Economic Evaluation of Digital Suppliers for Manufacturing SMEs Using Pythagorean Neutrosophic TOPSIS and VIKOR with a Flexible Distance Metric
DOI:
https://doi.org/10.31181/ijes1412025187Keywords:
Pythagorean neutrosophic set, MCDM, TOPSIS, VIKOR, Economic evaluation, Manufacturing SMEs, Alternative ranking, Neutrosophic distance measureAbstract
Selecting digital suppliers is not only a technological issue but also a fundamental economic decision for manufacturing SMEs pursuing digital transformation. Supplier choice directly affects cost efficiency, resource allocation, and long-term competitiveness. This study develops a quantitative multi-criteria decision-making (MCDM) framework that integrates Pythagorean Neutrosophic TOPSIS (PNTOPSIS) and VIKOR (PNVIKOR), strengthened by a novel distance measure, the Flexible Indeterminacy Quantifier (FIQ). The framework explicitly addresses incomplete and uncertain evaluations that complicate procurement under financial and operational constraints. A real-world case with five digital suppliers is analyzed across criteria including system capability, vendor support, total cost, and risk of disruption. The findings highlight the supplier that delivers the highest economic value by aligning affordability with operational reliability. FIQ improves score differentiation and ranking stability, enabling SMEs to make more economically rational choices. Sensitivity analysis confirms that the model produces consistent outcomes under varying budgetary assumptions, demonstrating its robustness for strategic procurement. Overall, this research provides SMEs with an uncertainty-aware, cost-sensitive tool that reduces financial risks, enhances transparency in supplier selection, and supports sustainable economic performance within digitally transforming industrial ecosystems.
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Taherdoost, H., & Madanchian, M. (2023). An effective compromising ranking technique for decision making. Macro Management & Public Policies, 5(2), 27–33. https://doi.org/10.30564/mmpp.v5i2.5578
Shanmugasundar, G., Kalita, K., Čep, R., & Chohan, J. S. (2023). Decision models for selection of industrial robots—A comprehensive comparison of multi-criteria decision making. Processes, 11(6). https://doi.org/10.3390/pr11061681
Ismail, J. N., Rodzi, Z., Hashim, H., Sulaiman, N. H., Al-Sharqi, F., Al-Quran, A., & Ahmad, G. (2023). Enhancing decision accuracy in DEMATEL using Bonferroni mean aggregation under Pythagorean neutrosophic environment. Journal of Fuzzy Extension and Applications, 4(4), 281–298. https://doi.org/10.22105/jfea.2023.422582.1318
Smarandache, F. (2023). Several new types of neutrosophic set. Infinite Study.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.
Ghobakhloo, M., & Iranmanesh, M. (2021). Digital transformation success under Industry 4.0: a strategic guideline for manufacturing SMEs. Journal of Manufacturing Technology Management, 32(8), 1533-1556. https://doi.org/10.1108/JMTM-11-2020-0455
Pramanik, H. S., Kirtania, M., & Pani, A. K. (2019). Essence of digital transformation—Manifestations at large financial institutions from North America. Future Generation Computer Systems, 95, 323-343. https://doi.org/10.1016/j.future.2018.12.003
Attaran, M. (2020). Digital technology enablers and their implications for supply chain management. Supply Chain Forum: An International Journal, 21(3), 158-172. https://doi.org/10.1080/16258312.2020.1751568
Garay-Rondero, C. L., Martinez-Flores, J. L., Smith, N. R., Morales, S. O. C., & Aldrette-Malacara, A. (2020). Digital supply chain model in Industry 4.0. Journal of Manufacturing Technology Management, 31(5), 887-933. https://doi.org/10.1108/JMTM-08-2018-0280
Sharma, M., & Joshi, S. (2023). Digital supplier selection reinforcing supply chain quality management systems to enhance firm's performance. The TQM Journal, 35(1), 102-130. https://doi.org/10.1108/TQM-07-2020-0160
Dutta, G., Kumar, R., Sindhwani, R., & Singh, R. K. (2020). Digital transformation priorities of India’s discrete manufacturing SMEs–a conceptual study in perspective of Industry 4.0. Competitiveness Review, 30(3), 289-314. https://doi.org/10.1108/CR-03-2019-0031
Hong, J., Liao, Y., Zhang, Y., & Yu, Z. (2019). The effect of supply chain quality management practices and capabilities on operational and innovation performance: Evidence from Chinese manufacturers. International Journal of Production Economics, 212, 227-235. https://doi.org/10.1016/j.ijpe.2019.01.036
Özbek, A., & Yıldız, A. (2020). Digital supplier selection for a garment business using interval type-2 fuzzy topsis. Textile and Apparel, 30(1), 61-72. https://doi.org/10.32710/tekstilvekonfeksiyon.569884
Tavana, M., Shaabani, A., Di Caprio, D., & Amiri, M. (2021). An integrated and comprehensive fuzzy multicriteria model for supplier selection in digital supply chains. Sustainable Operations and Computers, 2, 149-169. https://doi.org/10.1016/j.susoc.2021.07.008
Erbay, H., & Yıldırım, N. (2022). Combined technology selection model for digital transformation in manufacturing: a case study from the automotive supplier industry. International Journal of Innovation and Technology Management, 19(7), 2250023. https://doi.org/10.1142/S0219877022500237
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96.
