Abstract
A rough set, an extension of a crisp set, is a mathematical tool to understand and model uncertainty without much prior information, additional adjustments or pre-defined membership functions. To manipulate the subjectivity and vagueness of decision-making problems, rough models provide more objective description of given information using upper and lower approximations. In this research paper, we study the absurdity and falsity of existing definition of rough graph. Based on rough relations, we introduce the concepts of rough graphs and rough digraphs and establish certain formulae, lower and upper bounds of color energy of rough graphs. Using D numbers, rough weights and rough entropy weights, we develop rough D-TOPSIS method which incorporates the capability to analyze uncertain and vague information without additional assumptions. We study the importance of rough information for the evaluation of water requirement in agricultural farming, investment analysis in organic and inorganic farming systems and illegal communication networks.
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References
Adiga C, Sampathkumar E, Sriraj MA (2013) Color energy of a graph. Proc Jangjeon Math Soc 16(3):335–351
Akram M, Luqman A (2020) Granulation of ecological networks under fuzzy soft environment. Soft Comput 24:11867–11892
Akram M, Zafar F (2020) Hybrid soft computing models applied to graph theory. Studies in fuzziness and soft computing. Springer, Berlin. https://doi.org/10.1007/978-3-030-16020-3
Akram M, Shumaiza Smarandache F (2018) Decision-making with bipolar neutrosophic TOPSIS and bipolar neutrosophic ELECTRE-I. Axioms 7(2:33. https://doi.org/10.3390/axioms7020033
Akram M, Habib A, Alcantud JCR (2020) An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers. Neural Comput Appl. https://doi.org/10.1007/s00521-020-05034-y
Alghamdi MA, Alshehri NO, Akram M (2018) Multi-criteria decision-making methods in bipolar fuzzy environment. Int J Fuzzy Syst 20(6):2057–2064
Anjali N, Mathew S (2013) Energy of a fuzzy graph. Ann Fuzzy Maths Inform 6:455–65
Chang TW, Lo HW, Chen KY, Liou JJ (2019) A novel FMEA model based on rough BWM and rough TOPSIS-AL for risk assessment. Mathematics 7(10):874. https://doi.org/10.3390/math7100874
Chen CT (2000) Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst 114(1):1–9
Deng Y (2012) \(D\) numbers: theory and applications. J Inf Comput Sci 9(9):2421–2428
Fei L, Hu Y, Xiao F, Chen L, Deng Y (2016) A modified topsis method based on \(D\) numbers and its applications in human resources selection. Math Probl Eng 2016:14 (Article ID 6145196)
Feng F (2011) Soft rough sets applied to multicriteria group decision making. Ann Fuzzy Math Inform 2(1):69–80
Feng F, Liu X, Leoreanu-Fotea V, Jun YB (2011) Soft sets and soft rough sets. Inf Sci 181(6):1125–1137
Gutman I (1978) The energy of a graph. Ber Math Stat Sekt Forschungsz Graz 103:1–22
Gutman I (2001) The energy of a graph: old and new results. Algebraic combinatorics and applications. Springer, Berlin, pp 196–211
He T, Shi K (2006) Rough graph and its structure. J Shandong Univ 41(6):46–50
Hwang CL, Yoon K (1981) Methods for multiple attribute decision making. Multiple attribute decision making. Springer, Berlin, pp 58–191
Jiang H, Zhan J, Sun B, Alcantud JCR (2020) An MADM approach to covering-based variable precision fuzzy rough sets: an application to medical diagnosis. Int J Mach Learn Cybern 11:2181–2207. https://doi.org/10.1007/s13042-020-01109-3
Joshi D, Kumar S (2014) Intuitionistic fuzzy entropy and distance measure based TOPSIS method for multi-criteria decision making. Egypt Inform J 15(2):97–104
Leoreanu-Fotea V (2008) The lower and upper approximations in a hypergroup. Inf Sci 178(18):3605–3615
Li D-F, Nan J-X (2011) Extension of the TOPSIS for multi-attribute group decision making under Atanassov IFS environments. Int J Fuzzy Syst Appl (IJFSA) 1(4):47–61
Li J, Fang H, Song W (2019) Sustainable supplier selection based on SSCM practices: a rough cloud TOPSIS approach. J Clean Prod 222:606–621
