Skip to main content

On type-2 fuzzy weighted minimum spanning tree


This article investigates the notion of a minimum spanning tree (MST) of an undirected type-2 fuzzy weighted connected graph (UT2FWCG), where the edge weights are represented as discrete type-2 fuzzy variables. A modified type-2 fuzzy Borüvka’s algorithm (MT2FBA) is proposed to determine an MST of UT2FWCG. The effective weight of the MST is calculated, and its defuzzified value is compared with an equivalent MST with deterministic weights. The working principle of the algorithm includes ranking, addition operation and defuzzification of discrete type-2 fuzzy variables. The defuzzified effective weight of the MST is determined using the critical value (CV)-based type reduction technique and centroid method. The proposed algorithm is numerically illustrated with suitable examples. Moreover, the simulation results of nine random instances of UT2FWCG are also studied to analyze the proposed MT2FBA properly.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. Antonelli M, Bernardo D, Hagras H (2017) Multi-objective evolutionary optimization of type-2 fuzzy rule-based systems for financial data classification. IEEE Trans Fuzzy Syst 25(2):249–264

    Article  Google Scholar 

  2. Anusuya V, Sathya R (2014) A new approach for solving type-2 fuzzy shortest path problem. Ann Pure Appl Math 8(1):83–92

    Google Scholar 

  3. Assad A, Xu W (1992) The quadratic minimum spanning tree problem. Nav Res Logist 39(3):399–417

    MathSciNet  MATH  Article  Google Scholar 

  4. Blue M, Bush B, Puckett J (2002) Unified approach to fuzzy graph problems. Fuzzy Sets Syst 125:355–368

    MathSciNet  MATH  Article  Google Scholar 

  5. Borüvka O (1930) O jistém problem minimálním. Práce Moravské Přírodovĕdecké Společnosti V I(4):57–63

    Google Scholar 

  6. Chiang TC, Liu CH, Huang YM (2007) A near-optimal multicast scheme for mobile ad hoc networks using a hybrid genetic algorithm. Expert Syst Appl 33(3):734–742

    Article  Google Scholar 

  7. Dhamdhere K, Ravi R, Singh M (2005) On two-stage stochastic minimum spanning trees. In: Jünger M, Kaibel V (eds) Integer programming and combinatorial optimization IPCO 2005. lecture notes in computer science, vol 3509. Springer, Heidelberg, pp 321–334.

    Chapter  Google Scholar 

  8. Di Puglia PL, Guerriero F, Santos JL (2015) Dynamic programming for spanning tree problems: application to the multi-objective case. Optim Lett 9(3):437–450

    MathSciNet  MATH  Article  Google Scholar 

  9. Gao X, Jia L (2017) Degree constrained minimum spanning tree problem with uncertain edge weights. Appl Soft Comput 56:580–588

    Article  Google Scholar 

  10. Gao J, Lu M (2005) Fuzzy quadratic minimum spanning tree problem. Appl Math Comput 164(3):773–788

    MathSciNet  MATH  Google Scholar 

  11. Gower JC, Ross GJS (1969) Minimum spanning trees and single linkage cluster analysis. J R Statist Soc Ser C Appl Statist 18(1):54–64

    MathSciNet  Google Scholar 

  12. Ishii H, Shiode S, Nishida T, Namasuya Y (1981) Stochastic spanning tree problem. Discret Appl Math 3(4):263–273

    MathSciNet  MATH  Article  Google Scholar 

  13. Ishii H, Matsutomi T (1995) Confidence regional method of stochastic spanning tree problem. Math Comput Model 22(10):77–82

    MathSciNet  MATH  Article  Google Scholar 

  14. Itoh T, Ishii H (1996) An approach based on necessity measure to the fuzzy spanning tree problems. J Oper Res Soc Japan 39(2):247–257

    MathSciNet  MATH  Google Scholar 

  15. Janiak A, Kasperski A (2008) The minimum spanning tree problem with fuzzy costs. Fuzzy Optim Decis Mak 7(2):105–118

    MathSciNet  MATH  Article  Google Scholar 

  16. Kayacan E, Sarabakha A, Coupland S, John R, Khanesar MA (2018) Type-2 fuzzy elliptic membership functions for modeling uncertainty. Eng Appl Artif Intell 70:170–183

    Article  Google Scholar 

  17. Kruskal JB Jr (1956) On the shortest spanning subtree of a graph and traveling salesman problem. Proc Am Math Soc 7(1):48–50

    MathSciNet  MATH  Article  Google Scholar 

  18. Kumar R, Jha S, Singh S (2017) Shortest path problem in network with type-2 triangular fuzzy arc length. J Appl Res Ind Eng 4(1):1–7

    Google Scholar 

  19. Kundu P, Kar S, Maiti M (2014) Fixed charge transportation problem with type-2 fuzzy variables. Inf Sci 255:170–186

    MathSciNet  MATH  Article  Google Scholar 

  20. Lee S, Lee KH (2001) Shortest path problem in a type–2 weighted graph. J Korea Fuzzy Intell Syst Soc 11(6):528–531

    Google Scholar 

  21. Lee S, Lee KH, Lee D (2004) Comparison of type-2 fuzzy values with satisfaction function. Int J Uncert Fuzziness Knowl Based Syst 12(5):601–611

    MathSciNet  MATH  Article  Google Scholar 

  22. Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  23. Liu B, Liu Y-K (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans Fuzzy Syst 10(4):445–450

    Article  Google Scholar 

  24. Liu ZQ, Liu Y-K (2007) Fuzzy possibility space and type-2 fuzzy variable. In: FOCI 2007 IEEE symposium on foundations of computational intelligence, Honolulu, HI, USA pp. 616–621.

  25. Liu ZQ, Liu Y-K (2010) Type-2 fuzzy variables and their arithmetic. Soft Comput 14:729–747

    MATH  Article  Google Scholar 

  26. Majumder S (2019) Some network optimization models under diverse uncertain environments. Doctoral dissertation, National Institute of Technology Durgapur, Durgapur, West Bengal, India, arxiv.2103.08327

  27. Majumder S, Kar S, Pal T (2018) Mean-entropy model of uncertain portfolio selection problem. In: Mandal J, Mukhopadhyay S, Dutta P (eds) Multi-Objective Optimization. Springer, Singapore

    Google Scholar 

  28. Majumder S, Kar S, Pal T (2019) Rough-fuzzy quadratic minimum spanning tree problem. Expert Syst 36(2):1–29

    Article  Google Scholar 

  29. Majumder S, Kar S, Pal T (2019) Uncertain multi-objective Chinese postman problem. Soft Comput 23:11557–11572

    MATH  Article  Google Scholar 

  30. McDonald R, Pereira F, Ribarov K, Hajič J (2005) Non-projective dependency parsing using spanning tree algorithms. In: Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing, pp. 523–530. Association for Computational Linguistics, Stroudsburg, PA, USA.

  31. Mendel JM (2007a) Computing with words: Zadeh, Turing, Popper and Occam. IEEE Comput Intell Mag 2(4):10–17

    Article  Google Scholar 

  32. Mendel JM (2007b) Advances in type-2 fuzzy sets and systems. Inf Sci 177(1):84–110

    MathSciNet  MATH  Article  Google Scholar 

  33. Mendel JM, John RIB (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10(2):117–127

    Article  Google Scholar 

  34. Mendel JM, John RI, Liu FL (2006) Interval type-2 fuzzy logical systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821

    Article  Google Scholar 

  35. Nguyen T, Khosravi A, Creighton D, Nahavnadi S (2015) Medical data classification using interval type-2 fuzzy logic system and wavelets. Appl Soft Comput 30:812–822

    Article  Google Scholar 

  36. Öncan T (2007) Design of capacitated minimum spanning tree with uncertain cost and demand parameters. Inf Sci 177(20):4354–4367

    MathSciNet  Article  Google Scholar 

  37. Prim RC (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36(6):1389–1401

    Article  Google Scholar 

  38. Shukla AK, Muhuri PK (2019) Big data clustering with internal type-2 fuzzy Uncertainty modelling in gene expression datasets. Eng Appl Artif Intell 77:268–282

    Article  Google Scholar 

  39. Stam CJ, Tewarie CJ, Van Straaten ECW, Hillebrand A, Van Mieghem P (2014) The trees and the forest: Characterization of complex brain networks with minimum spanning trees. Int J Psychophysiol 92(3):129–138

    Article  Google Scholar 

  40. Torkestani JA (2013) Degree constrained minimum spanning tree problem: a learning automata approach. J Supercomput 64(1):226–249

    MATH  Article  Google Scholar 

  41. Torkestani JA, Meybodi MR (2012) A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs. J Supercomput 59(2):1035–1054

    Article  Google Scholar 

  42. Türkşen IB (2002) Type 2 representation and reasoning for CWW. Fuzzy Sets Syst 127(1):17–36

    MathSciNet  MATH  Article  Google Scholar 

  43. Qin R, Liu Y-K, Liu Z-Q (2011) Methods of critical value reduction for type-2 fuzzy variables and their applications. J Comput Appl Math 235(5):1454–1481

    MathSciNet  MATH  Article  Google Scholar 

  44. Xu Y, Uberbache EC (1997) 2D image segmentation using minimum spanning trees. Image vis Comput 15(1):47–57

    Article  Google Scholar 

  45. Yager RR (1981) A procedure for ordering fuzzy subsets of the unit interval. Inf Sci 24(2):143–161

    MathSciNet  MATH  Article  Google Scholar 

  46. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci 8(3):199–249

    MathSciNet  MATH  Article  Google Scholar 

  47. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28

    MathSciNet  MATH  Article  Google Scholar 

  48. Zhou J, Chen L, Wang K (2015) Path optimality conditions for minimum spanning tree problem with uncertain edge weights. Int J Uncert Fuzziness Knowl-Based Syst 23(1):49–71

    MathSciNet  MATH  Article  Google Scholar 

  49. Zhou J, Chen L, Wang K, Yang F (2016a) Fuzzy α-minimum spanning tree problem: definition and solutions. Int J Gen Syst 45(3):311–335

    MathSciNet  MATH  Article  Google Scholar 

  50. Zhou J, Yi X, Wang K, Liu J (2016b) Uncertain distribution-minimum spanning tree problem. Int J Uncert Fuzziness Knowl Based Syst 24(4):537–560

    MathSciNet  MATH  Article  Google Scholar 

Download references


The authors would like to express their gratitude and indebtedness to the Editor-in-Chief, Managing Editor, and the anonymous referees for their careful reading and valuable comments.

Author information



Corresponding author

Correspondence to Samarjit Kar.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Human and animal rights

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

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher's Note

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



We present some fundamental concepts of T1FS in this section.

A.1 Type-1 fuzzy variable

Zadeh (1975) introduced the possibility theory to deal with the uncertainty involved in the occurrence of an element in a fuzzy set. The membership function \({\mu }_{\tilde{A }}(x)\) of an element \(x\) in a fuzzy set \(\tilde{A }\) is then termed as degree of possibility that the element belongs to the set. In other words, the possibility of occurrence of an element \(x\) in \(\tilde{A }\) is characterized by a membership function \({\mu }_{\tilde{A }}(x)\), where \(x\in X\) and \({\mu }_{\tilde{A }}\left(x\right)\in \left[\mathrm{0,1}\right]\).

A type-1 fuzzy variable (T1FV) is defined as a function from the possibility space (\(U\), \(\mathcal{A}\),\(Pos\)) to the set of real numbers to describe fuzzy phenomena, where \(U\) is the non-empty set of points, \(\mathcal{A}\) is power set of \(U\). Furthermore, for a fuzzy event \(\left\{\tilde{\gamma }\in P\right\}, P\subset \) ℜ, the possibility measure \(\left(Pos\right)\) (Zadeh 1978), necessity measure \(\left(Nec\right)\) and credibility measure \(\left(Cr\right)\) (Liu and Liu 2002) are, respectively, defined in Eqs. (A1A3) as follows.

$${Pos}\left\{\tilde{\gamma }\in P\right\}={}_{x\in P}{}^{{sup}}{ \mu }_{\tilde{\gamma }}(x)$$
$${Nec}\left\{\tilde{\gamma }\in P\right\}=1-{}_{x\in {P}^{c}}{}^{{sup}}{ \mu }_{\tilde{\gamma }}(x)$$


$$Cr\left\{\tilde{\gamma }\in P\right\}=\frac{1}{2}\left({}_{x\in P}{}^{{sup}}{ \mu }_{\tilde{\gamma }}(x)+1-{}_{x\in {P}^{c}}{}^{{sup}}{ \mu }_{\tilde{\gamma }}(x)\right).$$

A.2 Fuzzy possibility measure

(Liu and Liu 2007) Let \(\widetilde{Pos }:\) \(\mathcal{A}\)\(\mathfrak{R}\left(\left[0,1\right]\right)\) be a set of functions defined on \(\mathcal{A}\) such that \(\{\tilde{{Pos}}({\varvec{A}})|\) \(\mathcal{A}\) \(\ni A \mathrm{atom}\}\) is a collection of mutually independent RFVs. \(\widetilde{Pos }\) is said to be the fuzzy possibility measure if it satisfies the following conditions: (i) \(\tilde{{Pos}}\left(\varnothing \right)=\tilde{0 }\) (ii) For any subclass \(\left\{{A}_{i}|i\in {\mathbb{I}}\right\}\mathrm{ of}\) \(\mathcal{A}\) (finite, countable or uncountable), \(\widetilde {{{~Pos}}}\left( {\bigcup\nolimits_{i \in } {{A_i}} } \right) = \begin{array}{*{20}{c}}{{{sup}}}\\{i \in }\end{array}\widetilde {{{Pos}}}\left( {{A_i}} \right)\). Then, the triplet \((U,\) \(\mathcal{A}\)\(,\widetilde{Pos}\)) is called fuzzy possibility space.

Remark A.1

Let \(U\) be the universe of discourse and \(\mathcal{A}\) the power set of \(U\); then, an atoms of \(\mathcal{A}\) are all single-point sets \(\left\{\delta \right\}, \delta \in U.\)

Remark A.2

(Liu and Liu 2007) If \({\mu }_{\tilde{{Pos}}\left(U\right)}\left(1\right)=1\) then \(\widetilde{Pos}\) becomes a regular fuzzy possibility measure, which is the generalization of scalar possibility measure, i.e., if \(B\in \)\(\mathcal{A}\) , then \(\widetilde{Pos}(B)\) is a crisp value within the interval \(\left[\mathrm{0,1}\right]\) instead of an RF, then \(\widetilde{Pos}\) is just a possibility measure.

Remark A.3

(Liu and Liu 2007) If \(U\) is a finite set, then \(\widetilde {{{Pos}}}\left( {\bigcup\nolimits_{i \in } {{A_i}} } \right) = \begin{array}{*{20}{c}}{{{sup}}}\\{1 \le i \le n}\end{array}\widetilde {{{Pos}}}\left( {{A_i}} \right)\) for any finite subclass \(\left\{{A}_{i}, i=\mathrm{1,2},\ldots ,n\right\}\) of \(\mathcal{A}\)

Centroid method

(Yager 1981) Centroid method is one of the defuzzification techniques of TIFVs. This method is used to obtain the center of area for continuous and discrete T1FVs, which are defined, respectively, in (A4) and (A5).

$${x}^{*}=\frac{\int x\times \mu \left(x\right)\mathrm{d}x}{\int \mu \left(x\right)\mathrm{d}x}$$


$${x}^{*}=\frac{\sum x\times \mu \left(x\right)}{\sum \mu \left(x\right)}.$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dan, S., Majumder, S., Kar, M.B. et al. On type-2 fuzzy weighted minimum spanning tree. Soft Comput 25, 14873–14892 (2021).

Download citation


  • Minimum spanning tree
  • Type-2 fuzzy variable
  • Satisfaction function
  • CV-based type reduction
  • Defuzzification
  • Borüvka’s algorithm