He has published more than 400 articles in journals, books and conference proceedings. He is on the editorial board of several journals. He has also been a guest editor for several international journals, member of the scientific committee of several conferences and reviewer in a wide range of international journals. Recently (2015–2018), Clarivate Analytics (previously Thomson & Reuters) has distinguished him as a highly cited researcher in computer science. He is currently interested in decision making, aggregation operators, computational intelligence, bibliometrics and applications in business and economics.
In this paper, we develop a new flexible method for interval-valued intuitionistic fuzzy decision-making problems with cosine similarity measure. We first introduce the interval-valued intuitionistic fuzzy cosine similarity measure based on the notion of the weighted reduced intuitionistic fuzzy sets. With this cosine similarity measure, we are able to accommodate the attitudinal character of decision-makers in the similarity measuring process. We study some of its essential properties and propose the weighted interval-valued intuitionistic fuzzy cosine similarity measure.
Further, the work uses the idea of GOWA operator to develop the ordered weighted interval-valued intuitionistic fuzzy cosine similarity (OWIVIFCS) measure based on the weighted reduced intuitionistic fuzzy sets. The main advantage of the OWIVIFCS measure is that it provides a parameterized family of cosine similarity measures for interval-valued intuitionistic fuzzy sets and considers different scenarios depending on the attitude of the decision-makers. The measure is demonstrated to satisfy some essential properties, which prepare the ground for applications in different areas. In addition, we define the quasi-ordered weighted interval-valued intuitionistic fuzzy cosine similarity (quasi-OWIVIFCS) measure. It includes a wide range of particular cases such as OWIVIFCS measure, trigonometric-OWIVIFCS measure, exponential-OWIVIFCS measure, radical-OWIVIFCS measure. Finally, the study uses the OWIVIFCS measure to develop a new decision-making method to solve real-world decision problems with interval-valued intuitionistic fuzzy information. A real-life numerical example of contractor selection is also given to demonstrate the effectiveness of the developed approach in solving real-life problems.
Atanassov (
A similarity measure is an essential tool for determining the degree of similarity between two objects. In 2002, Denfeng and Chuntian (
In many complex decision-making problems, the preference information provided by the decision-makers is often imprecise or uncertain due to the increasing complexity of the social-economic environment or a lack of data about the problem domain or the expert’s lack of expertise to precisely express their preferences over the considered objects. In such cases, it is suitable and convenient to express the decision-maker’s preference information in terms of IVIFSs. Therefore, it is necessary to pay attention to the study of the similarity measure for IVIFSs. There is some progress in this direction. Xu (
Note that the cosine similarity measures introduced by Singh (
To do so, we first propose a new cosine similarity measure for IVIFSs based on the weighted reduced intuitionistic fuzzy sets (Ye,
The paper is organized as follows. Section
In this section, we present some basic concepts related to fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and OWA operators, which will be needed in the following analysis.
A fuzzy set
A cosine similarity measure is defined as the inner product of two vectors divided by the product of their lengths. This is nothing but the cosine of the angle between the vectors representation of two fuzzy sets.
Let
Atanassov (
An intuitionistic fuzzy set
For convenience, we abbreviate the set of all IFSs defined in
In 2011, Ye (
Let
A cosine similarity measure between two intuitionistic fuzzy sets
Atanassov and Gargov (
Let
For any
Clearly, if
In the study of IVIFSs, the set-theoretic operations are defined as follows:
Let
where ∨, ∧ stand for max. and min. operators, respectively.
Singh ( Let If If If
The OWA operator was introduced by Yager (
An OWA operator of dimension
The OWA operator is commutative, monotonic, bounded, and idempotent. Especially, if
Furthermore, in 2004, Yager (
A GOWA operator of dimension
The quasi-arithmetic means are an important class of parameterized aggregation operators that have been used extensively in different application areas. It includes a wide range of aggregation operators such as arithmetic, quadratic, geometric, harmonic, root-power, and exponential. Fodor
A quasi-OWA operator of dimension
The quasi-OWA operator is monotonic, commutative, bounded, and idempotent. If we consider different functions
In the next section, using the idea of weighted reduced IFS of an IVIFS, we propose a new similarity measure on interval-valued intuitionistic fuzzy sets, called ‘
We proceed with the following formal definition:
Let
Analogous to the cosine similarity measure for IFSs given in Eq. (
The new measure
(a) It is evident that the property is true according to the cosine value of Eq. (
(b) This follows from the symmetry of
(c) First, let
This proves the theorem. □
For proof of the further properties, we will consider separation of
Using Definition
We prove (i) only, (ii) can be proved analogously.
