Research Article Application of Heuristic and Metaheuristic Algorithms in Solving Constrained Weber Problem with Feasible Region Bounded by Arcs Igor StojanoviT, 1 Ivona BrajeviT, 2 Predrag S. StanimiroviT, 2 Lev A. Kazakovtsev, 3 and Zoran Zdravev 1 1 Faculty of Computer Science, Goce Delˇ cev University, Goce Delˇ cev 89, 2000 ˇ Stip, Macedonia 2 Department of Mathematics and Informatics, Faculty of Science and Mathematics, University of Niˇ s, Viˇ segradska 33, 18000 Niˇ s, Serbia 3 Department of Systems Analysis and Operations Research, Reshetnev University, Prosp. Krasnoyarskiy Rabochiy 31, Krasnoyarsk 660037, Russia CorrespondenceshouldbeaddressedtoPredragS.Stanimirovi´ c;pecko@pmf.ni.ac.rs Received 26 February 2017; Accepted 15 May 2017; Published 14 June 2017 AcademicEditor:DomenicoQuagliarella Copyright©2017IgorStojanovi´ cetal.TisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited. Tecontinuousplanarfacilitylocationproblemwiththeconnectedregionoffeasiblesolutionsboundedbyarcsisaparticular caseoftheconstrainedWeberproblem.Tisproblemisacontinuousoptimizationproblemwhichhasanonconvexfeasibleset ofconstraints.Tispapersuggestsappropriatemodifcationsoffourmetaheuristicalgorithmswhicharedefnedwiththeaimof solvingthistypeofnonconvexoptimizationproblems.Also,acomparisonofthesealgorithmstoeachotheraswellastotheheuristic algorithmispresented.Teartifcialbeecolonyalgorithm,frefyalgorithm,andtheirrecentlyproposedimprovedversions for constrained optimization are appropriately modifed and applied to the case study. Te heuristic algorithm based on modifed Weiszfeldprocedureisalsoimplementedforthepurposeofcomparisonwiththemetaheuristicapproaches.Obtainednumerical resultsshowthatmetaheuristicalgorithmscanbesuccessfullyappliedtosolvetheinstancesofthisproblemofupto500constraints. Amongthesefouralgorithms,theimprovedversionofartifcialbeealgorithmisthemostefcientwithrespecttothequalityofthe solution,robustness,andthecomputationalefciency. 1. Introduction TeWeberproblemisoneofthemoststudiedproblemsin location theory [1–3]. Tis optimization problem searches foranoptimalfacilitylocation  ∗ ∈ R 2 onaplane,which satisfes  ∗ = argmin ∈R 2 ()= argmin ∈R 2  ∑ =1         −     . (1) In(1),itisassumedthat   ∈ R 2 , ∈{1,...,} areknown demandpoints,   ∈ R and   ≥0 areweightcoefcients, and ‖⋅‖ isamatrixnorm,usedasthedistancefunction. Te basic Weber problem is stated with the Euclidean normunderlyingthedefnitionofthedistancefunction.Also, manyothertypesofdistanceshavebeenusedinthefacility location problems [3–5]. In general, a lot of extensions and modifcations of the Weber location problem are known. Detailedreviewsoftheseproblemscanbefoundin[3,6]. TemostpopularmethodforsolvingtheWeberproblem withEuclideandistancesisgivenbyaone-pointiterativepro- cedurewhichwasfrstproposedbyWeiszfeld[7].Later,Vardi and Zhang developed a diferent extension of Weiszfeld’s algorithm [8], while Szegedy partially extended Weiszfeld’s algorithmtoamoregeneralproblem[9].Inparticular,some variants of the continuous Weber problem represent non- convex optimization problems which are hard to be solved exactly [10]. A nonconvex optimization problem may have multiplefeasibleregionsandmultiplelocallyoptimalpoints Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 8306732, 13 pages https://doi.org/10.1155/2017/8306732