International Journal of Fuzzy Logic and Intelligent Systems 2014; 14(3): 181-187
Published online September 30, 2014
https://doi.org/10.5391/IJFIS.2014.14.3.181
© The Korean Institute of Intelligent Systems
Ismat Beg1 and Tabasam Rashid2
1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan
2Department of Mathematics, University of Management and Technology, Lahore, Pakistan
Correspondence to :
Ismat Beg (ibeg@lahoreschool.edu.pk)
Dealing with uncertainty is always a challenging problem. Intuitionistic fuzzy sets was presented to manage situations in which experts have some membership and non-membership value to assess an alternative. Hesitant fuzzy sets was used to handle such situations in which experts hesitate between several possible membership values to assess an alternative. In this paper, the concept of intuitionistic hesitant fuzzy set is introduced to provide computational basis to manage the situations in which experts assess an alternative in possible membership values and non-membership values. Distance measure is defined between any two intuitionistic hesitant fuzzy elements. Fuzzy technique for order preference by similarity to ideal solution is developed for intuitionistic hesitant fuzzy set to solve multi-criteria decision making problem in group decision environment. An example is given to illustrate this technique.
Keywords: Hesitant fuzzy set,Intuitionistic fuzzy set,Multiple attribute group decisionmaking,Technique for order preference by similarity to ideal solution
Group decision making is the process of finding the best option among a set of feasible alternatives. Basic problem is how to aggregate several inputs into a single representative output [1–3]. Problems that are defined under uncertain situations are common in real world decision making. Zadeh [4] in his seminal paper introduced the notion of fuzzy sets to handle uncertainty. Bellman and Zadeh [5] used fuzzy sets in decision making for the solution of ambiguity in information obtained from human preferences. Recently, Dubois [6] presented a beautiful comparison about old and new techniques for fuzzy decision analysis. Atanassov [7, 8] introduced the concept of intuitionistic fuzzy sets (IFS) characterized by a membership function and a non-membership function, which is more suitable for dealing with fuzziness and uncertainty than the fuzzy set. The IFS is highly useful in depicting uncertainty and vagueness of an object, and thus can be used as a powerful tool to express data information under various different fuzzy environments which has attracted great attentions. Recently, the intuitionistic fuzzy set has been widely applied to decision making problems [9–12]. IFSs have been found to be a particularly useful tool to deal with vagueness. Torra [13] extended the concept of fuzzy sets to hesitant fuzzy sets. Hesitant fuzzy set theory tries to manage those situations where a set of values are possible in the definition process of the membership of an element. Group decision making problems are solved by using hesitant fuzzy sets and with aggregation operators in [1–3, 14, 15].
Hwang and Yoon [16] developed technique for order preference by similarity to ideal solution (TOPSIS) for multi-attribute/multi-criteria decision making (MADM/MCDM) problems. Shih et al. [17] addressed four advantages of TOPSIS: first is that a sound logic represents the rationale of human choice; secondly a scalar value considers the best and worst alternative simultaneously; third advantage is that it has a simple computation process and can be easily programmed and the last but not the least advantage is that it has ability of the performance measures of all alternatives on attributes to be visualized on a polyhedron, at least for any two dimensions. Fuzzy numbers are applied to establish a fuzzy TOPSIS [18].
The aim of this paper is to introduce the concept of intuitionistic hesitant fuzzy sets (IHFS) by merging the concept of IFS and HFS. It helps to manage those situations of uncertainty in which some values are possible as membership values of element as well as non-membership values of the same element. Additionally, we also develop fuzzy TOPSIS for IHFS. This article is organized as follows: In Section 2, we introduce the concept of IHFS and notion of distance between any two elements of IHFS. In Section 3, TOPSIS is constructed for IHFS. Then in Section 4, this fuzzy TOPSIS method is applied for the ranking of alternatives in an example, to demonstrate its feasibility. Conclusion is given in the last section.
In this section, we introduce an extension of IFS to manage those situations in which several values are possible for the definition of a membership function and non-membership function. We propose the concept of IHFS by keeping in view the importance of IFS and HFS. IHFS is defined in terms of a function that returns a set of membership values and a set of non-membership values for each element in the domain.
An IHFS on
where
Examples of IHFS are given below where
It is noted that the number of values in different IHFEs may be different, let
Consider an IHFS
Then
max
max
min
min
It is clear that
Let
It is easy to show that this distance ‘
1)
2)
In this section, we give construction of TOPSIS for IHFS. This TOPSIS is used for multi-criteria group decision making where the opinions about the criteria values are expressed in IHFS. We suppose that in this group decision making problem,
and
Performance of alternative
and
where
Graphical representation of this proposed technique is given in Figure 1.
In this section, we give an example. We utilized the method proposed in Section 4 to get the most desirable alternative as well as rank the alternatives from the best to worst or vice versa. Consider five schools (School of Business and Economics (
For cost criteria
Negative ideal separation matrix (
Thus the most desirable alternative is
IHFS is the best way to deal with uncertainty when fuzzy set theory is not able to cope with the situation. Decision makers gave their opinions about the criteria of alternatives by IHFS. Multi-criteria analysis provides an effective frame work for the evaluation of alternatives. Fuzzy TOPSIS method is proposed for IHFS to solve multi-criteria decision-making problem in group decision environment. The RC coefficient has ranked the alternatives from the best to worst by considering the smallest distance from the PIS and also the largest distance from the NIS. In future, we plan to continue to study Choquet integral based TOPSIS for IHFS. Furthermore, algebraic operations for IHFS will also be develop.
No potential conflict of interest relevant to this article was reported.
Table 1. Decision matrix (
((0.5,0.6,0.8),(0.1,0.2)) | |
((0.1,0.3),(0.3,0.4,0.5)) | |
((0.5,0.7),(0.2,0.25)) | |
((0.7,0.9),(0.05,0.1)) | |
((1),(0)) | |
((0.6,0.8),(0.1,0.2)) | |
((0.5,0.7,0.8),(0.1)) | |
((0.5,0.6),(0.2,0.35)) | |
((0.1,0.2),(0.6,0.7)) | |
((0.1,0.3),(0.5,0.65)) | |
((0.1,0.3),(0.6,0.7)) | |
((0.5,0.6),(0.1,0.3)) | |
((0.7,0.9),(0.05,0.1)) | |
((0.1,0.3),(0.6,0.7)) | |
((0,0.2),(0.7,0.8)) | |
((0.1,0.3),(0.5,0.6)) | |
((0.5,0.6),(0.2,0.3)) | |
((0.1,0.2),(0.6,0.7)) | |
((0.5,0.6,0.7),(0.2)) | |
((0.4,0.7),(0.1,0.2)) |
Table 2. Decision matrix (
((0.1,0.2),(0.5,0.6)) | |
((0,0.2),(0.6,0.7)) | |
((0.4,0.6),(0.1,0.3)) | |
((0.6,1),(0)) | |
((0.5,0.7),(0.1,0.2)) | |
((0.4,0.9),(0,0.1)) | |
((0.1,0.3),(0.5,0.6)) | |
((0.1,0.2),(0.6,0.7)) | |
((0.4,0.7),(0.1,0.2)) | |
((0.4,0.6),(0.2,0.3)) | |
((0,0.2),(0.6,0.7)) | |
((0.4,0.5),(0.3,0.4)) | |
((0.4,0.6),(0.3,0.4)) | |
((0,0.1),(0.6,0.8)) | |
((0,0.1),(0.7,0.8)) | |
((0.4,0.6),(0.1,0.3)) | |
((0.6,1),(0)) | |
((0,0.2),(0.5,0.7)) | |
((0.5,0.7),(0.2,0.3)) | |
((0.6,1),(0)) |
Table 3. Decision matrix (
((0.4,0.6),(0.2,0.3)) | |
((0.3,0.6),(0.2,0.3)) | |
((0.1,0.3),(0.5,0.6)) | |
((0.6,0.9),(0,0.1)) | |
((0.5,0.6),(0.3,0.4)) | |
((0.6,1),(0)) | |
((0.1,0.3),(0.6,0.7)) | |
((0.6,0.9),(0,0.1)) | |
((0.5,0.7),(0.1,0.2)) | |
((0.1,0.3),(0.6,0.7)) | |
((0.3,0.5),(0.3,0.4)) | |
((0.5,0.9),(0,0.05)) | |
((0.3,0.7),(0.1,0.2)) | |
((0,0.2,0.4),(0.4,0.5)) | |
((0.2,0.4),(0.4,0.5)) | |
((0,0.3),(0.5,0.6)) | |
((0.3,0.5),(0.3,0.4)) | |
((0,0.1),(0.7,0.8)) | |
((0.5,0.6,0.8),(0)) | |
((1),(0)) |
Table 4. Decision matrix (
((0.2,0.4,0.5),(0.2,0.3,0.5)) | |
((0.2,0.3),(0.3,0.4,0.5,0.6)) | |
((0.3,0.4,0.5),(0.25,0.3,0.5)) | |
((0.7,0.9),(0,0.05)) | |
((0.6,0.7,1),(0,0.1,0.2,0.3)) | |
((0.6,0.8),(0,0.1)) | |
((0.3,0.5),(0.1,0.5,0.6)) | |
((0.2,0.5,0.6),(0.1,0.2,0.35,0.6)) | |
((0.2,0.4,0.5),(0.2,0.6)) | |
((0.3,0.4),(0.3,0.5,0.6)) | |
((0.2,0.3),(0.4,0.6)) | |
((0.5),(0.05,0.1,0.3)) | |
((0.6,0.7),(0.1,0.2,0.3)) | |
((0.1),(0.5,0.6)) | |
((0.1,0.2),(0.5,0.7)) | |
((0.3,0.4),(0.3,0.5)) | |
((0.5,0.6),(0,0.2,0.3)) | |
((0.1),(0.7)) | |
((0.5,0.6,0.7),(0,0.2)) | |
((0.7,1),(0,0.1)) |
International Journal of Fuzzy Logic and Intelligent Systems 2014; 14(3): 181-187
Published online September 30, 2014 https://doi.org/10.5391/IJFIS.2014.14.3.181
Copyright © The Korean Institute of Intelligent Systems.
Ismat Beg1 and Tabasam Rashid2
1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan
2Department of Mathematics, University of Management and Technology, Lahore, Pakistan
Correspondence to:Ismat Beg (ibeg@lahoreschool.edu.pk)
Dealing with uncertainty is always a challenging problem. Intuitionistic fuzzy sets was presented to manage situations in which experts have some membership and non-membership value to assess an alternative. Hesitant fuzzy sets was used to handle such situations in which experts hesitate between several possible membership values to assess an alternative. In this paper, the concept of intuitionistic hesitant fuzzy set is introduced to provide computational basis to manage the situations in which experts assess an alternative in possible membership values and non-membership values. Distance measure is defined between any two intuitionistic hesitant fuzzy elements. Fuzzy technique for order preference by similarity to ideal solution is developed for intuitionistic hesitant fuzzy set to solve multi-criteria decision making problem in group decision environment. An example is given to illustrate this technique.
Keywords: Hesitant fuzzy set,Intuitionistic fuzzy set,Multiple attribute group decisionmaking,Technique for order preference by similarity to ideal solution
Group decision making is the process of finding the best option among a set of feasible alternatives. Basic problem is how to aggregate several inputs into a single representative output [1–3]. Problems that are defined under uncertain situations are common in real world decision making. Zadeh [4] in his seminal paper introduced the notion of fuzzy sets to handle uncertainty. Bellman and Zadeh [5] used fuzzy sets in decision making for the solution of ambiguity in information obtained from human preferences. Recently, Dubois [6] presented a beautiful comparison about old and new techniques for fuzzy decision analysis. Atanassov [7, 8] introduced the concept of intuitionistic fuzzy sets (IFS) characterized by a membership function and a non-membership function, which is more suitable for dealing with fuzziness and uncertainty than the fuzzy set. The IFS is highly useful in depicting uncertainty and vagueness of an object, and thus can be used as a powerful tool to express data information under various different fuzzy environments which has attracted great attentions. Recently, the intuitionistic fuzzy set has been widely applied to decision making problems [9–12]. IFSs have been found to be a particularly useful tool to deal with vagueness. Torra [13] extended the concept of fuzzy sets to hesitant fuzzy sets. Hesitant fuzzy set theory tries to manage those situations where a set of values are possible in the definition process of the membership of an element. Group decision making problems are solved by using hesitant fuzzy sets and with aggregation operators in [1–3, 14, 15].
Hwang and Yoon [16] developed technique for order preference by similarity to ideal solution (TOPSIS) for multi-attribute/multi-criteria decision making (MADM/MCDM) problems. Shih et al. [17] addressed four advantages of TOPSIS: first is that a sound logic represents the rationale of human choice; secondly a scalar value considers the best and worst alternative simultaneously; third advantage is that it has a simple computation process and can be easily programmed and the last but not the least advantage is that it has ability of the performance measures of all alternatives on attributes to be visualized on a polyhedron, at least for any two dimensions. Fuzzy numbers are applied to establish a fuzzy TOPSIS [18].
The aim of this paper is to introduce the concept of intuitionistic hesitant fuzzy sets (IHFS) by merging the concept of IFS and HFS. It helps to manage those situations of uncertainty in which some values are possible as membership values of element as well as non-membership values of the same element. Additionally, we also develop fuzzy TOPSIS for IHFS. This article is organized as follows: In Section 2, we introduce the concept of IHFS and notion of distance between any two elements of IHFS. In Section 3, TOPSIS is constructed for IHFS. Then in Section 4, this fuzzy TOPSIS method is applied for the ranking of alternatives in an example, to demonstrate its feasibility. Conclusion is given in the last section.
In this section, we introduce an extension of IFS to manage those situations in which several values are possible for the definition of a membership function and non-membership function. We propose the concept of IHFS by keeping in view the importance of IFS and HFS. IHFS is defined in terms of a function that returns a set of membership values and a set of non-membership values for each element in the domain.
An IHFS on
where
Examples of IHFS are given below where
It is noted that the number of values in different IHFEs may be different, let
Consider an IHFS
Then
max
max
min
min
It is clear that
Let
It is easy to show that this distance ‘
1)
2)
In this section, we give construction of TOPSIS for IHFS. This TOPSIS is used for multi-criteria group decision making where the opinions about the criteria values are expressed in IHFS. We suppose that in this group decision making problem,
and
Performance of alternative
and
where
Graphical representation of this proposed technique is given in Figure 1.
In this section, we give an example. We utilized the method proposed in Section 4 to get the most desirable alternative as well as rank the alternatives from the best to worst or vice versa. Consider five schools (School of Business and Economics (
For cost criteria
Negative ideal separation matrix (
Thus the most desirable alternative is
IHFS is the best way to deal with uncertainty when fuzzy set theory is not able to cope with the situation. Decision makers gave their opinions about the criteria of alternatives by IHFS. Multi-criteria analysis provides an effective frame work for the evaluation of alternatives. Fuzzy TOPSIS method is proposed for IHFS to solve multi-criteria decision-making problem in group decision environment. The RC coefficient has ranked the alternatives from the best to worst by considering the smallest distance from the PIS and also the largest distance from the NIS. In future, we plan to continue to study Choquet integral based TOPSIS for IHFS. Furthermore, algebraic operations for IHFS will also be develop.
Graphical structure of technique for order preference by similarity to ideal solution.
