Abstract
Fuzzy geometric programming (GP) is an important optimization type. In this paper, first the origin of fuzzy GP is introduced, some recent research findings of its theory have been summarized. Then a lot of progress achieved in application is introduced in fuzzy GP, including applications in power system, environmental engineering, economic management, etc. The range of its application expects to be further expanded. Besides, as a branch of fuzzy GP, the fuzzy relation GP, rough (extension) and grey one have also been mentioned. Finally, the new research direction of fuzzy GP will be put forward.
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References
Peterson E L (2001) The origins of geometric programming. Annals of Operations Research 105: 15–19
Yang J H, Cao B Y (2006) The origin and its application of geometric programming. Proceedings of the Eighth National Conference of Operations Research Society of China, Global-Link Publishing Company, Hong Kong: 358–363 (Chinese)
Cao B Y (1987) Solution and theory of question for a kind of fuzzy positive geometric program. Proceedings of the 2nd IFSA Congress, Tokyo, Japan 1: 205–208
Cao B Y (1989) Study of fuzzy positive geometric programming dual form. Proceedings 3rd IFSA Congress, 1989, August 6–August 11, Seattle: 775–778
Cao B Y (2002) Fuzzy geometric programming. Dordrecht/Boston/London: Kluwer Academic Publishers
Cao B Y, Yang J H (2007) Advances in fuzzy geometric programming. Springer Advances in Soft Computing 40, Fuzzy Information and Engineering: 497–502
Cao B Y (1993) Fuzzy geometric programming (I). Fuzzy Sets and Systems 53(2): 135–154
Cao B Y (1998) Further research of solution to fuzzy posynomial geometric programming. Academic Periodical Abstracts of China 4(12): 1435–1437
Cao B Y (2007) Fuzzy reversed posynomial geometric programming and its dual form. Lecture Notes in Computer Science, Springer Berlin, Heidelberg 4529: 553–562
Verma R K (1990) Fuzzy geometric programming with several objective functions. Fuzzy Sets and Systems 35(1): 115–120
Maleki H R, Mashinchi M (2007) Multiobjective geometric programming with fuzzy parameters. International Journal of Information Science & Technology 5(2): 35–45
Cao B Y (1995) Classification of fuzzy posynomial geometric programming and corresponding class properties. Fuzzy Systems and Mathematics 9(4): 60–64
Cao B Y (2000) Parameterized solution to a fractional geometric programming. Proceedings of the sixth National Conference of Operations Research Society of China, Global-Link Publishing Company, Hong Kong: 362–366
Cao B Y (2001) Extension posynomial geometric programming. Journal of Guangdong University of Technology 18(1): 61–64
Yang J H, Cao B Y (2005) Geometric programming with fuzzy relation equation constraints. Proceedings of the IEEE International Conference on Fuzzy Systems: 557–560
Cao B Y (1993) Extended fuzzy geometric programming. The Journal of Fuzzy Mathematics 2: 285–293
Cao B Y (1995) The study of geometric programming with (·, c)-fuzzy parameters. Journal of Changsha Univ. of Electric Power (Natural Sci. Ed.) 1: 15–21
Cao B Y (1994) Posynomial geometric programming with L-R fuzzy coefficients. Fuzzy Sets and System 64: 267–276
Cao B Y (2000) Research of posynomial geometric programming with flat fuzzy coefficients. Journal of Shantou University (Natural Sci. Ed.) 15(1): 13–19
Cao B Y (1992) Further study of posynomial geometric programming with fuzzy coefficients. Mathematics Applicata 5(4): 119–120
Cao B Y (1997) Research for a geometric programming model with T-fuzzy variable. The Journal of Fuzzy Mathematics 5(3): 625–632
Cao B Y (2001) Primal algorithm of fuzzy posynomial geometric programming. Joint 9th IFSA World Congress and 20th NAFIPS International Conference Proceedings, July 25–July 28, Vancouver: 31–34 (Also to see: Direct algorithm of fuzzy posynomial geometric programming. Fuzzy Systems and Mathematics 15(4): 81–86
Cao B Y (1996) Another proof of fuzzy posynomial geometric programming dual theorem. BUSEFAL 66:43–47
Cao B Y (1996) Fuzzy geometric programming (II)-Fuzzy strong dual results for fuzzy posynomial geometric programming. The Journal of Fuzzy Mathematics 4(1): 119–129
Cao B Y (2005) Lagrange problem in fuzzy reversed posynomial geometric programming. Lecture Notes in Artificial Intelligence, Springer 3614: 546–550
Cao B Y (2004) Antinomy in posynomial geometric programming. Advances in Systems Science and Applications, USA, 4(1): 7–12
Cao B Y (1999) Fuzzy geometric programming optimum seeking in power supply radius of transformer substation. Proceedings of the Fuzz-IEEE’ 99 Conference, Seoul, Korea, 3: 1749–1753
Cao B Y (1995) Fuzzy geometric programming optimum seeking of scheme for waste-water disposal in power plant. Proceedings of the Fuzz-IEEE /IFES’ 95 Conference, Yokohama, Japan, 5: 793–798
Biswal M P (1992) Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets and Systems 51(1): 67–71
