An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for Single-Valued Neutrosophic Sets
Abstract
:1. Introduction
2. Preliminaries
- (i)
- is contained inifandfor allThis relationship is denoted as
- (ii)
- andare said to be equal ifand
- (iii)
- for all
- (iv)
- for all
- (v)
- for all
- (i)
- (ii)
- (iii)
- where
- (iv)
- where.
- (i)
- The Hamming distance betweenandare defined as:
- (ii)
- The normalized Hamming distance betweenandare defined as:
- (ii)
- The Euclidean distance betweenandare defined as:
- (iv)
- The normalized Euclidean distance betweenandare defined as:
3. A TOPSIS Method for Single-Valued Neutrosophic Sets
3.1. Description of Problem
3.2. The Maximizing Deviation Method for Computing Incomplete or Completely Unknown Attribute Weights
- (i)
- The deviation value of all alternatives to other alternatives for the parameter denoted by is defined as:
- (ii)
- The total deviation value of all parameters to all alternatives, denoted by is defined as:
- (iii)
- The individual objective weight of each parameter denoted by is defined as:
3.3. TOPSIS Method for MADM Problems with Incomplete Weight Information
3.3.1. The Proposed TOPSIS Method for SVNSs
3.3.2. Attribute Weight Determination Method: An Integrated WEIGHT MEASure
Algorithm 1. (based on a modified TOPSIS approach). |
Step 1. Input the SVNS which represents the information pertaining to the problem. Step 2. Input the subjective weight for each of the attributes as given by the decision makers. Step 3. Compute the objective weight for each of the attributes using Equation (9). Step 4. The integrated weight coefficient for each of the attributes is computed using Equation as follow: Step 6. The difference between each alternative and the RNPIS, and the RNNIS are computed using Equations (12) and (13), respectively. Step 7. The relative closeness coefficient for each alternative is calculated using Equation (14). Step 8. Choose the optimal alternative based on the principal of maximum closeness coefficient. |
4. Application of the Topsis Method in a Made Problem
4.1. Illustrative Example
Algorithm 2. (based on the modified TOPSIS approach). |
Step 1. The SVNS constructed for this problem is given in tabular form in Table 2 Step 2. The subjective weight for each attribute as given by the procurement team (the decision makers) are Step 3. The objective weight for each attribute is computed using Equation (9) are as given below: Step 4. The integrated weight for each attribute is computed using Equation (15). The integrated weight coefficent obtained for each attribute is: Step 5. Use Equations (10) and (11) to compute the values of and from the neutrosophic numbers given in Table 2. The values are as given below: and Step 6. Use Equations (12) and (13) to compute the difference between each alternative and the RNPIS and the RNNIS, respectively. The values of and are as given below: and Step 7. Using Equation (14), the closeness coefficient for each alternative is: . Step 8. The ranking of the alternatives obtained from the closeness coefficient is as given below: |
4.2. Adaptation of the Algorithm to Non-Integrated Weight Measure
4.2.1. Objective-Only Adaptation of Our Algorithm
4.2.2. Subjective-Only Adaptation of Our Algorithm
5. Comparatives Studies
5.1. Comparison of Results Obtained Through Different Methods
5.2. Discussion of Results
- (i)
- The method proposed in this paper uses an integrated weight measure which considers both the subjective and objective weights of the attributes, as opposed to some of the methods that only consider the subjective weights or objective weights.
- (ii)
- Different operators emphasizes different aspects of the information which ultimately leads to different rankings. For example, in [40], the GSNNWA operator used is based on an arithmetic average which emphasizes the characteristics of the group (i.e., the whole information), whereas the GSNNWG operator is based on a geometric operator which emphasizes the characteristics of each individual alternative and attribute. As our method places more importance on the characteristics of the individual alternatives and attributes, instead of the entire information as a whole, our method produces the same ranking as the GSNNWG operator but different results from the GSNNWA operator.
5.3. Analysis of the Performance and Reliability of Different Methods
Analysis
- (i)
- (ii)
- The method proposed in [44] also uses the data given in above as inputs but ignores the opinions of the decision makers as it does not take into account the subjective weights of the attributes. The algorithm from this paper yields the ranking ofTo fit this data into our algorithm, we randomly assigned the subjective weights of the attributes as for . A ranking of was nonetheless obtained from our algorithm.
