Skip to main content

Advertisement

Springer Nature Link
Log in

Resolution of the sec-max-product fuzzy relation inequalities system and its lexicographic minimum solution

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper we consider the wireless Server-to-Client (S2C) network system with server peers and user (client) peers. The server peers provide the data resources to the user peers. In order to describe the wireless S2C network system in which there exists at least one candidate server for each user, we define and investigate the so-called sec-max-product fuzzy relation inequalities. Basic properties and the structure of the solution set have been studied for the sec-max-product fuzzy relation inequalities system. Moreover, the optimal solution under a lexicographic order, namely the lexicographic minimum solution, is further proposed and studied in the sec-max-product system, for minimizing all the variables under some fixed priority grade. An efficient resolution algorithm is developed for computing the lexicographic minimum solution of the sec-max-product system, with several illustrative examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

My manuscript has no associated data.

References

  • Barbarà S (1988) Maximin, leximin, and the protective criterion: characterizations and comparisons. J Econ Theory 46:34–44

    MathSciNet  Google Scholar 

  • Bartl E, Belohlavek R (2015) Hardness of solving relational equations. IEEE Trans Fuzzy Syst 23(6):2435–2438

    Google Scholar 

  • Bouveret S, Lematre M (2009) Computing leximin-optimal solutions in constraint networks. Artif Intell 173:343–364

    MathSciNet  Google Scholar 

  • Bräuning M, Hüllermeier E, Keller T, Glaum M (2017) Lexicographic preferences for predictive modeling of human decision making: a new machine learning method with an application in accounting. Eur J Oper Res 258:295–306

    MathSciNet  Google Scholar 

  • Chen L, Wang P-P (2002) Fuzzy relation equations (I): the general and specialized solving algorithms. Soft Comput 6:428–435

    Google Scholar 

  • Chiu Y-L, Guu S-M, Yu J, Wu Y-K (2019) A single-variable method for solving min-max programming problem with addition-min fuzzy relational inequalities. Fuzzy Optim Decis Making 18:433–449

    MathSciNet  Google Scholar 

  • Cococcioni M, Pappalardo M, Sergeyev YD (2018) Lexicographic multi-objective linear programming using grossone methodology: theory and algorithm. Appl Math Comput 318:298–311

    Google Scholar 

  • Cornejo M, Lobo D, Medina J (2019) On the solvability of bipolar max-product fuzzy relation equations with the product negation. J Comput Appl Math 354:520–532

    MathSciNet  Google Scholar 

  • Cornejo M, Lobo D, Medina J (2019) Bipolar fuzzy relation equations systems based on the product t-norm. Math Methods Appl Sci 30:5779–5793

    MathSciNet  Google Scholar 

  • Deschamps R, Gevers L (1978) Leximin and utilitarian rules: a joint characterization. J Econ Theory 17:143–163

    MathSciNet  Google Scholar 

  • Di Nola A, Russo C (2007) Lukasiewicz transform and its application to compression and reconstruction of digital images. Inf Sci 177:1481–1498

    Google Scholar 

  • Dubois D, Fortemps P (1999) Computing improved optimal solutions to max-min flexible constraint satisfaction problems. Eur J Oper Res 118:95–126

    Google Scholar 

  • Dubois D, Fargier H, Prade H (1996) Refinements of the maximin approach to decision-making in fuzzy environment. Fuzzy Sets Syst 81:103–122

    MathSciNet  Google Scholar 

  • Fang S-C, Li G (1999) Solving fuzzy relation equations with a linear objective function. Fuzzy Sets Syst 103:107–113

    MathSciNet  Google Scholar 

  • Fishburn PC (1974) Lexicographic orders, utilities and decision rules: a survey. Manage Sci 20:1442–1471

    MathSciNet  Google Scholar 

  • Ghodousian A (2019) Optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by Dubois-Prade family of t-norms. Inf Sci 503:291–306

    MathSciNet  Google Scholar 

  • Ghodousian A, Babalhavaeji A (2018) An efficient genetic algorithm for solving nonlinear optimization problems defined with fuzzy relational equations and max-Lukasiewicz composition. Appl Soft Comput 69:475–492

    Google Scholar 

  • Ghodousian A, Khorram E (2008) Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition. Inf Sci 178:501–519

    MathSciNet  Google Scholar 

  • Ghodousian A, Raeisian Parvari M (2017) A modified PSO algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints (FRI-FC). Inf Sci 418–419:317–345

    Google Scholar 

  • Ghodousian A, Naeeimi M, Babalhavaeji A (2018) Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois–Prade family of t-norms. Comput Ind Eng 119:167–180

