The p-Dispersion Problem with Distance Constraints
In the (maxmin) p-dispersion problem we seek to locate a set of facilities in an area so that the minimum distance between any pair of facilities is maximized. We study a variant of this problem where there exist constraints specifying the minimum allowed distances between the facilities. This type of problem, which we call PDDP, has not received much attention within the literature on location and dispersion problems, despite its relevance to real scenarios. We propose both ILP and CP methods to solve the PDDP. Regarding ILP, we give two formulations derived from a classic and a state-of-the-art model for p-dispersion, respectively. Regarding CP, we first give a generic model that can be implemented within any standard CP solver, and we then propose a specialized heuristic Branch&Bound method. Experiments demonstrate that the ILP formulations are more efficient than the CP model, as the latter is unable to prove optimality in reasonable time, except for small problems, and is usually slower in finding solutions of the same quality than the ILP models. However, although the ILP approach displays good performance on small to medium size problems, it cannot efficiently handle larger ones. The heuristic CP-based method can be very efficient on larger problems and is able to quickly discover solutions to problems that are very hard for an ILP solver.
Facility location
distance constraints
optimization
Theory of computation~Constraint and logic programming
30:1-30:18
Regular Paper
Nikolaos
Ploskas
Nikolaos Ploskas
University of Western Macedonia, Kozani, Greece
https://orcid.org/0000-0001-5876-9945
Kostas
Stergiou
Kostas Stergiou
University of Western Macedonia, Kozani, Greece
https://orcid.org/0000-0002-5702-9096
Dimosthenis C.
Tsouros
Dimosthenis C. Tsouros
KU Leuven, Belgium
https://orcid.org/0000-0002-3040-0959
10.4230/LIPIcs.CP.2023.30
T. Argo and E. Sandstrom. Separation distances in NFPA codes and standards (tech. rep.). Fire Protection Research Foundation. 2014.
O. Berman and R. Huang. The minimum weighted covering location problem with distance constraints. Computers and Operations Research, 35(12):356-372, 2008.
F. Boussemart, F. Heremy, C. Lecoutre, and L. Sais. Boosting systematic search by weighting constraints. In Proceedings of ECAI'04, pages 482-486, 2004.
J. Brimberg and H. Juel. A minisum model with forbidden regions for locating a semi-desirable facility in the plane. Location Science, 6(1):109-120, 1998.
H. Cambazard, D. Mehta, B. O'Sullivan, and L. Quesada. A computational geometry-based local search algorithm for planar location problems. In Proceedings of the 9th International Conference on the Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR 2012), pages 97-112, 2012.
S. Chaudhry, T. McCormick, and I. D. Moon. Locating independent facilities with maximum weight: Greedy heuristics. International Journal of Management Science, 14(5):383-389, 1986.
R. L. Church and M. E. Meadows. Results of a new approach to solving the p-median problem with maximum distance constraints. Geographical Analysis, 9(4):364-378, 1977.
V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979.
W. Comley. The location of ambivalent facilities: Use of a quadratic zero-one programming algorithm. Applied Mathematical Modeling, 19(1):26-29, 1995.
Z. Dai, K. Xu, and M. Ornik. Repulsion-based p-dispersion with distance constraints in non-convex polygons. Annals of Operations Research, 307:75-91, 2021.
Choco development team. An Open-Source java library for constraint programming. https://choco-solver.org/.
OR-Tools development team. OR-Tools, CP-SAT solver. https://developers.google.com/optimization/cp/cp_solver.
T. Drezner, Z. Drezner, and A. Schöbel. The weber obnoxious facility location model: A big arc small arc approach. Computers and Operations Research, 98:240-250, 2018.
Z. Drezner, P. Kalczynski, and S. Salhi. The planar multiple obnoxious facilities location problem: A Voronoi based heuristic. Omega, 87:105-116, 2019.
E. Erkut. The discrete p-dispersion problem. European Journal of Operational Research, 46(1):48-60, 1990.
E. Erkut and S. Neuman. Analytical models for locating undesirable facilities. European Journal of Operational Research, 40(3):275-291, 1989.
M. M. Fazel-Zarandi and J. C. Beck. Solving a location-allocation problem with logic-based benders' decomposition. In Proceedings of the 15th International Conference on Principles and Practice of Constraint Programming (CP 2009), pages 344-351, 2009.
