 Previous Article in Journal
Acknowledgment to Reviewers of Entropy in 2021
Article

# An Intuitionistic Extension of the Simple WISP Method

1
Institute of Sustainable Construction, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania
2
3
*
Author to whom correspondence should be addressed.
Entropy 2022, 24(2), 218; https://doi.org/10.3390/e24020218
Received: 22 December 2021 / Revised: 24 January 2022 / Accepted: 27 January 2022 / Published: 29 January 2022

## Abstract

In this article, we present a new extension of the Integrated Simple Weighted Sum-Product (WISP) method, adapted for intuitionistic numbers. The extension takes advantage of intuitionistic fuzzy sets for solving complex decision-making problems. The example of contractor selection demonstrates the use of the proposed extension.

## 1. Introduction

Many decision-making problems are related to inaccuracies, unreliability, or predictions. Therefore, the significant development and use of multiple criteria decision making (MCDM) occurred after Zadeh  proposed the fuzzy set theory. Based on the fuzzy set theory, Bellman and Zadeh  proposed decision making in a fuzzy environment and thus enabled the use of MCDM for solving more complex decision-making problems. Indeed, the use of MCDM methods for solving decision-making problems in a fuzzy environment also required their adaptation to fuzzy sets.
The possibilities of fuzzy sets to apply crisp numbers influenced newly proposed extensions of the fuzzy set theory, such as interval-valued fuzzy (IVF) sets , intuitionistic fuzzy (IF) sets , neutrosophic set theory , and others. Based on the IVF and IF sets, Atanassov and Gargov  introduced interval-valued intuitionistic fuzzy (IVIF) sets. In the fuzzy set theory, Zadeh  introduced the membership function $μ A ( x )$, which represents the belonging to the set, In IF set theory, Atanassov  extended the fuzzy set theory by introducing the non-membership function $μ A ( x )$, , with the following restriction The introduction of the non-membership function enabled the IF set theory to solve some decision-making problems that could not be easily solved by applying the FS theory.
Decision makers introduced many MCDM methods to solve complicated MCDM problems over time, such as ELECTRE , AHP , TOPSIS , COPRAS , VIKOR , MULTIMOORA [12,13], ARAS , WASPAS , and others. In addition to well-known MCDM methods, there are also newly proposed ones such as the EDAS , CODAS , CoCoSo , and MULTIMOOSRAL methods . A comprehensive overview of the newly proposed MCDM methods, as well as their applications, can be found in Mardani et al. [20,21], Hafezalkotob et al. , Chandrawati et al. , and Liu and Xu .
IF sets had success in many problems, such as selecting knowledge management systems , assessing and ranking the risk of failure modes , choosing the right supplier , monitoring and continuous improving of an end-of-life vehicle management system , analyzing failure mode effects , and assessing solid waste management techniques . Decision makers, to solve a much more comprehensive range of problems, proposed many extensions for almost all MCDM methods, such as TOPSIS [31,32], VIKOR , MULTIMOORA , and ARAS .
Decision makers have used the IVIF sets to assess reservoir flood control management , select the proper facility location , choose proper sustainable material , evaluate public transportation options , prioritize risks , rank choices of sustainable organizational development of companies , evaluate malicious code threats , and prioritize government roles in a merger and acquisition process . Moreover, decision makers have used the IVIF sets to determine criteria weights [44,45,46]. Roszkowska et al.  also adopted the intuitionistic fuzzy TOPSIS for assessing social and economic phenomena. Similar to IFS, appropriate IVIF extensions are available for many MCDM methods, such as the COPRAS , WASPAS [36,49], ELECTRE , CODAS [37,38], TOPSIS [51,52], VIKOR , and CoCoSo  methods.
Stanujkic et al.  proposed the Integrated Simple Weighted Sum-Product (WISP) method. So far, there is no extension proposed for this method that allows its usage with IF sets, i.e., IF numbers.
Therefore, in this article, we suggest an extension of the WISP method, enabling IF numbers. The rest of the article is structured as follows. Section 2 explains the basic elements of IF sets. Section 3 presents the WISP method. Section 4 introduces an intuitionistic extension of the WISP method and proposes an IF-WISP method. Section 5 considers an example of contractor selection to illustrate the usage of the proposed extension. Section 6 compares the results obtained using the proposed approach and similar extensions of MCDM methods. The final section presents conclusions.

