本論文旨在根據我們所提之區間直覺模糊值之得分函數、分數矩陣、及我們所提之非線性規劃法提出一個新的多屬性決策方法。首先，我們根據所提之新的得分函數以計算決策矩陣中之每一個區間直覺模糊值之得分值來建構分數矩陣，其中我們所提之區間直覺模糊值之新的得分函數能夠克服目前已存在之區間直覺模糊值得分函數的缺點。然後，我們分別計算在分數矩陣中每一行得分值之平均值。然後，我們根據所得之分數矩陣、在分數矩陣中每一行得分值之平均值、偏差變量的概念、及決策者所提供之每一個屬性的區間直覺模糊權重來構建非線性規劃模型。然後，我們求解非線性規劃模型以獲得每一個屬性的最佳權重。最後，我們根據所得之分數矩陣及每一個屬性的最佳權重計算每一個方案的加權分數，且根據所得之每一個方案的加權分數對每一個方案進行排序，其中加權分數越大的方案，其偏好順序越好。我們所提之多屬性決策方法能夠克服目前已存在之多屬性決策方法的缺點。我們所提之多屬性決策方法提供一個很有用的方法以在區間直覺模糊環境下作多屬性決策。

In this thesis, we propose a new multiple attribute decision making method on the basis of the proposed score function of interval-valued intuitionistic fuzzy values, the score matrix, and the proposed nonlinear programming model. Firstly, we apply the proposed score function of interval-valued intuitionistic fuzzy values to build the score matrix, where the proposed score function of interval-valued intuitionistic fuzzy values can conquer the shortcomings of the existing score functions of interval-valued intuitionistic fuzzy values. Then, we compute the average value of the values appeared at each column of the score matrix. Then, we construct the nonlinear programming model based on the obtained score matrix, the obtained average value of the values appeared at each column of the score matrix, the concept of deviation variables, and the interval-valued intuitionistic fuzzy weight of each attribute offered by the decision maker. Then, we solve the nonlinear programming model to obtain the optimal weight of each attribute. Then, we compute the weighted score of each alternative using the obtained score matrix and the optimal weights of the attributes. Finally, the alternatives are ranked according to the obtained weighted scores. The larger the weighted score of an alternative, the better the preference order of the alternative. The proposed multiple attribute decision making method can overcome the drawbacks of the existing multiple attribute decision making methods. It offers us a very useful way for multiple attribute decision making in interval-valued intuitionistic fuzzy environments.

[1] M. Akram, G. Shahzadi, and X. Peng, “Extension of Einstein geometric operators to multi-attribute decision making under q-rung orthopair fuzzy information,” Granular Computing, vol. 6, no. 4, pp. 779-795, 2021
[2] K. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343-349, 1989.
[3] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87-96, 1986.
[4] Z. Bai, “An interval-valued intuitionistic fuzzy TOPSIS method based on an improved score function,” The Scientific World Journal, vol. 2013, Article ID 879089, 6 pages, 2013.
[5] E. Bas, U. Yolcu, and E. Egrioglu, “Intuitionistic fuzzy time series functions approach for Time Series forecasting,” Granular Computing, vol. 6, no. 3, pp. 619-629, 2021.
[6] S. M. Chen and C. H. Chiou, “Multiattribute decision making based on interval-valued intuitionistic fuzzy sets, PSO techniques, and evidential reasoning methodology,” IEEE Transactions on Fuzzy Systems, vol. 23, no. 6, pp. 1905-1916, 2015.
[7] S. M. Chen and Y. C. Chu, “Multiattribute decision making based on U-quadratic distribution of intervals and the transformed matrix in interval-valued intuitionistic fuzzy environments,” Information Sciences, vol. 537, pp. 30-45, 2020.
[8] S. M. Chen and H. L. Deng, “Multiattribute decision making based on nonlinear programming methodology and novel score function of interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 607, pp. 1348-1371, 2022.
[9] S. M. Chen and K. Y. Fan, “Multiattribute decision making based on probability density functions and the variances and standard deviations of largest ranges of evaluating interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 490, pp. 329-343, 2019.
[10] S. M. Chen and W. H. Han, “A new multiattribute decision making method based on multiplication operations of interval-valued intuitionistic fuzzy values and linear programming methodology,” Information Sciences, vol. 429, pp. 421-432, 2018.
[11] S. M. Chen and W. H. Han, “An improved MADM method using interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 467, pp. 489-505, 2018.