- (iii)
- (iv)
- It can be observed that for the methods introduced in [10,11,39,44], we have for all the entries. A similar trend can be observed in [14,43,45], where for all the entries. Therefore, we are not certain about the results obtained through the decision making algorithms in these papers when the value of deviates very far from 1.
- the track record of the suppliers that is approved by the committee
- the track record of the suppliers that the committee feels is questionable
- the track record of the suppliers that is rejected by the committee
6. Conclusions
- (i)
- A novel TOPSIS method for the SVNS model is introduced, with the maximizing deviation method used to determine the objective weight of the attributes. Through thorough analysis, we have proven that our algorithm is compliant with all of the three tests that were discussed in Section 5.3. This clearly indicates that our proposed decision-making algorithm is not only an effective algorithm but one that produces the most reliable and accurate results in all the different types of situation and data inputs.
- (ii)
- Unlike other methods in the existing literature which reduces the elements from single-valued neutrosophic numbers (SVNNs) to fuzzy numbers, or interval neutrosophic numbers (INNs) to neutrosophic numbers or fuzzy numbers, in our version of the TOPSIS method the input data is in the form of SVNNs and this form is maintained throughout the decision-making process. This prevents information loss and enables the original information to be retained, thereby ensuring a higher level of accuracy for the results that are obtained.
- (iii)
- The objective weighting method (e.g., the ones used in [10,11,14,39,40,43,45]) only takes into consideration the values of the membership functions while ignoring the preferences of the decision makers. Through the subjective weighting method (e.g., the ones used in [42,44]), the attribute weights are given by the decision makers based on their individual preferences and experiences. Very few approaches in the existing literature (e.g., [15]) consider both the objective and subjective weighting methods. Our proposed method uses an integrated weighting model that considers both the objective and subjective weights of the attributes, and this accurately reflects the input values of the alternatives as well as the preferences and risk attitude of the decision makers.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
- Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Gorzalczany, M.B. A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 1987, 21, 1–17. [Google Scholar] [CrossRef]
- Gau, W.L.; Buehrer, D.J. Vague sets. IEEE Trans. Syst. Man Cybern. 1993, 23, 610–614. [Google Scholar] [CrossRef]
- Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst. 2010, 25, 529–539. [Google Scholar] [CrossRef]
- Atanassov, K.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
- Ezhilmaran, D.; Sankar, K. Morphism of bipolar intuitionistic fuzzy graphs. J. Discret. Math. Sci. Cryptogr. 2015, 18, 605–621. [Google Scholar] [CrossRef]
- Smarandache, F. Neutrosophy. Neutrosophic Probability, Set, and Logic; ProQuest Information & Learning: Ann Arbor, MI, USA, 1998; 105p, Available online: http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (accessed on 7 June 2018).
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multisp. Multistruct. 2010, 4, 410–413. [Google Scholar]
- Ye, J. Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int. J. Gen. Syst. 2013, 42, 386–394. [Google Scholar] [CrossRef]
- Ye, J. Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making. J. Intell. Fuzzy Syst. 2014, 27, 2453–2462. [Google Scholar]
- Ye, J. Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 23, 379–389. [Google Scholar] [CrossRef]
