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The Origin and Development of Fuzzy Geometric Programming

The Origin and Development of Fuzzy Geometric Programming FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 203–211 https://doi.org/10.1080/16168658.2019.1643078 The Origin and Development of Fuzzy Geometric Programming a,b b Bing-yuan Cao and Pei-hua Wang a b Foshan University, Foshan, People’s Republic of China; Guangzhou Vocational and Technical University of Science and Technology, Guangzhou , People’s Republic of China ABSTRACT KEYWORDS Geometric programming; Fuzzy Geometric Programming has been in discussion for 32 years development process; past; since 1987. According to Cao, the author, he in the paper introduces present and future; its development process, aiming to promote this new branch, attract- conjecture ing scholars home and abroad to join in ranks of the research, and helping to solve the three conjectures of Fuzzy Geometric Program- ming proposed by Cao [Three guess of fuzzy geometric program- ming. Vol. 147. Springer; 2012. p. 591–594. (Advances in intelligent and soft computing)]. At present, many books and teaching materials of it have been translated into Persian with some published. 1. Introduction From introduction of its generation, application and achievements, it is confirmed that it has developed into a new branch of discipline. Firstly, we in this paper trace back the origin of fuzzy geometric programming, besides we introduce its present situation, and finally, according to its present situation, we look forward to its development prospect, and once again put forward three conjectures of fuzzy geometric programming, which provides a difficult and promising path for the development of this branch. 2. The Past of Geometric Programming In 1987, B. Y. Cao proposed the fuzzy geometric programming (GP) theory in International Fuzzy Systems Association (IFSA) Congress for the first time [1,2]; its general form is (GP) min g ˜ (x) s.t. g (x)  1(1  i  p ), g (x)  1 (p + 1  i  p), x > 0 CONTACT Pei-hua Wang phwang321@163.com © 2019 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 204 B.-Y. CAO AND P.-H. WANG it is a reversed posynomial GP, where all of g (x)(0  i  p) are posynomial functions of variable x with g (x) = v (x); here ik k=1 ⎪ ikl c x , (1  k  J ;0  i  p ) ⎪ ik l=1 v (x) = (1) ik ⎪ −γ ikl c x , (1  k  J ; p + 1  i  p) ik i ⎩ l l=1 is a monomial function of variable x, variable x = (x , x , ... , x ) , and coefficients c > 0, 1 2 n ik exponents γ (1 ≤ k ≤ J ,0 ≤ i ≤ p,1 ≤ l ≤ n) are arbitrary real numbers. ikl i We have done a lot of work in the research of fuzzy GP [3–12], putting forward its orig- inal algorithm, and using fuzzy arithmetic geometric inequality and setting value theory to obtain its dual form and dual algorithm. Then, we proposed the geometric program- ming with a Fuzzy coefficient and Fuzzy variable and a multi-objective fuzzy geometric programming model [10,13–21]. At the same time, it is applied in power system, man- agement optimization and decision-making [22–27]. Based on the published fuzzy GP, another monograph of Optimal Models and Methods with Fuzzy Quantities was published in Springer in 2010, in which he introduced the work, such as GP models and algorithms with fuzzy coefficients and fuzzy variables [28,29]. The author has been awarded the National Natural Science Foundation of China for three consecutive times: (1) Fuzzy generalized GP decision model and method, 2008–2010; (2) Universal fuzzy GP and optimization techniques in management, 2003–2005; (3) Theory and method of fuzzy GP in power system management, 1997–1999; and has achieved rich results. In 2002, Kluwer Academic Publishers published a series of fuzzy GP in the Applied Mathematics series. In 2005, Fuzzy GP was awarded the third prize of Guangdong Science and Technology Award. The American Mathematical Review MathSciNet? and the West Ger- man Mathematical Digest Zentralblatt MATH commented on its original research. In 1993, Liu Yingming, an academician of the Chinese Academy of Sciences, called this work an international frontier. In 2004, Academician Wang Zikun, an academician of the Chinese Academy of Sciences, wrote an article in the Science Bulletin, saying that the monograph was at a high level in academic value with his