In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article

Authors

1 ‎Shahrood University‎ of Technology

2 Shahrood University of Technology

Abstract

Location theory is an interstice field of optimization and operations research‎. ‎In the classic location models‎, ‎the goal is finding the location of one or more facilities such that some criteria such as transportation cost‎, ‎the sum of distances passed by clients‎, ‎total service time, and cost of servicing are minimized‎. ‎The goal Weber location problem is a special case of location models that have been considered recently by some researchers‎. ‎In this problem, the ideal is locating the facility in the distance $r_i$‎, ‎from the $i$-th client‎. ‎However‎, ‎in most instances‎, ‎the solution to this problem doesn't exist‎. ‎Therefore‎, ‎the minimizing sum of errors is considered‎. ‎In the previous versions of the goal location problem, the penalty functions have been considered by some symmetric functions such as square and absolute errors of distances between clients and ideal point‎. ‎In this paper‎, ‎we consider the asymmetric linex function as the error function‎. ‎We consider the case that the distances are measured by $L_p$ norm‎. ‎Some iterative methods are used to solve the problem and the results are compared with some previously examined methods.

Keywords

bibitem{A18}‎
‎Andrei N‎. ‎(2018)‎. ‎``An adaptive scaled BFGS method for unconstrained optimization"‎, ‎Numerical Algorithms‎, ‎77‎, ‎413--432‎.
 
‎bibitem{A2-18} Andrei N‎. ‎(2018)‎. ‎``A double parameter scaled BFGS method for unconstrained optimization"‎, ‎Journal of Computational and Applied Mathematics‎, ‎332‎, ‎26--44‎.
 
‎bibitem{B16} Babaie-Kafaki S‎. ‎(2016)‎. ‎``A modified scaling parameter for the memoryless BFGS updating formula"‎, ‎Numerical Algorithms‎, ‎72‎, ‎425--433‎.
 
‎bibitem{BA19} Babaie-Kafaki S.‎, ‎Aminifard Z‎. ‎(2019)‎. ‎``Two–parameter scaled memoryless BFGS methods with a nonmonotone choice for the initial step length"‎, ‎Numerical Algorithms‎, ‎82‎, ‎1345--1357‎.
 
‎bibitem{B95} Brimberg J‎. ‎(1995)‎. ‎``The Fermat-Weber location problem revisited"‎, ‎Mathematical Programming‎, ‎71‎, ‎71--76‎.
 
‎bibitem{C11} Chen R‎. ‎(2011)‎. ‎``Noniterative solution of some Fermat-Weber location problems"‎, ‎Advances in Operations Research‎, ‎Volume 2011‎, ‎Article ID 379505‎, ‎10 pages‎.
 
‎bibitem{DM02} Dolan E. D‎. ‎and Mor´e J. J‎. ‎(2002)‎. ‎``Benchmarking optimization software with performance profiles"‎, ‎Math‎. ‎Program‎, ‎91‎, ‎201--213‎.
 
‎bibitem{d08} Drezner Z‎. ‎(2008)‎. ‎``On convergence of the generalized Weiszfeld algorithm"‎, ‎Ann Oper Res.‎, ‎167‎, ‎327--336‎.
 
‎bibitem{F15} Fathali J‎. ‎(2015)‎. ‎``Backup multifacility location problem with Lp norm"‎,
‎OPSEARCH‎, ‎52‎, ‎382--391‎.
 
‎bibitem{FJ17} Fathali J.‎, ‎Jamalian A‎. ‎(2017)‎. ‎``Efficient methods for goal square Weber location problem"‎, ‎Iranian Journal of Numerical Analysis and Optimization‎, ‎7‎, ‎65--82‎.
‎bibitem{FZN09} Fathali J.‎, ‎Zaferanieh M.‎, ‎Nezakati A‎. ‎(2009)‎. ‎``A BSSS algorithm for the location problem with minimum square error"‎, ‎Advances In Operations Rresearch‎, ‎Volume 2009‎ , ‎10 pages‎.
 
‎bibitem{JF09} Jamalian A.‎, ‎Fathali J‎. ‎(2009)‎. ‎``Linear programming for the location problem with minimum absolute error"‎, ‎World Applied Sciences Journal‎, ‎7‎, ‎1423--1427‎.
 
‎bibitem{NW06} Nocedal J.‎, ‎Wright S‎. ‎J‎. ‎(2006)‎. ‎``Numerical Optimization''‎, ‎Springer‎.
 
‎bibitem{OL74} Oren S. S.‎, ‎Luenberger D. G‎. ‎(1974)‎. ‎``Self-scaling variable metric (ssvm) algorithms‎: ‎Part i‎: ‎Criteria and sufficient conditions for scaling a class of algorithms"‎, ‎Management Science‎, ‎20‎, ‎845--862‎.
 
‎bibitem{SFN18} Soleimani A.‎, ‎Fathali J.‎, ‎Nazari M‎. ‎(2019)‎. ‎``Single facility goal location problems with Lp norm"‎, ‎Modern Research in Decision Making‎, ‎3‎, ‎125--152‎.
 
‎bibitem{TLA15} Trinh M. H.‎, ‎Lee B. H.‎, ‎Ahn H. S‎. ‎(2015)‎. ‎``The Fermat–Weber location problem in single integrator dynamics using only local bearing angles''‎, ‎Automatica‎, ‎59‎, ‎90--96‎.
 
‎bibitem{V75} Varian H. R‎. ‎(1975)‎. ‎``A Bayesian approach to real estate assessment‎" ‎In‎: ‎S‎. ‎E‎. ‎Fienberg‎, ‎A‎. ‎Zellner‎, ‎(eds.)‎, ‎Studies in Bayesian Econometrics and Statistics in Honour of Leonard J‎. ‎Savage‎, ‎North-Holland‎, ‎Amesterdam‎, ‎195--208‎.
 
 
‎bibitem{W37} Weiszfeld E‎. ‎(1937)‎. ‎``Sur Le Point Pour Lequel La Somme Des Distances De N Points Donnes Est Minimum"‎, ‎Tohoku Mathematical Journal‎, ‎60‎, ‎355--386‎ .
 
‎bibitem{YZ10} Yuan G.‎, ‎Zengxin W‎. ‎(2010)‎. ‎``Convergence analysis of a modified BFGS method on convex minimizations"‎, ‎Computational Optimization and Applications‎, ‎47‎, ‎237--255‎.