    Google Scholar 

  • Guo H-M, Zheng C-F, Zhu T-X, Lin H-T, Yang X-P (2018) Min-product fuzzy relation inequalities with application in supply chain. In: Proceedings of the 2018 14th international conference on natural computation, fuzzy systems and knowledge discovery (ICNC-FSKD), Huangshan, China, pp 554–560

  • Guu S-M, Wu Y-K (2010) Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint. Fuzzy Sets Syst 161:285–297

    MathSciNet  Google Scholar 

  • Guu S-M, Wu Y-K (2017) A Linear programming approach for minimizing a linear function subject to fuzzy relational inequalities with addition-min composition. IEEE Trans Fuzzy Syst 25(4):985–992

    Google Scholar 

  • Guu S-M, Yu J, Wu Y-K (2018) A two-phase approach to finding a better managerial solution for systems with addition-min fuzzy relational inequalities. IEEE Trans Fuzzy Syst 16(4):2251–2260

    Google Scholar 

  • Hedayatfar B, Molai AA, Aliannezhadi S (2019) Separable programming problems with the max-product fuzzy relation equation constraints. Iran J Fuzzy Syst 16(1):1–15

    MathSciNet  Google Scholar 

  • Li PK, Fang SC (2008) On the resolution and optimization of a system of fuzzy relational equations with sup-t composition. Fuzzy Optim Decis Making 7:169–214

    MathSciNet  Google Scholar 

  • Li J-X, Yang S-J (2012) Fuzzy relation inequalities about the data transmission mechanism in BitTorrent-like Peer-to-Peer file sharing systems. In: Proceedings of the 2012 9th international conference on fuzzy systems and knowledge discovery, FSKD , pp 452–456

  • Lin J-L, Wu Y-K, Guu S-M (2011) On fuzzy relational equations and the covering problem. Inf Sci 181:2951–2963

    MathSciNet  Google Scholar 

  • Lin H, Yang X, Guo H, Zheng C, Yang X (2019) Maximin optimization problem subject to min-product fuzzy relation inequalities with application in supply and demand scheme. Complexity 4960638

  • Loetamonphong J, Fang S-C (2001) Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets Syst 118:509–517

    MathSciNet  Google Scholar 

  • Loia V, Sessa S (2005) Fuzzy relation equations for coding/decoding processes of images and videos. Inf Sci 171:145–172

    MathSciNet  Google Scholar 

  • Markovskii AV (2005) On the relation between equations with max-product composition and the covering problem. Fuzzy Sets Syst 153:261–273

    MathSciNet  Google Scholar 

  • Matusiewicz Z, Drewniak J (2013) Increasing continuous operations in fuzzy max-\({*}\) equations and inequalities. Fuzzy Sets Syst 232:120–133

    MathSciNet  Google Scholar 

  • Molai AA (2010) Fuzzy linear objective function optimization with fuzzy valued max-product fuzzy relation inequality constraints. Math Comput Model 51:1240–1250

    MathSciNet  Google Scholar 

  • Molai AA (2013) Resolution of a system of the max-product fuzzy relation equations using L\(\circ \)U-factorization. Inf Sci 234:86–96

    MathSciNet  Google Scholar 

  • Molai AA (2014) A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints. Comput Ind Eng 72:306–314

    Google Scholar 

  • Nobuhara H, Bede B, Hirota K (2006) On various eigen fuzzy sets and their application to image reconstruction. Inf Sci 176:2988–3010

    MathSciNet  Google Scholar 

  • Nobuhara H, Pedrycz W, Sessa S, Hirota K (2006) A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations. Inf Sci 176:2526–2552

    Google Scholar 

  • Peeva K, Kyosev Y (2007) Algorithm for solving max-product fuzzy relational equations. Soft Comput 11(7):593–605

    Google Scholar 

  • Salles RM, Barria JA (2008) Lexicographic maximin optimisation for fair bandwidth allocation in computer networks. Eur J Oper Res 185:778–794

    Google Scholar 

  • Wang PZ, Zhang DZ, Sanchez E, Lee ES (1991) Latticized linear programming and fuzzy relation inequalities. J Math Anal Appl 159(1):72–87

    MathSciNet  Google Scholar 

  • Wu Y-K (2007) Optimization of fuzzy relational equations with max-av composition. Inf Sci 177:4216–4229

    MathSciNet  Google Scholar 

  • Wu Y-K, Guu S-M (2004) A note on fuzzy relation programming problems with max-strict-t-norm composition. Fuzzy Optim Decis Making 3:271–278

