Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-13T02:47:25.107Z Has data issue: false hasContentIssue false
  • English
  • Français

SOLVING A SPECIAL CLASS OF MULTIPLE OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEMS

Published online by Cambridge University Press:  09 October 2014

S. F. TANTAWY
S. F. TANTAWY*
Affiliation:
Mathematics Department, Faculty of Science, Helwan University (11795), Cairo, Egypt email saidtantawy@hotmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper a feasible direction method is presented to find all efficient extreme points for a special class of multiple objective linear fractional programming problems, when all denominators are equal. This method is based on the conjugate gradient projection method, so that we start with a feasible point and then a sequence of feasible directions towards all efficient adjacent extremes of the problem can be generated. Since methods based on vertex information may encounter difficulties as the problem size increases, we expect that this method will be less sensitive to problem size. A simple production example is given to illustrate this method.

Keywords

multiple objectivefractional programmingefficient solutionnondominated solutionconjugate gradient

MSC classification

Secondary: 90C05: Linear programming 90C29: Multi-objective and goal programming
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 1 , July 2014 , pp. 91 - 103
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Benson, H. P., “Finding certain weakly-efficient vertices in multiple objective linear fractional programming”, Manag. Sci. 31 (1985) 240–245; doi:10.1137/0132004.CrossRefGoogle Scholar
Benson, H. P. and Morin, T. L., “The vector maximization problem proper efficiency and stability”, SIAM J. Appl. Math. 32 (1977) 64–72; doi:10.1137/0132004.CrossRefGoogle Scholar
Charnes, A. and Cooper, W., “Programming with linear fractional functions”, Naval. Res. Logist. Quart. 9 (1962) 181–186; doi:10.1002/nav.3800090303.CrossRefGoogle Scholar
Choo, E. U., “Technical note – Proper efficiency and the linear fractional vector maximum problem”, J. Oper. Res. 32 (1984) 216–220; doi:10.1287/opre.32.1.216.CrossRefGoogle Scholar
Ecker, J. G. and Kauda, I. A., “Finding all efficient extreme points for multiple objective linear programs”, Math. Program 14 (1978) 249–261; doi:10.1007/BF01588968.CrossRefGoogle Scholar
Ecker, J. G., Hegren, H. S. and Kauda, I. A., “Generating maximal efficient faces for multiple objective linear programs”, J. Optim. Theory Appl. 30 (1980) 353–361; doi:10.1007/BF00935493.CrossRefGoogle Scholar
Evans, J. P. and Steuer, R. F., “Generating efficient extreme points in linear multiple objective programming: two algorithms and computing experience”, in: Multiple criteria decision making (eds Cochran, J. L. and Zeleny, M.), (University of South Carolina Press, Columbia, 1973).Google Scholar
Gal, T., “A general method for determining the set of all efficient solutions to a linear vector maximum problem”, European J. Oper. Res. 1 (1977) 307–322; doi:10.1016/0377-2217(77)90063-7.CrossRefGoogle Scholar
Gandibleux, X., Sevaux, M., Sörensen, K. and T’kindt, V., “Meta-heuristics for multi-objective optimization series”, Volumne 535 of Lecture Notes in Economics and Mathematical Systems, (2004) 82.Google Scholar
Geoffrion, A. H., “Proper efficiency and the theory of vector maximization”, J. Math. Anal. Appl. 22 (1968) 618–630; doi:10.1016/0022-247X(68)90201-1.CrossRefGoogle Scholar
Goldfarb, D., “Extension of Davidson’s variable metric method to maximization under linear inequality and equality constraints”, SIAM J. Appl. Math. 17 (1969) 739–764; doi:10.1137/0117067.CrossRefGoogle Scholar
Goldfarb, D. and Lapiduo, L., “Conjugate gradient method for non linear programming problems with linear constraints”, Find & Eng. Chem. Fund. 7 (1968) 148–151.Google Scholar
Greig, D. M., Optimization (Longman, London and New York, 1980).Google Scholar
Hillier, F. S. and Lieberman, G. J., Introduction to operations research, 5th edn (McGraw-Hill, New York, 1990).Google Scholar
Isermann, H., “The enumeration of the set of all efficient solution for a linear multiple objective program”, Oper. Res. Quart. 28 (1977) 711–725; doi:10.2307/3008921.CrossRefGoogle Scholar
Jones, D. F., Mirrazavi, S. K. and Tamiz, M., “Multi-objective meta-heuristics: an overview of the current state-of-the-art”, European J. Oper. Res. 137 (2002) 1–9; doi:10.1016/S0377-2217(01)00123-0.CrossRefGoogle Scholar
Jozefowiez, N., Glover, F. and Laguna, M., “Multi-objective meta-heuristics for the traveling salesman problem with profits”, J. Math. Model. Algorithms 7 (2008) 177–195; doi:10.1007/s10852-008-9080-2.CrossRefGoogle Scholar
Kornbluth, J. S. H. and Steuer, R. E., “Multiple linear fractional programming”, Manag. Sci. 27 (1987) 1024–1039.CrossRefGoogle Scholar
Stoer, J. and Witzgall, C., Convexity and optimization in finite dimension I (Springer, Berlin, 1970).CrossRefGoogle Scholar
Stummer, C. and Sun, M., “New multiobjective metaheuristic solution procedures for capital investment planning”, J. Heuristics 11 (2005) 183–199; doi:10.1007/s10732-005-0970-4.CrossRefGoogle Scholar
Sun, M., “Some issues in measuring and reporting solution quality of interactive multiple objective programming procedures”, European J. Oper. Res. 162 (2005) 468–483; doi:10.1016/j.ejor.2003.08.058.CrossRefGoogle Scholar
Wets, R. J. B. and Witzgall, C., “Algorithms for frames and linearity spaces of cones”, J. Res. Natl. Bur. Stand. 71B (1967) 1–7.CrossRefGoogle Scholar
Zeleny, M., Linear multiobjective programming (Springer, Berlin, 1974).CrossRefGoogle Scholar