identification of fuzzy GP reaching the inter- national advanced level. In 2007, Guo Bolin, another academician of the Chinese Academy of Sciences, who served as the director of the appraisal committee, identified this work as the advanced level of the international frontier, saying that it partially reached the leading level. In 2008, specialists in the international academic circle confirmed that the fuzzy GP proposed by the applicant is the seventh kind of optimization model [30]. While studying the theory and application of fuzzy GP, Cao and others are actively expanding in other direction. In 2001, the author proposed the extension GP. In 2005, Luo Dang proposed the gray positive GP. In 2009, the author proposed the rough posynomial GP. In 2014, Cao’s doc- toral student Zeinab Kheiri wrote the doctoral thesis with intuitionistic fuzzy posynomial GP [31–36]. In 2005, the author and his doctoral student Yang Jihui, in the IEEE Fuzzy System Annual Meeting held in the United States, first proposed the fuzzy relation GP, and its form was FUZZY INFORMATION AND ENGINEERING 205 described as follows: (PGPF) min z(x) = c x (2) k=1 j=1 s.t. A ◦ x = b, where A = (a ) , x = (x ) , b = (b ) , a , x , b ∈ [0, 1], c , γ ∈ R, c > 0, i ∈ I = ij m×n j n×1 i m×1 ij j i k k k {1, 2, ... , m}, j ∈ J ={1, 2, ... , n}, k ∈ K ={1, 2, ... , p}, and for given j ∈ J, γ (k ∈ K) are either all non-positive real numbers or all non-negative real numbers. Without loss of generality, we assume that problem (2) satisfies the following inequalities: 1  b  b  ···  b  0. 1 2 m Otherwise, rearrange the components of b in decreasing order and adjust the rows of A according to b [37–45]. So far, research in this direction has begun to heat up. 3. The Present of Geometric Programming From 30th July to 1st August 2016, Cao Bingyuan’s students, family and relatives gathered in the Applied Mathematics Conference Room on the 7th floor of the Computer Experiment Building of Guangzhou University to celebrate the 30 years of fuzzy geometry planning and 40 years of his teaching job, and published a conference proceeding by Chinese Science and Education Press in 2019. The book, containing the main articles of Cao and his students, has been published in the fuzzy GP for 30 years, reporting their projects, achievements and rewards. At present, the research on fuzzy relational linear programming and fuzzy rela- tional GP is becoming a hot topic. Nearly 100 papers have been published in the magazine; recently, they have headed to some new research directions [46–49], and Cao’s book of fuzzy relationship programming as well as the book of Indian scholars’ fuzzy relational GP and its application will be published by Springer-Nature. Now Cao’s colleagues and he are already preparing to collaborate with the team on optimized secured sharing of documents, which was proposed in Jana Wyżykowskiego Uni- versity, Poland, on fuzzy relationships and their programming. At the same time, the three conjectures of fuzzy GP proposed in 2012 still remain to be solved. They are expanding our propaganda and taking strong measures to attract more scholars to discuss them. And they will use the 9th International Information and Engineering Conference of Kish Island to prepare for the establishment of the International Fuzzy Information and Engineering Association. We will use the EI collection and ISC conference proceedings and the maga- zine ‘Operational Management and Fuzzy Mathematics’ to exchange ideas. The excellently chosen papers in this direction will be published in the magazine Fuzzy Information and Engineering, which is included in the Web of Sciences. At present, copies of works in Fuzzy geometric programming researches have been pub- lished by famous Springer publishing house, including essays on Fuzzy geometric program- ming, Fuzzy relational geometric programming and Fuzzy programming. Besides, China Science and Education Press and Fuzzy Information and Engineering Journal will strongly support the development in this direction. Two monographs [29,30] will be translated into Persian language for publication (Figure 1). 206 B.-Y. CAO AND P.