Fuzzy Systems and its Applications

Fuzzy Systems and its Applications

A Credibility-Based Fuzzy Mathematical Programming Approach for Z-Number Data Envelopment Analysis Model

Document Type : Original Article

Authors
Pejman Peykani 1
Jafar Gheidar-Kheljani 2
Sanly Ghanidel 3
Fatemeh Ramezanlou 3
Seyed Ehsan Shojaie 4
1 Department of Industrial Engineering, Faculty of Engineering, Khatam University, Tehran, Iran
2 Management and Industrial Engineering Department, Malek Ashtar University of Technology, Tehran, Iran
3 Department of Computer Science, Science and Research Branch, Islamic Azad University, Tehran, Iran
4 School of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran
10.22034/jfsa.2025.515296.1268
Abstract
This study aims to propose a novel and effective approach in data envelopment analysis (DEA) capable of evaluating the performance of homogeneous decision-making units (DMUs) in the presence of negative data and under uncertain conditions. The directional distance function model, one of the most widely used models for analyzing negative data, serves as the foundational model for this research. Additionally, Z-number theory, credibility theory, and chance-constrained programming have been employed to address data uncertainty. Given the prevalence of negative and uncertain data in real-world problems and applications, the insurance industry in the Tehran Stock Exchange was selected as a case study to examine the applicability and efficacy of the proposed approach. The empirical results obtained from implementing the credibility-based Z-number data envelopment analysis (ZDEA) approach demonstrate the proposed methodology's effectiveness and capability in evaluating and ranking stocks under conditions of negative and uncertain data.
Keywords
Fuzzy Data Envelopment Analysis
Z-Number Theory
Credibility Approach
Negative Data
Stock Evaluation
Subjects

[1]    Ramanathan, R. (2003). An Introduction to Data Envelopment Analysis: A Tool for Performance Measurement. Sage.
[2]    Cook, W. D., & Zhu, J. (2005). Modeling Performance Measurement: Applications and Implementation Issues in DEA. Boston, MA: Springer US.
[3]    Cooper, W. W., Seiford, L. M., Tone, K., & Zhu, J. (2007). Some Models and Measures for Evaluating Performances with DEA: Past Accomplishments and Future Prospects. Journal of Productivity Analysis, 28(3), 151-163.
[4]    Liu, J. S., Lu, L. Y., Lu, W. M., & Lin, B. J. (2013). A Survey of DEA Applications. Omega, 41(5), 893-902.
[5]    Mergoni, A., Emrouznejad, A., & De Witte, K. (2025). Fifty Years of Data Envelopment Analysis. European Journal of Operational Research, 326(3), 389-412.
[6]    Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the Efficiency of Decision Making Units. European Journal of Operational Research, 2(6), 429-444.
[7]    Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science, 30(9), 1078-1092.
[8]    Charnes, A., Cooper, W., Lewin, A. Y., & Seiford, L. M. (1997). Data Envelopment Analysis Theory, Methodology and Applications. Journal of the Operational Research society, 48(3), 332-333.
[9]    Cooper, W. W., Seiford, L. M., & Zhu, J. (2011). Handbook on Data Envelopment Analysis. Springer New York, NY.
[10]    Peykani, P., & Pishvaee, M. S. (2024). Performance Evaluation of Hospitals under Data Uncertainty: An Uncertain Common-Weights Data Envelopment Analysis. Healthcare, 12(6), 611.
[11]    Garner, C., Holder, A., & Hurtig, N. (2025). Uncertain Data Envelopment Analysis: Theory, Algorithms, and Applications. Decision Making Optimization Models for Business Partnerships, 38-78. Chapman and Hall/CRC.
 
