Soft Computing

, Volume 23, Issue 9, pp 2887–2898 | Cite as

Sparse analytic hierarchy process: an experimental analysis

  • Gabriele OlivaEmail author
  • Roberto Setola
  • Antonio Scala
  • Paolo Dell’Olmo


The aim of the sparse analytic hierarchy process (SAHP) problem is to rank a set of alternatives based on their utility/importance; this task is accomplished by asking human decision-makers to compare selected pairs of alternatives and to specify relative preference information, in the form of ratios of utilities. However, such an information is often affected by subjective biases or inconsistencies. Moreover, there is no general consent on the best approach to accomplish this task, and in the literature several techniques have been proposed. Finally, when more than one decision-maker is involved in the process, there is a need to provide adequate methodologies to aggregate the available information. In this view, the contribution of this paper to the SAHP body of knowledge is twofold. From one side, it develops a novel methodology to aggregate sparse data given by multiple sources of information. From another side, the paper undertakes an experimental validation of the most popular techniques to solve the SAHP problem, discussing the strength points and shortcomings of the different methodology with respect to a real case study.


Analytic hierarchy process Sparse information Decision-making 


Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.

Ethical approval

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.

Informed consent

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


  1. Achlioptas D, Molloy M, Moore C, Van Bussel F (2005) Rapid mixing for lattice colourings with fewer colours. J Stat Mech Theory Exp 2005(10):P10012CrossRefGoogle Scholar
  2. Aczél J, Saaty TL (1983) Procedures for synthesizing ratio judgements. J Math Psychol 27(1):93–102MathSciNetCrossRefzbMATHGoogle Scholar
  3. Alcaraz C, Lopez J (2014) WASAM: A dynamic wide-area situational awareness model for critical domains in smart grids. Future Gener Comput Syst 30:146–154CrossRefGoogle Scholar
  4. Barzilai J, Golany B (1994) Ahp rank reversal, normalization and aggregation rules. Inf Syst Oper Res 32(2):57–64zbMATHGoogle Scholar
  5. Barzilai J, Cook WD, Golany B (1987) Consistent weights for judgements matrices of the relative importance of alternatives. Oper Res Lett 6(3):131–134MathSciNetCrossRefzbMATHGoogle Scholar
  6. Beg I, Rashid T (2014) Multi-criteria trapezoidal valued intuitionistic fuzzy decision making with Choquet integral based TOPSIS. Opsearch 51(1):98–129MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bessi A, Coletto M, Davidescu GA, Scala A, Caldarelli G, Quattrociocchi W (2015) Science vs conspiracy: collective narratives in the age of misinformation. PLoS ONE 10(2):e0118093CrossRefGoogle Scholar
  8. Bozóki S, Tsyganok V (2017) The logarithmic least squares optimality of the geometric mean of weight vectors calculated from all spanning trees for (in) complete pairwise comparison matrices. arXiv preprint arXiv:1701.04265
  9. Carmone FJ, Kara A, Zanakis SH (1997) A monte carlo investigation of incomplete pairwise comparison matrices in AHP. Eur J Oper Res 102(3):538–553CrossRefzbMATHGoogle Scholar
  10. Chen C-T (2000) Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst 114(1):1–9CrossRefzbMATHGoogle Scholar
  11. Chen S-J, Hwang C-L, Hwang FP (2011) Fuzzy multiple attribute decision making (methods and applications). Lecture notes in economics and mathematical systemsGoogle Scholar
  12. Crawford GB (1987) The geometric mean procedure for estimating the scale of a judgement matrix. Math Model 9(3–5):327–334CrossRefzbMATHGoogle Scholar
  13. Davis JM (1958) The transitivity of preferences. Behav Sci 3(1):26–33CrossRefGoogle Scholar
  14. Dolan JG, Isselhardt BJ, Cappuccio JD (1989) The analytic hierarchy process in medical decision making: a tutorial. Med Decis Mak 9(1):40–50CrossRefGoogle Scholar