(i) Let us consider the expression
From Definition
This proves the theorem. □
(a) It follows from the relation of membership and non-membership of an element in a set and its complement.
(b) It directly follows from Definition
(c) It simply follows (a) and (b).
This proves the theorem. □
By adjusting the values of If If If If
Assume that the elements in the universe of discourse
(i) If
Obviously, the
These properties can be proved easily on lines similar to the proof of Theorem
Note that the cosine similarity measures defined in Eq. (
Let
A OWIVIFCS measure based on the weighted reduced IFSs of IVIFSs is a mapping
The main advantages of the OWIVIFCS measure are that it is not only a straightforward generalization of measure defined in Eq. (
Now consider the following numerical example to understand the computation procedure more clearly.
Let
Taking different values of
Values of
0.2 | 0.7 | 1 | 2 | 5 | 7 | 9 | 15 | 25 | |
0.9303 | 0.9304 | 0.9305 | 0.9309 | 0.9317 | 0.9325 | 0.9331 | 0.9349 | 0.9374 |
The OWIVIFCS measure is commutative, monotonic, bounded, idempotent, non-negative, and reflexive. These properties can be proved with the following theorems:
Note that the proofs of these theorems are straightforward and thus omitted here.
By using the different manifestation of the weighting vector
When we consider different values of the parameter If If If If
By considering the different selections of the weighting vector, we are able to analyse the cosine similarity measure between two interval-valued intuitionistic fuzzy sets from min. similarity to max. similarity.
If If More generally, if If The WIVIFCS measure is obtained when the ordered position of the If If If If A quasi-OWIVIFCS measure based on the weighted reduced IFSs of IVIFSs is a mapping As we can see, when Further, by assigning different functions to
It is interesting to note that the OWIVIFCS measure can be further generalized by using the quasi-OWA operator in place of GOWA. We call it quasi-OWIVIFCS measure. It can be defined as follows:
when
When
If
The OWIVIFCS measure can be applied to solve different problems, including decision-making, medical diagnosis, pattern recognition, engineering, and economics. In the next section, we present an application of the proposed OWIVIFCS measure to solve the multiple criteria decision-making problem with the interval-valued intuitionistic fuzzy information.
IVIFS is a suitable tool for better modelling the imperfectly defined facts and data, as well as imprecise knowledge. In this section, we present a 5-step method to solve a multiple criteria decision-making problem under an interval-valued intuitionistic fuzzy environment.
Let
Using the OWIVIFCS measure defined in Eq. (
In the following, we are going to consider a real-life numerical example to demonstrate the applicability of the proposed method to multiple criteria decision making. To do so, we consider below a contractor selection decision-making problem for road development. Chile is a South American country occupying a long, narrow strip of land between the Andes to the east and the Pacific Ocean to the west. In Chile, tourism has become one of the main sources of income for the people, especially living in its most extreme areas. In 2018, a record of a total of 7 million international tourists visited Chile. The online guestbook Lonely Planet listed ‘ The step-wise decision-making process as follows: Values of Values of Values of Values of Values of Values of
Ranking of options based on different similarity measures. As we can see, depending on the cosine similarity measure used, the ranking order of the available options is different. Therefore, depending on the similarity measure employed, the results may lead to different decisions. In this problem, the IVIFMAXCS is the most optimistic cosine similarity measure because it considers only the highest similarity value. On the other hand, IVIFMINCS is the most pessimistic one. The IVIFNORCS is a neutral measure because it gives the same weights to all the characteristics. From Table Further, in order to validate the performance of the developed different cosine similarity measures, a comparative study has been conducted and analysed in detail. Based on the normal distribution method (Xu,
Ranking of options based on different cosine similarity measures under IVIF environment. From Table
0.7468
0.9883
0.8986
1.0000
0.6623
0.9883
1.0000
0.9437
0.9883
0.9910
0.7349
0.9883
0.9492
0.9437
0.9437
0.9883
0.9272
0.8547
1.0000
0.9883
1.0000
0.7218
0.9832
1.0000
0.9832
0.8022
0.9883
1.0000
0.9815
0.9650
0.7175
0.9832
0.8619
1.0000
0.5870
0.9492
1.0000
0.9492
0.9832
0.9487
0.8043
0.8165
0.9338
1.0000
0.9239
0.9832
0.9338
0.7235
1.0000
0.9492
1.0000
0.6623
0.9806
1.0000
0.9783
0.7285
0.9832
0.9847
1.0000
0.9239
0.7276
0.9858
0.8867
1.0000
0.6261
0.9766
1.0000
0.9492
0.9858
0.9866
0.7674
0.9358
0.9410
1.0000
0.9337
0.9913
0.9329
0.7995
1.0000
0.9772
1.0000
0.6910
0.9815
1.0000
0.9805
0.7670
0.9913
0.9970
0.9949
0.9509
IVIFMAXCS
IVIFMINCS
IVIFNORCS
IVIFMEDCS
IVIFOLMCS
IVIFWINCS
1.0000
0.6623
0.8723
0.9883
0.9010
0.9010
1.0000
0.7349
0.9364
0.9883
0.9770
0.9770
1.0000
0.8547
0.9430
0.9492
0.9521
0.9521
1.0000
0.7218
0.9439
1.0000
0.9929
0.9929
1.0000
0.8022
0.9512
0.9832
0.9795
0.9795
1.0000
0.6623
0.8755
0.9883
0.9027
0.9027
1.0000
0.7349
0.9380
0.9883
0.9771
0.9771
1.0000
0.8547
0.9434
0.9492
0.9522
0.9522
1.0000
0.7218
0.9458
1.0000
0.9929
0.9929
1.0000
0.8022
0.9521
0.9832
0.9795
0.9795
1.0000
0.6623
0.8807
0.9883
0.9055
0.9055
1.0000
0.7349
0.9406
0.9883
0.9771
0.9771
1.0000
0.8547
0.9440
0.9492
0.9524
0.9524
1.0000
0.7218
0.9489
1.0000
0.9929
0.9929
1.0000
0.8022
0.9534
0.9832
0.9795
0.9795
1.0000
0.6623
0.8904
0.9883
0.9109
0.9109
1.0000
0.7349
0.9452
0.9883
0.9773
0.9773
1.0000
0.8547
0.9452
0.9492
0.9526
0.9526
1.0000
0.7218
0.9543
1.0000
0.9929
0.9929
1.0000
0.8022
0.9558
0.9832
0.9795
0.9795
1.0000
0.6623
0.9142
0.9883
0.9247
0.9247
1.0000
0.7349
0.9560
0.9883
0.9779
0.9779
1.0000
0.8547
0.9486
0.9492
0.9535
0.9535
1.0000
0.7218
0.9664
1.0000
0.9930
0.9930
1.0000
0.8022
0.9619
0.9832
0.9797
0.9797
1.0000
0.6623
0.9388
0.9883
0.9409
0.9409
1.0000
0.7349
0.9666
0.9883
0.9788
0.9788
1.0000
0.8547
0.9536
0.9492
0.9549
0.9549
1.0000
0.7218
0.9774
1.0000
0.9931
0.9931
1.0000
0.8022
0.9689
0.9832
0.9798
0.9798
IVIFMAXCS
IVIFMINCS
IVIFNORCS
IVIFMEDCS
IVIFOLMCS
IVIFWINCS
1.0000
0.5870
0.8380
0.9492
0.8729
0.8729
1.0000
0.8043
0.9200
0.9832
0.9313
0.9313
1.0000
0.7235
0.9124
0.9338
0.9435
0.9435
1.0000
0.6623
0.9246
1.0000
0.9823
0.9823
1.0000
0.7285
0.9288
0.9832
0.9673
0.9673
1.0000
0.5870
0.8426
0.9492
0.8748
0.8748
1.0000
0.8043
0.9211
0.9832
0.9321
0.9321
1.0000
0.7235
0.9139
0.9338
0.9435
0.9435
1.0000
0.6623
0.9275
1.0000
0.9823