Yager, R. R. (2013). Pythagorean fuzzy subsets. In 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) (pp. 57–61). IEEE. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
Smarandache, F. (1998). Neutrosophy: Neutrosophic probability, set, and logic: Analytic synthesis & synthetic analysis. American Research Press.
Smarandache, F. (1999). A unifying field in logics: Neutrosophic logic. In Philosophy (pp. 1-141). American Research Press.
Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Infinite Study.
Wang, H., Smarandache, F., Zhang, Y.-Q., & Sunderraman, R. (2005). Interval neutrosophic sets and logic: Theory and applications in computing (Vol. 5). Infinite Study.
Ye, J. (2014). A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. Journal of Intelligent and Fuzzy Systems, 26(5), 2459–2466. https://doi.org/10.3233/IFS-130916
Maji, P. K. (2013). Neutrosophic soft set. Annals of Fuzzy Mathematics and Informatics, 5(1), 157–168.
Wang, J. Q., & Li, X. E. (2015). An application of the TODIM method with multi-valued neutrosophic set. Control and Decision, 30, 1139–1142. http://dx.doi.org/10.13195/j.kzyjc.2014.0467
Broumi, S., Smarandache, F., & Dhar, M. (2014). Rough neutrosophic sets. Neutrosophic Sets and Systems, 3.
Al-Quran, A., Al-Sharqi, F., Rahman, A. U., & Rodzi, Z. M. (2024). The q-rung orthopair fuzzy-valued neutrosophic sets: Axiomatic properties, aggregation operators and applications. AIMS Mathematics, 9(2), 5038–5070. https://doi.org/10.3934/math.2024245
Jansi, R., Mohana, K., & Smarandache, F. (2019). Correlation measure for Pythagorean neutrosophic fuzzy sets with T and F as dependent neutrosophic components. Neutrosophic Sets and Systems, 30.
Saini, R. K., Ahirwar, A., Smarandache, F., & Kushwaha, M. (2024). Multi-criteria group decision-making with ELECTRE-III method for selection of female spouse in Pythagorean neutrosophic environment. Plithogenic Logic and Computation, 1, 36–53. https://doi.org/10.61356/j.plc.2024.18960
Kamari, M. S. B. M., Rodzi, Z. B. M., Al-Obaidi, R. H., Al-Sharqi, F., Al-Quran, A., & A. shlaka, R. (2024). Deciphering the geometric Bonferroni mean operator in Pythagorean neutrosophic sets framework. Neutrosophic Sets and Systems, 75(1), 7. http://dx.doi.org/10.5281/zenodo.13932052
Ahmad, N., Rodzi, Z., Al-Sharqi, F., Al-Quran, A., Lutfi, A., Yusof, Z. M., & Hassanuddin, N. A. (2024). Innovative theoretical approach: Bipolar Pythagorean neutrosophic sets (BPNSs) in decision-making. International Journal of Neutrosophic Science, 23(1), 249–256. https://doi.org/10.54216/IJNS.230122
Ahmad, N., Rodzi, Z. M., Ahmad, N. H. F., & Razak, S. A. (2024). Bipolar Pythagorean fuzzy neutrosophic set (BPNS) integrated with AHP express (BPNS-AHP express) with linguistic variable: A new approach. International Journal of the Analytic Hierarchy Process, 16(2), 1182–1182. https://doi.org/10.13033/ijahp.v16i2.1181
Razak, S. A., Rodzi, Z. M., Ahmad, N., & Ahmad, G. (2024). Exploring the boundaries of uncertainty: Interval valued Pythagorean neutrosophic set and their properties. Malaysian Journal of Fundamental and Applied Sciences, 20(4), 813–824. https://doi.org/10.11113/mjfas.v20n4.3482
Razak, S. A., Rodzi, Z. M., Al-Sharqi, F., & Ramli, N. (2025). Revolutionizing decision-making in e-commerce and IT procurement: An IVPNS-COBRA linguistic variable framework for enhanced multi-criteria analysis. International Journal of Economic Sciences, 14(1), 1–31. https://doi.org/10.31181/ijes1412025176
Ye, J. (2014). Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. Journal of Intelligent Systems, 23(4), 379–389. https://doi.org/10.1515/jisys-2013-0091
Biswas, P., Bibhas, S. P., Giri, C., Biswas, S., Pramanik, B., Chandra, G., Pramanik, S., & Giri, B. C. (2019). Distance measure based MADM strategy with interval trapezoidal neutrosophic numbers. Neutrosophic Sets and Systems, 19.