  15. Dyer JS (1990) Remarks on the analytic hierarchy process. Manag Sci 36(3):249–258MathSciNetCrossRefGoogle Scholar
  16. Dyer M, Greenhill C, Ullrich M (2014) Structure and eigenvalues of heat-bath markov chains. Linear Algebra Appl 454:57–71MathSciNetCrossRefzbMATHGoogle Scholar
  17. Escobar MT, Aguarón J, Moreno-Jiménez JM (2004) A note on AHP group consistency for the row geometric mean priorization procedure. Eur J Oper Res 153(2):318–322CrossRefzbMATHGoogle Scholar
  18. Fax AJ, Murray RM (2004) Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control 49(9):1465–1476MathSciNetCrossRefzbMATHGoogle Scholar
  19. Fedrizzi M, Giove S (2007) Incomplete pairwise comparison and consistency optimization. Eur J Oper Res 183(1):303–313CrossRefzbMATHGoogle Scholar
  20. Forman EH (1990) Multi criteria decision making and the analytic hierarchy process. Springer, Berlin, pp 295–318Google Scholar
  21. Gilks WR, Richardson S, Spiegelhalter D (1995) Markov chain Monte Carlo in practice. CRC Press, LondonCrossRefzbMATHGoogle Scholar
  22. Häggström O (2002) Finite Markov chains and algorithmic applications, vol 52. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  23. Harker PT (1987a) Alternative modes of questioning in the analytic hierarchy process. Math Model 9(3–5):353–360MathSciNetCrossRefzbMATHGoogle Scholar
  24. Harker PT (1987b) Incomplete pairwise comparisons in the analytic hierarchy process. Math Model 9(11):837–848MathSciNetCrossRefGoogle Scholar
  25. Hastings KW (1970) Monte carlo sampling methods using markov chains and their applications. Biometrika 57(1):97–109MathSciNetCrossRefzbMATHGoogle Scholar
  26. Hummel JM, IJzermann MJ (2009) The use of the analytic hierarchy process in health care decision making. University of Twente, EnschedeGoogle Scholar
  27. Kendall MG (1938) A new measure of rank correlation. Biometrika 30(1/2):81–93CrossRefzbMATHGoogle Scholar
  28. Liang L, Wang G, Hua Z, Zhang B (2008) Mapping verbal responses to numerical scales in the analytic hierarchy process. Socio-Econ Plan Sci 42(1):46–55CrossRefGoogle Scholar
  29. Liberatore MJ, Nydick RL (2008) The analytic hierarchy process in medical and health care decision making: a literature review. Eur J Oper Res 189(1):194–207CrossRefzbMATHGoogle Scholar
  30. Linstone HA, Turoff M et al (1975) The delphi method. Addison-Wesley, ReadingzbMATHGoogle Scholar
  31. Menci M, Oliva G, Papi M, Setola R, Scala A (2018) A suite of distributed methodologies to solve the sparse analytic hierarchy process problem. In: 2018th European control conferenceGoogle Scholar
  32. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092CrossRefGoogle Scholar
  33. Olfati-Saber R, Fax JA, Murray RM (2007) Consensus and cooperation in networked multi-agent systems. Proc IEEE 95(1):215–233CrossRefzbMATHGoogle Scholar
  34. Oliva G, Setola R, Scala A (2017) Sparse and distributed analytic hierarchy process. Automatica 85:211–220MathSciNetCrossRefzbMATHGoogle Scholar
  35. Rubio JE, Alcaraz C, Lopez J (2017) Preventing advanced persistent threats in complex control networks. In: European symposium on research in computer security. Springer, pp 402–418Google Scholar
  36. Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 15(3):234–281MathSciNetCrossRefzbMATHGoogle Scholar
  37. Saaty TL (1990) An exposition of the AHP in reply to the paper “remarks on the analytic hierarchy process”. Manag Sci 36(3):259–268CrossRefGoogle Scholar
  38. Shiraishi S, Obata T, Daigo M (1998) Properties of a positive reciprocal matrix and their application to AHP. J Oper Res Soc Jpn 41(3):404–414MathSciNetCrossRefzbMATHGoogle Scholar
  39. Van Brummelen G (2012) Heavenly mathematics: the forgotten art of spherical trigonometry. Princeton University Press, PrincetonCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Unit of Automatic ControlUniversity Campus Bio-Medico di RomaRomeItaly
  2. 2.ISC-CNR UoS “Sapienza”RomeItaly
  3. 3.Dipartimento di Scienze StatisticheUniversitá degli Studi di Roma “La Sapienza”RomeItaly

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