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Department

Position held

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.

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

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

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

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 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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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). 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

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

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

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

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

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

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

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)
**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

Williams, H. Paul
(1986)
**Fourier's method of linear programming and its dual**
*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

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.

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

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

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

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

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

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

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

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)
**Linear programming II: Open University unit for numerical computation course M351**

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

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

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

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.

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

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

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

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