-H. WANG Figure 1. Book titled ‘Optimal models and methods with fuzzy quantity’ has been translated into Persian language publishing. The world’s first high school fuzzy mathematics textbook ‘Fuzzy sets theory preliminary’ was officially published in February 2018 by the world-famous publishing houses Springer- Nature and China Science and Education Publishing House, which draws strong attention of scholars and readers from all over the world. In April this year, it was selected as the fourth of the eight most popular teaching materials of Springer-Nature, which exerts a great international influence (Figure 2). 4. The Future of Geometric Programming For 32 years, we have witnessed the development of fuzzy GP. At present, its research has entered a critical phrase. More research scholars are involved in it and infiltration fields are constantly expanding, and its research needs more talents to participate in and sup- port for. The three conjectures we have to solve are (i) the local optimal (satisfactory) FUZZY INFORMATION AND ENGINEERING 207 Figure 2. ‘Fuzzy sets theory preliminary’ has been selected as the fourth of the eight most popular textbooks in Springer-Nature. solution to fuzzy GP is still its global optimal (satisfactory) solution; (ii) after replacing the operator (+, ·) in (1) with other logical operators, the fuzzy relational still holds with its dual programming (2) established; (iii) confirmation of the existence of fuzzy GP taxon- omy and identification. As for the above three, they will continue to organize the team to deal with them, strive to make a breakthrough in theory, further find the background in the application, and make arduous efforts to establish a branch of the fuzzy geometric programming. The fuzzy GP will attract all of us to further research because many aspects remain untouched. In the basic field, we shall consider the following topics. (i) Fuzzy reverse GP, including a GP problem with mixed sign-terms, is much more com- plex than the fuzzy convex (resp. concave) GP and fuzzy PGP ; we want to continue to explore their properties. (ii) Fuzzy fractionation, extension, gray, and rough GP still need to be studied. (iii) GPs with intuitionistic fuzzy coefficients and fuzzy variables have yet to be further refined and expanded. (iv) Further solve real-world problems paradox with fuzzy GP. (v) Solving fuzzy relation GP. (vi) Explore GP with discrete variables and coefficients. (vii) Fuzzy GP in application in BitTorrent-like Peer-to-Peer file sharing system. (viii) GP problem subject to max-product fuzzy relation inequalities. The local optimal solu- tion of the GP problem subject to fuzzy relation inequalities is also its global optimal solution. (ix) Study of fuzzy GP classification. (x) The export of fuzzy GP’s genetic algorithm. 208 B.-Y. CAO AND P.-H. WANG Figure 3. Book titled ‘Application of fuzzy mathematics and systems’ in Persian language publishing Copyright Export Signing Ceremony Site. On 26 April 2019, the signing ceremony for the Persian copyright export of the monograph ‘Applied fuzzy mathematics and systems’ was jointly held by Shokoh-e-Sahel Publishing House and Scientific Publishing at the China Guest of Honor booth at the Iran International Book Fair. The translation project of this book has been included in the national ‘the belt and road initiative’ Silk Road Fund Plan applied by Science Press (Figure 3). Colleagues are welcome to participate and work hard to establish a new branch of fuzzy GP. 5. Conclusion The research on the theory and algorithm to Fuzzy geometric programming has achieved phased results, and the application background was first found in power systems. The monographs and applications of Fuzzy Optimization have been and are being disseminated in the world through Persian, English, Chinese languages, etc. As long as scholars in many other countries persist in their research and experiments, the three conjectures of Fuzzy geometric programming [50] could be overcome. Acknowledgments This article was presented at the 9th International Conference of Fuzzy Information and Engineering. Disclosure statement No potential conflict of interest was reported by the authors. FUZZY INFORMATION AND ENGINEERING 209 Funding The work was supported by the Natural Science Foundation of Guangdong Province (No. 2016A030313552). Notes on contributors Bing-yuan Cao, Changde, Hunan is second-level Chair Professor of Lingnan of Foshan University and Dean & Professor of School of Finance and Economics, Guangzhou Vocational