## 2. Preliminaries

This section presents some basic elements of IF sets.

#### 2.1. The Basic Elements of Intuitionistic Fuzzy Sets

Definition 1.
Let X be the universe of discourse. The IF set I in X is as follows :
where $μ I ( x )$denotes the extent of the membershipand $ν I ( x )$denotes the extent of the non-membership of the element x to the set I, , and .
Membership and non-membership functions can have different shapes such as trapezoidal, triangular, Gaussian, or the less commonly used singleton.
Definition 2.
A singleton intuitionistic fuzzy (SIF) number$i = < t i , f i >$, shown in Figure 1, is as follows:
$μ I ( x ) = { t i x = m 0 o t h e r w i s e ,$
$ν I ( x ) = { f i x = m 0 o t h e r w i s e ,$
where$m ∈ ℜ .$
Definition 3.
Let and $i 2 = < t 2 , f 2 >$be two IF numbers and $λ > 0$. The basic operations on IF numbers are as follows:
Definition 4.
Letbe an IF number. The score function s(i) of i is as follows :
$s ( i ) = t i − f i ,$
where $s ( i ) ∈ [ 1 , − 1 ]$.
Definition 5.
Letbe a collection of n SIF numbers. The intuitionistic fuzzy weighted arithmetic mean (IFWA) operator of Ij is as follows :
where wj denotes the weight of element j of the collection Aj, and = 1.
Definition 6.
Let be a collection of n SIF numbers. The intuitionistic fuzzy weighted geometric (IFWG) operator of Ij is as follows :
where wj denotes the weight of element j of the collection Aj, and = 1.

#### 2.2. Deintuitionistification

At some stage in the MCDM process, it is necessary to transform the IF number into a crisp value. Decision makers can perform such a transformation using Equation (8). However, to perform a different analysis and consider different scenarios, a new approach for deintuitionistification, based on Equation (8), is proposed, as follows:
where λ represents coefficients, and .

## 3. The Simple Weighted Sum-Product Method

The procedure of the WISP method for a decision-making problem involving m alternatives that are evaluated based on n criteria is systemic procedure, the steps of which are as follows:
Step 1. Form a decision-making matrix and determine criteria weights.
Step 2. Construct a normalized decision-making matrix as follows:
where $r i j$ denotes a dimensionless number representing normalized alternative i regarding criterion j.
Step 3. Calculate the values of four indicators, as follows:
$u i s d = ∑ j ∈ Ω max r i j w j − ∑ j ∈ Ω min r i j w j ,$
$u i p d = ∏ j ∈ Ω max r i j w j − ∏ j ∈ Ω min r i j w j ,$
$u i p r = ∏ j ∈ Ω max r i j w j ∏ j ∈ Ω min r i j w j ,$
where $u i s d$ and denote differences between the weighted sum and weighted product of normalized ratings of alternative i, respectively, and $Ω max$ and $Ω min$ denote sets of maximization and minimization criteria, respectively. Similar to the previous one, $u i sr$ and $u i pr$ denote ratios between the weighted sum and weighted product of normalized ratings of alternative i, respectively.
Step 4. Recalculate values of four indicators, as follows:
where $u ¯ i s d$, , and $u ¯ i p r$ denote recalculated values of $u i s d$, $u i p d$, $u i s r$, and $u i p r$.
Step 5. Determine the overall utility of the considered alternative as follows:
$u i = 1 4 ( u ¯ i s d + u ¯ i p d + u ¯ i s r + u ¯ i p r ) .$
Step 6. Rank the alternatives and select the most suitable one. In this approach, the alternative with the highest value of ui is the most preferable.
The authors of the WISP method initially proposed using it to solve decision-making problems that contain both benefit- and cost-type criteria. However, the WISP method can also solve MCDM problems that contain only beneficial or only non-beneficial criteria, but in these cases, Equations (15) and (16) must be modified as follows:
$u i p r = ∏ j ∈ Ω max r i j w j ,$
when $Ω m i n = ∅$, that is:
$u i p r = 1 ∏ j ∈ Ω min r i j w j ,$
when $Ω m a x = ∅$.

