VIKOR-based group decision-making method for software quality assessment

Document Type : Research Paper


1 College of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, China

2 Southern Marine Science and Engineering Guangdong Laboratory (Zhanjiang), Zhanjiang, Guangdong, China


Software quality is an important research direction in software engineering domain,
  which is a multi-dimen-sional evaluation problem.
  The VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) technique is a comprehensive
  method for handing the multi-dimensional evaluation problems. The projection is commonly used to measure the closeness between two decision objects in decision science.
  However, two research questions are found in this work: (1) There is no specific concrete regret
  matrix in current VIKOR technique; (2) The existing projection measures are not
  always reasonable in interval-valued intuitionistic fuzzy setting. Two research gaps are filled
  systematically in this paper: (1) A specific concrete regret matrix is provided in this extended VIKOR-based group decision-making method; (2) A new normalized projection measure is provided in order to measure the closeness between two decision matrices.
  The decision procedures are also provided in this paper. A practical application to
  the software quality assessment is introduced.
  Some experimental comparisons are provided in order to illustrate the feasibility and practicability of introduced method.


[1] M. Amiri, S. A. Ayazi, L. Olfat, J. S. Moradi, Group decision making process for supplier selection with VIKOR under fuzzy circumstance case study: An Iranian car parts supplier, International Bulletin of Business Administration, 10(6) (2011), 66-75.
[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87-96.
[3] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31(3) (1989), 343-349.
[4] D. Azar, H. Harmanani, R. Korkmaz, A hybrid heuristic approach to optimize rule-based software quality estimation models, Information and Software Technology, 51(9) (2009), 1365-1376.
[5] T. K. Biswas, A. Abbasi, R. K. Chakrabortty, A two-stage VIKOR assisted multi-operator differential evolution approach for influence maximization in social networks, Expert Systems with Applications, 192 (2022), 116342.
[6] G. Buy¨uk¨ozkan, F. G¨o¸cer, Y. Karabulut, A new group decision making approach with IF AHP and IF VIKOR for selecting hazardous waste carriers, Measurement, 134 (2019), 66-82.
[7] S. C¸ alı, S¸. Y. Balaman, A novel outranking based multi criteria group decision making methodology integrating ELECTRE and VIKOR under intuitionistic fuzzy environment, Expert Systems with Applications, 119 (2019), 36-50.
[8] S. T. Chae, E. S. Chung, J. Jiang, Robust siting of permeable pavement in highly urbanized watersheds considering climate change using a combination of fuzzy-TOPSIS and the VIKOR method, Water Resources Management, 36 (2022), 951-969.
[9] T. Y. Chen, An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its application to financial decision making, Applied Soft Computing, 71 (2018), 460-487.
[10] T. Y. Chen, Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis, Information Fusion, 41 (2018), 129-150.
[11] T. Y. Chen, A novel VIKOR method with an application to multiple criteria decision analysis for hospital-based post-acute care within a highly complex uncertain environment, Neural Computing and Applications, 31(8) (2019), 3969-3999
[12] S. M. Chen, Z. C. Huang, Multiattribute decision making based on interval-valued intuitionistic fuzzy values and particle swarm optimization techniques, Information Sciences, 397 (2017), 206-218.
[13] S. H. Cheng Autocratic multiattribute group decision making for hotel location selection based on interval-valued intuitionistic fuzzy sets, Information Sciences, 427 (2018), 77-87.
[14] J. Chu, X. Liu, L. Wang, Y. Wang, A group decision making approach based on newly defined additively consistent interval-valued intuitionistic preference relations, International Journal of Fuzzy Systems, 20(3) (2018), 1027-1046.
[15] P. Gupta, M. K. Mehlawat, N. Grover, Intuitionistic fuzzy multi-attribute group decision-making with an application to plant location selection based on a new extended VIKOR method, Information Sciences, 370 (2016), 184-203.
[16] P. Gupta, M. K. Mehlawat, N. Grover, W. Pedrycz, Multi-attribute group decision making based on extended TOPSIS method under interval-valued intuitionistic fuzzy environment, Applied Soft Computing, 69 (2018), 554-567.
[17] P. H´ajek, W. Froelich, Integrating TOPSIS with interval-valued intuitionistic fuzzy cognitive maps for effective group decision making, Information Sciences, 485 (2019), 394-412.