and Technical Uni- versity of Science and Technology and Doctoral & Postdoctoral Supervisor of School of Guangzhou University, China. Pei-hua Wang, Huaron, Hunan is associate professor, Guangzhou Vocational and Technical University of Science and Technology, China. References [1] Cao BY. Solution and theory of question for a kind of fuzzy positive geometric program. Proceed- ings of the 2nd IFSA Conference, Tokyo, Japan, Vol. 1; 1987. p. 205–208. [2] Cao BY. Fuzzy geometric programming (I). Int J Fuzzy Sets Syst. 1993;53(2):135–153. [3] Cao BY. Classification of fuzzy posynomial geometric programming and corresponding class properties. J Fuzzy Syst Math. 1995;9(4):60–64. [4] Cao BY. Fuzzy geometric programming(II). J Fuzzy Math. 1996;4(1):119–129. [5] Cao BY. Primal algorithm of fuzzy posynomial geometric programming. Annual Conference of the North American Fuzzy Information Processing Society – NAFIPS, Vol. 1; Vancouver: 2001. p. 31–34. [6] Cao BY. Antinomy in posynomial geometric programming. Adv Syst Sci Appl. 2004;4(1):7–12. [7] Cao BY. Lagrange problem in fuzzy reversed posynomial geometric programming. Fuzzy Sys- tems and Knowledge Discovery – Second International Conference, FSKD 2005 Aug, Proceed- ings; Changsha: 2005, p. 546–550. (Lecture notes in computer science; vol. 3614 LNAI 2006). [8] Yang JH, Cao BY. The origin and its application of geometric programming. Proc. of the Eighth National Conference of Operations Research Society of China. Hong Kong: Global-Link Publish- ing Company; 2006. p. 358–363. [9] Cao BY. New proof to first dual theorem on fuzzy posynomial geometric programming. J Fuzzy Math. 2006;14(1):1–14. [10] Cao BY. Dual method to geometric programming with fuzzy variables. Chongqing: Springer; 2009. p. 1293–1301. (Advances in intelligent and soft computing; vol. 62). [11] Cao BY. The more-for-less paradox in fuzzy posynomial geometric programming. Inf Sci. 2012;211:81–92. [12] Zahmatkesh F, Cao BY. On the fuzzy fractional posynomial geometric programming problems. Adv Intell Syst Comput. 2016;367:97–108. [13] Cao BY. Posynomial geometry programming with L-R fuzzy coefficients. Int J Fuzzy Sets Syst. 1994;67(3):267–276. [14] Cao BY. Research for a geometric programming model with T-fuzzy variations. J Fuzzy Math. 1997;5(3):625–632. [15] Cao BY. Multi-objective geometric programming with T-fuzzy variables. 22nd Int. Conference of NAFIPS Proceedings; Chicago, Illinois: 2003. p. 456–461. [16] Cao BY. Geometric programming with trapezoidal fuzzy variables. Annual Conference of the North American Fuzzy Information Processing Society-NAFIPS, Vol. 2; Canadian Rockies: 2004. p. 826–831. [17] Baykasoglu A, Gocken T. A review and classification of fuzzy mathematical programs. J Intell Fuzzy Syst. 2008;19(3):205–229. [18] Cao BY. Extension posynomial geometric programming. J Guangdong Univ Techonol. 2001;18(1):61–64. 210 B.-Y. CAO AND P.-H. WANG [19] Cao BY. Types of non-distinct multiobjective geometric programming. Hunan Ann Math. 1995;15(1):99–106. [20] Verma RK. Fuzzy geometric programming with several objective functions. Fuzzy Sets Syst. 1990;35(1):115–120. [21] Biswal MP. Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets Syst. 1992;51(1):67–71. [22] Mandal NK, Roy TK, Maiti M. Multi-objective fuzzy inventory model with three constraints: a geometric programming approach. Fuzzy Sets Syst. 2005;150(1):87–106. [23] Cao BY. Fuzzy geometric programming optimum seeking of scheme for waste-water disposal in power plant. Proceedings of the Fuzz-IEEE/IFES95 Conference, Yokohama, Japan, Vol. 5; 1995. p. 793–798. [24] Cao BY. Fuzzy geometric programming optimum seeking in power supply radius of transformer substation. Proceedings of the Fuzz-IEEE99 Conference, Seoul, Korea, Vol. 3; 1999. p. 1749–1753. [25] Cao BY. Reverse geometric programming with fuzzy coefficient and its application in chemical industry production cost analysis. IEEE Int Conf Fuzzy Syst. 2003;2:1311–1316. [26] Liu ST. Fuzzy geometric programming approach to a fuzzy machining economics model. Int J Prod Res. 2004;42(16):3253–3269. [27] Islam S, Roy TK. A new fuzzy multi-objective programming: entropy based geometric program- ming and its application of transportation problems. Eur J Oper Res. 2006;173(2):387–404. [28] Cao BY. Power supply radius optimized with fuzzy geometric program in substation. Fuzzy Optim Decis Mak. 2006;5(2):123–139. [29] Cao BY. Fuzzy geometric programming. Dordrecht: Kluwer Academic Publishers; 2001. [30] Cao BY. Optimal models and methods with fuzzy quantity. Heidelberg, Berlin: Springer; 2010. [31] Cao BY. Extensional positive geometric programming. J Guangdong Univ Technol. 