[12]    Sadjadi, S. J., & Omrani, H. (2008). Data Envelopment Analysis with Uncertain Data: An Application for Iranian Electricity Distribution Companies. Energy Policy, 36(11), 4247-4254.
[13]    Peykani, P., Soltani, R., Tanasescu, C., Shojaie, S. E., & Jandaghian, A. (2025). The Robust Malmquist Productivity Index: A Framework for Measuring Productivity Changes over Time Under Uncertainty. Mathematics, 13(11), 1727.
[14]    Olesen, O. B., & Petersen, N. C. (2016). Stochastic Data Envelopment Analysis—A Review. Euro- pean Journal of Operational Research, 251(1), 2-21.
[15]    Zhou, W., & Xu, Z. (2020). An Overview of the Fuzzy Data Envelopment Analysis Research and Its Successful Applications. International Journal of Fuzzy Systems, 22(4), 1037-1055.
[16]    Peykani, P., Mohammadi, E., Farzipoor Saen, R., Sadjadi, S. J., & Rostamy-Malkhalifeh, M. (2020). Data Envelopment Analysis and Robust Optimization: A Review. Expert Systems, 37(4), e12534.
[17]    Despotis, D. K., & Smirlis, Y. G. (2002). Data envelopment analysis with imprecise data. European journal of operational research, 140(1), 24-36.
[18]    Hatami-Marbini, A., Emrouznejad, A., & Tavana, M. (2011). A Taxonomy and Review of the Fuzzy Data Envelopment Analysis Literature: Two Decades in the Making. European Journal of Operational Research, 214(3), 457-472.
[19]    Emrouznejad, A., Tavana, M., & Hatami-Marbini, A. (2014). The State of the Art in Fuzzy Data Envelopment Analysis. Performance Measurement with Fuzzy Data Envelopment Analysis, 1-45. Springer Berlin, Heidelberg.
[20]    Peykani, P., Hosseinzadeh Lotfi, F., Sadjadi, S. J., Ebrahimnejad, A., & Mohammadi, E. (2022). Fuzzy Chance-Constrained Data Envelopment Analysis: A Structured Literature Review, Current Trends, and Future Directions. Fuzzy Optimization and Decision Making, 21(2), 197-261.
[21]    Emrouznejad, A., Anouze, A. L., & Thanassoulis, E. (2010). A Semi-Oriented Radial Measure for Measuring the Efficiency of Decision Making Units with Negative Data, Using DEA. European Journal of Operational Research, 200(1), 297-304.
[22]    Kerstens, K., & Van de Woestyne, I. (2011). Negative Data in DEA: A Simple Proportional Distance Function Approach. Journal of the Operational Research Society, 62(7), 1413-1419.
[23]    Guo, P., Tanaka, H., & Inuiguchi, M. (2000). Self-Organizing Fuzzy Aggregation Models to Rank the Objects with Multiple Attributes. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 30(5), 573-580.
[24]    Lertworasirikul, S., Fang, S. C., Joines, J. A., & Nuttle, H. L.W. (2003). Fuzzy Data Envelopment Analysis (DEA): A Possibility Approach. Fuzzy Sets and Systems, 139(2), 379-394.
[25]    Lertworasirikul, S., Fang, S. C., Nuttle, H. L. W., & Joines, J. A. (2003). Fuzzy BCC Model for Data Envelopment Analysis. Fuzzy Optimization and Decision Making, 2(4), 337-358.
 
[26]    Garcia, P. A. A., Schirru, R., & Frutuoso E Melo, P. F. (2005). A Fuzzy Data Envelopment Analysis Approach for FMEA. Progress in Nuclear Energy, 46(3-4), 359-373.
[27]    Wu, D. D., Yang, Z., & Liang, L. (2006). Efficiency Analysis of Cross-Region Bank Branches Using Fuzzy Data Envelopment Analysis. Applied Mathematics and Computation, 181(1), 271-281.
[28]    Wen, M., & Li, H. (2009). Fuzzy Data Envelopment Analysis (DEA): Model and Ranking Method. Journal of Computational and Applied Mathematics, 223(2), 872-878.
[29]    Khodabakhshi, M., Gholami, Y., & Kheirollahi, H. (2010). An Additive Model Approach for Estimating Returns to Scale in Imprecise Data Envelopment Analysis. Applied Mathematical Modelling, 34(5), 1247-1257.
[30]    Lin, H. T. (2010). Personnel Selection Using Analytic Network Process and Fuzzy Data Envelopment Analysis Approaches. Computers & Industrial Engineering, 59(4), 937-944.
[31]    Wen, M., You, C., & Kang, R. (2010). A New Ranking Method to Fuzzy Data Envelopment Analysis. Computers & Mathematics with Applications, 59(11), 3398-3404.
[32]    Hosseinzadeh Lotfi, F., Jahanshahloo, G. R., Khodabakhshi, M., & Moradi, F. (2011). A Fuzzy Chance Constraint Multi Objective Programming Method in Data Envelopment Analysis. African Journal of Business Management, 5(33), 12873.
[33]    Wen, M., Qin, Z., & Kang, R. (2011). Sensitivity and Stability Analysis in Fuzzy Data Envelopment Analysis. Fuzzy Optimization and Decision Making, 10(1), 1-10.
[34]    Nedeljković, R. R., & Drenovac, D. (2012). Efficiency Measurement of Delivery Post Offices Using Fuzzy Data Envelopment Analysis (Possibility Approach). International Journal for Traffic and Transport Engineering, 2(1), 22-29.
[35]    Zhao, X., & Yue, W. (2012). A Multi-Subsystem Fuzzy DEA Model with Its Application in Mutual Funds Management Companies’ Competence Evaluation. Procedia Computer Science, 1(1), 2469- 2478.
[36]    Payan, A., & Shariff, M. (2013). Scrutiny Malmquist Productivity Index on Fuzzy Data by Credibility Theory with an Application to Social Security Organizations. Journal of Uncertain Systems, 7(1), 36-49.
[37]    Tavana, M., Shiraz, R. K., Hatami-Marbini, A., Agrell, P. J., & Paryab, K. (2013). Chance- Constrained DEA Models with Random Fuzzy Inputs and Outputs. Knowledge-Based Systems, 52, 32-52.
[38]    Azadi, M., Jafarian, M., Farzipoor Saen, R., & Mirhedayatian, S. M. (2015). A New Fuzzy DEA Model for Evaluation of Efficiency and Effectiveness of Suppliers in Sustainable Supply Chain Management Context. Computers & Operations Research, 54, 274-285.
 