## 4. An Intuitionistic Extension of the WISP Method

To enable using the IFWG operator in the proposed IF extension of the WISP (IF-WISP) method, Equations (14) and (16), in the computational procedure of the standard WISP method, should be modified as follows:
$u i p d = ∏ j ∈ Ω max r i j w j − ∏ j ∈ Ω min r i j w j ,$
$u i p r = ∏ j ∈ Ω max r i j w j ∏ j ∈ Ω min r i j w j ,$
After that, decision makers use the procedure of the IF-WISP method presented in the following steps:
Step 1. Construct an initial decision-making matrix. In this step, decision makers create an initial decision-making matrix that expresses the ratings of alternatives using IF numbers.
Step 2. Determine criteria weights. In this step, the criteria weights can be determined using any MCDM method primarily intended for determining the criteria weights, such as the AHP method , the SWARA method , or the Best-Worst method .
Step 3. Calculate the sum and product of the weighted intuitionistic ratings of each alternative for the maximization and minimization criteria, using Equations (9) and (10), as follows:
where and denote the sum of the weighted intuitionistic rating of alternative i, achieved based on maximization and minimization criteria, respectively, and and denote the product of the weighted intuitionistic ratings of alternative i, achieved based on maximization and minimization criteria, respectively.
Step 4. Deintuitionistification. The subtraction and division operations required for determining utility measures used in the WISP method are not primarily defined for IF set and IF numbers. Therefore, $S i + ,$ $S i − ,$ $P i + ,$ and should be transformed into crisp values using Equation (8) or Equation (11).
Step 5. Calculate the values of four indicators, $u i sd$, $u i pd$, $u i sr$, and $u i pr$, as follows:
$u i s d = S i + − S i − ,$
$u i p d = P i + − P i − ,$
Step 6. Recalculate values of four indicators, as follows:
$u ¯ i p r = 1 + u i p r 1 + max i u i p r ,$
where $u ¯ i s d$, $u ¯ i p d$, $u ¯ i s r$, and $u ¯ i p r$ denote recalculated values of $u i s d$, $u i p d$, $u i s r$, and $u i p r$.
Step 5. Determine the overall utility of each alternative as follows:
$u i = 1 4 ( u ¯ i s d + u ¯ i p d + u ¯ i s r + u ¯ i p r ) .$
Step 6. Rank the alternatives and select the most suitable one. Decision makers rank the alternatives in descending order and select the best with the highest ui.

## 5. A Numerical Example

In this section, we discuss the application of the proposed extension of the WISP method on the example of contractor selection.
Based on the example discussed in Turskis and Zavadskas , in this case, the evaluation of four contractors was performed based on the following criteria: production specifications (C1), financial position (C2), standards and relevant certificates (C3), commercial strength (C4), performance (C5), and delivery price (C6).
Table 1 shows an initial intuitionistic decision-making matrix.
Table 1 also shows the criteria weights and optimization directions.
Table 2 shows the weighted intuitionistic ratings of the maximization and minimization criteria for considered alternatives.
Table 3 shows crisp sums and products of the weighted intuitionistic ratings. In this case, decision makers used Equation (8) to deintuitionistificate, i.e., transform IF numbers into crisp values, but they can also use Equation (11).
Table 3 shows the values of four utility measures, $u i sd$, $u i pd$, $u i sr$, and $u i pr$, calculated using Equations (31)–(34).
Table 4 shows the recalculated values of four utility measures, $u ¯ i sd$, $u ¯ i pd$, $u ¯ i sr$, and $u ¯ i pr$, calculated using Equations (36)–(39), as well as the overall utility measures, calculated using Equation (40).
As can be concluded from Table 4, the alternative denoted as A3 is the most appropriate alternative.
In addition to selecting the most appropriate alternative, the IF-WISP method allows analysis of the impact of membership and non-membership functions on the overall utility measures, using Equation (11). Table 5 and Figure 2 show the values of overall utility measures and ranks of alternatives for several selected values of the coefficient λ.
Based on the above, it is evident that the proposed IF-WISP extension decision makers can analyze different scenarios, thus making better use of the benefits that IF set theory provides for solving complex decision-making problems.