Ma X, Akram M, Zahid K, Alcantud JCR (2020a) Group decision-making framework using complex Pythagorean fuzzy information. Neural Comput Appl. https://doi.org/10.1007/s00521-020-05100-5
Ma X, Zhan J, Sun B, Alcantud JCR (2020b) Novel classes of coverings based multigranulation fuzzy rough sets and corresponding applications to multiple attribute group decision-making. Artif Intell Rev. https://doi.org/10.1007/s10462-020-09846-1
Mahapatra T, Ghorai G, Pal M (2020) Fuzzy fractional coloring of fuzzy graph with its application. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s10462-020-09846-1
Nagarani A, Vimala S (2017) Energy of fuzzy regular and graceful graphs. Asian Res J Math 4(2):1–8
Nicoletti MC, Uchôa JQ, Baptistini MT (2001) Rough relation properties. Int J Appl Math Comput Sci 11(3):621–635
Poulik S, Ghorai G (2020) Detour g-interior nodes and detour g-boundary nodes in bipolar fuzzy graph with applications. Hacettepe J Math Stat 49(1):106–119
Poulik S, Ghorai G (2020) Certain indices of graphs under bipolar fuzzy environment with applications. Soft Comput 24:5119–5131
Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11(5):341–356
Pawlak Z (1996) Rough sets, rough relations and rough functions. Fundam Inform 27(2–3):103–108
Sarwar M, Akram M, Zafar F (2018) Decision making approach based on competition graphs and extended TOPSIS method under bipolar fuzzy environment. Math Comput Appl 23:68. https://doi.org/10.3390/mca23040068
Sarwar M, Akram M, Alshehri NO (2018) A new method to decision-making with fuzzy competition hypergraphs. Symmetry 10:404. https://doi.org/10.3390/sym10090404
Sarwar M, Akram M, Ali U (2020) Double dominating energy of \(m\)-polar fuzzy graphs. J Intell Fuzzy Syst 38(2):1997–2008
Sharbaf SR, Fayazi F (2014) Laplacian energy of a fuzzy graph. Iran J Math Chem 5(1):1–10
Skiena S (1990) Implementing discrete mathematics: combinatorics and graph theory with mathematica. Addison-Wesley, Reading
Song W, Ming X, Wu Z, Zhu B (2014) A rough TOPSIS approach for failure mode and effects analysis in uncertain environments. Qual Reliab Eng Int 30(4):473–486
Suder A, Kahraman C (2018) Multiattribute evaluation of organic and inorganic agricultural food investments using fuzzy TOPSIS. Technol Econ Dev Econ 24(3):844–858
Sudha AS, Anitha S (2017) Applying Topsis method for evaluating the water requirement of agricultural crops. Int J Future Revolut Comput Sci Commun Eng 3(8):197–202
Wan N, Li L, Ye C, Wang B (2019) Risk assessment in intelligent manufacturing process: a case study of an optical cable automatic arranging robot. IEEE Access 7:105892–901
Zafar F, Akram M (2018) A novel decision-making method based on rough fuzzy information. Int J Fuzzy Syst 20(3):1000–1014
Zhan J, Jiang H, Yao Y (2020) Three-way multi-attribute decision-making based on outranking relations. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2020.3007423
Zhan J, Sun B, Zhang X (2020) PF-TOPSIS method based on CPFRS models: an application to unconventional emergency events. Comput Ind Eng 139:106192
Zhang K, Zhan J, Wu WZ (2020) On multi-criteria decision-making method based on a fuzzy rough set model with fuzzy \(\alpha -\)neighborhoods. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2020.3001670
Zhang K, Zhan J, Wu WZ (2020) Novel fuzzy rough set models and corresponding applications to multi-criteria decision-making. Fuzzy Sets Syst 383:92–126
Zhang L, Zhan J, Yao Y (2020) Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: an application to biomedical problems. Inf Sci 517:315–339
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Communicated by Marcos Eduardo Valle.
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Sarwar, M. Decision-making approaches based on color spectrum and D-TOPSIS method under rough environment. Comp. Appl. Math. 39, 291 (2020). https://doi.org/10.1007/s40314-020-01284-7
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DOI: https://doi.org/10.1007/s40314-020-01284-7
Keywords
- Rough graph
- Rough digraph
- Spectrum
- Color energy
- Rough D-TOPSIS method
- Agricultural farming investments
- Illegal communication networks