2001;18(1): 61–64. [32] Cao BY. Multi-objective geometric programming with T-fuzzy variables. Annual Conference of the North American Fuzzy Information Processing Society – NAFIPS, Vol. 2003, January; Paso, Texas: 2003. p. 456–461. [33] Dang L. Study on the gery posynomial geometric programming. Chinese Quart J Math. 2005;20(1):34–41. [34] Cao BY. Rough posynomial geometric programming. Fuzzy Inf Eng. 2009;1(1):37–57. [35] Kheiri Z, Zahmatkesh F, Cao BY. A new ranking approach to fuzzy posynomial geometric pro- gramming with trapezoidal fuzzy number; Berlin Heidelberg; Springer-Verlag: 2012. p. 517–523. (Advances in intelligent and soft computing, AISC; vol. 147). [36] Zahmatkesh F, Cao BY. On the solution of fractional geometric programming problem with fuzzy coefficient. 4th Iranian Joint Congress on Fuzzy and Intelligent Systems, CFIS 2015; Zahedan, Iran: [37] Burnwal AP, Mukherjee SN, Singh D. Fuzzy geometric programming with nonequivalent objec- tives. Ranchi Univ Math J. 1996;27:53–58. [38] Yang JH, Cao BY. Geometric programming with fuzzy relation equation constraints. Proceedings of the IEEE International Conference on Fuzzy Systems; Reno, Nevada: 2005. p. 557–560. [39] Yang JH, Cao BY. Posynomial fuzzy relation geometric programming. IFSA 2007, Proceedings; Cancun, Mexico: 2007. p. 563–572. (Lecture notes in computer science; vol. 4529 LNAI). [40] Yang J, Cao BY. Monomial geometric programming with fuzzy relation equation constraints. Fuzzy Optim Decis Mak. 2007;6(4):337–349. [41] Zhou XG, Cao BY. Optimizing the geometric programming problem with single-term exponents subject to max-product fuzzy relational equation. Proceedings – 5th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2008, Vol. 1; Jinan, China: 2008. p. 621–625. [42] Yang XP, Zhou XG, Cao BY. Single-variable term semi-latticized fuzzy relation geometric pro- gramming with max-product operator. Inf Sci. 2015;325:271–287. [43] Zhou XG, Yang XP, Cao BY. Posynomial geometric programming problem subject to max-min fuzzy relation equations. Inf Sci. 2016;328:15–25. [44] Yang XP, Zhou XG, Cao BY. Min-max programming problem subject to addition-min fuzzy relation inequalities. IEEE Trans Fuzzy Syst. 2016;24(1):111–119. FUZZY INFORMATION AND ENGINEERING 211 [45] Zhou XG, Cao BY, Yang XP. The set of optimal solutions of geometric programming problem with max-product fuzzy relational equations constraints. Int J Fuzzy Syst. 2016;18(3):436–447. [46] Kheiri Z, Cao BY. Posynomial geometric programming with intuitionistic fuzzy coefficients. Adv Intell Syst Comput. 2016;367:15–30. [47] Qin ZJ, Cao BY, Fang SC. Geometric programming with discrete variables subject to max-product fuzzy relation constraints. Discrete Dyn Nature Soc. 2018;2018. Article ID 1610349. [48] Yang XP, Lin HT, Zhou XG, et al. Addition-min fuzzy relation inequalities with application in BitTorrent-like Peer-to-Peer file sharing system. Fuzzy Sets Syst. 2018;343(SI):126–140. [49] Yang XP, Yuan DH, Cao BY. Lexicographic optimal solution of the multi-objective programming problem subject to max-product fuzzy relation inequalities. Fuzzy Sets Syst. 2018;341:92–112. [50] Cao BY. Three guess of fuzzy geometric programming. Springer; 2012. p. 591–594. (Advances in intelligent and soft computing; vol. 147). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

The Origin and Development of Fuzzy Geometric Programming

Fuzzy Information and Engineering , Volume 11 (2): 9 – Apr 3, 2019

The Origin and Development of Fuzzy Geometric Programming

Abstract

Fuzzy Geometric Programming has been in discussion for 32 years since 1987. According to Cao, the author, he in the paper introduces its development process, aiming to promote this new branch, attracting scholars home and abroad to join in ranks of the research, and helping to solve the three conjectures of Fuzzy Geometric Programming proposed by Cao [Three guess of fuzzy geometric programming. Vol. 147. Springer; 2012. p. 591–594. (Advances in intelligent and soft computing)]. At...