## 6. A Comparison of the Proposed Extension with Similar Extensions of Some MCDM Methods

In this section, we present tests of the proposed extension of the WISP method. We compared the obtained ranking results using the proposed extension with the results obtained using the neutrosophic WASPAS, CoCoSo, and SAW methods.
The authors chose the example discussed by Stanujkic et al.  to compare the ranking results. This example evaluated three alternatives based on four beneficial criteria: environment (En), content (Co), graphics (Gr), and authority (Au). Table 6 shows the ratings of the alternatives according to the evaluation criteria and the weights of the criteria.
Table 7 shows the ratings and ranking orders of alternatives obtained using intuitionistic extensions of the WASPAS, CoCoSo, SAW, and WISP methods.
As can be seen from Table 7, the ranking order of alternatives obtained using the proposed intuitionistic extension of the WISP method is the same as the ranking orders of alternatives obtained using the extensions mentioned above, which confirms the usability of the proposed extension.

## 7. Conclusions

Intuitionistic fuzzy sets provide an opportunity to solve more complex decision-making problems. The use of singleton intuitionistic fuzzy numbers is more straightforward than other intuitionistic fuzzy numbers (trapezoidal, triangular, or bell-shaped). However, they are still adequate to solve complex decision-making problems.
Therefore, we propose an extension of the WISP method adapted to use singleton intuitionistic fuzzy numbers (IF-WISP). The contractor selection problem demonstrates the usability of the newly proposed IF-WISP extension.
Finally, developing an interval-valued intuitionistic fuzzy extension of the WISP method can be stated as the future development direction. Furthermore, the development of similar fuzzy extensions, such as spherical, picture, and Pythagorean, can be mentioned as possible directions for further development of the WISP method.

## Author Contributions

Conceptualization, E.K.Z. and D.S.; methodology, E.K.Z., D.S. and Z.T.; validation, Z.T. and D.K.; writing—original draft preparation, E.K.Z. and D.S.; writing—review and editing, D.S. and Z.T. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

## Institutional Review Board Statement

Informed consent was obtained from all subjects involved in the study.

## Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

## Acknowledgments

The authors wish to thank the anonymous reviewers for the valuable suggestions and comments, which improved the quality of this article.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
2. Bellman, R.E.; Zadeh, L.A. Decision Making in a Fuzzy Environment. J. Manag. Sci. 1970, 17, 141–164. [Google Scholar] [CrossRef]
3. Turksen, I.B. Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 1986, 20, 191–210. [Google Scholar] [CrossRef]
4. Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
5. Smarandache, F. Neutrosophy Probability Set and Logic; American Research Press: Rehoboth, DE, USA, 1998. [Google Scholar]
6. Atanassov, K.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
7. Benayoun, R.; Roy, B.; Sussman, N. Manual de Reference du Programme ELECTRE; Note de Synthese et Formation; 25. Direction Scientifique SEMA: Paris, France, 1966. [Google Scholar]
8. Saaty, L.T. The Analytic Hierarchy Process; McGraw Hill Company: New York, NY, USA, 1980. [Google Scholar]
9. Hwang, C.L.; Yoon, K. Multiple Attribute Decision Making Methods and Applications; Springer: Berlin/Heidelberg, Germany, 1981. [Google Scholar]
10. Zavadskas, E.K.; Kaklauskas, A.; Sarka, V. The new method of multicriteria complex proportional assessment of projects. Technol. Econ. Dev. 1994, 1, 131–139. [Google Scholar]
11. Opricovic, S.; Tzeng, G.-H. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. Eur. J. Oper. Res. 2004, 156, 445–455. [Google Scholar] [CrossRef]
12. Brauers, W.K.M.; Zavadskas, E.K. Project management by MULTIMOORA as an instrument for transition economies. Technol. Econ. Dev. 2010, 16, 5–24. [Google Scholar] [CrossRef]
13. Zavadskas, E.K.; Baušys, R.; Leščauskienė, I.; Omran, J. M-generalised q-neutrosophic MULTIMOORA for decision making. Stud. Inform. Control 2020, 29, 389–398. [Google Scholar] [CrossRef]
14. Zavadskas, E.K.; Turskis, Z. A new additive ratio assessment (ARAS) method in multicriteria decision making. Technol. Econ. Dev. 2010, 16, 159–172. [Google Scholar] [CrossRef]
15. Zavadskas, E.K.; Turskis, Z.; Antucheviciene, J.; Zakarevicius, A. Optimization of weighted aggregated sum product assessment. Elektron Elektrotech. 2012, 122, 3–6. [Google Scholar] [CrossRef]
16. Keshavarz Ghorabaee, M.; Zavadskas, E.K.; Olfat, L.; Turskis, Z. Multi-criteria inventory classification using a new method of Evaluation Based on Distance from Average Solution (EDAS). Informatica 2015, 26, 435–451. [Google Scholar] [CrossRef]
17. Keshavarz Ghorabaee, M.; Zavadskas, E.K.; Turskis, Z.; Antucheviciene, J. A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Econ. Comput. Econ. Cybern. Stud. Res. 2016, 50, 25–44. [Google Scholar]
18. Yazdani, M.; Zarate, P.; Zavadskas, E.K.; Turskis, Z. A Combined Compromise Solution (CoCoSo) method for multi-criteria decision-making problems. Manag. Decis. 2018, 57, 2501–2519. [Google Scholar] [CrossRef]
19. Ulutaş, A.; Stanujkic, D.; Karabasevic, D.; Popovic, G.; Zavadskas, E.K.; Smarandache, F.; Brauers, W.K. Developing of a Novel Integrated MCDM MULTIMOOSRAL Approach for Supplier Selection. Informatica 2021, 32, 145–161. [Google Scholar] [CrossRef]
20. Mardani, A.; Nilashi, M.; Zakuan, N.; Loganathan, N.; Soheilirad, S.; Saman, M.Z.M.; Ibrahim, O. A systematic review and meta-Analysis of SWARA and WASPAS methods: Theory and applications with recent fuzzy developments. Appl. Soft Comput. 2017, 57, 265–292. [Google Scholar] [CrossRef]
21. Mardani, A.; Jusoh, A.; Halicka, K.; Ejdys, J.; Magruk, A.; Ahmad, U.U.N. Determining the utility in management by using multi-criteria decision support tools: A review. Econ. Res.-Ekon. Istraz. 2018, 31, 1666–1716. [Google Scholar] [CrossRef]
