Abstract
In this paper, we propose a heuristic algorithm to solve a new variant of the partial set covering problem. In this variant, each element \(e_i\) has a gain \(g_i\) (i.e., a positive profit), each set \(s_j\) has a cost \(c_j\) (i.e., a negative profit), and each set \(s_j\) is part of a unique group \(G_k\) that has a fixed cost \(f_k\) (i.e., a negative profit). The objective is to maximize profit and it is not necessary to cover all of the elements. We present an industrial application of the model and propose a hybrid heuristic algorithm to solve it; the proposed algorithm is an iterated-local-search algorithm that uses two levels of perturbations and a tabu-search heuristic. Whereas the first level of perturbation diversifies the search around the current local optimum, the second level of perturbation performs long jumps in the search space to help escape from local optima with large basins of attraction. The proposed algorithm is evaluated on thirty real-world problems and compared to a memetic algorithm. Computational results show that most of the solutions found by ITS are either optimal or very close to optimality.
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Athanassopoulos, S., Caragiannis, I., Kaklamanis, C.: Analysis of approximation algorithms for k-set cover using factor-revealing linear programs. Theory Comput. Syst. 45(3), 555–576 (2009)
Azim, G.A., Ben Othman, M.: Hybrid iterated local search algorithm for solving multiple sequences alignment problem. Far East J. Exp. Theor. Intell. 5(1–2), 1–17 (2010)
Balas, E.: A class of location, distribution and scheduling problems: modeling and solution methods. Proceedings of the Chinese-US Symposium on Systems Analysis, pp. 323–346. Wiley, New York (1983)
Balas, E., Vazacopoulos, A.: Guided local search with shifting bottleneck for job shop scheduling. Manag. Sci. 44(2), 262–275 (1998)
Battiti, R., Protasi, M.: Reactive search, a history-sensitive heuristic for max-sat. J. Exp. Algorithmics (JEA) 2(2), 1–31 (1997)
Beasley, J., Chu, P.: A genetic algorithm for the set covering problem. Eur. J. Oper. Res. 94(2), 392–404 (1996)
Caprara, A., Fischetti, M., Toth, P.: A heuristic method for the set covering problem. Oper. Res. 47(5), 730–743 (1999)
Chen, P., Qu, Y., Huang, H., Dong, X.: A new hybrid iterated local search for the open vehicle routing problem. In: Computational Intelligence and Industrial Application, 2008. PACIIA’08. Pacific-Asia Workshop on, IEEE vol. 1, pp. 891–895 (2008)
Congram, R., Potts, C., Van De Velde, S.: An iterated dynasearch algorithm for the single-machine total weighted tardiness scheduling problem. Inf. J. Comput. 14(1), 52–67 (2002)
Farahani, R.Z., Hekmatfar, M.: Facility location. Springer, Dordrecht, Heidelberg, London, New York, Chap 7: 3 (2009)
Garey, M., Johnson, D.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman, Oxford (1979)
Glover, F., Taillard, E.: A user’s guide to tabu search. Annals Oper. Res. 41(1), 1–28 (1993)
Glover, F., et al.: Tabu search-part I. ORSA J. Comput. 1(3), 190–206 (1989)
Johnson, D.: Local optimization and the traveling salesman problem. In: Paterson, M. (ed.) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 443, pp. 446–461. Springer, Berlin (1990)
Johnson, D., McGeoch, L.: The traveling salesman problem: a case study in local optimization. In: Aarts, E.H.L., Lenstra, J.K. (eds.) Local Search in Combinatorial Optimization, vol. 1, pp. 215–310. Wiley, Chichester (1997)
Katayama K, Narihisa H, et al. Iterated local search approach using genetic transformation to the traveling salesman problem. In: Proceedings of GECCO 99, vol. 1, pp. 321–328 (1999)
Khuller, S., Moss, A., Naor, J.S.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)
Könemann, J., Parekh, O., Segev, D.: A unified approach to approximating partial covering problems. In: Algorithms-ESA 2006, pp. 468–479. Springer, Berlin (2006)
Kreipl, S.: A large step random walk for minimizing total weighted tardiness in a job shop. J. Sched. 3(3), 125–138 (2000)
Lan, G., DePuy, G., Whitehouse, G.: An effective and simple heuristic for the set covering problem. Eur. J. Oper. Res. 176(3), 1387–1403 (2007)
Lourenço HR, Zwijnenburg M (1996) Combining the large-step optimization with tabu-search: Application to the job-shop scheduling problem. In: Meta-Heuristics, pp 219–236. Springer, Berlin (1996).
Lourenço, H.R., Martin, O.C., Stützle, T.: Iterated local search. In: Handbook of Metaheuristics, vol. 57 of International Series in Operations Research and Management Science, pp. 321–353. Kluwer Academic Publishers, Norwell (2002).
Martin, O., Otto, S.: Partitioning of unstructured meshes for load balancing. Concurr. Pract. Experience 7(4), 303–314 (1995)
Martin, O., Otto, S.: Combining simulated annealing with local search heuristics. Annals Oper. Res. 63(1), 57–75 (1996)
Martin, O., Otto, S., Felten, E.: Large-step Markov chains for the traveling salesman problem. Technical report, Oregon Graduate Institute of Science and Technology. Department of Computer Science and Engineering, (1991)
Misevičius, A.: Using iterated tabu search for the travelling salesman problem. Inf. Technol. Control 32(3), 29–40 (2004)
Misevičius, A., Lenkevicius, A., Rubliauskas, D.: Iterated tabu search: an improvement to standard tabu search. Inf. Technol. Control 35(3), 187–197 (2006)
Palubeckis, G.: Iterated tabu search for the maximum diversity problem. Appl. Math. Comput. 189(1), 371–383 (2007)
Pan, G.: Geostatistical design of infill drilling programs. Society of Mining Engineers of AIME 142 (1995)
Radcliffe, N.J., Surry, P.D.: Formal memetic algorithms. In: Evolutionary Computing, pp. 1–16. Springer, Berlin (1994)
Smyth, K., Hoos, H.H., Stützle, T.: Iterated robust tabu search for max-sat. In: Advances in Artificial Intelligence, Lecture Notes in Computer Science vol. 2671, pp. 129–144. Springer, Berlin (2003)
Stützle, T.: Applying iterated local search to the permutation flow shop problem. Technical Report, FG Intellektik, TU Darmstadt (1998)
Subramanian, M.: A hybrid heuristic, based on iterated local search and genius, for the vehicle routing problem with simultaneous pickup and delivery. Int. J. Logist. Syst. Manag. 10(2), 142–157 (2011)
Zäpfel, G., Braune, R., Bögl, M.: Metaheuristic Search Concepts: A Tutorial with Applications to Production and Logistics. Springer, Berlin (2010)
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Bilal, N., Galinier, P. & Guibault, F. An iterated-tabu-search heuristic for a variant of the partial set covering problem. J Heuristics 20, 143–164 (2014). https://doi.org/10.1007/s10732-013-9235-9
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DOI: https://doi.org/10.1007/s10732-013-9235-9