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FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 203–211 https://doi.org/10.1080/16168658.2019.1643078 The Origin and Development of Fuzzy Geometric Programming a,b b Bing-yuan Cao and Pei-hua Wang a b Foshan University, Foshan, People’s Republic of China; Guangzhou Vocational and Technical University of Science and Technology, Guangzhou , People’s Republic of China ABSTRACT KEYWORDS Geometric programming; Fuzzy Geometric Programming has been in discussion for 32 years development process; past; since 1987. According to Cao, the author, he in the paper introduces present and future; its development process, aiming to promote this new branch, attract- conjecture ing scholars home and abroad to join in ranks of the research, and helping to solve the three conjectures of Fuzzy Geometric Program- ming proposed by Cao [Three guess of fuzzy geometric program- ming. Vol. 147. Springer; 2012. p. 591–594. (Advances in intelligent and soft computing)]. At present, many books and teaching materials of it have been translated into Persian with some published. 1. Introduction From introduction of its generation, application and achievements, it is confirmed that it has developed into a new branch of discipline. Firstly, we in this paper trace back the origin of fuzzy geometric programming, besides we introduce its present situation, and finally, according to its present situation, we look forward to its development prospect, and once again put forward three conjectures of fuzzy geometric programming, which provides a difficult and promising path for the development of this branch. 2. The Past of Geometric Programming In 1987, B. Y. Cao proposed the fuzzy geometric programming (GP) theory in International Fuzzy Systems Association (IFSA) Congress for the first time [1,2]; its general form is (GP) min g ˜ (x) s.t. g (x)  1(1  i  p ), g (x)  1 (p + 1  i  p), x > 0 CONTACT Pei-hua Wang phwang321@163.com © 2019 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 204 B.-Y. CAO AND P.-H. WANG it is a reversed posynomial GP, where all of g (x)(0  i  p) are posynomial functions of variable x with g (x) = v (x); here ik k=1 ⎪ ikl c x , (1  k  J ;0  i  p ) ⎪ ik l=1 v (x) = (1) ik ⎪ −γ ikl c x , (1  k  J ; p + 1  i  p) ik i ⎩ l l=1 is a monomial function of variable x, variable x = (x , x , ... , x ) , and coefficients c > 0, 1 2 n ik exponents γ (1 ≤ k ≤ J ,0 ≤ i ≤ p,1 ≤ l ≤ n) are arbitrary real numbers. ikl i We have done a lot of work in the research of fuzzy GP [3–12], putting forward its orig- inal algorithm, and using fuzzy arithmetic geometric inequality and setting value theory to obtain its dual form and dual algorithm. Then, we proposed the geometric program- ming with a Fuzzy coefficient and Fuzzy variable and a multi-objective fuzzy geometric programming model [10,13–21]. At the same time, it is applied in power system, man- agement optimization and decision-making [22–27]. Based on the published fuzzy GP, another monograph of Optimal Models and Methods with Fuzzy Quantities was published in Springer in 2010, in which he introduced the work, such as GP models and algorithms with fuzzy coefficients and fuzzy variables [28,29]. The author has been awarded the National Natural Science Foundation of China for three consecutive times: (1) Fuzzy generalized GP decision model and method, 2008–2010; (2) Universal fuzzy GP and optimization techniques in management, 2003–2005; (3) Theory and method of fuzzy GP in power system management, 1997–1999; and has achieved rich results. In 2002, Kluwer Academic Publishers published a series of fuzzy GP in the Applied Mathematics series. In 2005, Fuzzy GP was awarded the third prize of Guangdong Science and Technology Award. The American Mathematical Review MathSciNet? and the West Ger- man Mathematical Digest Zentralblatt MATH commented on its original research. In 1993, Liu Yingming, an academician of the Chinese Academy of Sciences, called this work an international frontier. In 2004, Academician Wang Zikun, an academician of the Chinese Academy of Sciences, wrote an article in the Science Bulletin, saying that the monograph was at a high level in academic value with his identification of fuzzy GP reaching the inter- national advanced level. In 2007, Guo Bolin, another academician of the Chinese Academy of Sciences, who served as the director of the