22. Hafezalkotob, A.; Hafezalkotob, A.; Liao, H.; Herrera, F. An overview of MULTIMOORA for multi-criteria decision-making: Theory, developments, applications, and challenges. Inf. Fusion. 2019, 51, 145–177. [Google Scholar] [CrossRef]
23. Chandrawati, T.B.; Ratna, A.A.P.; Sari, R.F. Path Selection using Fuzzy Weight Aggregated Sum Product Assessment. Int. J. Comput. Commun. Control 2020, 15, 1–19. [Google Scholar]
24. Liu, N.; Xu, Z. An overview of ARAS method: Theory development, application extension, and future challenge. Int. J. Intell. Syst. 2021, 36, 3524–3565. [Google Scholar] [CrossRef]
25. Li, M.; Jin, L.; Wang, J. A new MCDM method combining QFD with TOPSIS for knowledge management system selection from the user’s perspective in intuitionistic fuzzy environment. Appl. Soft Comput. 2014, 21, 28–37. [Google Scholar] [CrossRef]
26. Wang, L.E.; Liu, H.C.; Quan, M.Y. Evaluating the risk of failure modes with a hybrid MCDM model under interval-valued intuitionistic fuzzy environments. Comput. Ind. Eng. 2016, 102, 175–185. [Google Scholar] [CrossRef]
27. Büyüközkan, G.; Göçer, F. Application of a new combined intuitionistic fuzzy MCDM approach based on axiomatic design methodology for the supplier selection problem. Appl. Soft Comput. 2017, 52, 1222–1238. [Google Scholar] [CrossRef]
28. Karagoz, S.; Deveci, M.; Simic, V.; Aydin, N.; Bolukbas, U. A novel intuitionistic fuzzy MCDM-based CODAS approach for locating an authorized dismantling center: A case study of Istanbul. Waste Manag. Res. 2020, 38, 660–672. [Google Scholar] [CrossRef]
29. Kushwaha, D.K.; Panchal, D.; Sachdeva, A. Risk analysis of cutting system under intuitionistic fuzzy environment. Rep. Mech. Eng. 2020, 1, 162–173. [Google Scholar] [CrossRef]
30. Garg, H.; Rani, D. An efficient intuitionistic fuzzy MULTIMOORA approach based on novel aggregation operators for the assessment of solid waste management techniques. Appl. Intell. 2021, 1–34. [Google Scholar] [CrossRef]
31. Zhang, L.; Zhan, J.; Yao, Y. Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: An application to biomedical problems. Inf. Sci. 2020, 517, 315–339. [Google Scholar] [CrossRef]
32. Rouyendegh, B.D.; Yildizbasi, A.; Üstünyer, P. Intuitionistic fuzzy TOPSIS method for green supplier selection problem. Soft Comput. 2020, 24, 2215–2228. [Google Scholar] [CrossRef]
33. Krishankumar, R.; Premaladha, J.; Ravichandran, K.S.; Sekar, K.R.; Manikandan, R.; Gao, X.Z. A novel extension to VIKOR method under intuitionistic fuzzy context for solving personnel selection problem. Soft Comput. 2020, 24, 1063–1081. [Google Scholar] [CrossRef]
34. Zhang, C.; Chen, C.; Streimikiene, D.; Balezentis, T. Intuitionistic fuzzy MULTIMOORA approach for multi-criteria assessment of the energy storage technologies. Appl. Soft Comput. 2019, 79, 410–423. [Google Scholar] [CrossRef]
35. Raj Mishra, A.; Sisodia, G.; Raj Pardasani, K.; Sharma, K. Multi-criteria IT personnel selection on intuitionistic fuzzy information measures and ARAS methodology. Iran. J. Fuzzy Syst. 2020, 17, 55–68. [Google Scholar]
36. Mishra, A.R.; Rani, P. Interval-valued intuitionistic fuzzy WASPAS method: Application in reservoir flood control management policy. Group Decis. Negot. 2018, 27, 1047–1078. [Google Scholar] [CrossRef]
37. Bolturk, E.; Kahraman, C. Interval-valued intuitionistic fuzzy CODAS method and its application to wave energy facility location selection problem. J. Intell. Fuzzy Syst. 2018, 35, 4865–4877. [Google Scholar] [CrossRef]
38. Roy, J.; Das, S.; Kar, S.; Pamučar, D. An extension of the CODAS approach using interval-valued intuitionistic fuzzy set for sustainable material selection in construction projects with incomplete weight information. Symmetry 2019, 11, 393. [Google Scholar] [CrossRef]