- Ye, J. Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single valued neutrosophic environment. J. Intell. Fuzzy Syst. 2014, 27, 2927–2935. [Google Scholar]
- Huang, H.L. New distance measure of single-valued neutrosophic sets and its application. Int. J. Gen. Syst. 2016, 31, 1021–1032. [Google Scholar] [CrossRef]
- Peng, X.; Liu, C. Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set. J. Intell. Fuzzy Syst. 2017, 32, 955–968. [Google Scholar] [CrossRef]
- Yang, H.L.; Guo, Z.L.; She, Y.H.; Liao, X.W. On single valued neutrosophic relations. J. Intell. Fuzzy Syst. 2016, 30, 1045–1056. [Google Scholar] [CrossRef]
- Broumi, S.; Smarandache, F.; Talea, M.; Bakali, A. Single valued neutrosophic graph: Degree, order and size. In Proceedings of the IEEE International Conference on Fuzzy Systems, Vancouver, BC, Canada, 24–29 July 2016; pp. 2444–2451. [Google Scholar]
- Broumi, S.; Bakali, A.; Talea, M.; Smarandache, F. Isolated single valued neutrosophic graphs. Neutrosophic Sets Syst. 2016, 11, 74–78. [Google Scholar]
- Broumi, S.; Talea, M.; Bakali, A.; Smarandache, F. Single valued neutrosophic graphs. J. New Theory 2016, 10, 86–101. [Google Scholar]
- Broumi, S.; Smarandache, F.; Talea, M.; Bakali, A. An introduction to bipolar single valued neutrosophic graph theory. Appl. Mech. Mater. 2016, 841, 184–191. [Google Scholar] [CrossRef]
- Broumi, S.; Talea, M.; Bakali, A.; Smarandache, F. On bipolar single valued neutrosophic graphs. J. New Theory 2016, 11, 84–102. [Google Scholar]
- Hassan, A.; Malik, M.A.; Broumi, S.; Bakali, A.; Talea, M.; Smarandache, F. Special types of bipolar single valued neutrosophic graphs. Ann. Fuzzy Math. Inform. 2017, 14, 55–73. [Google Scholar]
- Tian, Z.P.; Wang, J.; Zhang, H.Y.; Wang, J.Q. Multi-criteria decision-making based on generalized prioritized aggregation operators under simplified neutrosophic uncertain linguistic environment. Int. J. Mach. Learn. Cybern. 2016. [Google Scholar] [CrossRef]
- Wu, X.H.; Wang, J.; Peng, J.J.; Chen, X.H. Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. Int. J. Fuzzy Syst. 2016, 18, 1104–1116. [Google Scholar] [CrossRef]
- Sahin, R.; Kucuk, A. Subsethood measure for single valued neutrosophic sets. J. Intell. Fuzzy Syst. 2015, 29, 525–530. [Google Scholar] [CrossRef]
- Ye, J. An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers. J. Intell. Fuzzy Syst. 2015, 28, 247–255. [Google Scholar]
- Biswas, P.; Pramanik, S.; Giri, B.C. TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 2016, 27, 727–737. [Google Scholar] [CrossRef]
- Majumdar, P.; Samanta, S.K. On similarity and entropy of neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 26, 1245–1252. [Google Scholar]
- Wang, Y.M. Using the method of maximizing deviations to make decision for multiindices. Syst. Eng. Electron. 1997, 8, 21–26. [Google Scholar]
- Hwang, C.L.; Yoon, K. Multiple Attribute Decision Making: Methods and Applications; Springer-Verlag: New York, NY, USA, 1981. [Google Scholar]
- Chen, M.F.; Tzeng, G.H. Combining grey relation and TOPSIS concepts for selecting an expatriate host country. Math. Comput. Model. 2004, 40, 1473–1490. [Google Scholar] [CrossRef]
- Shaw, K.; Shankar, R.; Yadav, S.S.; Thakur, L.S. Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain. Expert Syst. Appl. 2012, 39, 8182–8192. [Google Scholar] [CrossRef]
- Rouyendegh, B.D.; Saputro, T.E. Supplier selection using fuzzy TOPSIS and MCGP: A case study. Procedia Soc. Behav. Sci. 2014, 116, 3957–3970. [Google Scholar] [CrossRef]
- Dargi, A.; Anjomshoae, A.; Galankashi, M.R.; Memari, A.; Tap, M.B.M. Supplier selection: A fuzzy-ANP approach. Procedia Comput. Sci. 2014, 31, 691–700. [Google Scholar] [CrossRef]
- Kaur, P. Selection of vendor based on intuitionistic fuzzy analytical hierarchy process. Adv. Oper. Res. 2014, 2014. [Google Scholar] [CrossRef]