    MathSciNet  Google Scholar 

  • Wu Y-K, Guu S-M (2005) Minimizing a linear function under a fuzzy max-min relational equation constraint. Fuzzy Sets Syst 150:147–162

    MathSciNet  Google Scholar 

  • Wu Y-K, Guu S-M (2008) An efficient procedure for solving a fuzzy relation equation with max-Archimedean t-norm composition. IEEE Trans Fuzzy Syst 16:73–84

    Google Scholar 

  • Wu Y-K, Guu S-M, Liu JY-C (2002) An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Trans Fuzzy Syst 10(4):552–558

    Google Scholar 

  • Xiong Q-Q, Wang X-P (2012) Fuzzy relational equations on complete Brouwerian lattices. Inf Sci 193:141–152

    MathSciNet  Google Scholar 

  • Yang S-J (2014) An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition. Fuzzy Sets Syst 255:41–51

    MathSciNet  Google Scholar 

  • Yang X-P (2017) Optimal-vector-based algorithm for solving min-max programming subject to addition-min fuzzy relation inequality. IEEE Trans Fuzzy Syst 25(5):1127–1140

    Google Scholar 

  • Yang X-P (2020) Solutions and strong solutions of min-product fuzzy relation inequalities with application in supply chain, Fuzzy Sets and Systems. Fuzzy Sets Syst 384:54–74

    Google Scholar 

  • Yang X-P, Zhou X-G, Cao B-Y (2016) Min-max programming problem subject to addition-min fuzzy relation inequalities. IEEE Trans Fuzzy Syst 24:1–9

    Google Scholar 

  • Yang X-P, Zhou X-G, Cao B-Y (2016) Latticized linear programming subject to max-product fuzzy relation inequalities with application in wireless communication. Inf Sci 358–359:44–55

    Google Scholar 

  • Yang X-P, Zheng G-Z, Zhou X-G, Cao B-Y (2017) Lexicography minimum solution of fuzzy relation inequalities: applied to optimal control in P2P file sharing system. Int J Mach Learn Cybern 8(5):1555–1563

    Google Scholar 

  • Yang X-Y, Lin H-T, Zhou X-G, Cao B-Y (2018) Addition-min fuzzy relation inequalities with application in BitTorrent-like Peer-to-Peer file sharing system. Fuzzy Sets Syst 343:126–140

    MathSciNet  Google Scholar 

  • Yang X-P, Yuan D-H, Cao B-Y (2018) Lexicographic optimal solution of the multi-objective programming problem subject to max-product fuzzy relation inequalities. Fuzzy Sets Syst 341:92–112

    MathSciNet  Google Scholar 

  • Zhou X, Zhong X, Lin H, Qin Z, Yang X (2018) Lexicographic maximum solution of min-product fuzzy relation inequalities for modeling the optimal pricing with fixed priority grade in supply chain. IEEE Access 6:71306–71316

    Google Scholar 

Download references

Funding

This study was funded by the National Natural Science Foundation of China (12471441, 12271132), the Guangdong Basic and Applied Basic Research Foundation (2024A1515010532, 2023A1515011093, 2022A1515011460) and the characteristic innovation project of Guangdong Universities (2022KTSCX074, 2023KQNCX041, 2024ZDZX1027).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, 524048, China

    Jianjun Qiu

  2. School of Humanities Education, Guangdong Innovative Technical College, Dongguan, 523960, China

    Guocheng Zhu

  3. School of Mathematical Sciences, Sichuan Normal University, Chengdu, 610066, China

    Qianyu Shu

  4. School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China

    Xiaopeng Yang

Authors
  1. Jianjun Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guocheng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qianyu Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaopeng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaopeng Yang.

Ethics declarations

Conflict of interest

Xiaopeng Yang has received research grants from the National Natural Science Foundation of China. The authors declare that they have no Conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by the National Natural Science Foundation of China (12471441, 12271132), the Guangdong Basic and Applied Basic Research Foundation (2024A1515010532, 2023A1515011093, 2022A1515011460) and the characteristic innovation project of Guangdong Universities (2022KTSCX074, 2023KQNCX041, 2024ZDZX1027).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, J., Zhu, G., Shu, Q. et al. Resolution of the sec-max-product fuzzy relation inequalities system and its lexicographic minimum solution. Comp. Appl. Math. 43, 439 (2024). https://doi.org/10.1007/s40314-024-02945-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-024-02945-7

Keywords

Mathematics Subject Classification

Search

Navigation