Liu S T (2004) Fuzzy geometric programming approach to a fuzzy machining economics model. International Journal of Production Research 42(16): 3253–3269
Panda D, Kar S, Maiti M (2008) Multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzy stochastic resource constraints: A geometric programming approach. Computers & Mathematics with Applications 56(11): 2970–2985
Islam S. and Roy T K (2006) A new fuzzy multi-objective programming: entropy based geometric programming and its application of transportation problems. European Journal of Operational Research 173(2): 387–404
Mandal N K, Roy T K, Maiti M (2005) Multi-objective fuzzy inventory model with three constraints: a geometric programming approach. Fuzzy Sets and Systems 150(1): 87–106
Liu S T (2007) Geometric programming with fuzzy parameters in engineering optimization. International Journal of Approximate Reasoning 46(3): 484–498
Roy T K (2008) Fuzzy geometric programming with numerical examples. Springer Optimization and Its Applications 16: Fuzzy Multi-Criteria Decision Making Theory and Applications with Recent Developments: 567–587
Bellman R E, Zadeh L A (1970) Decision-making in a fuzzy environment. Management Science 17(4): B 141–164
Gen M, Yun Y S (2006) Soft computing approach for reliability optimization: State-of-the-art survey. Reliability Engineering & System Safety 91(9): 1008–1026
Yang J H (2008) A Brief description on fuzzy relation programming. Proceedings of the ninth National Conference of Operations Research Society of China, Global-Link Publishing Company, Hong Kong: 554–558 (Chinese)
Yang J H, Cao B Y (2007) Posynomial fuzzy relation geometric programming. Lecture Notes in Computer Science, Springer Berlin, Heidelberg, 4529: 563–572
Yang J H, Cao B Y (2007) Monomial geometric programming with fuzzy relation equation constraints. Fuzzy Optimization and Decision Making 6(4): 337–349
Wu Y.K. (2008) Optimizing the geometric programming problem with single-term exponents subject to max-min fuzzy relational equation constraints. Mathematical and Computer Modelling 47(3–4): 352–362
Yang J H, Cao B Y (2005) Geometric programming with max-product fuzzy relation equation constraints. Proceedings of the 24th North American Fuzzy Information Processing Society, Ann Arbor, Michigan, June 22–25, 650–653
Lu J J, Fang S C (2001) Solving nonlinear optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 119(1): 1–20
Cao B Y (2009) Rough posynomial geometric programming. Fuzzy Information and Engineering 1(1): 37–55
Pawlak Z (1982) Rough sets. International Journal of Information and Computer Sciences 11(5): 341–356
Khorram E, Hassanzadeh R (2008) Solving nonlinear optimization problems subjected to fuzzy relation equation constraints with max-average composition using a modified genetic algorithm. Computers & Industrial Engineering 55(1): 1–14
Cai W (1983) The extension set and incompatible problem. J. of Scientific Exploration 1: 81–93
Luo D (2005) Study on the grey polynomial geometric programming. Chin. Quart. J. of Math 20(1): 34–41
Cao B Y (1990) Extension convex set. Acta Science Naturalium Univ. Norm Hunanenist 13(1): 18–24
Zadeh L A (1965) Fuzzy sets. Inform. and Control. 8: 338–353
Shivanian E, Khorram E (2009) Monomial geometric programming with fuzzy relation inequality constraints with max-product composition. Computers and Industrial Engineering 56(4): 1386–1392
Yousef S, Badra N, Yazied Abu-El T G (2009) Geometric programming problems with fuzzy parameters and its application to crane load sway. World Applied Sciences Journal 7(1): 94–101
Mahapatra G S, Roy T K (2009) Single and multi container maintenance model: A fuzzy geometric programming approach. Journal of Mathematics Research 1(2): 47–60
Sadjadi S J, Ghazanfari M, Yousefli A (2010) Fuzzy pricing and marketing planning model: A possibilistic geometric programming approach. Expert Systems with Applications 37: 3392–3397
Mandal N K, Roy T K (2006) A displayed inventory model with L-R fuzzy number. Fuzzy Optimiza tion and Decision Making 5(3): 227–243
Cao B Y (2010) Optimal models and methods with fuzzy quantities. Springer-Verlag, Berlin
Das K, Roy T K, Maiti M (2000) Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: a geometric programming approach. Production Planning & Control 11(8): 781–788
Jumarie G (1995) Possibility, probability and relative information-A unified approach via geometricprogramming. Kybernetes 24(1): 18–33 (CA)
Li P K, Fang S C (2009) Latticized linear optimization on the unit interval. IEEE Transactions on Fuzzy Systems 17(6), 1353–1365
Aggarwal A, Singh H (2005) Optimization of machining techniques — A retrospective and literature review. Sadhana-Academy Proceedings in Engineering Sciences 30: 699–711 Part 6
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Yang, Jh., Cao, By. Fuzzy geometric programming and its application. Fuzzy Inf. Eng. 2, 101–112 (2010). https://doi.org/10.1007/s12543-010-0040-y
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DOI: https://doi.org/10.1007/s12543-010-0040-y