Saqlain, M., Riaz, M., Saleem, M. A., & Yang, M. S. (2021). Distance and similarity measures for neutrosophic hypersoft set (NHSS) with construction of NHSS-TOPSIS and applications. IEEE Access, 9, 30803–30816. https://doi.org/10.1109/ACCESS.2021.3059712
Kamari, M. S. B. M., Rodzi, Z. B. M., & Kamis, N. H. (2025). Pythagorean neutrosophic method based on the removal effects of criteria (PNMEREC): An innovative approach for establishing objective weights in multi-criteria decision-making challenges. Malaysian Journal of Fundamental and Applied Sciences, 21(1), 1678–1696. https://doi.org/10.11113/mjfas.v21n1.3600
Biswas, P., Pramanik, S., & Giri, B. C. (2016). TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Computing and Applications, 27(3), 727–737. https://doi.org/10.1007/s00521-015-1891-2
Ali, M., Hussain, Z., & Yang, M. S. (2023). Hausdorff distance and similarity measures for single-valued neutrosophic sets with application in multi-criteria decision making. Electronics (Switzerland), 12(1). https://doi.org/10.3390/electronics12010201
Dasan, M. A., Bementa, E., Aslam, M., & Flower, V. F. L. (2024). Multi-attribute decision-making problem in career determination using single-valued neutrosophic distance measure. Complex and Intelligent Systems, 10(4), 5411–5425. https://doi.org/10.1007/s40747-024-01433-z
Štilić, A., & Puška, A. (2023). Integrating multi-criteria decision-making methods with sustainable engineering: A comprehensive review of current practices. Eng, 4(2), 1536–1549. https://doi.org/10.3390/eng4020088
Kumar, R., & Pamucar, D. (2025). A comprehensive and systematic review of multi-criteria decision-making (MCDM) methods to solve decision-making problems: Two decades from 2004 to 2024. Spectrum of Decision Making and Applications, 2(1), 178–197. https://doi.org/10.31181/sdmap21202524
Odeyinka, O. F., Okandeji, A. A., & Sogbesan, A. A. (2022). A fuzzy TOPSIS model for selecting raw material suppliers in a manufacturing company. Nigerian Journal of Technology, 41(4), 797–804. https://doi.org/10.4314/njt.v41i4.17
Manzoor, S., Mustafa, S., Gulzar, K., Gulzar, A., Kazmi, S. N., Akber, S. M. A., Bukhsh, R., Aslam, S., & Mohsin, S. M. (2024). MultiFuzzTOPS: A fuzzy multi-criteria decision-making model using Type-2 soft sets and TOPSIS. Symmetry, 16(6). https://doi.org/10.3390/sym16060655
Omidi, K., Afzali, A., Vahidi, H., & Mahnam, S. (2022). Ranking of suitable areas for establishing industries in Kashan city using VIKOR and TOPSIS methods in fuzzy environment. Journal of Advances in Environmental Health Research, 10(2), 133–148. https://doi.org/10.32598/JAEHR.10.2.1241
Zaman, M. M. K., Rodzi, Z. M., Andu, Y., Shafie, N. A., Sanusi, Z. M., Ghazali, A. W., & Mahyideen, J. M. (2025). Adaptive Utility Ranking Algorithm for Evaluating Blockchain-Enabled Microfinance in Emerging-A New MCDM Perspective. International Journal of Economic Sciences, 14(1), 123-146. https://doi.org/10.31181/ijes1412025182
Abdalla, M. E. M., Uzair, A., Ishtiaq, A., Tahir, M., & Kamran, M. (2025). Algebraic Structures and Practical Implications of Interval-Valued Fermatean Neutrosophic Super HyperSoft Sets in Healthcare. Spectrum of Operational Research, 2(1), 199-218. https://doi.org/10.31181/sor21202523
Biswas, A., Gazi, K. H., Bhaduri, P., & Mondal, S. P. (2024). Neutrosophic fuzzy decision-making framework for site selection. Journal of Decision Analytics and Intelligent Computing, 4(1), 187–215. https://doi.org/10.31181/jdaic10004122024b
Basuri, T., Gazi, K. H., Bhaduri, P., Das, S. G., & Mondal, S. P. (2025). Decision-analytics-based Sustainable Location Problem - Neutrosophic CRITIC-COPRAS Assessment Model. Management Science Advances, 2(1), 19-58. https://doi.org/10.31181/msa2120257
Hwang, C. L., & Yoon, K. (1981). Multiple attribute decision making: Methods and applications. Springer.