39. Seker, S.; Aydin, N. Sustainable public transportation system evaluation: A novel two-stage hybrid method based on IVIF-AHP and CODAS. Int. J. Fuzzy Syst. 2020, 22, 257–272. [Google Scholar] [CrossRef]
40. Fu, Y.; Qin, Y.; Wang, W.; Liu, X.; Jia, L. An Extended FMEA Model Based on Cumulative Prospect Theory and Type-2 Intuitionistic Fuzzy VIKOR for the Railway Train Risk Prioritization. Entropy 2020, 22, 1418. [Google Scholar] [CrossRef] [PubMed]
41. Alimohammadlou, M.; Khoshsepehr, Z. Investigating organizational sustainable development through an integrated method of interval-valued intuitionistic fuzzy AHP and WASPAS. Environ. Dev. Sustain. 2022, 24, 2193–2224. [Google Scholar] [CrossRef]
42. Wu, X.; Song, Y.; Wang, Y. Distance-Based Knowledge Measure for Intuitionistic Fuzzy Sets with Its Application in Decision Making. Entropy 2021, 23, 1119. [Google Scholar] [CrossRef]
43. Opoku-Mensah, E.; Yin, Y.; Asiedu-Ayeh, L.O.; Asante, D.; Tuffour, P.; Ampofo, S.A. Exploring governments’ role in mergers and acquisitions using IVIF MULTIMOORA-COPRAS technique. Int. J. Emerg. Mark. 2021, in press. [Google Scholar] [CrossRef]
44. Abdullah, L.; Najib, L. A new preference scale MCDM method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process. Soft Comput. 2016, 20, 511–523. [Google Scholar] [CrossRef]
45. Sahu, M.; Gupta, A.; Mehra, A. Hierarchical clustering of interval-valued intuitionistic fuzzy relations and its application to elicit criteria weights in MCDM problems. Opsearch 2017, 54, 388–416. [Google Scholar] [CrossRef]
46. Liu, X.; Qian, F.; Lin, L.; Zhang, K.; Zhu, L. Intuitionistic fuzzy entropy for group decision making of water engineering project delivery system selection. Entropy 2019, 21, 1101. [Google Scholar] [CrossRef]
47. Roszkowska, E.; Kusterka-Jefmańska, M.; Jefmański, B. Intuitionistic Fuzzy TOPSIS as a Method for Assessing Socioeconomic Phenomena on the Basis of Survey Data. Entropy 2021, 23, 563. [Google Scholar] [CrossRef] [PubMed]
48. Razavi Hajiagha, S.H.; Hashemi, S.S.; Zavadskas, E.K. A complex proportional assessment method for group decision making in an interval-valued intuitionistic fuzzy environment. Technol. Econ. Dev. 2013, 19, 22–37. [Google Scholar] [CrossRef]
49. Zavadskas, E.K.; Antucheviciene, J.; Hajiagha, S.H.R.; Hashemi, S.S. Extension of weighted aggregated sum product assessment with interval-valued intuitionistic fuzzy numbers (WASPAS-IVIF). Appl. Soft Comput. 2014, 24, 1013–1021. [Google Scholar] [CrossRef]
50. Chen, T.Y. An IVIF-ELECTRE outranking method for multiple criteria decision-making with interval-valued intuitionistic fuzzy sets. Technol. Econ. Dev. 2016, 22, 416–452. [Google Scholar] [CrossRef]
51. Dammak, F.; Baccour, L.; Alimi, A.M. A new ranking method for TOPSIS and VIKOR under interval valued intuitionistic fuzzy sets and possibility measures. J. Intell. Fuzzy Syst. 2020, 38, 4459–4469. [Google Scholar] [CrossRef]
52. Alrasheedi, M.; Mardani, A.; Mishra, A.R.; Streimikiene, D.; Liao, H.; Al-nefaie, A.H. Evaluating the green growth indicators to achieve sustainable development: A novel extended interval-valued intuitionistic fuzzy-combined compromise solution approach. J. Sustain. Dev. 2021, 29, 120–142. [Google Scholar] [CrossRef]
53. Stanujkic, D.; Popovic, G.; Karabasevic, D.; Meidute-Kavaliauskiene, I.; Ulutaş, A. An Integrated Simple Weighted Sum Product Method–WISP. IEEE Trans. Eng. Manag. 2021, 1–12. [Google Scholar] [CrossRef]
54. Chen, S.M.; Tan, J.M. Handling multicriteria fuzzy decision-making problems based on vague set theory. J. Intell. Fuzzy Syst. 1994, 67, 163–172. [Google Scholar] [CrossRef]
55. Tikhonenko-Kędziak, A.; Kurkowski, M. An approach to exponentiation with interval-valued power. J. Appl. Math. Comput. Mech. 2016, 15, 157–169. [Google Scholar] [CrossRef]
