duals. Journal of the Operational Research Society, 33 (2). pp. 161-169. ISSN 0160-5682
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
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 (1979) Four examples of artless modelling. Omega, 7 (2). pp. 163-165. ISSN 0305-0483
Williams, H. Paul (1978) The reformulation of two mixed integer programming problems. Mathematical Programming, 14 (1). pp. 325-331. ISSN 0025-5610
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
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
Brearley, A. L., 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, 3 (5). pp. 551-556. ISSN 0305-0483
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 0163-5786
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
Williams, H. Paul (1968) The synthesis of logical nets consisting of NOR units. Computer Journal, 11 (2). pp. 173-176. ISSN 0010-4620
Book Sections
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). Springer Berlin / Heidelberg, Berlin, Germany, pp. 93-108. ISBN 9783540366256
Williams, H. Paul (1998) Recent developments in combinatorial optimisation. In: OR Keynote Papers 1998. Operational Research Society (Great Britain), Birmingham, UK.
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, pp. 249-281. ISBN 9780198538561
Smith, Barbara M., Brailsford, Sally C., 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 / Heidelberg, Berlin, Germany, pp. 36-52. ISBN 9783540602996
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. John Wiley & Sons, Chichester, UK, pp. 25-32.
Williams, H. Paul (1989) Mathematical programming modelling. In: Shahani, Arjan K. and Stainton, Roy, (eds.) Tutorial Papers in Operational Research. Operational Research Society (Great Britain), Birmingham, UK, pp. 41-56.
Williams, H. Paul (1987) Recent developments in mathematical programming. In: Thomas, L. C., (ed.) Recent Developments in Operational Research. Pergamon Press (Oxford, England), Oxford, UK, pp. 33-44.
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 / Heidelberg, Berlin, Germany, pp. 25-35.
Williams, H. Paul (1983) A reduction procedure for linear and integer programming models. In: Karwan, M. H., Lotfi, V., 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 Berlin / Heidelberg, Berlin, Germany, pp. 87-107. ISBN 0387115528
Williams, H. Paul (1980) Integer programming. In: Lederman, Walter and Vajda, Steven, (eds.) Handbook of Applicable Mathematics: Vol. 1, Algebra. John Wiley & Sons, Chichester, UK, pp. 339-390. ISBN 0471277045
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, pp. 567-586. ISBN 0444850333
Williams, H. Paul (1974) Experiments in the formulation of integer programming problems. In: Balinski, M. L., (ed.) Approaches to Integer Programming. Mathematical Programming Studies. Springer Berlin / Heidelberg, 180 - 197.
Monographs
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.
Williams, H. Paul (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 linearprogramme. Working paper (LSEOR 13.138). London School of Economics and Political Science, London, UK.
Williams, H. Paul (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.
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.
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
Appa, Gautam, 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 (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.
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 (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.
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 (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 (1973) Decision procedures in formal logic and mathematical programming algorithms. Research report (73-5). Operational Research Group, University of Sussex, Brighton, UK.
Williams, H. Paul (1973) Simplifying linear programming problems. Research report (73-2). Operational Research Group, University of Sussex, Brighton, UK.
Williams, H. Paul (1970) An algorithm for the solution of linear programming problems. IBM working papers. IBM, UK.
Conference or Workshop Items
Williams, H. Paul (1998) Tutorials and recent reviews. In: 16th EURO Conference on Operational Analysis, 1998-07-01, Brussels, Belgium, BEL. (Submitted)
Williams, H. Paul (1986) Optimisation is best. In: Inaugural lecture as Professor of Operational Research, 1986-11-01, Southampton, United Kingdom, GBR. (Submitted)
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, 1978-01-01, Edinburgh, United Kingdom, GBR. (Submitted)
Online Resources
Williams, H. Paul (1996) Obituary of Professor Steven Vajda. Times (03 Jan 1996). Website.
Other
Williams, H. Paul (1976) Linear programming II: Open University unit for numerical computation course M351. Open University.