=Paper=
{{Paper
|id=Vol-2387/20190479
|storemode=property
|title=None
|pdfUrl=https://ceur-ws.org/Vol-2387/20190479.pdf
|volume=Vol-2387
|dblpUrl=https://dblp.org/rec/conf/icteri/KondratenkoKS19
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==None==
https://ceur-ws.org/Vol-2387/20190479.pdf
Hesitant Fuzzy Information Processing Based on the
Generalized Aggregation of Resulting Trapezoidal
Linguistic Terms
Yuriy Kondratenko, Galyna Kondratenko, Ievgen Sidenko
Intelligent Information Systems Department, Petro Mohyla Black Sea National University,
68th Desantnykiv Str., 10, Mykolaiv, 54003, Ukraine,
(yuriy.kondratenko,halyna.kondratenko,ievgen.sidenko)@chmnu.edu.ua
Abstract. This paper discuss the results of the analysis of multi-criteria deci-
sion-making algorithms based on expert evaluations, which are presented in the
form of hesitant linguistic terms (LTs). Authors propose to form the resulting
trapezoidal linguistic terms for any pairs of hesitant triangular fuzzy numbers.
To increase the efficiency of the multi-criteria decision making process with
hesitant input data, the authors suggest a new approach for fuzzy aggregation of
generalized trapezoidal LTs based on combination of pessimistic and optimistic
views of decision-makers. Simulation results proves high efficiency of the pro-
posed hesitant fuzzy information processing approach, in particular in solving
multi-criteria problem for selection of the most efficient transport company
from the set of the existing alternatives.
Keywords: hesitant fuzzy set, linguistic term, aggregation, pessimistic position,
optimistic position, transport company.
1 Introduction
The process of multi-criteria decision-making consists in choosing the best solution
among alternatives according to a certain list of criteria. Modeling on the basis of
expert knowledge of the system is an approach based on the knowledge and experi-
ence of the person who acts as an expert in matters related to the real system [1].
Finding a solution to a multi-criterial problem does not pose any particular difficul-
ties, if the advantage of one of the criteria leads to the same advantage by another
criterion, that is, if the criteria are co-operated [2]. Difficulty arises when evaluating
the decision according to the relevant criteria is not comparable. In this case, situa-
tions arise in which it is difficult for an expert to evaluate an alternative solution using
only a quantitative assessment scale. This is due to the fact that the expert's judgment
in most cases takes the linguistic form in the form of fuzzy sets, rules and grammar
[3]. For example, it is easier for an expert to evaluate a transport company by the
criterion of "insurance level of cargo" in a fuzzy form (with the help of linguistic
estimates - terms), for example, "the level of cargo insurance - low or lower than av-
erage" than giving an exact quantification. Solving the problems of multi-criteria
decision-making under uncertainty conditions, in which expert evaluations fluctuate
within the limits of several linguistic values of the estimating parameter, is an urgent
problem for today. This is due to the complexity of developing models that take into
account the relevant conditions of uncertainty, since expert assessments in most cases
are presented as fuzzy interval intervals [2, 3]. In this case, the expert usually indi-
cates an assessment within the limits of several TLs for each alternative solution ac-
cording to the relevant criterion.
2 Related Works and Problem Statement
More and more urgent is the need for the processing of fuzzy, that is, qualitative in-
formation, the process of formalization which is quite complicated [1, 2]. In addition,
the automation of decision support processes in conditions of uncertainty becomes
especially important in conditions of rapid and dynamic growth of the functional ca-
pabilities [3].
Different estimation methods are used to measure customer expectations, including
questionnaires, expert assessments, statistical methods, etc. The difficulty lies in the
fact that most of the parameters of the system cannot be quantified, that is, it is diffi-
cult to obtain formalized estimates. An instrument for formalizing the fuzzy expecta-
tions of consumers is a mathematical apparatus based on the theory of fuzzy sets [4,
5]. The motivation to use fuzzy logic to solve multi-criteria decision-making tasks is
the possibility of a convenient and understandable linguistic interpretation of the pro-
cesses of constructing models of human judgment, which facilitates their implementa-
tion in interactive computer decision support systems [2].
In the decision-making process, expert assessments may vary within several LTs.
In paper [6] authors presented several approaches to assess the degree of comparison
of vibrational fuzzy sets. The authors of the study [7] proposed the use of various
types of distances between vague fuzzy sets. In research [8], the authors proposed a
method of multi-criteria decision-making based on a comparison of the values of the
probability of fuzzy vibrational sets. Y. Tang and J. Zheng [9] introduced the concept
of the sequence of LT in multi-criteria decision-making problems and presented a
fuzzy model in which expert estimates are expressed in terms of several LTs. The
process of transformation of LT triangular form into a trapezoidal way of their previ-
ous aggregation is considered [10], while the transfer of trapezoidal LTs to fuzzy
intervals is carried out on the basis of α-cross models, and the choice of the best solu-
tion is based on a comparative analysis of alternatives in terms of the probability that
the interval is longer than (or equal to) the pessimistic and optimistic position of the
decision maker (DM) [3, 11, 12].
The purpose of this study is to develop a new approach for fuzzy aggregation of
generalized trapezoidal LTs based on combination of pessimistic and optimistic views
of DM and prove high efficiency of the proposed approach, in particular in solving
multi-criteria problem for selection of the most efficient transport company from the
set of the existing alternatives.
3 Modeling Results
Let's consider the solution of the multi-criteria decision-making problem on the ex-
ample of the task of choosing the best transportation company for delivery of cargo
[13-15]. The method proposed by the authors will be illustrated by the example of
choosing the best solution from 4 alternatives according to 8 criteria. In particular, the
expert is asked to assess the quality of transport services of the relevant 4 transport
companies ( x1 is a transport company A, x2 is a transport company B, x3 is a
transport company C and x4 is a transport company D) according to the following
criteria: C1 is a company image, C2 is a cargo support, C3 is an insurance level, C4
is a vehicle traffic monitoring, C5 is a conservancy of cargo by quantity, C6 is a
consistency of cargo in quality, C7 is a timeliness of delivery, C8 is a flexibility of
payment service system. The assessment scale is presented in the form of fuzzy trian-
gular LTs {L – low, LM – lower than medium, M – medium, HM – higher than medi-
um, H – high}. Grammar of the formation of expert assessments allows the use of
operators {within, lower, and higher} to represent judgments of experts within the
limits of several triangular forms.
Let’s formulate in more detail the main steps of the method of the generalized ag-
gregation of resulting trapezoidal LTs proposed by the authors.
Step 1. Formation of the matrix of expert assessments. At this step, the expert eval-
uates each alternative solution in relation to a certain list of criteria according to the
linguistic scale of assessment, presented in the form of a corresponding LTs, for ex-
ample, a triangular form. In this case, the expert score may vary within several LTs
(Table 1).
Table 1. Expert evaluation of alternatives by a certain criteria according to the linguistic scale
Criteria
C1 C2 C3 C8
x1 M within HM and H HM within M and HM
x2 LM within LM and M LM … within LM and M
x3 M within M and HM M within M and HM
x4 within LM and M LM lower LM M
Step 2. Transformation of the matrix of expert estimates (Table 1) into the matrix of
interval estimates. At this stage, expert assessments that were within defined limits of
LT were transformed into interval-type estimates. If, for example, the expert evaluat-
ed the decision x1 on the criterion C8 as "within M and HM," then the corresponding
estimate is transformed into an interval {M, HM}.
Step 3. Aggregation of LTs into generalized trapezoidal terms. In this case, the
combination of interval estimates (LTs triangular form) is combined into generalized
trapezoidal terms. The model of LT trapezoidal form can be represented in the form
Si j a1 , a2 , a3 , a4 , where i is the number of the alternative; j is the number of the
criterion (Table 2), for example S18 0.25,0.5,0.75,1 (Fig. 1).
Fig. 1. Aggregation of LTs for the alternative x1 according to the criterion C8 into a general-
ized trapezoidal LT S18
Table 2. The matrix of generalized trapezoidal LTs
Criteria
C1 C2 C3 C8
x1 (0.25, 0.5, 0.5, 0.75) (0.5, 0.75, 1, 1) (0.5, 0.75, 0.75, 1) (0.25, 0.5, 0.75, 1)
x2 (0, 0.25, 0.25, 0.5) (0, 0.25, 0.5, 0.75) (0, 0.25, 0.25, 0.5) … (0, 0.25, 0.5, 0.75)
x3 (0.25, 0.5, 0.5, 0.75) (0.25, 0.5, 0.75, 1) (0.25, 0.5, 0.5, 0.75) (0.25, 0.5, 0.75, 1)
x4 (0, 0.25, 0.5, 0.75) (0, 0.25, 0.25, 0.5) (0, 0, 0.25, 0.5) (0.25, 0.5, 0.5, 0.75)
Step 4. Aggregation of generalized trapezoidal LTs Si j into averaged (combined)
trapezoidal LTs GSi . This allows to take into account both the minimum (pessimistic
position) and the maximum (optimistic position) expert assessments simultaneously
(Fig. 2). It eliminates the need to define intervals for all individual generalized trape-
zoidal LTs, in particular for the pessimistic and optimistic positions of DM. Averaged
trapezoidal LTs GSi for all alternatives: GS1 0.25, 0.5, 1, 1 ;
GS2 0, 0.25, 0.75, 1 ; GS3 0.25, 0.5, 0.75, 1 ; GS4 0, 0, 0.5, 0.75 .
j
Fig. 2. Aggregation of generalized trapezoidal LTs S1 for the alternative x1 according to all
criteria into the averaged trapezoidal LT GS1
Step 5. Formulating the average trapezoidal LTs of each alternative to fuzzy intervals.
At this stage, the formula (1) transforms the averaged trapezoidal LTs into fuzzy in-
tervals [2, 15, 17] with the appropriate choice of parameter 0,1 .
I xi I L , I R a2 a1 a1 , a4 a4 a3 , i 1,..., m . (1)
Step 6. Determine the probability indicator [9, 10, 16] for each alternative (2). Rank-
ing decisions by the appropriate indicator.
1 IL
p I xi [0,1] max 1 max , 0 , 0 , i 1,..., m . (2)
IR IL 1
According to Step 5 and Step 6 fuzzy intervals I xi and probability indicators
p I xi [0,1] for all alternatives: I x1 0.375, 1 and p I x1 [0,1] 0.62 ;
I x2 0.125, 0.875 and p I x2 [0,1] 0.5 ; I x3 0.375, 0.875 and
p I x3 [0,1] 0.58 ; I x4 0, 0.625 and p I x4 [0,1] 0.38 .
As a result of the implementation of the proposed method of the generalized ag-
gregation of resulting trapezoidal LTs, the best alternative is x1 (company A),
p I x1 [0,1] p I x3 [0,1] p I x2 [0,1] p I x4 [0,1] , since ranking
of the relevant probability indicators allows to determine the next order of priority of
the alternativess x1 x3 x2 x4 .
The results of the application of existing methods for multi-criteria decision-
making based on the pessimistic ( x1 x3 x2 x4 , execution time is 2300 ms) and
optimistic ( x1 x2 x3 x4 , execution time is 2300 ms) positions of DM [6, 10] and
the proposed method of aggregation of generalized trapezoidal LTs
( x1 x3 x2 x4 , execution time is 1350 ms) prove the performance and effective-
ness of the proposed method.
4 Conclusions
The proposed method of aggregation of generalized trapezoidal LTs makes it possible
to simplify the process of choosing the best alternative (in comparison with existing
methods) and to increase the efficiency, in particular, the speed of the processes of
multi-criteria decision-making. This statement is based on a comparative analysis of
the time duration of computing operations in the implementation of the appropriate
methods. In particular, the implementation time of the program code for the imple-
mentation of the method of aggregation of generalized trapezoidal LTs is 1350 micro-
seconds, and for the method of multi-criteria decision-making based on hesitant fuzzy
terms using pessimistic and optimistic positions of DM is 2300 microseconds.
References
1. Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338-353 (1965).
2. Piegat, A.: Fuzzy Modeling and Control. Springer-Verlag, 2001.
3. Torra, V.: Hesitant fuzzy sets. International Journal of Intelligent Systems 25(6), 529-539
(2010). DOI: 10.1002/int.20418.
4. Gozhyj, A., Kalinina, I., Vysotska, V., Gozhyj, V.: The method of web-resources man-
agement under conditions of uncertainty based on fuzzy logic. In: IEEE 13th International
Scientific and Technical Conference on Computer Sciences and Information Technologies
(CSIT), pp. 347-352. Lviv, Ukraine (2018). DOI: 10.1109/STC-CSIT.2018.8526761.
5. Kondratenko, Y., Kondratenko, N.: Universal direct analytic models for the minimum of
triangular fuzzy numbers. In: Ermolaev, V. et al (eds.) International Conference on ICT in
Education, Research and Industrial Applications. Integration, Harmonization and
Knowledge Transfer, vol. 2, pp. 100-115. Kyiv, Ukraine (2018).
6. Farhadinia, B. Information measures for hesitant fuzzy sets and interval-valued hesitant
fuzzy sets. Information Sciences 240(13), 129-144 (2013). DOI: 10.1016/j.ins.2013.03.034
7. Xu, Z., Xia, M.: Distance and similarity measures for hesitant fuzzy sets. Information Sci-
ences 181(11), 2128-2138 (2011). DOI: 10.1016/j.ins.2011.01.028.
8. Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: IEEE International Con-
ference on Fuzzy Systems, pp. 1378-1382. Jeju Island, South Korea (2009).
9. Tang, Y., Zheng, J.: Linguistic modelling based on semantic similarity relation among
linguistic labels. Fuzzy Sets and Systems 157(12), 1662-1673 (2006).
10. Lee, L.W., Chen, S.M.: Fuzzy decision making based on hesitant fuzzy linguistic term
sets. In: Selamat, A., Nguyen, N.T., Haron, H. (eds.) Intelligent Information and Database
Systems (ACIIDS), LNCS, vol. 7802, pp. 21-30. Springer, Berlin, Heidelberg (2013).
11. Kondratenko, Y., Kondratenko, G., Sidenko, I.: Two-stage method of fuzzy rule base cor-
rection for variable structure of input vector. In: IEEE First Ukraine Conference on Elec-
trical and Computer Engineering (UKRCON), pp. 1043-1049. Kiev, Ukraine (2017), DOI:
10.1109/UKRCON.2017.8100409.
12. Wei, C., Zhao, N., Tang, X.: Operators and comparisons of hesitant fuzzy linguistic term
sets. IEEE Transactions on Fuzzy Systems 22(3), 575-585 (2013).
13. Kondratenko, Y.P., Sidenko, Ie.V.: Decision-Making Based on Fuzzy Estimation of Quali-
ty Level for Cargo Delivery. In: Zadeh, L.A. et al. (eds.) Recent Developments and New
Directions in Soft Computing. Studies in Fuzziness and Soft Computing, vol. 317, pp. 331-
344. Springer, Switzerland (2014). DOI: 10.1007/978-3-319-06323-2_21.
14. Kondratenko, Y.P., Klymenko, L.P., Sidenko, I.V.: Comparative Analysis of Evaluation
Algorithms for Decision-Making in Transport Logistics. In: Jamshidi, M., Kreinovich, V.,
Kazprzyk, J. (eds.) Advance Trends in Soft Computing, vol. 312, pp. 203-217. Springer,
Cham (2014). DOI: 10.1007/978-3-319-03674-8_20.
15. Solesvik, M., Kondratenko, Y., Kondratenko, G., Sidenko, I., Kharchenko, V., Boyarchuk,
A.: Fuzzy decision support systems in marine practice. In: IEEE International Conference
on Fuzzy Systems (FUZZ-IEEE), pp. 1-6. Naples, Italy (2017). DOI: 10.1109/FUZZ-
IEEE.2017.8015471.
16. Rodriguez, R.M., Martinez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision
making. IEEE Transactions on Fuzzy Systems 20(1), 109-119 (2012).
17. Chen, S.M., Hong, J.: Multicriteria Linguistic Decision Making Based on Hesitant Fuzzy
Linguistic Term Sets and the Aggregation of Fuzzy Sets. Information Sciences 286, 63-74
(2014). DOI: 10.1016/j.ins.2014.06.020.