J. B. Ghosh. Computational aspects of the maximum diversity problem. Operations Research Letters, 19(4):175-181, 1996.
C. Gomes and M. Sellmann. Streamlined constraint reasoning. In Proceedings of the International Conference on Principles and Practice of Constraint Programming (CP 2004), pages 274-289, 2004.
T. Guns. Increasing modeling language convenience with a universal n-dimensional array, CPpy as python-embedded example. In Proceedings of the 18th workshop on Constraint Modelling and Reformulation, 2019.
LLC Gurobi Optimization. Gurobi optimizer reference manual. , 2023. URL: https://www.gurobi.com.
https://www.gurobi.com
N. Isoart and J.-C. Régin. A k-Opt Based Constraint for the TSP. In Proceedings of the 27th International Conference on Principles and Practice of Constraint Programming (CP 2021), 2021.
B. M. Khumawala. An efficient algorithm for the p-median problem with maximum distance constraints. Geographical Analysis, 5(4):309-321, 1973.
R. K. Kincaid. Good solutions to discrete noxious location problems via meta-heuristics. Annals of Operations Research, 40(1):265-281, 1992.
M. J. Kuby. Programming models for facility dispersion: The p-dispersion and maxisum dispersion problems. Mathematical and Computer Modelling, 10(10):792, 1988.
Mikael Zayenz Lagerkvist and Magnus Rattfeldt. Half-checking propagators. In Proceedings of the 19th workshop on Constraint Modelling and Reformulation, 2020.
A. Maier and H.W. Hamacher. Complexity results on planar multifacility location problems with forbidden regions. Mathematical Methods of Operations Research, 89:433-484, 2019.
R. Marti, A. Martinez-Gavara, S. Perez-Pelo, and J. Sanchez-Oro. A review on discrete diversity and dispersion maximization from an OR perspective. European Journal of Operational Research, 299(3):795-813, 2022.
I. D. Moon and S. Chaudhry. An analysis of network location problems with distance constraints. Management Science, 30(3):290-307, 1984.
I. D. Moon and L. Papayanopoulos. Minimax location of two facilities with minimum separation: Interactive graphical solutions. Journal of the Operations Research Society, 42:685-694, 1991.
C.S. Orloff. A theoretical model of net accessibility in public facility location. Geographical Analysis, 9:244-256, 1977.
M. G. Resende, R. Marti, M. Gallego, and A. Duarte. Grasp and path relinking for the max-min diversity problem. Computers and Operations Research, 37(3):498-508, 2010.
B. Saboonchi, P. Hansen, and S. Perron. MaxMinMin p-dispersion problem: A variable neighborhood search approach. Computer & Operations Research, 52:251-259, 2014.
D. Sayah and S. Irnich. A new compact formulation for the discrete p-dispersion problem. European Journal of Operational Research, 256(1):62-67, 2017.
F. Sayyady and Y Fathi. An integer programming approach for solving the p-dispersion problem. European Journal of Operational Research, 253(1):216-225, 2016.
D. R. Shier. A min-max theorem for p-center problems on a tree. Transportation Science, 11(3):243-252, 1977.
S.Y.D. Sorkhabi, D.A. Romero, J.C. Beck, and C. Amon. Constrained multi-objective wind farm layout optimization: Novel constraint handling approach based on constraint programming. Renewable Energy, 126(C):341-353, 2018.
B.C. Tansel, R.L. Francis, T.J. Lowe, and M.L. Chen. Duality and distance constraints for the nonlinear p-center problem and covering problem on a tree network. Operations Research, 30(4):725-744, 1982.
S.B. Welch and S. Salhi. The obnoxious p facility network location problem with facility interaction. European Journal of Operations Research, 102:302-319, 1997.
49 C.F.R. §175.701. Separation distance requirements for packages containing class 7 (radioactive) materials in passenger-carrying aircraft. Title 49 Code of Federal Regulations, Part 175. 2021.
29 C.F.R. §1910.157. Portable fire extinguishers. Title 29 Code of Federal Regulations, Part 157. 2021.
Nikolaos Ploskas, Kostas Stergiou, and Dimosthenis C. Tsouros
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode