Soft Computing

, Volume 22, Issue 3, pp 1003–1012 | Cite as

Uncertain weighted dominating set: a prototype application on natural disaster relief management

  • Mehdi Djahangiri
  • Alireza Ghaffari-Hadigheh
Methodologies and Application


Indeterminacy is an intrinsic characteristics of real-world data. Where they originate from credible experiments, probability theory is a robust tool to manipulate this type of indeterminacy. However, this is not always the case, and referring to the domain expert belief is an alternative approach. Baoding Liu initiated an axiomatic basis of uncertainty theory to answer this kind of indeterminacy. Dominating set with its different versions has a wide range of applications in many fields, while the practice suffers indeterminacy with no reliable data in most cases. In this paper, we investigate the minimum weighted dominating set with indeterministic weights in two cases. The weights in the first one have probability distribution and in the other one uncertainty distribution which they are based on the belief degree of the domain expert. In both cases, the objective function of model is not defined. To overcome this difficulty, based on probability and uncertainty theory, deterministically two different models are constructed. The first model considers an \(\alpha \)-chance method, and the second exploits the expected value of the uncertain variables. Both models are converted to deterministic ones resulting to the so-called \(\alpha \)-minimum weighted dominating set, and the uncertain minimum weighted dominating set, respectively. A prototype application in earthquake relief management is provided, and the performance of models is experimented in a concrete illustrative example.


Uncertainty theory Uncertain programming \(\alpha \)-minimum weighted dominating set Uncertain minimum weighted dominating set 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.

Informed consent

Further, the research involves no human participants and animals and consequently no need for informed consent.


  1. Alber J, Betzler N, Niedermeier R (2006) Experiments on data reduction for optimal domination in networks. Ann Oper Res 146(1):105–117MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chen L, Peng J, Zhang B, Li S (2014) Uncertain programming model for uncertain minimum weight vertex covering problem. J Intell Manuf 1–8. doi: 10.1007/s10845-014-1009-1. ISSN: 1572-8145
  3. Cooper C, Klasing R, Zito M (2005) Lower bounds and algorithms for dominating sets in web graphs. Internet Math 2(3):275–300MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dai F, Wu J (2005) On constructing \(k\)-connected \(k\)-dominating set in wireless networks. In: 19th IEEE international parallel and distributed processing symposium. IEEE, p 81aGoogle Scholar
  5. Gao Y (2011) Shortest path problem with uncertain arc lengths. Comput Math Appl 62(6):2591–2600MathSciNetCrossRefzbMATHGoogle Scholar
  6. Han S, Peng Z, Wang S (2014) The maximum flow problem of uncertain network. Inf Sci 265:167–175MathSciNetCrossRefzbMATHGoogle Scholar
  7. Haynes TW, Hedetniemi S, Slater P (1998) Fundamentals of domination in graphs. CRC Press, Boca RatonzbMATHGoogle Scholar
  8. Kelleher LL, Cozzens MB (1988) Dominating sets in social network graphs. Math Soc Sci 16(3):267–279MathSciNetCrossRefzbMATHGoogle Scholar
  9. Liu B (2002) Theory and practice of uncertain programming. Springer, BerlinCrossRefzbMATHGoogle Scholar
  10. Liu B (2007) Uncertainty theory. In: Studies in fuzziness and soft computing, vol 154. SpringerGoogle Scholar
  11. Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
  12. Liu B (2010) Uncertain risk analysis and uncertain reliability analysis. J Uncertain Syst 4(3):163–170Google Scholar
  13. Liu B (2011) Uncertainty theory: a branch of mathematics for modeling human uncertainty, vol 300. Springer, BerlinGoogle Scholar
  14. Liu B (2013) Toward uncertain finance theory. J Uncertainty Anal Appl 1(1):1–15MathSciNetCrossRefGoogle Scholar
  15. Liu B (2015) Uncertainty theory, vol 24. Springer, BerlinGoogle Scholar
  16. Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst Int J 41(7):535–547CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Azarbaijan Shahid Madani UniversityTabrizIran

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