0.9823
1.0000
0.7285
0.9305
0.9832
0.9674
0.9674
1.0000
0.5870
0.8498
0.9492
0.8780
0.8780
1.0000
0.8043
0.9230
0.9832
0.9334
0.9334
1.0000
0.7235
0.9164
0.9338
0.9437
0.9437
1.0000
0.6623
0.9320
1.0000
0.9825
0.9825
1.0000
0.7285
0.9331
0.9832
0.9675
0.9675
1.0000
0.5870
0.8631
0.9492
0.8839
0.8839
1.0000
0.8043
0.9265
0.9832
0.9539
0.9359
1.0000
0.7235
0.9208
0.9338
0.9440
0.9440
1.0000
0.6623
0.9400
1.0000
0.9827
0.9827
1.0000
0.7285
0.9379
0.9832
0.9679
0.9679
1.0000
0.5870
0.8939
0.9492
0.8995
0.8995
1.0000
0.8043
0.9362
0.9832
0.9427
0.9427
1.0000
0.7235
0.9316
0.9338
0.9448
0.9448
1.0000
0.6623
0.9565
1.0000
0.9833
0.9833
1.0000
0.7285
0.9490
0.9832
0.9688
0.9688
1.0000
0.5870
0.9234
0.9492
0.9180
0.9180
1.0000
0.8043
0.9486
0.9832
0.9513
0.9513
1.0000
0.7235
0.9432
0.9338
0.9464
0.9464
1.0000
0.6623
0.9702
1.0000
0.9843
0.9843
1.0000
0.7285
0.9603
0.9832
0.9703
0.9703
IVIFMAXCS
IVIFMINCS
IVIFNORCS
IVIFMEDCS
IVIFOLMCS
IVIFWINCS
1.0000
0.6261
0.8575
0.9766
0.8898
0.8898
1.0000
0.7674
0.9346
0.9866
0.9641
0.9641
1.0000
0.7995
0.9311
0.9410
0.9495
0.9495
1.0000
0.6910
0.9454
1.0000
0.9896
0.9896
0.9970
0.7670
0.9438
0.9913
0.9793
0.9793
1.0000
0.6261
0.8614
0.9766
0.8918
0.8918
1.0000
0.7674
0.9357
0.9866
0.9642
0.9642
1.0000
0.7995
0.9319
0.9410
0.9496
0.9496
1.0000
0.6910
0.9378
1.0000
0.9896
0.9896
0.9970
0.7670
0.9450
0.9913
0.9793
0.9793
1.0000
0.6261
0.8676
0.9766
0.8949
0.8949
1.0000
0.7674
0.9375
0.9866
0.9643
0.9643
1.0000
0.7995
0.9331
0.9410
0.9497
0.9497
1.0000
0.6910
0.9416
1.0000
0.9897
0.9897
0.9970
0.7670
0.9469
0.9913
0.9794
0.9794
1.0000
0.6261
0.8787
0.9766
0.9009
0.9009
1.0000
0.7674
0.9408
0.9866
0.9646
0.9646
1.0000
0.7995
0.9354
0.9410
0.9500
0.9500
1.0000
0.6910
0.9483
1.0000
0.9897
0.9897
0.9970
0.7670
0.9505
0.9913
0.9796
0.9796
1.0000
0.6261
0.9067
0.9766
0.9161
0.9161
1.0000
0.7674
0.9491
0.9866
0.9654
0.9654
1.0000
0.7995
0.9451
0.9410
0.9510
0.9510
1.0000
0.6910
0.9626
1.0000
0.9899
0.9899
0.9970
0.7670
0.9590
0.9913
0.9800
0.9800
1.0000
0.6261
0.9336
0.9766
0.9334
0.9334
1.0000
0.7674
0.9583
0.9866
0.9666
0.9666
1.0000
0.7995
0.9494
0.9410
0.9527
0.9527
1.0000
0.6910
0.9748
1.0000
0.9902
0.9902
0.9970
0.7670
0.9680
0.9913
0.9807
0.9807
Optimistic case
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
Pessimistic case
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
Neutral case
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
IVIFMAXCS
IVIFMAXCS
IVIFMINCS
IVIFMINCS
IVIFNORCS
IVIFNORCS
IVIFMEDCS
IVIFMEDCS
IVIFOLMCS
IVIFOLMCS
IVIFWINCS
IVIFWINCS
Ranking order
OWIVIFACS
0.8860
0.9468
0.9451
0.9560
0.9577
OWQIVIFCS
0.8950
0.9509
0.9462
0.9606
0.9598
OWCIVIFCS
0.9031
0.9544
0.9472
0.9646
0.9617
OWIVIFGCS
0.8763
0.9422
0.9441
0.9505
0.9553
0.8310
0.8983
0.9296
0.8963
0.9239
0.8889
0.9479
0.9453
0.9572
0.9581
1.0000
1.0000
1.0000
1.0000
1.0000
0.8915
0.9494
0.9458
0.9589
0.9590
0.8566
0.9324
0.9421
0.9385
0.9507
OWIVIFCS
0.8553
0.9258
0.9204
0.9404
0.9391
OWQIVIFCS
0.8673
0.9292
0.9242
0.9472
0.9432
OWQIVIFCS
0.8779
0.9323
0.9276
0.9529
0.9468
OWIVIFGCS
0.8418
0.9223
0.9161
0.9321
0.9344
0.7976
0.8921
0.8848
0.8729
0.8934
0.8599
0.9265
0.9215
0.9425
0.9402
1.0000
1.0000
1.0000
1.0000
1.0000
0.8623
0.9280
0.9228
0.9446
0.9417
0.8128
0.9158
0.9071
0.9128
0.9244
OWIVIFCS
0.8734
0.9419
0.9353
0.9493
0.9524
OWQIVIFCS
0.8840
0.9448
0.9373
0.9551
0.9554
OWQIVIFCS
0.8934
0.9474
0.9391