Arıcan, O. H., & Kara E., G. E. (2024). Selection model of chemical tanker ships for cargo types using fuzzy AHP and fuzzy TOPSIS. Regional Studies in Marine Science, 78, 103724. https://doi.org/10.1016/j.rsma.2024.103724
He, Q., Attan, S. A., Zhang, J., Shang, R., & He, D. (2024). Evaluating music education interventions for mental health in Chinese university student: A dual fuzzy analytic method. Scientific Reports, 14(1), 19727. https://doi.org/10.1038/s41598-024-70753-4
Otay, İ., Çevik Onar, S., Öztayşi, B., & Kahraman, C. (2024). Evaluation of sustainable energy systems in smart cities using a multi-expert Pythagorean fuzzy BWM & TOPSIS methodology. Expert Systems with Applications, 250, 123874. https://doi.org/10.1016/j.eswa.2024.123874
Usun, S. O., Bas, S. A., Meniz, B., & Ozkok, B. A. (2024). Passenger satisfaction assessment in the aviation industry using Type-2 fuzzy TOPSIS. Journal of Air Transport Management, 119, 102630. https://doi.org/10.1016/j.jairtraman.2024.102630
Quynh, V. T. N. (2024). An extension of fuzzy TOPSIS approach using integral values for banking performance evaluation. Multidisciplinary Science Journal, 6(8), 2024155. https://doi.org/10.31893/multiscience.2024155
Sharma, H., Tandon, A., Kapur, P. K., & Aggarwal, A. G. (2019). Ranking hotels using aspect ratings based sentiment classification and interval-valued neutrosophic TOPSIS. International Journal of System Assurance Engineering and Management, 10(5), 973–983. https://doi.org/10.1007/s13198-019-00827-4
Argilagos, M. R., Herrera, A. V., & Valdiviezo, W. V. (2022). Evaluation of nutritional education strategies in schools in Ecuador using neutrosophic TOPSIS. International Journal of Neutrosophic Science, 18(3), 208–217. https://doi.org/10.54216/IJNS.1803018
Recalde, M. B., Rojas, S. R., Llerena, M. Z., & León, A. S. (2023). Evaluation of the effectiveness of preventive dental education in primary schools. Neutrosophic Sets and Systems, 62, 319–326.
Anwar, K., Zafar, A., & Iqbal, A. (2023). Neutrosophic MCDM approach for performance evaluation and recommendation of best players in sports league. International Journal of Neutrosophic Science, 20(1), 128–149. https://doi.org/10.54216/IJNS.200111
Opricovic, S. (1998). Multicriteria optimization of civil engineering systems. Faculty of Civil Engineering, Belgrade, 2(1), 5–21.