[12] S. M. Chen and M. H. Hsu, “Multiple attribute decision making based on novel score function of interval-valued intuitionistic fuzzy values, score matrix, and nonlinear programming model,” Information Sciences, vol. 645, 119332, 2023.
[13] S. M. Chen and Z. C. Huang, “Multiattribute decision making based on interval-valued intuitionistic fuzzy values and linear programming methodology,” Information Sciences, vol. 381, pp. 341-351, 2017.
[14] S. M. Chen and L. W. Kuo, “Multiattribute decision making based on non-linear programming methodology with hyperbolic function and interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 453, pp. 379-388, 2018.
[15] S. M. Chen, L. W. Kuo, and X. Y. Zou, “Multiattribute decision making based on Shannon’s information entropy, non-linear programming methodology, and interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 465, pp. 404-424, 2018.
[16] S. M. Chen and W. T. Liao, “Multiple attribute decision making using Beta distribution of intervals, expected values of intervals, and new score function of interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 579, pp. 863-887, 2021.
[17] S. M. Chen and B. D. H. Phuong, “Fuzzy time series forecasting based on optimal partitions of intervals and optimal weighting vectors,” Knowledge-Based Systems, vol. 118, pp. 204-216, 2017.
[18] S. M. Chen and C. A. Tsai, “Multiattribute decision making using novel score function of interval-valued intuitionistic fuzzy values and the means and the variances of score matrices,” Information Sciences, vol. 577, pp. 748-768, 2021.
[19] S. M. Chen and K. Y. Tsai, “Multiattribute decision making based on new score function of interval-valued intuitionistic fuzzy values and normalized score matrices,” Information Sciences, vol. 575, pp. 714-731, 2021.
[20] S. M. Chen and C. H. Wang, “Fuzzy risk analysis based on ranking fuzzy numbers using α-cuts, belief features and signal/noise ratios,” Expert Systems with Applications, vol. 36, no. 3, pp. 5576-5581, 2009.
[21] S. M. Chen and S. H. Yu, “Multiattribute decision making based on novel score function and the power operator of interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 606, pp. 763-785, 2022.
[22] T. Y. Chen, “A comparative analysis of score functions for multiple criteria decision making in intuitionistic fuzzy settings,” Information Sciences, vol. 181, no. 17, pp. 3652-3676, 2011.
[23] J. Dong, S. Wan, and S. M. Chen, “Fuzzy best-worst method based on triangular fuzzy numbers for multi-criteria decision-making,” Information Sciences, vol. 547, pp. 1080-1104, 2021.
[24] P. Dutta and D. Doley, “Fuzzy decision making for medical diagnosis using arithmetic of generalised parabolic fuzzy numbers,” Granular Computing, vol. 6, no. 2, pp. 377-388, 2021.
[25] P. Dutta, “Multi-criteria decision making under uncertainty via the operations of generalized intuitionistic fuzzy numbers,” Granular Computing, vol. 6, no. 2, pp. 321-337, 2021.
[26] F. Feng, C. Zhang, M. Akram, and J. Zhang, “Multiple attribute decision making based on probabilistic generalized orthopair fuzzy sets,” Granular Computing, vol. 8, no. 4, pp. 863-891, 2023.
[27] R. Gupta and S. Kumar, “Intuitionistic fuzzy scale-invariant entropy with correlation coefficients-based VIKOR approach for multi-criteria decision-making,” Granular Computing, vol. 7, no. 1, pp. 77-93, 2022.
[28] Y. T. İç and M. Yurdakul, “Development of a new trapezoidal fuzzy AHP-TOPSIS hybrid approach for manufacturing firm performance measurement,” Granular Computing, vol. 6, no. 4, pp. 915-929, 2021.
[29] R. Kadian and S. Kumar, “A novel intuitionistic Renyi’s-Tsallis discriminant information measure and its applications in decision-making,” Granular Computing, vol. 6, no. 4, pp. 901-913, 2021.
[30] K. Kumar and S. M. Chen, “Multiattribute decision making based on interval-valued intuitionistic fuzzy values, score function of connection numbers, and the set pair analysis theory,” Information Sciences, vol. 551, pp. 100-112, 2021.
[31] K. Kumar and S. M. Chen, “Multiattribute decision making based on the improved intuitionistic Fuzzy Einstein weighted averaging operator of intuitionistic fuzzy values,” Information Sciences, vol. 568, pp. 369-383, 2021.