appraisal committee, identified this work as the advanced level of the international frontier, saying that it partially reached the leading level. In 2008, specialists in the international academic circle confirmed that the fuzzy GP proposed by the applicant is the seventh kind of optimization model [30]. While studying the theory and application of fuzzy GP, Cao and others are actively expanding in other direction. In 2001, the author proposed the extension GP. In 2005, Luo Dang proposed the gray positive GP. In 2009, the author proposed the rough posynomial GP. In 2014, Cao’s doc- toral student Zeinab Kheiri wrote the doctoral thesis with intuitionistic fuzzy posynomial GP [31–36]. In 2005, the author and his doctoral student Yang Jihui, in the IEEE Fuzzy System Annual Meeting held in the United States, first proposed the fuzzy relation GP, and its form was FUZZY INFORMATION AND ENGINEERING 205 described as follows: (PGPF) min z(x) = c x (2) k=1 j=1 s.t. A ◦ x = b, where A = (a ) , x = (x ) , b = (b ) , a , x , b ∈ [0, 1], c , γ ∈ R, c > 0, i ∈ I = ij m×n j n×1 i m×1 ij j i k k k {1, 2, ... , m}, j ∈ J ={1, 2, ... , n}, k ∈ K ={1, 2, ... , p}, and for given j ∈ J, γ (k ∈ K) are either all non-positive real numbers or all non-negative real numbers. Without loss of generality, we assume that problem (2) satisfies the following inequalities: 1  b  b  ···  b  0. 1 2 m Otherwise, rearrange the components of b in decreasing order and adjust the rows of A according to b [37–45]. So far, research in this direction has begun to heat up. 3. The Present of Geometric Programming From 30th July to 1st August 2016, Cao Bingyuan’s students, family and relatives gathered in the Applied Mathematics Conference Room on the 7th floor of the Computer Experiment Building of Guangzhou University to celebrate the 30 years of fuzzy geometry planning and 40 years of his teaching job, and published a conference proceeding by Chinese Science and Education Press in 2019. The book, containing the main articles of Cao and his students, has been published in the fuzzy GP for 30 years, reporting their projects, achievements and rewards. At present, the research on fuzzy relational linear programming and fuzzy rela- tional GP is becoming a hot topic. Nearly 100 papers have been published in the magazine; recently, they have headed to some new research directions [46–49], and Cao’s book of fuzzy relationship programming as well as the book of Indian scholars’ fuzzy relational GP and its application will be published by Springer-Nature. Now Cao’s colleagues and he are already preparing to collaborate with the team on optimized secured sharing of documents, which was proposed in Jana Wyżykowskiego Uni- versity, Poland, on fuzzy relationships and their programming. At the same time, the three conjectures of fuzzy GP proposed in 2012 still remain to be solved. They are expanding our propaganda and taking strong measures to attract more scholars to discuss them. And they will use the 9th International Information and Engineering Conference of Kish Island to prepare for the establishment of the International Fuzzy Information and Engineering Association. We will use the EI collection and ISC conference proceedings and the maga- zine ‘Operational Management and Fuzzy Mathematics’ to exchange ideas. The excellently chosen papers in this direction will be published in the magazine Fuzzy Information and Engineering, which is included in the Web of Sciences. At present, copies of works in Fuzzy geometric programming researches have been pub- lished by famous Springer publishing house, including essays on Fuzzy geometric program- ming, Fuzzy relational geometric programming and Fuzzy programming. Besides, China Science and Education Press and Fuzzy Information and Engineering Journal will strongly support the development in this direction. Two monographs [29,30] will be translated into Persian language for publication (Figure 1). 206 B.-Y. CAO AND P.-H. WANG Figure 1. Book titled ‘Optimal models and methods with fuzzy quantity’ has been translated into Persian language publishing. The world’s first high school fuzzy mathematics textbook ‘Fuzzy sets theory preliminary’ was officially published in February 2018 by the world-famous publishing houses Springer- Nature and China Science and Education Publishing House, which draws strong attention of scholars and readers from all over the world. In April this year, it was selected as the fourth of the eight most popular teaching materials of Springer-Nature, which exerts a great international influence (Figure 2). 