Abstract
In this paper we propose the big cube small cube (BCSC) technique for multicriteria optimization problems. The output of our algorithm results in a set which consists of epsilon efficient solutions and which contains all efficient solutions.
Furthermore, the method is demonstrated on several semi-obnoxious location problems which are reported in the literature. Computational results and an illustrative example are given.
Similar content being viewed by others
References
Blanquero R, Carrizosa E (2002) A DC biobjective location model. J Glob Optim 23:139–154
Blanquero R, Carrizosa E (2008) Continuous location problems and big triangle small triangle: constructing better bounds. J Glob Optim. doi:10.1007/s10898-008-9381-z
Brimberg J, Juel H (1998a) A bicriteria model for locating a semi-desirable facility in the plane. Eur J Oper Res 106:144–151
Brimberg J, Juel H (1998b) On locating a semi-desirable facility on the continuous plane. Int Trans Oper Res 5:59–66
Carrizosa E, Plastria F (1999) Location of semi-obnoxious facilities. Stud Locat Anal 12:1–27
Chalmet LG, Francis RL, Kolen A (1981) Finding efficient solutions for rectilinear distance location problems efficiently. Eur J Oper Res 6:117–124
Drezner Z (2007) A general global optimization approach for solving location problems in the plane. J Glob Optim 37:305–319
Ehrgott M (2000) Multicriteria optimization, 1st edn. Springer, Berlin
Engau A, Wiecek MW (2007a) Exact generation of epsilon-efficient solutions in multiple objective programming. OR Sprectrum 29:335–350
Engau A, Wiecek MW (2007b) Generating epsilon-efficient solution in multiobjective programming. Eur J Oper Res 177:1566–1579
Fernández J, Tóth B (2007a) Obtaining an outer approximation of the efficient set of nonlinear biobjective problems. J Glob Optim 38:315–331
Fernández J, Tóth B (2007b) Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods. Comput Optim Appl 42:393–419
Hamacher HW, Nickel S (1996) Multicriteria planar location problems. Eur J Oper Res 94:66–86
Hansen P, Thisse JF (1981) The generalized Weber–Rawls problem. In: Brans JP (ed) Operations research. North Holland, Amsterdam, pp 487–495
Hansen P, Peeters D, Richard D, Thisse JF (1985) The minisum and minimax location problems revisited. Oper Res 33:1251–1265
Ichida K, Fujii Y (1990) Multicriterion optimization using interval analysis. Computing 44:47–57
Loridan P (1984) ε-solutions in vector minimization problems. J Optim Theory Appl 43:265–276
McGinnis LF, White JA (1978) A single facility rectilinear location problem with multiple criteria. Transp Sci 12:217–231
Melachrinoudis E, Xanthopulos Z (2003) Semi-obnoxious single facility location in Euclidean space. Comput Oper Res 30:2191–2209
Nickel S, Puerto J, Rodríguez-Chía AM, Weißler A (1997) General continuous multicriteria location problems. Technical report, University of Kaiserslautern, Department of Mathematics
Ohsawa Y (2000) Bicriteria Euclidean location associated with maximin and minimax criteria. Nav Res Logist 47:581–592
Ohsawa Y, Tamura K (2003) Efficient location for a semi-obnoxious facility. Ann Oper Res 123:173–188
Ohsawa Y, Plastria F, Tamura K (2006) Euclidean push–pull partial covering problems. Comput Oper Res 33:3566–3582
Ohsawa Y, Ozaki N, Plastria F (2008) Equity-efficiency bicriteria location with squared Euclidean distances. Oper Res 56:79–87
Plastria F (1992) GBSSS: the generalized big square small square method for planar single-facility location. Eur J Oper Res 62:163–174
Schöbel A, Scholz D (2009) The big cube small cube solution method for multidimensional facility location problems. Comput Oper Res. doi:10.1016/j.cor.2009.03.031
Skriver AJV, Anderson KA (2003) The bicriterion semi-obnoxious location (BSL) problem solved by an ε-approximation. Eur J Oper Res 146:517–528
White DJ (1986) Epsilon efficiency. J Optim Theory Appl 49:319–337
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Scholz, D. The multicriteria big cube small cube method. TOP 18, 286–302 (2010). https://doi.org/10.1007/s11750-009-0105-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-009-0105-4
Keywords
- Multicriteria optimization
- Bicriteria optimization
- Approximation algorithms
- Facility location problems
- Continuous location
- Global optimization
- Nondifferentiable optimization
- Semi-obnoxious location