56. Keršuliene, V.; Zavadskas, E.K.; Turskis, Z. Selection of rational dispute resolution method by applying new step-wise weight assessment ratio analysis (SWARA). J. Bus. Econ. Manag. 2010, 11, 243–258. [Google Scholar] [CrossRef]
57. Rezaei, J. Best-worst multi-criteria decision-making method. Omega 2015, 53, 49–57. [Google Scholar] [CrossRef]
58. Turskis, Z.; Zavadskas, E.K. A novel method for multiple criteria analysis: Grey additive ratio assessment (ARAS-G) method. Informatica 2010, 21, 597–610. [Google Scholar] [CrossRef]
59. Stanujkic, D.; Karabasevic, D. An extension of the WASPAS method for decision-making problems with intuitionistic fuzzy numbers: A case of website evaluation. Oper. Res. Eng. Sci. Theor. Appl. 2018, 1, 29–39. [Google Scholar] [CrossRef]
Figure 1. An SIF number.
Figure 1. An SIF number. Figure 2. The ranking order of alternatives for different values of λ.
Figure 2. The ranking order of alternatives for different values of λ. Table 1. An initial decision-making matrix.
Table 1. An initial decision-making matrix.
C1C2C3C4C5C6
wj0.2100.1370.1370.1310.1750.210
Optimizationmaxmaxmaxmaxmaxmin
A1<0.9, 0.0><0.7, 0.0><0.9, 0.0><1.0, 0.1><1.0, 0.0><1.0, 0.1>
A2<0.9, 0.1><0.8, 0.1><1.0, 0.1><0.9, 0.0><0.8, 0.0><0.9, 0.1>
A3<0.7, 0.0><1.0, 0.0><1.0, 0.0><1.0, 0.0><0.9, 1.0><0.9, 0.0>
A4<0.8, 0.0><0.8, 0.1><0.9, 0.1><1.0, 0.0><1.0, 0.0><1.0, 0.2>
Table 2. Sums and products of weighted intuitionistic ratings of alternatives.
Table 2. Sums and products of weighted intuitionistic ratings of alternatives.
$S i +$$S i −$$P i +$$P i −$
A1<0.08, 0.00><0.00, 0.62><0.92, 1.00><1.00, 0.38>
A2<0.10, 0.00><0.02, 0.62><0.90, 1.00><0.98, 0.38>
A3<0.09, 0.00><0.02, 0.00><0.91, 1.00><0.98, 1.00>
A4<0.09, 0.00><0.00, 0.71><0.91, 1.00><1.00, 0.29>
Table 3. Crisp values of sums and products of weighted intuitionistic ratings.
Table 3. Crisp values of sums and products of weighted intuitionistic ratings.
$S i +$$S i −$$P i +$$P i −$$u i s d$$u i p d$$u i s r$$u i p r$
A10.08−0.62−0.080.620.70−0.70−0.13−0.13
A20.10−0.59−0.100.590.69−0.69−0.17−0.17
A30.090.02−0.09−0.020.07−0.074.074.07
A40.09−0.71−0.090.710.80−0.80−0.12−0.12
Table 4. The recalculated values of four utility measures, overall utility measures, and ranking order of alternatives.
Table 4. The recalculated values of four utility measures, overall utility measures, and ranking order of alternatives.
$u ¯ i s d$$u ¯ i p d$$u ¯ i s r$$u ¯ i p r$$u i$Rank
A10.940.320.170.170.4022
A20.940.330.160.160.3993
A30.591.001.001.000.8981
A41.000.210.170.170.3904
Table 5. The overall utility measures and ranking order of alternatives for different values of λ.
Table 5. The overall utility measures and ranking order of alternatives for different values of λ.
λ0.010.250.50.750.999
$u i$Rank$u i$Rank$u i$Rank$u i$Rank$u i$Rank
A10.58130.66620.49420.7002−5.0214
A20.58130.63830.49130.69340.9931
A30.75510.69710.93410.96110.9672
A40.62220.17540.49130.6963−4.5963
Table 6. The ratings of alternatives and criteria weights.
Table 6. The ratings of alternatives and criteria weights.
C1C2C3C4
EnCoGrAu
wj0.280.250.240.23
Optimizationmaxmaxmaxmax
A1<0.742, 0.125><0.625, 0.375><0.590, 0.250><0.375, 0.250>
A2<0.595, 0.327><0.750, 0.158><0.590, 0.125><0.500, 0.250>
A3<0.717, 0.155><0.500, 0.125><0.586, 0.327><0.339, 0.176>
Table 7. The overall utility measures and ranking order of alternatives obtained using intuitionistic extensions of some MCDM methods.
Table 7. The overall utility measures and ranking order of alternatives obtained using intuitionistic extensions of some MCDM methods.
WASPASRankCoCoSoRankSAWRankWISPRank
A10.32531.88430.38030.9633
A20.30012.16410.41911.0001
A30.32321.90220.38120.9662
 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.