zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A heuristic algorithm for constrained multi-source Weber problem - the variational inequality approach. (English) Zbl 1149.90091
Summary: For solving the well-known multi-source Weber problem (MWP), each iteration of the heuristic alternate location-allocation algorithm consists of a location phase and an allocation phase. The task of the location phase is to solve finitely many single-source Weber problems (SWP), which are reduced by the heuristic of nearest center reclassification for the customers in the previous allocation phase. This paper considers the more general and practical case - the MWP with constraints (CMWP). In particular, a variational inequality approach is contributed to solving the involved constrained SWP (CSWP), and thus a new heuristic algorithm for CMWP is presented. The involved CSWP in the location phases are reformulated into some linear variational inequalities, whose special structures lead to a new projection-contraction (PC) method. Global convergence of the PC method is proved under mild assumptions. The new heuristic algorithm using the PC method in the location phases approaches to the heuristic solution of CMWP efficiently, which is verified by the preliminary numerical results reported in this paper.

90B80Discrete location and assignment
90C59Approximation methods and heuristics
Full Text: DOI
[1] Brimberg, J.; Love, R. F.: Global convergence of a generalized iterate procedure for the minisum location problem with lp distances, Operations research 41/6, 1153-1163 (1993) · Zbl 0795.90037 · doi:10.1287/opre.41.6.1153
[2] Burachik, R. S.; Iusem, A. N.: A generalized proximal point algorithm for the variational inequality problem in a Hilbert space, SIAM journal on optimization 8, 197-216 (1998) · Zbl 0911.90273 · doi:10.1137/S1052623495286302
[3] Chandrasekaran, R.; Tamir, A.: Open questions concerning weiszfeld’s algorithm for the Fermat -- Weber location problem, Mathematical programming 44, 293-295 (1989) · Zbl 0683.90026 · doi:10.1007/BF01587094
[4] Cooper, L.: Location -- allocation problems, Operations research 11, 331-343 (1963) · Zbl 0113.14201 · doi:10.1287/opre.11.3.331
[5] Cooper, L.: Heuristic methods for location -- allocation problems, SIAM review 6, 37-53 (1964) · Zbl 0956.90014
[6] Drezner, Z.: Facility location: A survey of applications and methods, (1995) · Zbl 0917.90224
[7] Durier, R.; Michelot, C.: Geometrical properties of the Fermat -- Weber problem, European journal of operational research 20, 332-343 (1985) · Zbl 0564.90013 · doi:10.1016/0377-2217(85)90006-2
[8] Eaves, B. C.: On the basic theorem of complementarity, Mathematical programming 1, 68-75 (1971) · Zbl 0227.90044 · doi:10.1007/BF01584073
[9] Ferris, M. C.; Kanzow, C.: Engineering and economic applications of complementarity problems, SIAM review 39, 669-713 (1997) · Zbl 0891.90158 · doi:10.1137/S0036144595285963
[10] Ghosh, A.; Rushton, G.: Spatial analysis and location -- allocation models, (1987)
[11] He, B. S.: A new method for a class of linear variational inequalities, Mathematical programming 66, 137-144 (1994) · Zbl 0813.49009 · doi:10.1007/BF01581141
[12] He, B. S.: A modified projection and contraction method for a class of linear complementarity problems, Journal of computational mathematics 14, 54-63 (1996) · Zbl 0854.65047
[13] Kinderlehrer, D.; Stampacchia, G.: An introduction to variational inequalities and their application, (1980) · Zbl 0457.35001
[14] Kuhn, H. W.: A note on Fermat’s problem, Mathematical programming 4, 98-107 (1973) · Zbl 0255.90063 · doi:10.1007/BF01584648
[15] Idriss, H. F.; Lefebvre, O.; Michelot, C.: A primal -- dual algorithm for a constrained Fermat -- Weber problem involving mixed norms, Recherche operationelle -- operations research 22, No. 4, 313-330 (1988) · Zbl 0663.90026
[16] Levin, Y.; Ben-Israel, A.: A heuristic method for large-scale multi-facility location problems, Computers & operations research 31, 257-272 (2004) · Zbl 1088.90036 · doi:10.1016/S0305-0548(02)00191-0
[17] Love, R. F.; Morris, J. G.; Wesolowsky, G. O.: Facilities location: models & methods, (1988) · Zbl 0685.90036
[18] Martinet, B.: Regularision d’inéquations variationnelles par approximations successive, Revue francaise d’automatique et informatique recherche opérationnelle 126, 154-159 (1970)
[19] Jr., L. M. Ostresh: An efficient algorithm for solving the two-center location -- allocation problem, Journal of regional science 15, 209-216 (1975)
[20] Jr., L. M. Ostresh: On the convergence of a class of iterative methods for solving the Weber location problem, Operations research 26, 597-609 (1978) · Zbl 0396.90073 · doi:10.1287/opre.26.4.597
[21] Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM journal on control and optimization 14, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[22] Rosing, K. E.: An optimal method for solving the (generalized) multi-Weber problem, European journal of operational research 58, 414-426 (1992) · Zbl 0760.90064 · doi:10.1016/0377-2217(92)90072-H
[23] Uzawa, H.: Iterative methods for concave programming, Studies in linear and nonlinear programming, 154-165 (1958)
[24] Vardi, Y.; Zhang, C. H.: A modified weiszfeld algorithm for the Fermat -- Weber location problem, Mathematical programming 90, 559-566 (2001) · Zbl 0990.65064 · doi:10.1007/s101070100222
[25] Weiszfeld, E.: Sur le point pour lequel la somme des distances de n points donnés est minimum, Tohoku mathematical journal 43, 355-386 (1937) · Zbl 0017.18007 · doi:10.2748/tmj/1178227459