Table 1 . Decision matrix (
((0.5,0.6,0.8),(0.1,0.2)) | |
((0.1,0.3),(0.3,0.4,0.5)) | |
((0.5,0.7),(0.2,0.25)) | |
((0.7,0.9),(0.05,0.1)) | |
((1),(0)) | |
((0.6,0.8),(0.1,0.2)) | |
((0.5,0.7,0.8),(0.1)) | |
((0.5,0.6),(0.2,0.35)) | |
((0.1,0.2),(0.6,0.7)) | |
((0.1,0.3),(0.5,0.65)) | |
((0.1,0.3),(0.6,0.7)) | |
((0.5,0.6),(0.1,0.3)) | |
((0.7,0.9),(0.05,0.1)) | |
((0.1,0.3),(0.6,0.7)) | |
((0,0.2),(0.7,0.8)) | |
((0.1,0.3),(0.5,0.6)) | |
((0.5,0.6),(0.2,0.3)) | |
((0.1,0.2),(0.6,0.7)) | |
((0.5,0.6,0.7),(0.2)) | |
((0.4,0.7),(0.1,0.2)) |
Table 2 . Decision matrix (
((0.1,0.2),(0.5,0.6)) | |
((0,0.2),(0.6,0.7)) | |
((0.4,0.6),(0.1,0.3)) | |
((0.6,1),(0)) | |
((0.5,0.7),(0.1,0.2)) | |
((0.4,0.9),(0,0.1)) | |
((0.1,0.3),(0.5,0.6)) | |
((0.1,0.2),(0.6,0.7)) | |
((0.4,0.7),(0.1,0.2)) | |
((0.4,0.6),(0.2,0.3)) | |
((0,0.2),(0.6,0.7)) | |
((0.4,0.5),(0.3,0.4)) | |
((0.4,0.6),(0.3,0.4)) | |
((0,0.1),(0.6,0.8)) | |
((0,0.1),(0.7,0.8)) | |
((0.4,0.6),(0.1,0.3)) | |
((0.6,1),(0)) | |
((0,0.2),(0.5,0.7)) | |
((0.5,0.7),(0.2,0.3)) | |
((0.6,1),(0)) |
Table 3 . Decision matrix (
((0.4,0.6),(0.2,0.3)) | |
((0.3,0.6),(0.2,0.3)) | |
((0.1,0.3),(0.5,0.6)) | |
((0.6,0.9),(0,0.1)) | |
((0.5,0.6),(0.3,0.4)) | |
((0.6,1),(0)) | |
((0.1,0.3),(0.6,0.7)) | |
((0.6,0.9),(0,0.1)) | |
((0.5,0.7),(0.1,0.2)) | |
((0.1,0.3),(0.6,0.7)) | |
((0.3,0.5),(0.3,0.4)) | |
((0.5,0.9),(0,0.05)) | |
((0.3,0.7),(0.1,0.2)) | |
((0,0.2,0.4),(0.4,0.5)) | |
((0.2,0.4),(0.4,0.5)) | |
((0,0.3),(0.5,0.6)) | |
((0.3,0.5),(0.3,0.4)) | |
((0,0.1),(0.7,0.8)) | |
((0.5,0.6,0.8),(0)) | |
((1),(0)) |
Table 4 . Decision matrix (
((0.2,0.4,0.5),(0.2,0.3,0.5)) | |
((0.2,0.3),(0.3,0.4,0.5,0.6)) | |
((0.3,0.4,0.5),(0.25,0.3,0.5)) | |
((0.7,0.9),(0,0.05)) | |
((0.6,0.7,1),(0,0.1,0.2,0.3)) | |
((0.6,0.8),(0,0.1)) | |
((0.3,0.5),(0.1,0.5,0.6)) | |
((0.2,0.5,0.6),(0.1,0.2,0.35,0.6)) | |
((0.2,0.4,0.5),(0.2,0.6)) | |
((0.3,0.4),(0.3,0.5,0.6)) | |
((0.2,0.3),(0.4,0.6)) | |
((0.5),(0.05,0.1,0.3)) | |
((0.6,0.7),(0.1,0.2,0.3)) | |
((0.1),(0.5,0.6)) | |
((0.1,0.2),(0.5,0.7)) | |
((0.3,0.4),(0.3,0.5)) | |
((0.5,0.6),(0,0.2,0.3)) | |
((0.1),(0.7)) | |
((0.5,0.6,0.7),(0,0.2)) | |
((0.7,1),(0,0.1)) |
Graphical structure of technique for order preference by similarity to ideal solution.