4. The Future of Geometric Programming For 32 years, we have witnessed the development of fuzzy GP. At present, its research has entered a critical phrase. More research scholars are involved in it and infiltration fields are constantly expanding, and its research needs more talents to participate in and sup- port for. The three conjectures we have to solve are (i) the local optimal (satisfactory) FUZZY INFORMATION AND ENGINEERING 207 Figure 2. ‘Fuzzy sets theory preliminary’ has been selected as the fourth of the eight most popular textbooks in Springer-Nature. solution to fuzzy GP is still its global optimal (satisfactory) solution; (ii) after replacing the operator (+, ·) in (1) with other logical operators, the fuzzy relational still holds with its dual programming (2) established; (iii) confirmation of the existence of fuzzy GP taxon- omy and identification. As for the above three, they will continue to organize the team to deal with them, strive to make a breakthrough in theory, further find the background in the application, and make arduous efforts to establish a branch of the fuzzy geometric programming. The fuzzy GP will attract all of us to further research because many aspects remain untouched. In the basic field, we shall consider the following topics. (i) Fuzzy reverse GP, including a GP problem with mixed sign-terms, is much more com- plex than the fuzzy convex (resp. concave) GP and fuzzy PGP ; we want to continue to explore their properties. (ii) Fuzzy fractionation, extension, gray, and rough GP still need to be studied. (iii) GPs with intuitionistic fuzzy coefficients and fuzzy variables have yet to be further refined and expanded. (iv) Further solve real-world problems paradox with fuzzy GP. (v) Solving fuzzy relation GP. (vi) Explore GP with discrete variables and coefficients. (vii) Fuzzy GP in application in BitTorrent-like Peer-to-Peer file sharing system. (viii) GP problem subject to max-product fuzzy relation inequalities. The local optimal solu- tion of the GP problem subject to fuzzy relation inequalities is also its global optimal solution. (ix) Study of fuzzy GP classification. (x) The export of fuzzy GP’s genetic algorithm. 208 B.-Y. CAO AND P.-H. WANG Figure 3. Book titled ‘Application of fuzzy mathematics and systems’ in Persian language publishing Copyright Export Signing Ceremony Site. On 26 April 2019, the signing ceremony for the Persian copyright export of the monograph ‘Applied fuzzy mathematics and systems’ was jointly held by Shokoh-e-Sahel Publishing House and Scientific Publishing at the China Guest of Honor booth at the Iran International Book Fair. The translation project of this book has been included in the national ‘the belt and road initiative’ Silk Road Fund Plan applied by Science Press (Figure 3). Colleagues are welcome to participate and work hard to establish a new branch of fuzzy GP. 5. Conclusion The research on the theory and algorithm to Fuzzy geometric programming has achieved phased results, and the application background was first found in power systems. The monographs and applications of Fuzzy Optimization have been and are being disseminated in the world through Persian, English, Chinese languages, etc. As long as scholars in many other countries persist in their research and experiments, the three conjectures of Fuzzy geometric programming [50] could be overcome. Acknowledgments This article was presented at the 9th International Conference of Fuzzy Information and Engineering. Disclosure statement No potential conflict of interest was reported by the authors. FUZZY INFORMATION AND ENGINEERING 209 Funding The work was supported by the Natural Science Foundation of Guangdong Province (No. 2016A030313552). Notes on contributors Bing-yuan Cao, Changde, Hunan is second-level Chair Professor of Lingnan of Foshan University and Dean & Professor of School of Finance and Economics, Guangzhou Vocational and Technical Uni- versity of Science and Technology and Doctoral & Postdoctoral Supervisor of School of Guangzhou University, China. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Apr 3, 2019

Keywords: Geometric programming; development process; past; present and future; conjecture

References