An algorithm for approximating the Pareto set of the multiobjective set covering problem

The multiobjective set covering problem (MOSCP), a challenging combinatorial optimization problem, has received limited attention in the literature. This paper presents a heuristic algorithm to approximate the Pareto set of the MOSCP. The proposed algorithm applies a local branching approach on a tree structure and is enhanced with a node exploration strategy specially developed for the MOSCP. The main idea is to partition the search region into smaller subregions based on the neighbors of a reference solution and then to explore each subregion for the Pareto points of the MOSCP. Numerical experiments for instances with two, three and four objectives set covering problems are reported. Results on a performance comparison with benchmark algorithms from the literature are also included and show that the new algorithm is competitive and performs best on some instances.

[1]  Douglas R. Shier,et al.  A Hierarchical Approach to Designing Compact Ecological Reserve Systems , 2014, Environmental Modeling & Assessment.

[2]  Yi-Hsin Liu,et al.  A heuristic algorithm for the multi-criteria set-covering problems , 1993 .

[3]  Dirk Thierens,et al.  The balance between proximity and diversity in multiobjective evolutionary algorithms , 2003, IEEE Trans. Evol. Comput..

[4]  Matteo Fischetti,et al.  A Heuristic Method for the Set Covering Problem , 1999, Oper. Res..

[5]  Andrzej Jaszkiewicz,et al.  On the performance of multiple-objective genetic local search on the 0/1 knapsack problem - a comparative experiment , 2002, IEEE Trans. Evol. Comput..

[6]  R. S. Laundy,et al.  Multiple Criteria Optimisation: Theory, Computation and Application , 1989 .

[7]  Jacques Teghem,et al.  Very Large-Scale Neighborhood Search for Solving Multiobjective Combinatorial Optimization Problems , 2011, EMO.

[8]  Christian Prins,et al.  Two-phase method and Lagrangian relaxation to solve the Bi-Objective Set Covering Problem , 2006, Ann. Oper. Res..

[9]  Richard W. Hamming,et al.  Error detecting and error correcting codes , 1950 .

[10]  Mark D. McDonnell,et al.  Mathematical Methods for Spatially Cohesive Reserve Design , 2002 .

[11]  Andrzej Jaszkiewicz,et al.  A Comparative Study of Multiple-Objective Metaheuristics on the Bi-Objective Set Covering Problem and the Pareto Memetic Algorithm , 2004, Ann. Oper. Res..

[12]  E. L. Ulungu,et al.  Multi‐objective combinatorial optimization problems: A survey , 1994 .

[13]  Eckart Zitzler,et al.  Evolutionary algorithms for multiobjective optimization: methods and applications , 1999 .

[14]  J. Hooker,et al.  Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction , 2000 .

[15]  Claude Le Pape,et al.  Exploring relaxation induced neighborhoods to improve MIP solutions , 2005, Math. Program..

[16]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[17]  Nysret Musliu,et al.  Local Search Algorithm for Unicost Set Covering Problem , 2006, IEA/AIE.

[18]  Andrzej Jaszkiewicz,et al.  Do multiple-objective metaheuristics deliver on their promises? A computational experiment on the set-covering problem , 2003, IEEE Trans. Evol. Comput..

[19]  M. Ehrgott.  Approximation algorithms for combinatorial multicriteria optimization problems , 2000 .

[20]  George Mavrotas,et al.  Generation of the exact Pareto set in Multi-Objective Traveling Salesman and Set Covering Problems , 2014, Appl. Math. Comput..

[21]  Banu Soylu,et al.  Heuristic approaches for biobjective mixed 0-1 integer linear programming problems , 2015, Eur. J. Oper. Res..

[22]  Mark S. Daskin,et al.  A Hierarchical Objective Set Covering Model for Emergency Medical Service Vehicle Deployment , 1981 .

[23]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[24]  H. W. Corley,et al.  An existence result for maximizations with respect to cones , 1980 .

[25]  M. Hansen,et al.  Evaluating the quality of approximations to the non-dominated set , 1998 .

[26]  Xavier Gandibleux,et al.  A survey and annotated bibliography of multiobjective combinatorial optimization , 2000, OR Spectr..

[27]  Pierre Hansen,et al.  Variable neighborhood search and local branching , 2004, Comput. Oper. Res..

[28]  S. R. Arora,et al.  Linearization approach to multi-objective quadratic set covering problem , 1998 .

[29]  Andrzej J. Osiadacz.  Multiple criteria optimization; theory, computation, and application, Ralph E. Steuer, Wiley Series in Probability and Mathematical Statistics - Applied, Wiley, 1986, No. of pages 546, Price f5 1.40, $77.10 , 1989 .

[30]  Daniel Tuyttens,et al.  Variable and large neighborhood search to solve the multiobjective set covering problem , 2014, Journal of Heuristics.