[39]    Fasanghari, M., Amalnick, M. S., Anvari, R. T., & Razmi, J. (2015). A Novel Credibility-Based Group Decision Making Method for Enterprise Architecture Scenario Analysis Using Data Envelopment Analysis. Applied Soft Computing, 32, 347-368.
[40]    Payan, A. (2015). Common Set of Weights Approach in Fuzzy DEA with an Application. Journal of Intelligent & Fuzzy Systems, 29(1), 187-194.
[41]    Wen, M. (2015). Fuzzy DEA. Uncertain Data Envelopment Analysis, 83-116. Springer, Berlin, Heidelberg.
[42]    Zerafat Angiz, M., Mustafa, A., Ghadiri, M., & Tajaddini, A. (2015). Relationship Between Efficiency in the Traditional Data Envelopment Analysis and Possibility Sets. Computers & Industrial Engineering, 81, 140-146.
[43]    Ruiz, J. L., & Sirvent, I. (2017). Fuzzy Cross-Efficiency Evaluation: A Possibility Approach. Fuzzy Optimization and Decision Making, 16(1), 111-126.
[44]    Ahmadvand, S., & Pishvaee, M. S. (2018). An Efficient Method for Kidney Allocation Problem: A Credibility-Based Fuzzy Common Weights Data Envelopment Analysis Approach. Health Care Management Science, 21(4), 587-603.
[45]    Peykani, P., Mohammadi, E., Pishvaee, M. S., Rostamy-Malkhalifeh, M., & Jabbarzadeh, A. (2018). A Novel Fuzzy Data Envelopment Analysis Based on Robust Possibilistic Programming: Possibility, Necessity and Credibility-Based Approaches. RAIRO-Operations Research, 52(4-5), 1445-1463.
[46]    Peykani, P., Seyed Esmaeili, F. S., Rostamy-Malkhalifeh, M., & Hosseinzadeh Lotfi, F. (2018). Measuring Productivity Changes of Hospitals in Tehran: The Fuzzy Malmquist Productivity Index. International Journal of Hospital Research, 7(3), 1-17.
[47]    Nasseri, S. H., & Khatir, M. A. (2019). Fuzzy Stochastic Undesirable Two-Stage Data Envelopment Analysis Models with Application to Banking Industry. Journal of Intelligent & Fuzzy Systems, 37(5), 7047-7057.
[48]    Peykani, P., Mohammadi, E., Emrouznejad, A., Pishvaee, M. S., & Rostamy-Malkhalifeh, M. (2019). Fuzzy Data Envelopment Analysis: An Adjustable Approach. Expert Systems with Appli- cations, 136, 439-452.
[49]    Shiraz, R. K., Tavana, M., & Fukuyama, H. (2020). A Random-Fuzzy Portfolio Selection DEA Model Using Value-at-Risk and Conditional Value-at-Risk. Soft Computing, 24, 17167–17186.
[50]    Wardana, R. W., Masudin, I., & Restuputri, D. P. (2020). A Novel Group Decision-Making Method by P-Robust Fuzzy DEA Credibility Constraint for Welding Process Selection. Cogent Engineering, 7(1), 1728057.
[51]    Mahla, D., & Agarwal, S. (2021). A Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model. Iranian Journal of Fuzzy Systems, 18(3), 39-49.
 
[52]    Mahla, D., Agarwal, S., & Mathur, T. (2021). A Novel Fuzzy Non-Radial Data Envelopment Analysis: An Application in Transportation. RAIRO-Operations Research, 55(4), 2189-2202.
[53]    Peykani, P., Mohammadi, E., & Emrouznejad, A. (2021). An Adjustable Fuzzy Chance-Constrained Network DEA Approach with Application to Ranking Investment Firms. Expert Systems with Ap- plications, 166, 113938.
[54]    Peykani, P., & Seyed Esmaeili, F. S. (2021). Malmquist Productivity Index under Fuzzy Environ- ment. Fuzzy Optimization and Modeling Journal, 2(4), 10-19.
[55]    Arabjazi, N., Rostamy-Malkhalifeh, M., Hosseinzadeh Lotfi, F., & Behzadi, M. H. (2022). Stability Analysis with General Fuzzy Measure: An Application to Social Security Organizations. Plos One, 17(10), e0275594.
[56]    Peykani, P., Memar-Masjed, E., Arabjazi, N., & Mirmozaffari, M. (2022). Dynamic Performance Assessment of Hospitals by Applying Credibility-Based Fuzzy Window Data Envelopment Analysis. Healthcare, 10(5), 876.
[57]    Peykani, P., Namazi, M., & Mohammadi, E. (2022). Bridging the Knowledge Gap between Tech- nology and Business: An Innovation Strategy Perspective. Plos One, 17(4), e0266843.
[58]    Pourbabagol, H., Amiri, M., Taghavifard, M. T., & Hanafizadeh, P. (2023). A New Fuzzy DEA Net- work Based on Possibility and Necessity Measures for Agile Supply Chain Performance Evaluation: A Case Study. Expert Systems with Applications, 220, 119552.
[59]    Masudin, I., Mawarni, C. A., Wardana, R. W., & Restuputri, D. P. (2023). Supplier Selection Using Fuzzy DEA Credibility Constrained and relative Closeness Index: A Case of Indonesian Manufacturing Industry. Cogent Business & Management, 10(2), 2228555.
[60]    Zhou, J., & Liu, Y. (2023). Possibilistic Chance-Constrained Data Envelopment Analysis Frame- work for Failure Modes and Effects Analysis. Applied Soft Computing, 147, 110830.
[61]    Omrani, H., Alizadeh, A., Emrouznejad, A., & Teplova, T. (2024). Data Envelopment Analysis Model with Decision Makers’ Preferences: A Robust Credibility Approach. Annals of Operations Research, 339(3), 1269-1306.
[62]    Peykani, P., Sargolzaei, M., Hamidzadeh, F., Seyed Esmaeili, F. S., & Takaloo, A. (2024). Possibilistic Network DEA Approach for Performance Evaluation of Two-Stage Decision Making Units Under Uncertainty. Analytical Decision Making and Data Envelopment Analysis: Advances and Challenges, 59-79. Singapore: Springer Nature Singapore.
[63]    Shojaie, S. E., Sadjadi, S. J., & Tavakkoli-Moghaddam, R. (2024). Malmquist Productivity Index for Two-Stage Network Systems under Data Uncertainty: A Real-World Case Study. Plos One, 19(7), e0307277.
 
[64]    Mahmoodirad, A., Pamucar, D., Niroomand, S., & Simic, V. (2025). Data Envelopment Analysis Based Performance Evaluation of Hospitals–Implementation of Novel Picture Fuzzy BCC Model. Expert Systems with Applications, 263, 125775.
[65]    Peykani, P., Mahmoodirad, A., & Amirteimoori, A. (2025). A Novel Adjustable Intuitionistic Fuzzy Framework for Two-Stage Data Envelopment Analysis: An Application in the Banking Sector. In- formation Sciences, 718, 122372.
[66]    Cheng, G., Zervopoulos, P., & Qian, Z. (2013). A Variant of Radial Measure Capable of Dealing with Negative Inputs and Outputs in Data Envelopment Analysis. European Journal of Operational Research, 225(1), 100-105.
[67]    Portela, M. S., Thanassoulis, E., & Simpson, G. (2004). Negative Data in DEA: A Directional Distance Approach Applied to Bank Branches. Journal of the Operational Research Society, 55(10), 1111-1121.
[68]    Zadeh, L. A. (1978). Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems, 1(1), 3-28.
[69]    Zadeh, L. A. (2011). A Note on Z-Numbers. Information Sciences, 181(14), 2923-2932.
[70]    Rezaei Ahmadi, F., Rafiei, H., & Akbarzadeh Totonchi, M. R. (2023). An Overview of Z-Numbers and Its Applications. Fuzzy Systems and Its Applications, 6(2), 1-50.
[71]    Azadeh, A., & Kokabi, R. (2016). Z-Number DEA: A New Possibilistic DEA in the Context of Z-Numbers. Advanced Engineering Informatics, 30(3), 604-617.
[72]    Banerjee, R., Pal, S. K., & Pal, J. K. (2021). A Decade of the Z-Numbers. IEEE Transactions on Fuzzy Systems, 30(8), 2800-2812.
[73]    Alam, N. M. F. H. N. B., Ku Khalif, K. M. N., Jaini, N. I., & Gegov, A. (2023). The Application of Z-Numbers in Fuzzy Decision Making: The State of the Art. Information, 14(7), 400.
[74]    Liao, H., Liu, F., Xiao, Y., Wu, Z., & Zavadskas, E. K. (2024). A Survey on Z-Number-based Decision Analysis Methods and Applications: What’s Going on and How to Go Further?. Information Sciences, 663, 120234.
[75]    Kang, B., Wei, D., Li, Y., & Deng, Y. (2012). A Method of Converting Z-Number to Classical Fuzzy Number. Journal of Information & Computational Science, 9(3), 703-709.
[76]    Peykani, P., Nouri, M., Pishvaee, M. S., Oprean Stan, C., & Mohammadi, E. (2023). Credibilistic Multi-Period Mean-Entropy Rolling Portfolio Optimization Problem Based on Multi-Stage Scenario Tree. Mathematics, 11(18), 3889.
[77]    Charnes, A., & Cooper, W. W. (1959). Chance-Constrained Programming. Management Science, 6(1), 73-79.
 
[78]    Sun, C. C. (2010). A Performance Evaluation Model by Integrating Fuzzy AHP and Fuzzy TOPSIS Methods. Expert Systems with Applications, 37(12), 7745-7754.
[79]    Guo, S., & Zhao, H. (2017). Fuzzy Best-Worst Multi-Criteria Decision-Making Method and Its Applications. Knowledge-Based Systems, 121, 23-31.
[80]    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.
[81]    Hosseinzadeh Lotfi, F., Allahviranloo, T., Pedrycz, W., Shahriari, M., Sharafi, H., & Razipour- GhalehJough, S. (2023). Fuzzy Decision Analysis: Multi Attribute Decision Making Approach. Springer.
[82]    Peykani, P., Emrouznejad, A., & Nouri, M. (2026). Best-Worst Multi-Criteria Decision-Making Method: A Review of the Literature. Socio-Economic Planning Sciences, 104, 102345.
[83]    Kao, C. (2014). Network Data Envelopment Analysis: A Review. European Journal of Operational Research, 239(1), 1-16.
[84]    Mansouri, H., Mollaei, H. R., Mousavi-Nasab, S. H., Mohammadi, E., & Firoozi, Z. (2024). Performance Evaluation and Credit Rating of Mutual Funds Using the Optimistic-Pessimistic Fuzzy Network Data Envelopment Analysis Approach. Fuzzy Systems and Its Applications, 7(2), 149-180.
[85]    Peykani, P., Seyed Esmaeili, F. S., Pishvaee, M. S., Rostamy-Malkhalifeh, M., & Hosseinzadeh Lotfi, F. (2024). Matrix-Based Network Data Envelopment Analysis: A Common Set of Weights Approach. Socio-Economic Planning Sciences, 95, 102044.
[86]    Seyed Esmaeili, F. S., & Mohammadi, E. (2024). Z-Number Network Data Envelopment Analysis Approach: A Case Study on the Iranian Insurance Industry. Plos One, 19(7), e0306876.
[87]    Shojaie, S. E., Sadjadi, S. J., & Tavakkoli-Moghaddam, R. (2024). Malmquist Productivity Index under Network Structure and Negative Data: An Application to Banking Industry. Journal of System Management, 10(3), 107-118.
Fuzzy Systems and its Applications
Volume 8, Issue 2 - Serial Number 17
Open Access Statement
December 2025
Pages 65-90

  • Receive Date 06 April 2025
  • Revise Date 30 July 2025
  • Accept Date 17 August 2025