- Kaur, P.; Rachana, K.N.L. An intuitionistic fuzzy optimization approach to vendor selection problem. Perspect. Sci. 2016, 8, 348–350. [Google Scholar] [CrossRef]
- Dweiri, F.; Kumar, S.; Khan, S.A.; Jain, V. Designing an integrated AHP based decision support system for supplier selection in automotive industry. Expert Syst. Appl. 2016, 62, 273–283. [Google Scholar] [CrossRef]
- Junior, F.R.L.; Osiro, L.; Carpinetti, L.C.R. A comparison between fuzzy AHP and fuzzy TOPSIS methods to supplier selection. Appl. Soft Comput. 2014, 21, 194–209. [Google Scholar] [CrossRef]
- Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 26, 2459–2466. [Google Scholar]
- Peng, J.J.; Wang, J.; Wang, J.; Zhang, H.; Chen, X.H. Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int. J. Syst. Sci. 2016, 47, 2342–2358. [Google Scholar] [CrossRef]
- Maji, P.K. A neutrosophic soft set approach to a decision making problem. Ann. Fuzzy Math. Inform. 2012, 3, 313–319. [Google Scholar]
- Karaaslan, F. Neutrosophic soft sets with applications in decision making. Int. J. Inf. Sci. Intell. Syst. 2015, 4, 1–20. [Google Scholar]
- Ye, J. Single valued neutrosophic cross-entropy for multicriteria decision making problems. Appl. Math. Model. 2014, 38, 1170–1175. [Google Scholar] [CrossRef]
- Biswas, P.; Pramanik, S.; Giri, B.C. Entropy based grey relational analysis method for multi-attribute decision making under single valued neutrosophic assessments. Neutrosophic Sets Syst. 2014, 2, 102–110. [Google Scholar]
- Ye, J. Improved cross entropy measures of single valued neutrosophic sets and interval neutrosophic sets and their multicriteria decision making methods. Cybern. Inf. Technol. 2015, 15, 13–26. [Google Scholar] [CrossRef]
. |
Method | The Final Ranking | The Best Alternative |
---|---|---|
Ye [39] (i) WAAO * (ii) WGAO ** | ||
Ye [10] (i) Weighted correlation coefficient (ii) Weighted cosine similarity measure | ||
Ye [11] | ||
Huang [14] | ||
Peng et al. [40] (i) GSNNWA *** (ii) GSNNWG **** | ||
Peng & Liu [15] (i) EDAS (ii) Similarity measure | ||
Maji [41] | ||
Karaaslan [42] | ||
Ye [43] | ||
Biswas et al. [44] | ||
Ye [45] | ||
Adaptation of our algorithm (objective weights only) | ||
Adaptation of our algorithm (subjective weights only) | ||
Our proposed method (using integrated weight measure) |
Paper | Test 1 Compliance | Test 2 Compliance | Test 3 Compliance | |
---|---|---|---|---|
Ye [39] | WAAO * | Y | Y | N |
WGAO * | N | Y | N | |
Ye [10] | Weighted correlation coefficient | Y | Y | N |
Weighted cosine similarity measure | N | Y | N | |
Ye [11] | Y | Y | N | |
Huang [14] | Y | Y | N | |
Peng et al. [40] | GSNNWA ** | Y | Y | N |
GSNNWG ** | Y | Y | N | |
Peng & Liu [15] | EDAS | Y | Y | N |
Similarity measure | N | Y | Y | |
Maji [41] | N | N | N | |
Karaaslan [42] | Y | Y | N | |
Ye [43] | Y | Y | N | |
Biswas et al. [44] | Y | N | Y | |
Ye [45] | Y | Y | N | |
Adaptation of our proposed algorithm (objective weights only) | Y | N | Y | |
Adaptation of our proposed algorithm (subjective weights only) | Y | Y | N | |
Our proposed algorithm | Y | Y | Y |
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Selvachandran, G.; Quek, S.G.; Smarandache, F.; Broumi, S. An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for Single-Valued Neutrosophic Sets. Symmetry 2018, 10, 236. https://doi.org/10.3390/sym10070236
Selvachandran G, Quek SG, Smarandache F, Broumi S. An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for Single-Valued Neutrosophic Sets. Symmetry. 2018; 10(7):236. https://doi.org/10.3390/sym10070236
Chicago/Turabian StyleSelvachandran, Ganeshsree, Shio Gai Quek, Florentin Smarandache, and Said Broumi. 2018. "An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for Single-Valued Neutrosophic Sets" Symmetry 10, no. 7: 236. https://doi.org/10.3390/sym10070236