[18] P. Ji, H. Y. Zhang, J. Q. Wang, A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection, Neural Computing and Applications, 29(1) (2018), 221-234.
[19] D. K. Joshi, S. Kumar, Entropy of interval-valued intuitionistic hesitant fuzzy set and its application to group decision making problems, Granular Computing, 3(4) (2018), 367-381.
[20] M. J. Khan, M. I. Ali, P. Kumam, W. Kumam, M. Aslam, J. C. R. Alcantud, Improved generalized dissimilarity measure-based VIKOR method for Pythagorean fuzzy sets, International Journal of Intelligent Systems, 37(3) (2022), 1807-1845.
[21] E. Koksalmis, O. Kabak, Deriving decision makers’ weights in group decision making: An overview of objective methods, Information Fusion, 49 (2019), 146-160. 
[22] D. Kong, T. Chang, J. Pan, N. Hao, K. Xu, L. Zhang, G. Yang, A decision variable-based combinatorial optimization approach for interval-valued intuitionistic fuzzy MAGDM, Information Sciences, 484 (2019), 197-218.
[23] D. Kong, T. Chang, Q. Wang, H. Sun, W. Dai, A threat assessment method of group targets based on intervalvalued intuitionistic fuzzy multi-attribute group decision-making, Applied Soft Computing, 67 (2018), 350-369.
[24] D. Liang, Y. Zhang, Z. Xu, A. Jamaldeen, Pythagorean fuzzy VIKOR approaches based on TODIM for evaluating internet banking website quality of Ghanaian banking industry, Applied Soft Computing, 78 (2019), 583-594.
[25] P. Liu, H. Li, Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making, Cognitive Computation, 9(4) (2017), 494-512.
[26] J. A. Marshall, G. Brown, A. N. Radford, Individual confidence-weighting and group decision-making, Trends in Ecology and Evolution, 32(9) (2017), 636-645.
[27] V. Mohagheghi, S. M. Mousavi, B. Vahdani, Enhancing decision-making flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and pythagorean fuzzy sets, Applied Soft Computing, 61 (2017), 527-535.
[28] S. E. Mohammadi, A. Makui, Multi-attribute group decision making approach based on interval-valued intuitionistic fuzzy sets and evidential reasoning methodology, Soft Computing, 21(17) (2017), 5061-5080.
[29] S. Opricovic, G. Tzeng, Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, European Journal of Operational Research, 156(2) (2004), 445-455.
[30] N. Ploskas, J. Papathanasiou, A decision support system for multiple criteria alternative ranking using TOPSIS and VIKOR in fuzzy and nonfuzzy environments, Fuzzy Sets and Systems, 377 (2019), 1-30.
[31] F. Smarandache, α-discounting method for multi-criteria decision making (α-d MCDM), In 2010 13th International Conference on Information Fusion, IEEE, (2010), 1-7.
[32] F. Smarandache, α-discounting method for multi-criteria decision making (α-D MCDM), SCS AdSumus, Oradea, Romania and Educational Publisher, Columbus, USA, 2015.
[33] C. Y. Tsao, T. Y. Chen, A projection-based compromising method for multiple criteria decision analysis with interval-valued intuitionistic fuzzy information, Applied Soft Computing, 45 (2016), 207-223.
[34] B. Vahdani, S. M. Mousavi, H. Hashemi, M. Mousakhani, R. Tavakkoli-Moghaddam, A new compromise solution method for fuzzy group decision-making problems with an application to the contractor selection, Engineering Applications of Artificial Intelligence, 26(2) (2013), 779-788.
[35] B. Vahdani, S. M. Mousavi, R. Tavakkoli-Moghaddam, Group decision making based on novel fuzzy modified TOPSIS method, Applied Mathematical Modelling, 35(9) (2011), 4257-4269. 
[36] L. Wang, H. Y. Zhang, J. Q. Wang, L. Li, Picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project, Applied Soft Computing, 64 (2018), 216-226.
[37] G. Wei, F. E. Alsaadi, T. Hayat, A. Alsaedi, Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics, 9 (2018), 713-719.
[38] Z. Wu, J. Ahmad, J. Xu, A group decision making framework based on fuzzy VIKOR approach for machine tool selection with linguistic information, Applied Soft Computing, 42 (2016), 314-324.
[39] Q. Wu, W. Lin, L. Zhou, Y. Chen, H. Chen, Enhancing multiple attribute group decision making flexibility based on information fusion technique and hesitant Pythagorean fuzzy sets, Computers and Industrial Engineering, 127 (2019), 954-970.
[40] Z. Wu, J. Xu, X. Jiang, L. Zhong, Two MAGDM models based on hesitant fuzzy linguistic term sets with possibility distributions: VIKOR and TOPSIS, Information Sciences, 473 (2019), 101-120.
[41] H. Wu, Z. Xu, P. Ren, H. Liao, Hesitant fuzzy linguistic projection model to multi-criteria decision making for hospital decision support systems, Computers and Industrial Engineering, 115 (2018), 449-458.
[42] Q. Wu, L. Zhou, Y. Chen, H. Chen, An integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods, Information Sciences, 502 (2019), 394-417.
[43] Z. Xu, J. Chen, An approach to group decision making based on interval-valued intuitionistic judgment matrices, Systems Engineering: Theory and Practice, 27(4) (2007), 126-132.
[44] Z. Xu, H. Hu, Projection models for intuitionistic fuzzy multiple attribute decision making, International Journal of Information Technology and Decision Making, 9(2) (2010), 267-280.
[45] G. Xu, F. Liu, An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection, Applied Mathematical Modelling, 37(6) (2013), 3929-3943.
[46] X. Y. You, J. X. You, H. C. Liu, L. Zhen, Group multi-criteria supplier selection using an extended VIKOR method with interval 2-tuple linguistic information, Expert Systems with Applications, 42(4) (2015), 1906-1916.
[47] Z. Yue, Application of the projection method to determine weights of decision makers for group decision making, Scientia Iranica, 19(3) (2012), 872-878.
[48] Z. Yue, Approach to group decision making based on determining the weights of experts by using projection method, Applied Mathematical Modelling, 36(7) (2012), 2900-2910. 
[49] Z. Yue, An intuitionistic fuzzy projection-based approach for partner selection, Applied Mathematical Modelling, 37(23) (2013), 9538-9551.
[50] Z. Yue, TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting, Information Sciences, 277 (2014), 141-153.
[51] C. Yue, A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group decision-making, Computers and Industrial Engineering, 102 (2016), 233-245.
[52] C. Yue, Entropy-based weights on decision makers in group decision-making setting with hybrid preference representations, Applied Soft Computing, 60 (2017), 737-749.
[53] C. Yue, Two normalized projection modfels and application to group decision-making, Journal of Intelligent and Fuzzy Systems, 32(6) (2017), 4389-4402.
[54] C. Yue, Normalized projection approach to group decision-making with hybrid decision information, International Journal of Machine Learning and Cybernetics, 9(8) (2018), 1365-1375.
[55] C. Yue, A novel approach to interval comparison and application to software quality evaluation, Journal of Experimental and Theoretical Artificial Intelligence, 30(5) (2018), 583-602.
[56] C. Yue, Projection-based approach to group decision-making with hybrid information representations and application to software quality evaluation, Computers and Industrial Engineering, 132 (2018), 98-113.
[57] C. Yue, An interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge transfer effectiveness, Neural Computing and Applications, 31(11) (2019), 7685-7706.
[58] C. Yue, Normalization of attribute values with interval information in group decision-making setting: With an application to software quality evaluation, Journal of Experimental and Theoretical Artificial Intelligence, 31(3) (2019), 475-492.
[59] C. Yue, A normalized projection-based group decision-making method with heterogeneous decision information and application to software development effort assessment, Applied Intelligence, 49(10) (2019), 3587-3605.
[60] C. Yue, A projection-based approach to software quality evaluation from the users’ perspectives, International Journal of Machine Learning and Cybernetics, 10(9) (2019), 2341-2353.
[61] C. Yue, A symbol-based fuzzy decision-making approach to evaluate the user satisfaction on services in academic digital libraries, Iranian Journal of Fuzzy Systems, 16(2) (2019), 73-86.
[62] C. Yue, An intuitionistic fuzzy projection-based approach and application to software quality evaluation, Soft Computing, 24 (2020), 429-443.
[63] C. Yue, Picture fuzzy normalized projection and extended VIKOR approach to software reliability assessment, Applied Soft Computing, (2020), 106056.
[64] Z. Yue, Y. Jia, A group decision making model with hybrid intuitionistic fuzzy information, Computers and Industrial Engineering, 87 (2015), 202-212.
[65] Z. Yue, Y. Jia, A direct projection-based group decision-making methodology with crisp values and interval data, Soft Computing, 21(9) (2017), 2395-2405.
[66] Z. Yue, Y. Jia, A projection-based approach to intuitionistic fuzzy group decision making, Scientia Iranica, 24(3) (2017), 1505-1518.
[67] C. Yue, Z. Yue, Measuring the satisfaction and loyalty for Chinese smartphone users: A simple symbol-based decision making method, Scientia Iranica, 26(1) (2019), 589-604.
[68] S. Zeng, S. M. Chen, L. W. Kuo, Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method, Information Sciences, 488 (2019), 76-92.