0.9599
0.9581
OWIVIFGCS
0.8616
0.9387
0.9332
0.9425
0.9489
0.8151
0.9064
0.9113
0.8847
0.9104
0.8771
0.9426
0.9357
0.9510
0.9531
1.0000
1.0000
1.0000
1.0000
0.9914
0.8797
0.9437
0.9366
0.9530
0.9543
0.8372
0.9322
0.9291
0.9269
0.9419
In this paper, we have suggested a new and flexible method for measuring the similarity between interval-valued intuitionistic fuzzy sets. Using the idea of weighted reduced intuitionistic fuzzy sets, the work has developed a new interval-valued intuitionistic fuzzy cosine similarity measure and proved some of its basic and essential properties. Its fundamental advantage is the ability to combine the subjective knowledge and attitudinal character of the decision-maker in measuring the process of similarity degree. Further, we have defined the ordered weighted interval-valued intuitionistic fuzzy cosine similarity measure. It is a similarity measure that uses the notion of GOWA in the normalization process of interval-valued intuitionistic fuzzy cosine similarity based on reduced intuitionistic fuzzy sets. This approach alleviates the influence of unduly large (or small) similarity values on aggregation results by assigning them low (or high) weights. Moreover, it also provides a parameterized family of cosine similarity measures from minimum cosine similarity to maximum cosine similarity between two interval-valued intuitionistic fuzzy sets. We have studied some of its main properties and particular cases.
The use of quasi-arithmetic means under this framework has also been studied to obtain the quasi-ordered weighted interval-valued intuitionistic fuzzy cosine similarity measure. This cosine similarity measure includes a wide range of particular cases, including the OWIVIFCS measure, the trigonometric-OWIVIFCS measures, the exponential-OWIVIFCS measure, and the radical-OWIVIFCS measure.
The newly developed interval-valued intuitionistic cosine similarity measures can be applied in different real-world decision problems. This paper has focused on multiple criteria decision-making problems. We have developed a decision-making method based on OWIVIFCS to solve real-world decision problems with interval-valued intuitionistic fuzzy information. Finally, a numerical example has been provided to illustrate the multiple criteria decision-making process. We have seen that this approach provides more information for decision making because it can consider a wide range of situations depending on the interest of decision-makers. The proposed approach also has some limitations. The developed interval-valued intuitionistic fuzzy cosine similarity measures can be utilized in situations where the degrees of membership and non-membership values take interval numerical values. However, in many real-life situations, linguistic variables are used to represent qualitative information. These similarity measures cannot be utilized under the linguistic environment. So, we need a further study of these similarity measures with linguistic interval-valued intuitionistic fuzzy information.
In future research, we expect to develop further extensions by using more complex formulations, including the use of inducing variables, probabilities, moving averages, power averages, Bonferroni means, etc. Other important issues to consider are consensus (Chiclana
We thank the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper.