Fuzzy MCDM

Fuzzy MCDM

Criteria weights from one set of expert pairwise judgments, computed with crisp, fuzzy and interval type-2 fuzzy AHP, compared side by side and traced back to the published source of every step.

Ing. Akram Elsaied Elomiya Faculty of Transport Engineering, University of Pardubice

Download for Windows Read the documentation

Version 1.0.0 · 15 October 2026 · Windows 10 and 11, 64-bit · Works offline

Fuzzy MCDM showing the global weight of six criteria under five weighting methods as a dot chart, for a synthetic bus-fleet study.

What it does

Weighting criteria is the step of a multi-criteria decision where expert opinion enters the numbers. Different weighting methods can turn the same judgments into different weights, and published studies rarely show how. Fuzzy MCDM makes that comparison routine and checkable.

One input, five methods

Experts compare criteria in pairs on the Saaty 1 to 9 scale, once. The same judgments feed crisp AHP, two triangular fuzzy AHP methods and two interval type-2 fuzzy AHP methods, so differences in the weights come from the methods and not from the data.

Consistency before results

Every expert's matrix is checked with Saaty's consistency ratio. A matrix above 0.10 is excluded from that comparison set for every method alike, and the report states which experts were excluded and why.

Every number has a source

The calculation trace shows each matrix, aggregate and weight, the equation that produced it, and the page or table of the publication that defines the equation. Every run keeps fingerprints of its inputs and results.

Methods

Each method is fixed by a versioned profile with its scale, aggregation rule, consistency policy and sources, and is checked against published worked examples. Two equal-weight comparators are shown beside the methods as reference points; they are not weighting methods.

  1. AHP (Saaty 1980), eigenvector

    Crisp AHP

    Primary method

    Saaty (1980): the principal right eigenvector of the pairwise comparison matrix. Experts are combined by the weighted geometric mean of their judgments.

    Limitations: Crisp numbers only: the vagueness of a verbal judgment is not represented.

    Verified against its sources and published worked examples
    Saaty, T. L. (1980). The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill.
    Tan, X., Lin, J., Wang, F., Luo, H., Luo, L., Wu, L. (2007). Evaluation of the Effect of a Health Education Campaign of HIV by Using an Analytical Hierarchy Process Method. International Journal of Environmental Research and Public Health, 4(3), 254-259. https://doi.org/10.3390/ijerph2007030010
    Mortazavi, A., Kuzembayev, N. (2022). Development of a Weighting Procedure for Geomechanical Risk Assessment. Energies, 15(18), 6517. https://doi.org/10.3390/en15186517

  2. Fuzzy AHP, Buckley (1985) geometric mean

    Triangular fuzzy AHP

    Primary method

    Buckley (1985): judgments become triangular fuzzy numbers; the fuzzy geometric mean of each row gives a fuzzy weight, which is defuzzified and normalised. The equations follow the open-access transcription of Mortazavi and Kuzembayev (2022).

    Limitations: The original paper was checked through that transcription and Boender et al. (1989), not in full text. The spread of the triangular numbers is a fixed scale convention.

    Verified against its sources and published worked examples
    Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17(3), 233-247. https://doi.org/10.1016/0165-0114(85)90090-9
    Mortazavi, A., Kuzembayev, N. (2022). Development of a Weighting Procedure for Geomechanical Risk Assessment. Energies, 15(18), 6517. https://doi.org/10.3390/en15186517
    Boender, C. G. E., de Graan, J. G., Lootsma, F. A. (1989). Multi-criteria decision analysis with fuzzy pairwise comparisons. Fuzzy Sets and Systems, 29(2), 133-143. https://doi.org/10.1016/0165-0114(89)90187-5
    Liu, Y., Eckert, C. M., Earl, C. (2020). A review of fuzzy AHP methods for decision-making with subjective judgements. Expert Systems with Applications, 161, 113738. https://doi.org/10.1016/j.eswa.2020.113738

  3. Fuzzy AHP, FICSM (Ahmed & Kilic 2019)

    Triangular fuzzy AHP

    Robustness variant

    Ahmed and Kilic (2019): fuzzy weights from the fuzzy inverse of the column sums of the aggregated triangular fuzzy matrix.

    Limitations: No published worked example states its results, so it is verified on a published input matrix only. The multi-expert step is a documented extension. Offered as a robustness variant.