Mahmudah, R. S., Putri, D. I., Abdullah, A. G., Shafii, M. A., Hakim, D. L., & Setiadipura, T. (2024). Developing a multi-criteria decision-making model for nuclear power plant location selection using fuzzy analytic hierarchy process and fuzzy VIKOR methods focused on socio-economic factors. Cleaner Engineering and Technology, 19. https://doi.org/10.1016/j.clet.2024.100737
Phan, V. D. V., Huang, Y. F., Hoang, T. T., & Do, M. H. (2023). Evaluating barriers to supply chain resilience in Vietnamese SMEs: The fuzzy VIKOR approach. Systems, 11(3). https://doi.org/10.3390/systems11030121
Lam, W. S., Lam, W. H., Jaaman, S. H., & Liew, K. F. (2021). Performance evaluation of construction companies using integrated entropy–fuzzy VIKOR model. Entropy, 23(3), 1–16. https://doi.org/10.3390/e23030320
Ayouni, S., Menzli, L. J., Hajjej, F., Maddeh, M., & Al-Otaibi, S. (2021). Fuzzy VIKOR application for learning management systems evaluation in higher education. International Journal of Information and Communication Technology Education, 17(2), 17–35. https://doi.org/10.4018/IJICTE.2021040102
Hosseini, S. M., Paydar, M. M., & Hajiaghaei-Keshteli, M. (2021). Recovery solutions for ecotourism centers during the Covid-19 pandemic: Utilizing fuzzy DEMATEL and fuzzy VIKOR methods. Expert Systems with Applications, 185. https://doi.org/10.1016/j.eswa.2021.115594
Öztürk, F., & Kaya, G. K. (2020). Bulanik VIKOR ile personel seçimi: Otomotiv yan sanayinde uygulama. Gazi Üniversitesi Fen Bilimleri Dergisi Part C: Tasarım ve Teknoloji, 8(1), 94–108. https://doi.org/10.29109/gujsc.595288
Uluçay, V., Şahin, N. M., Toz, N. İ., & Bozkurt, E. (2023). VIKOR method for decision-making problems based on Q-single-valued neutrosophic sets: Law application. Journal of Fuzzy Extension and Applications, 4(4), 310–326. https://doi.org/10.22105/jfea.2023.426042.1331
Álvarez Enríquez, G. F., Aguirre Solórzano, J. F., Thumb Chávez, J. M., & Almeida Huertas, O. Y. (2024). Selection of adaptive strategies in child development through modeling of the neutrosophic VIKOR method. Journal of Fuzzy Extension and Applications, 5(Special Issue), 99–111. https://doi.org/10.22105/jfea.2024.468201.1549
Luo, X., Wang, Z., Yang, L., Lu, L., & Hu, S. (2023). Sustainable supplier selection based on VIKOR with single-valued neutrosophic sets. PLoS ONE, 18(9). https://doi.org/10.1371/journal.pone.0290093
Paronyan, H., Melendez Carballido, R., & Alfaro Matos, M. (2020). Neutrosophic VIKOR for proposal of reform to Article 189 of the Integral Criminal Code in Ecuador. Neutrosophic Sets and Systems, 37, 287–294.
Kamal, N. L. A. M., Abdullah, L., Yee, F. M., Abdullah, I., & Vafaei, N. (2021). Single valued neutrosophic VIKOR and its application to wastewater treatment selection. Neutrosophic Sets and Systems, 47.
Radha, R., Stanis, A., & Mary, A. (2021). Neutrosophic Pythagorean soft set with T and F as dependent neutrosophic components. Neutrosophic Sets and Systems, 42.
Ismail, J. N., Md Rodzi, Z., Al-Sharqi, F., Al-Quran, A., Hashim, H., & Sulaiman, N. H. (2023). Algebraic operations on Pythagorean neutrosophic sets (PNS): Extending applicability and decision-making capabilities. International Journal of Neutrosophic Science, 21(4), 127–134. https://doi.org/10.54216/IJNS.210412
Krishnan, A. R. (2022). Past efforts in determining suitable normalization methods for multi-criteria decision-making: A short survey. Frontiers in Big Data, 5, 990699. https://doi.org/10.3389/fdata.2022.990699
Mhlanga, S. T., & Lall, M. (2022). Influence of normalization techniques on multi-criteria decision-making methods. Journal of Physics: Conference Series, 2224(1). https://doi.org/10.1088/1742-6596/2224/1/012076
Aytekin, A. (2021). Comparative analysis of normalization techniques in the context of MCDM problems. Decision Making: Applications in Management and Engineering, 4(2), 1–25. https://doi.org/10.31181/dmame210402001a
Diakoulaki, D., Mavrotas, G., & Papayannakis, L. (1995). Determining objective weights in multiple criteria problems: The CRITIC method. Computers & Operations Research, 22(7), 763–770. http://dx.doi.org/10.1016/0305-0548(94)00059-H
Shannon, C. E., & Weaver, W. (1949). The mathematical theory of communication.
Pomerol, J. C., & Barba-Romero, S. (2000). Weighting methods and associated problems. In Multicriterion decision in management: Principles and practice (pp. 75–104). https://doi.org/10.1007/978-1-4615-4459-3_4
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