[32] D. F. Li, “TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 2, pp. 299-311, 2010.
[33] M. Pant and S. Kumar, “Particle swarm optimization and intuitionistic fuzzy set-based novel method for fuzzy time series forecasting,” Granular Computing, vol. 7, no. 2, pp. 285-303, 2022.
[34] K. Patra, “Fuzzy risk analysis using a new technique of ranking of generalized trapezoidal fuzzy numbers,” Granular Computing, vol. 7, no. 1, pp. 127-140, 2022.
[35] Y. Qin, X. Cui, M. Huang, Y. Zhong, Z. Tang, and P. Shi, “Multiple-attribute decision-making based on picture fuzzy Archimedean power Maclaurin symmetric mean operators,” Granular Computing, vol. 6, no. 3, pp. 737-761, 2021.
[36] K. Rahman, S. Ayub, and S. Abdullah, “Generalized intuitionistic fuzzy aggregation operators based on confidence levels for group decision making,” Granular Computing, vol. 6, no. 4, pp. 867-886, 2021.
[37] M. R. Seikh and U. Mandal, “Intuitionistic fuzzy Dombi aggregation operators and their application to multiple attribute decision-making,” Granular Computing, vol. 6, no. 3, pp. 473–488, 2021.
[38] M. R. Seikh and U. Mandal, “Multiple attribute decision-making based on 3,4-quasirung fuzzy sets,” Granular Computing, vol. 7, no. 4, pp. 965-978, 2022.
[39] M. R. Seikh and U. Mandal, “Multiple attribute group decision making based on quasirung orthopair fuzzy sets: Application to Electric Vehicle Charging Station site selection problem,” Engineering Applications of Artificial Intelligence, vol. 115, 105299, 2022.
[40] M. R. Seikh and U. Mandal, “Q-rung orthopair fuzzy Frank aggregation operators and its application in multiple attribute decision-making with unknown attribute weights,” Granular Computing, vol. 7, no. 3, pp. 709-730, 2022.
[41] S. Sen, K. Patra, and S. K. Mondal, “A new approach to similarity measure for generalized trapezoidal fuzzy numbers and its application to fuzzy risk analysis,” Granular Computing, vol. 6, no. 3, pp. 705-718, 2021.
[42] C. Y. Wang and S. M. Chen, “Multiple attribute decision making based on interval-valued intuitionistic fuzzy sets, linear programming methodology, and the extended TOPSIS method,” Information Sciences, vol. 397, pp. 155-167, 2017.
[43] C. Y. Wang and S. M. Chen, “An improved multiattribute decision making method based on new score function of interval-valued intuitionistic fuzzy values and linear programming methodology,” Information Sciences, vol. 411, pp. 176-184, 2017.
[44] C. Y. Wang and S. M. Chen, “A new multiple attribute decision making method based on linear programming methodology and novel score function and novel accuracy function of interval-valued intuitionistic fuzzy values,” Information Sciences, vol. 438, pp. 145-155, 2018.
[45] H. J. Wang, Y. Liu, F. Liu, and J. Lin, “Multiple attribute decision-making method based upon intuitionistic fuzzy partitioned dual Maclaurin symmetric mean operators,” International Journal of Computational Intelligence Systems, vol. 14, Article Number: 154, 2021.
[46] W. Wang, J. Zhan, and J. Mi, “A three-way decision approach with probabilistic dominance relations under intuitionistic fuzzy information,” Information Sciences, vol. 582, pp. 114-145.
[47] Z. S. Xu, “Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making,” Control and Decision, vol. 22, no. 2, pp. 215-219, 2007. (in Chinese)
[48] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.
[49] A. Zeb, A. Khan, M. Fayaz, and M. Izhar, “Aggregation operators of Pythagorean fuzzy bi-polar soft sets with application in multiple attribute decision making,” Granular Computing, vol. 7, no. 4, pp. 931-950, 2022.
[50] S. Zeng, S. M. Chen, and K. Y. Fan, “Interval-valued intuitionistic fuzzy multiple attribute decision making based on nonlinear programming methodology and TOPSIS method,” Information Sciences, vol. 506, pp. 424-442, 2020.
[51] Z. Zhao and Y. Zhang, “Multiple attribute decision making method in the frame of interval-valued intuitionistic fuzzy sets,” in: Proceedings of the 2011 Eighth International Conference on Fuzzy Systems and Knowledge Discovery, Shanghai, China, 2011, pp. 192-196.