    Verified against its sources and published worked examples
    Ahmed, F., Kilic, K. (2019). Fuzzy Analytic Hierarchy Process: A performance analysis of various algorithms. Fuzzy Sets and Systems, 362, 110-128. https://doi.org/10.1016/j.fss.2018.08.009
    Ahmed, F. (2014). Fuzzy analytic hierarchy process: a comparison of the existing algorithms with new proposals. MSc thesis, Sabancı University.
    Mortazavi, A., Kuzembayev, N. (2022). Development of a Weighting Procedure for Geomechanical Risk Assessment. Energies, 15(18), 6517. https://doi.org/10.3390/en15186517

  4. IT2 fuzzy AHP (Kahraman et al. 2014-based, full-9 composition)

    Interval type-2 fuzzy AHP

    Primary benchmark method

    Interval type-2 fuzzy AHP after Kahraman et al. (2014), with the full 1 to 9 trapezoidal scale of Beresneva and Gordenko (2018) and geometric-mean aggregation of experts. A composition of published components.

    Limitations: Not a reproduction of one paper. Scale levels 2, 4, 6 and 8 appear in no published worked example; they are covered by a transcription test and property tests.

    Verified against its sources and published worked examples
    Kahraman, C., Öztayşi, B., Uçal Sarı, İ., Turanoğlu, E. (2014). Fuzzy analytic hierarchy process with interval type-2 fuzzy sets. Knowledge-Based Systems, 59, 48-57. https://doi.org/10.1016/j.knosys.2014.02.001
    Beresneva, E., Gordenko, M. (2018). Applying the methods of system analysis to teaching assistants evaluation. Proceedings of the Institute for System Programming of the RAS, 30(3), 251-270. https://doi.org/10.15514/ISPRAS-2018-30(3)-18
    Liu, Y., Eckert, C. M., Earl, C. (2020). A review of fuzzy AHP methods for decision-making with subjective judgements. Expert Systems with Applications, 161, 113738. https://doi.org/10.1016/j.eswa.2020.113738
    Görener, A., Ayvaz, B., Kuşakcı, A. O., Altınok, E. (2017). A hybrid type-2 fuzzy based supplier performance evaluation methodology: The Turkish Airlines technic case. Applied Soft Computing, 56, 436-445. https://doi.org/10.1016/j.asoc.2017.03.026

  5. IT2 fuzzy AHP, historical frozen (Milošević 2019 scale)

    Interval type-2 fuzzy AHP

    Historical reference, kept frozen

    Interval type-2 fuzzy AHP with the operators and DTraT defuzzification of Kahraman et al. (2014) and the scale of Milošević et al. (2019, Table 3).

    Limitations: Kept unchanged as a historical reference, so that earlier results can be reproduced.

    Verified against its sources and published worked examples
    Milošević, M. R., Milošević, D. M., Stević, D. M., Stanojević, A. D. (2019). Smart City: Modeling Key Indicators in Serbia Using IT2FS. Sustainability, 11(13), 3536. https://doi.org/10.3390/su11133536
    Kahraman, C., Öztayşi, B., Uçal Sarı, İ., Turanoğlu, E. (2014). Fuzzy analytic hierarchy process with interval type-2 fuzzy sets. Knowledge-Based Systems, 59, 48-57. https://doi.org/10.1016/j.knosys.2014.02.001
    Liu, Y., Eckert, C. M., Earl, C. (2020). A review of fuzzy AHP methods for decision-making with subjective judgements. Expert Systems with Applications, 161, 113738. https://doi.org/10.1016/j.eswa.2020.113738
    Görener, A., Ayvaz, B., Kuşakcı, A. O., Altınok, E. (2017). A hybrid type-2 fuzzy based supplier performance evaluation methodology: The Turkish Airlines technic case. Applied Soft Computing, 56, 436-445. https://doi.org/10.1016/j.asoc.2017.03.026

  6. Equal weights (hierarchical)

    Comparator

    Equal weights within the hierarchy: each main criterion gets the same share, split equally among its subcriteria. Uses no expert information.

  7. Equal weights (flat)

    Comparator

    Equal weights over all leaf criteria, ignoring the hierarchy. Uses no expert information.

Analysis and evidence

Two-level criteria hierarchies
Main criteria with subcriteria, or a single flat list, with up to ten criteria per comparison set. Local weights are combined into global weights.
Group judgments
Any number of experts, with equal or user-defined influence. Judgments are typed in, pasted, or imported from CSV and Excel files, and aggregated by the weighted geometric mean or by the rule each fuzzy method defines.
Method comparison
Rank correlations, distances and rank changes between every pair of methods and the comparators, and a disagreement measure across experts.
Leave-one-expert-out
Recomputes the weights with each expert removed in turn, showing how much one person drives the result.
Resampling stability
A seeded bootstrap over experts gives the 5th to 95th percentile range of every weight and rank, a stability range rather than a confidence interval. The seed and replicate count are part of the run record, so the ranges can be reproduced exactly.
Reproducible runs
Each run is identified by a SHA-256 fingerprint of its canonical inputs, method profiles and settings, and stores a fingerprint of its results. The application shows whether a run still matches the study's current judgments, and the run bundle holds everything needed to repeat it.
Figures and exports
An Excel calculation book with every table and step at full precision, publication figures as SVG or PNG, a printable A4 calculation report in PDF, and a run bundle with everything needed to audit or repeat the run.
Offline and private
The application has no network code, no telemetry and no updater. Studies stay in local files. Links to publications open in your browser only when you click them.

Workflow

A study moves through the same steps every time. The left-hand navigation of the application shows where you are and what is still missing.

  1. Set up the study

    Name the decision goal, the criteria and subcriteria, and the experts.

  2. Enter judgments

    Type, paste or import each expert's pairwise comparisons on the 1 to 9 scale.

  3. Review quality

    See consistency ratios and missing judgments before anything is computed.

  4. Choose methods

    Select the methods, then leave-one-out and resampling if you need them.

  5. Run and compare

    Read the weights, rankings, comparisons and figures for each method.

  6. Trace and export

    Check any number in the trace, then export the calculation book, figures and report.

Screenshots

All screenshots show a synthetic example: a choice of urban bus fleet technology with six invented criteria and three invented experts. You can download the same study below and open it in the application.

Pairwise comparisons. One expert's matrix for the main criteria.
Quality review. Consistency ratios per expert and comparison set, before the run.
Methods. The five weighting methods and their profiles.
Formulas and sources. Each equation with the publication that defines it.
Method comparison. Rank correlations between every pair of methods.
Leave-one-expert-out. How the weights change without each expert.
Resampling stability. Bootstrap ranges with the seed recorded.
Global weights. Every leaf criterion under each method.
Figures. Ranks by method, ready for export as SVG or PNG.
Calculation trace. Each step with its equation and source page.
Provenance. The run fingerprint, profiles and version.
Export. Calculation book, figures, PDF report and run bundle.

Download

Fuzzy MCDM is free for research, teaching and other non-commercial use under the Fuzzy MCDM Research Licence.

Version
1.0.0
Released
15 October 2026
File
Fuzzy MCDM-1.0.0-win-x64.exe
Size
112.1 MiB
Platform
Windows 10 or 11, 64-bit (x64)
SHA-256
86438731b8503349a90c2a2f3f72585362e03334ecf8259d00f2ebce260584f0
Archive
https://doi.org/10.5281/zenodo.23380643
Download the installer Example study (.mcdmlab)

Install

  1. Download the installer and check its SHA-256 checksum (below).
  2. Run it. The installer is not code-signed, so Windows SmartScreen may show “Windows protected your PC”. Choose More info, check that the file name matches, then Run anyway.
  3. Choose the installation folder. No administrator rights are needed for a per-user installation.
  4. Start Fuzzy MCDM from the Start menu, then open the example study or create a new one.

Check the download in PowerShell; the output must equal the SHA-256 shown here:

Get-FileHash -Algorithm SHA256 '.\Fuzzy MCDM-1.0.0-win-x64.exe'

Uninstall from Settings › Apps. Your study files are kept; they are ordinary files in the folders you chose.

Documentation

  • Technical and Scientific Documentation

    The methods, equations, consistency policy, analyses, reproducibility design, verification against published examples, and the novelty of the software.

    PDF · 0.8 MB
  • User Guide

    Installation and a step-by-step walk through a complete study, with screenshots of every screen.

    PDF · 4.9 MB
  • Release notes

    What this version contains and what changed.

    HTML
  • Third-party notices

    The open-source components included in the application and their licences.

    HTML

How to cite

If Fuzzy MCDM contributed to your work, please cite the software and the version you used.

Elomiya, A. E. (2026). Fuzzy MCDM (Version 1.0.0) [Computer software]. University of Pardubice. https://doi.org/10.5281/zenodo.23380643

CITATION.cff · BibTeX

Licence

Free of charge for research, teaching and other non-commercial use; the unmodified installer may be redistributed with attribution; no warranty; commercial use needs written permission.

Read the full licence (Fuzzy MCDM Research Licence 1.0)

© 2026 University of Pardubice.

Funding

This software is supported by the project SGS_2026_008 ‘Modelling of Transport Technology and Management Selected Aspects VI’ at the University of Pardubice.

Student Grant Competition project at the Faculty of Transport Engineering, University of Pardubice, 1 January – 31 December 2026. Project page at the faculty.

People and contact

Ing. Akram Elsaied Elomiya

Author · PhD candidate

University of Pardubice, Faculty of Transport Engineering, Department of Transport Management, Marketing, and Logistics, Studentská 95, 532 10 Pardubice, Czech Republic

Tanta University, Faculty of Engineering, Department of Public Works Engineering, 31511, Tanta, Egypt

akramelsaied.elomiya@upce.cz

ORCID 0000-0002-7166-0239

doc. Ing. Jiří Křupka, Ph.D.

Supervisor

University of Pardubice, Faculty of Transport Engineering

Fuzzy MCDM was developed for the PhD thesis Decision Models with Uncertainty in Transport.