# Hydrogen production technologies evaluation based on interval-valued intuitionistic fuzzy multiattribute decision making method.

1. IntroductionOne of the important parts of multicriteria decision making is intuitionistic fuzzy multiattribute decision making, and it is an important branch of operations research and management sciences. Intuitionistic fuzzy set (IFS) is a useful technique to describe the fuzziness of the world and it was characterized by membership degree and nonmembership degree [1]. Three years later, Atanassov and Gargov [2] extended the IFS to a more generalized form and introduced the interval-valued intuitionistic fuzzy set (IIFS). IIFS is characterized by the membership degree range and nonmembership degree range. Therefore, IIFS is more powerful to depict the fuzziness of the world and has been utilized in many fields, especially in decision making [3-8].

During the interval-valued intuitionistic fuzzy multi-criteria decision making process, the experts often provide their evaluation information which should be aggregated by using the proper aggregation methods. Interval-valued intuitionistic fuzzy aggregation operators play an important role in multicriteria decision making. Up to now, there are many aggregation operators for IIFNs; the most basic interval-valued intuitionistic fuzzy aggregation operators are interval-valued intuitionistic fuzzy weighted average (IIFWA) operator and interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator proposed by Xu [9], based on which, a lot of extended operators are proposed by researchers, such as generalized interval-valued intuitionistic fuzzy geometric operator [10], interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-IVIFEOWG) operator proposed by Yang and Yuan [11], induced interval-valued intuitionistic fuzzy Hamacher ordered weighted geometric (I-IVIFHOWG) operator [12], the interval-valued intuitionistic fuzzy Einstein weighted geometric operator, interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator and intervalvalued intuitionistic fuzzy Einstein hybrid weighted geometric operator [13], and induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (IG-IVIFHSA) operator [14].

However, the basic aggregation operators IIFWA and IIFWG for aggregating IIFNs are not perfect since they cannot deal with some special cases. For example, suppose [[??].sub.i] = ([[a.sub.i], [b.sub.i]], [[c.sub.i], [d.sub.i]]) (i = 1, 2, ..., n) are a group of IIFNs, when one of the IIFNs' nonmembership degree ranges reduce to [0, 0], then the nonmembership degree of the aggregated IIFN (IIFWA([[??].sub.i])) must be [0, 0] without the consideration of other n - 1 nonmembership degree ranges which is unreasonable. Inspired by the idea of He et al. [15, 16], we propose some interactive interval-valued intuitionistic fuzzy aggregation operators for aggregating IIFNs which are good complement of the existing interval-valued intuitionistic fuzzy aggregation operators.

Hydrogen technologies evaluations using multicriteria decision making method is an important research area in energy management and has attracted much attention from researchers [17, 18]. Afgan et al. [19] used the multicriteria assessment technology to select the hydrogen energy systems from the performance, environment, and market criteria. McDowall and Eames [20] introduced a new methodology to assess the alternative future hydrogen energy systems for the UK. Ren et al. [21] developed a novel fuzzy multiactor decision making approach to assess the hydrogen technologies; the feature of the proposed method is that it can deal with the uncertainties and imprecision. Lee et al. [22] combined the AHP and DEA approaches and proposed a two-stage multicriteria decision making method for efficiently allocating energy R&D resources. However, IIFS is more powerful to express the uncertainties and imprecision in evaluating the hydrogen technologies which are the focus of this paper.

The remainder of this paper is organized as follows. Section 2 reviews the basic concept of interval-valued intuitionistic fuzzy set and the operations for IIFNs. Section 3 presents some new interval-valued intuitionistic fuzzy aggregation operators and numeric examples are presented. Comparative studies of these operators with the interval-valued intuitionistic fuzzy aggregation operators proposed by Xu [9] are illustrated. In Section 4, we develop a decision making method for dealing with interval-valued intuitionistic fuzzy information and we apply this approach to evaluate the hydrogen production technologies in China and compare it with other methods. Conclusion and the future research directions are discussed in Section 5.

2. Some Basic Concepts

Intuitionistic fuzzy set (IFS) proposed by Atanassov [1] is characterized by the ability of defining the membership degree [[mu].sub.A](x) and nonmembership degree [v.sub.A](x) of an element to a set simultaneously, and the [[mu].sub.A](x) and [v.sub.A](x) are the real numbers belonging to a set [0, 1]. Interval-valued intuitionistic fuzzy set (IIFS), proposed by Atanassov and Gargov [2], can express the experts' preference information more effectively since it uses the interval number instead of real number to express the membership degree and nonmembership degree. The definition of the IIFS is shown as follows.

Definition 1 (Atanassov and Gargov [2]). Let a set X be fixed; the concept of interval-valued intuitionistic fuzzy set (IIFS) [??] on X is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[??].sup.A](x) and [[??].sub.A](v) are the degree ranges of membership and nonmembership and satisfy the following condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

For convenience, an IIFN a can be denoted by ([a, b], [c, d]), where

[a, b] [subset] [0,1], [c, d] [subset] [0,1], b + d [less than or equal to] 1. (3)

Definition 2 (Xu [9]). Let [[??].sub.1] = ([[a.sub.1], [b.sub.1]], [[c.sub.1], [d.sub.1]]) and [[??].sub.2] = ([[a.sub.2], [b.sub.2]], [[c.sub.2], [d.sub.2]]) be any two IIFNs; then some operations of [[??].sub.1] and [[??].sub.2] can be defined as

(1) [[??].sub.1] [direct sum] [[??].sub.2] = ([[a.sub.1] + [a.sub.2] - [a.sub.1] [a.sub.2], [b.sub.1] + [b.sub.2] - [b.sub.1][b.sub.2]], [[c.sub.1][c.sub.2], [d.sub.1][d.sub.2]]);

(2) [[??].sub.1] [cross product] [[??].sub.2] = ([[a.sub.1][a.sub.2], [b.sub.1][b.sub.2]], [[c.sub.1] + [c.sub.2] - [c.sub.1][c.sub.2], [d.sub.1] + [d.sub.2] - [d.sub.1][d.sub.2]]);

(3) [lambda][[??].sub.1] = ([1 - [(1 - [a.sub.1]).sup.[lambda]], 1 - [(1 - [b.sub.1]).sup.[lambda]]], [[c.sup.[lambda].sub.1], [d.sup.[lambda].sub.1]]), [lambda] > 0;

(4) [[??].sup.[lambda].sub.1] = ([[a.sup.[lambda].sub.1], [b.sup.[lambda].sub.1]], [1 - [(1 - [c.sub.1]).sup.[lambda]], 1 - [(1 - [d.sub.1]).sup.[lambda]]]), [lambda] > 0.

Xu [9] introduced the score function s([??]) = (1/2)(a - c + b - d) to get the score of [??] and defined an accuracy function h([??]) = (1/2)(a + b + c + d) to evaluate the accuracy degree of [??]. Xu [9] gave an order relation between two IVIFNs, [??] and [??].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

(i) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(ii) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It should be noted that Definition 2 and the comparing laws for any IIFNs proposed by Xu [9] have been used and cited widely [3, 23-28]. In the other words, they had produced main effect to the development of IIFS theory.

3. Interval-Valued Intuitionistic Fuzzy Interactive Aggregation Operators

3.1. The New Operations for IIFNs. Though the operations defined by Xu [9] have been used and cited widely, they still have some shortcomings. The following examples illustrated this phenomenon.

Example 3. Suppose [[??].sub.1] = <[0.2, 0.4], [0.0, 0.0]>, [[??].sub.2] = <[0.1, 0.2], [0.2, 0.4]>, [[??].sub.3] = <[0.0, 0.1], [0.5, 0.8]>, and [[??].sub.4] = <[0.1, 0.3], [0.4, 0.6]) are four IIFNs; then, using operation (1) defined in Definition 2, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Example 3 shows that the nonmembership degrees range of the sum of the two IIFNs is totally decided by the nonmembership degree range of [[??].sub.1] without any consideration of other IIFNs, which is not reasonable in reality.

Example 4. Suppose [[??].sub.1] = <[0.0, 0.0], [0.3, 0.5]>, [[??].sub.2] = <[0.3, 0.5], [0.4, 0.5]>, [[??].sub.3] = <[0.2, 0.7], [0.1, 0.2]>, and [[??].sub.4] = <[0.5, 0.9], [0.0,0.1]> are four IIFNs; then, using operation (2) defined in Definition 2, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Example 4 shows that the membership degrees range of the product of the two IIFNs is totally decided by the membership degree range of [[??].sub.1] without any consideration of other IIFNs, which is not workable.

The above analysis indicates that the definition of IIFNs introduced by Xu [9] could be improved to some extent, and we defined some new operations for IIFNs motivated by the idea of He et al. [15, 16].

Definition 5. Suppose [??] = ([a, b], [c, d]), [[??].sub.1] = ([[a.sub.1], [b.sub.1]], [[c.sub.1], [d.sub.1]), and [[??].sub.2] = ([[a.sub.2], [b.sub.2]], [[c.sub.2], [d.sub.2]]) are three IIFNs; some new operations were defined as follows:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(3) [lambda][??] = <[1 - [(1 - a).sup.[lambda]], 1 - [(1 - b).sup.[lambda]]], [[(1 - a).sup.[lambda]] - [(1 - (a + c)).sup.[lambda]], [(1 - b).sup.[lambda]] - [(1 - (b + d)).sup.[lambda]]>;

(4) [([??]).sup.[lambda]] = <[[(1 - c).sup.[lambda]] - [(1 - (a + c)).sup.[lambda]], [(1 - d).sup.[lambda]] - [(1 - (b + d)).sup.[lambda]]], [1 - [(1 - c).sup.[lambda]], 1 - [(1 - d).sup.[lambda]]]}.

Example 6. Suppose [[??].sub.1] = <[0.2, 0.4], [0.0, 0.0]>, [[??].sub.2] = <[0.1, 0.2], [0.2, 0.4]>, [[??].sub.3] = <[0.0, 0.1], [0.5, 0.8]>, and [[??].sub.4] = <[0.1, 0.3], [0.4, 0.6]> are four IIFNs; then, using operation (1) defined in Definition 5, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Example 6 shows that the drawbacks described in Example 3 disappeared.

Example 7. Suppose [[??].sub.1] = <[0.0, 0.0], [0.3, 0.5]>, [[??].sub.2] = <[0.3, 0.5], [0.4, 0.5]>, [[??].sub.3] = <[0.2, 0.7], [0.1, 0.2]>, and [[??].sub.4] = <[0.5, 0.9], [0.0, 0.1]> are four IIFNs; then, using the operation (2) defined in Definition 5, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Example 7 shows that the drawbacks described in Example 4 disappeared.

The new operational laws for IIFNs defined in Definition 5 satisfy Theorem 8.

Theorem 8. Suppose [??] = ([a, b], [c, d]), [[??].sub.1] = ([[a.sub.1], [b.sub.1]], [[c.sub.1], [d.sub.1]]), and [[??].sub.2] = ([[a.sub.2], [b.sub.2]], [[c.sub.2], [d.sub.2]]) are three IIFNs; the following equations are valid:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The proof of Theorem 8 is very simple, omitted here. []

3.2. Interval-Valued Intuitionistic Fuzzy Interactive Aggregation Operators. In Section 3.1, we have introduced the new operations for IIFNs based on the analysis of the imperfections of the existing operations. The main advantage of the new operations is that it can handle the extreme cases better such as the nonmembership degree range or the membership degree range reduced to the [0, 0] . Furthermore, the new aggregation operators for IIFNs also need to be addressed. Therefore, we proposed a series of interaction interval-valued intuitionistic fuzzy aggregation operators for aggregating the IIFNs. The comparisons with the existing operators are also presented.

Definition 9. Suppose ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a group of IIFNs and w = ([w.sub.1], [w.sub.2], ..., [w.sub.n]) is the weight vector of them, such that [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1] [w.sub.j] = 1. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

is named an interval-valued intuitionistic fuzzy interactive weighted average (IIFIWA) operator.

Theorem 10. Suppose ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a group of IIFNs; then their aggregated value by using IIFIWA operator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] is the weight vector of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) with [w.sub.j] [member of] [0,1] and [[summation].sup.n.sub.j=1] [w.sub.j] = 1.

Proof. The mathematical induction method is applied to prove (9).

(1) When n = 2, according to the Definition 5, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

This portrays (9) is valid when n = 2.

(2) If (8) holds for n = k, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

then, when n = k + 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

In other words, (9) is valid when n = k + 1. Therefore, (9) is valid for all n. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

It should be noted that the above proof is largely inspired by the idea of Zhao et al. [29] and He et al. [15,16]. []

Example 11 shows the application of Theorem 10 in IIFNs aggregation problem.

Example 11. Let [[??].sub.1] = <[0.1, 0.2], [0.0, 0.0]>, [[??].sub.2] = <[0.2,0.4], [0.2, 0.4]>, [[??].sub.3] = <[0.2, 0.3], [0.4, 0.7]>, and [[??].sub.4] = <[0.1.0.4], [0.4, 0.6]> be four IIFNs and w = (0.14, 0.36, 0.32, 0.18) their weight. Use the IIFWA operator [9] to aggregate the four IIFNs; the result can be obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The aggregated result based on the IIFWA operator is <[0.1693,0.3438], [0,0]>, and the nonmembership degree range is [0, 0] which is totally determined by the nonmembership degree of IIFN [[??].sub.1]. Obviously, it is unreasonable.

Utilize the IIFIWA operator (Definition 9 and Theorem 10); the aggregated result is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The aggregated result based on the IIFIWA operator is <[0.1693, 0.3438], [0.1750, 0.4799]>, and the nonmembership degree range is [0.1750, 0.4799] which is not totally determined by the nonmembership degree of one of the single IIFNs. Obviously, the result is more reasonable than the result obtained by IIFWA operator.

Definition 12. Suppose ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a group of IIFNs and w = ([w.sub.1], [w.sub.2], ..., [w.sub.n]) is the weight vector of them, such that [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1] [w.sub.j] = 1. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

is named an interval-valued intuitionistic fuzzy interactive weighted geometric (IIFIWG) operator.

Theorem 13. Suppose ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a group of IIFNs; then their aggregated value by using IIFIWG operator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] is the weight vector of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) with [w.sub.j] [member of] [0,1] and [[summation].sup.n.sub.j=1][w.sub.j] = 1.

Like Example 11, here we illustrate Example 14 to show the application of IIFIWG operator in aggregating the IIFNs.

Example 14. Let [[??].sub.1] = <[0.0,0.0], [0.2,0.3]>, [[??].sub.2] = <[0.1,0.3], [0.5, 0.6]>, [[??].sub.3] = <[0.2, 0.5], [0.1,0.4]>, and [[??].sub.4] = <[0.2, 0.3], [0.2, 0.6]> be four IIFNs and w = (0.19, 0.23, 0.34, 0.24) their weight. Use the IIFWG operator [9] to aggregate the four IIFNs; the result can be obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

From Example 14, we can find out that the aggregated result based on the IIFWG operator is <[0,0], [0.2526, 0.4894]> and the membership degree range is [0,0] which is totally determined by the membership degree of IIFN [[??].sub.1]. This was obviously an unreasonable calculated result.

Based on the IIFIWG operator (Definition 12 and Theorem 13), the aggregated result is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Obviously, the membership degree range is [0.1390, 0.3659] rather than [0, 0] and was more reasonable than the result obtained by IIFWG operator.

3.3. Interval-Valued Intuitionistic Fuzzy Interactive Ordered Weighted Operator. In many real situations, the data should be ordered before application. In many sports events, such as gymnastics and diving, the biggest and the smallest evaluation results given by the experts should be deleted and the other evaluation results will be aggregated. In these situations, the evaluation results should be ordered. OWA operator, proposed by Yager [30], is a very useful aggregation technique to deal with this situation. The OWA operator has attracted the interest of many researchers [31-44]. In the following, based on the idea of OWA operator, we extended the IIFIWA and IIFIWG operators and proposed the IIFIOWA and IIFIOWG operators.

Definition 15. Suppose is a group of IIFNs expressed as ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]); the interval-valued intuitionistic fuzzy interactive ordered weighted average (IIFIOWA) operator and interval-valued intuitionistic fuzzy interactive ordered weighted geometric (IIFIOWG) operator are defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] is the associated weight vector such that [[omega].sub.j] [member of] [0,1] and [[summation].sup.n.sub.j=1] [[omega].sub.j] = 1. [delta]: (1, 2, ..., n) [right arrow](1, 2, ..., n), [[??].sub.[delta]](j)] is the jth largest of [[??].sub.j].

Theorem 16. Suppose ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a group of IIFNs; then their aggregated value by using IIFIOWA operator or IIFIOWG operator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] is the weight vector of ([[??].sub.1], [[??].sub.2], ..., [[??].sub.n]) with [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1] [w.sub.j] = 1.

Example 17. Let [[??].sub.1] = <[0.2, 0.3], [0.5, 0.6]>, [[??].sub.2] = <[0.5, 0.6], [0.2, 0.4]>, [[??].sub.3] = <[0.3, 0.4], [0.0, 0.0]>, and [[??].sub.4] = <[0.1, 0.2], [0.3, 0.5]> be four IIFNs and [omega] = (0.14, 0.36, 0.32, 0.18) be the associate weight of IIFIOWA and IIFIOWG operators. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Based on the IIFOWA operator proposed by Xu [9], we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

This aggregation result indicates that the nonmembership degree range of the [[??].sup.*] is determined by the IIFN [[??].sub.3].

Based on the IIFIOWA operator proposed in this paper, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Obviously, this result seems more reasonable.

Based on the IIFOWG operator proposed by Xu [9], we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

This aggregation result indicates that the membership degree range of the [[??].sup.*] is determined by the IIFN [[??].sub.1].

Based on the IIFIOWG operator proposed in this paper, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Obviously, this result seems more reasonable.

4. Application of the Proposed

Operators to Evaluate the Hydrogen Production Technologies

With China's sustained and rapid economic and social development, energy resources, and increasing pressure on the environment, developing light pollution and renewable energy is of great significance to China's sustainable development. Hydrogen is recognized as clean energy, low carbon, and zero carbon energy source which has attracted wide attention in various countries [45-47]. Hydrogen technologies evaluation involves multiattribute decision making and many attribute should be evaluated, such as environment, economic, and social [48].

One high-tech development company in Zhejiang Province, China, intends to invest in the hydrogen energy production. Three kinds of hydrogen production technologies have been identified according to their own business situation and the famous energy expert's suggestions, such as nuclear based high temperature electrolysis technology (NHTET), electrolysis of water technology by hydropower, and coal gasification technology, expressed by [A.sub.1], [A.sub.2], and [A.sub.3]. The company wants to find out the most suitable technique from the three alternatives mainly according to environment performance [C.sub.1], economic performance [C.sub.2], social performance [C.sub.3], and the support degree of government policies [C.sub.4]. Meanwhile, the four attributes have different importance weight and could be determined by many effective methods, such as AHP Here we suppose the weight of the four attributes is [(0.14, 0.36, 0.32, 0.18).sup.T]. The performance of the three alternatives on the four attributes is expressed by IIFNs and is shown in Table 1.

First, we use the IIFIWA operator to aggregate the performance of the four attributes for three kinds of hydrogen production technologies, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Next, according to the scores function of IIFNs given in Section 2, the scores s([[??].sub.i]) (1 = 1,2,3) can be calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

then

[A.sub.2] [??] [A.sub.3] [??] [A.sub.1]. (31)

Therefore, the most suitable hydrogen production technology is [A.sub.2].

4.1. Systematic Comparison with Other Research Results. Based on the IIFWA operator proposed by Xu [9], the result is inconsistent with the method in this paper

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Next, according to the scores function of IIFNs given in Section 2, the scores s([[??]'.sub.i]) (i = 1,2, 3) can be calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

then

[A.sub.1] [??] [A.sub.2] [??] [A.sub.3]. (35)

Therefore, the most suitable alternative is [A.sub.1].

The optimal selection of two different methods is changed. From the above research results, we can find that the most suitable alternative is [A.sub.1] when the IIFWA operator is selected and the most suitable alternative is [A.sub.2] when the IIFIWA operator is involved.

Table 2 showed the detailed comparison of [A.sub.1] and [A.sub.2]. From Table 2, we can easily find that the membership degree range of [A.sub.1] is worse than [A.sub.2] regarding the four attributes. Meanwhile, three of the four nonmembership degree ranges of [A.sub.1] are bigger than [A.sub.2] regarding the four attributes. Therefore, it is hard to accept the result that [A.sub.1] is better than [A.sub.2]. The main reason for this result is that the nonmembership degree range of [A.sub.1] regarding criteria [c.sub.1] is [0, 0]. This indicates that the IIFWA operator proposed by Xu [9] is too sensitive to the situation where the nonmembership degree is reduced to [0, 0].

5. Concluding Remarks and Future Works

In this paper, we have introduced some new aggregation operators for aggregating IIFNs, based on which a new MADM method has been proposed. Furthermore, we have used the MADM method to solve the problem of the evaluation of hydrogen production technologies. In order to find the effectiveness and superiority of the MADM method, we compared it with some existing methods. The MADM method proposed in this paper is meaningful because it can be used to solve some actual evaluation problems. However, just like all the existing MADM method, the MADM method proposed in this paper cannot be applied to deal with all decision making problems. Our method can be adapted from many aspects, such as considering the interconnection between the attributes. From the author's point of view, the future research should be the application of the MADM proposed in this paper with some necessary modifications, which is more suitable for concrete research problems.

http://dx.doi.org/10.1155/2014/751249

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 71301142) and Zhejiang Natural Science Foundation of China (no. LQ13G010004).

References

[1] K. T. Atanassov, "Intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87-96, 1986.

[2] K. Atanassov and G. Gargov, "Interval valued intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343-349, 1989.

[3] P. D. Liu, "Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making," IEEE Transactions on Fuzzy Systems, vol. 22, no. 1, pp. 83-97, 2014.

[4] L.-L. Wang, D.-F. Li, and S.-S. Zhang, "Mathematical programming methodology for multiattribute decision making using interval-valued intuitionistic fuzzy sets," Journal of Intelligent & Fuzzy Systems, vol. 24, no. 4, pp. 755-763, 2013.

[5] X. W. Qi, C. Y. Liang, and J. L. Zhang, "Some generalized dependent aggregation operators with interval-valued intuitionistic fuzzy information and their application to exploitation investment evaluation," Journal of Applied Mathematics, vol. 2013, Article ID 705159, 24 pages, 2013.

[6] Z. M. Zhang, "Interval-valued intuitionistic hesitant fuzzy aggregation operators and their application in group decision-making," Journal of Applied Mathematics, vol. 2013, Article ID 670285, 33 pages, 2013.

[7] D. Yu, Y. Wu, and T. Lu, "Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making," Knowledge-Based Systems, vol. 30, no. 6, pp. 57-66, 2012.

[8] D. J. Yu, J. M. Merigo, and L. G. Zhou, "Interval-valued multiplicative intuitionistic fuzzy preference relations," International Journal of Fuzzy Systems, vol. 15, no. 4, pp. 412-422, 2013.

[9] Z.-S. Xu, "Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making," Control and Decision, vol. 22, no. 2, pp. 215-219, 2007.

[10] D. Yu, "Decision making based on generalized geometric operator under interval-valued intuitionistic fuzzy environment," Journal of Intelligent & Fuzzy Systems, vol. 25, no. 2, pp. 471-480, 2013.

[11] Y. R. Yang and S. Yuan, "Induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator and their application to multiple attribute decision making," Journal of Intelligent and Fuzzy Systems. In press.

[12] S. Xiao, "Induced interval-valued intuitionistic fuzzy Hamacher ordered weighted geometric operator and their application to multiple attribute decision making," Journal of Intelligent and Fuzzy Systems. In press.

[13] W. Wang and X. Liu, "The multi-attribute decision making method based on interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric operator," Computers & Mathematics with Applications, vol. 66, no. 10, pp. 1845-1856, 2013.

[14] F. Y. Meng, C. Q. Tan, and Q. Zhang, "The induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging operator and its application in decision making," Knowledge-Based Systems, vol. 42, pp. 9-19, 2013.

[15] Y. D. He, H. Y. Chen, L. G. Zhou, J. P. Liu, and Z. F. Tao, "Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making," Information Sciences, vol. 259, pp. 142-159, 2014.

[16] Y. D. He, H. Y. Chen, L. G. Zhou, B. Han, Q. Y. Zhao, and J. P. Liu, "Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making," Expert Systems with Applications, vol. 41, no. 5, pp. 2484-2495, 2014.

[17] S. K. Lee, G. Mogi, Z. Li et al., "Measuring the relative efficiency of hydrogen energy technologies for implementing the hydrogen economy: an integrated fuzzy AHP/DEA approach," International Journal of Hydrogen Energy, vol. 36, no. 20, pp. 12655-12663, 2011.

[18] J. Z. Ren, A. Manzardo, S. Toniolo, and A. Scipioni, "Sustainability of hydrogen supply chain. Part II: prioritizing and classifying the sustainability of hydrogen supply chains based on the combination of extension theory and AHP," International Journal of Hydrogen Energy, vol. 38, no. 32, pp. 13845-13855, 2013.

[19] N. H. Afgan, A. Veziroglu, and M. G. Carvalho, "Multi-criteria evaluation of hydrogen system options," International Journal of Hydrogen Energy, vol. 32, no. 15, pp. 3183-3193, 2007.

[20] W. McDowall and M. Eames, "Towards a sustainable hydrogen economy: a multi-criteria sustainability appraisal of competing hydrogen futures," International Journal of Hydrogen Energy, vol. 32, no. 18, pp. 4611-4626, 2007.

[21] J. Z. Ren, A. Fedele, M. Mason, A. Manzardo, and A. Scipioni, "Fuzzy multi-actor multi-criteria decision making for sustainability assessment of biomass-based technologies for hydrogen production," International Journal of Hydrogen Energy, vol. 38, no. 22, pp. 9111-9120, 2013.

[22] S. K. Lee, G. Mogi, and K. S. Hui, "A fuzzy analytic hierarchy process (AHP)/data envelopment analysis (DEA) hybrid model for efficiently allocating energy R&D resources: in the case of energy technologies against high oil prices," Renewable and Sustainable Energy Reviews, vol. 21, pp. 347-355, 2013.

[23] X. Zhang and P. Liu, "Method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making," Technological and Economic Development of Economy, vol. 16, no. 2, pp. 280-290, 2010.

[24] Z. Xu and Q. Chen, "A multi-criteria decision making procedure based on interval-valued intuitionistic fuzzy bonferroni means," Journal of Systems Science and Systems Engineering, vol. 20, no. 2, pp. 217-228, 2011.

[25] Z. Xu, "A method based on distance measure for interval-valued intuitionistic fuzzy group decision making," Information Sciences, vol. 180, no. 1, pp. 181-190, 2010.

[26] T.-Y. Chen, "Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis," Information Sciences, vol. 261, no. 10, pp. 149-169, 2014.

[27] Z. L. Yue and Y. Y. Jia, "A method to aggregate crisp values into interval-valued intuitionistic fuzzy information for group decision making," Applied Soft Computing, vol. 13, no. 5, pp. 2304-2317, 2013.

[28] J. Wu and Y. J. Liu, "An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers," Computers & Industrial Engineering, vol. 66, no. 2, pp. 311-324, 2013.

[29] H. Zhao, Z. Xu, M. Ni, and S. Liu, "Generalized aggregation operators for intuitionistic fuzzy sets," International Journal of Intelligent Systems, vol. 25, no. 1, pp. 1-30, 2010.

[30] R. R. Yager, "On ordered weighted averaging aggregation operators in multi-criteria decision making," IEEE Transactions on Systems, Man, and Cybernetics, vol. 18, no. 1, pp. 183-190, 1988.

[31] R. R. Yager, "Generalized OWA aggregation operators," Fuzzy Optimization and Decision Making, vol. 3, no. 1, pp. 93-107, 2004.

[32] R. R. Yager, "Prioritized aggregation operators," International Journal of Approximate Reasoning, vol. 48, no. 1, pp. 263-274, 2008.

[33] R. R. Yager, "Norms induced from oWA operators," IEEE Transactions on Fuzzy Systems, vol. 18, no. 1, pp. 57-66, 2010.

[34] R. R. Yager and G. Beliakov, "OWA operators in regression problems," IEEE Transactions on Fuzzy Systems, vol. 18, no. 1, pp. 106-113, 2010.

[35] V Torra, "The weighted OWA operator," International Journal of Intelligent Systems, vol. 12, no. 2, pp. 153-166, 1997.

[36] J. M. Merigoo, M. Casanovas, and J.-B. Yang, "Group decision making with expertons and uncertain generalized probabilistic weighted aggregation operators," European Journal of Operational Research, vol. 235, no. 1, pp. 215-224, 2014.

[37] X. Sang and X. Liu, "An analytic approach to obtain the least square deviation OWA operator weights," Fuzzy Sets and Systems, vol. 240, pp. 103-116, 2014.

[38] W. Yang and Y. F. Pang, "The quasi-arithmetic triangular fuzzy OWA operator based on Dempster-Shafer theory," Journal of Intelligent and Fuzzy Systems, vol. 26, no. 3, pp. 1123-1135, 2014.

[39] W. Zhou and J.-M. He, "Intuitionistic fuzzy normalized weighted Bonferroni mean and its application in multicriteria decision making," Journal of Applied Mathematics, vol. 2012, Article ID 136254, 22 pages, 2012.

[40] J. M. Merigo and M. Casanovas, "The uncertain induced quasiarithmetic OWA operator," International Journal of Intelligent Systems, vol. 26, no. 1, pp. 1-24, 2011.

[41] J. M. Merigo and A. M. Gil-Lafuente, "Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making," Information Sciences, vol. 236, pp. 1-16, 2013.

[42] J. M. Merigo and R. R. Yager, "Generalized moving averages, distance measures and OWA operators," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 21, no. 4, pp. 533-559, 2013.

[43] S. Z. Zeng, J. M. Merigo, and W. H. Su, "Linguistic induced generalized aggregation distance operators and their application to decision making," Economic Computer and Economic Cybernetics Studies and Research, vol. 46, no. 2, pp. 155-172, 2012.

[44] S. Zeng, J. M. Merigo, and W. Su, "The uncertain probabilistic OWA distance operator and its application in group decision making," Applied Mathematical Modelling, vol. 37, no. 9, pp. 6266-6275, 2013.

[45] P.-L. Chang, C.-W. Hsu, and P.-C. Chang, "Fuzzy Delphi method for evaluating hydrogen production technologies," International Journal of Hydrogen Energy, vol. 36, no. 21, pp. 14172-14179, 2011.

[46] E. Cetinkaya, I. Dincer, and G. F. Naterer, "Life cycle assessment of various hydrogen production methods," International Journal of Hydrogen Energy, vol. 37, no. 3, pp. 2071-2080, 2012.

[47] M. Smitkova, F. Janocek, and J. Riccardi, "Life cycle analysis of processes for hydrogen production," International Journal of Hydrogen Energy, vol. 36, no. 13, pp. 7844-7851, 2011.

[48] A. Manzardo, J. Z. Ren, A. Mazzi, and A. Scipioni, "A grey-based group decision-making methodology for the selection of hydrogen technologies in life cycle sustainability perspective," International Journal of Hydrogen Energy, vol. 37, no. 23, pp. 17663-17670, 2013.

Dejian Yu

School of Information, Zhejiang University of Finance and Economics, Hangzhou, Zhejiang 310018, China

Correspondence should be addressed to Dejian Yu; yudejian62@126.com

Received 2 December 2013; Accepted 1 March 2014; Published 3 April 2014

Academic Editor: Kai Diethelm

TABLE 1: The interval-valued intuitionistic fuzzy decision matrix [??]. [c.sub.1] [c.sub.2] [A.sub.1] ([0.1,0.2], [0.0,0.0]) ([0.1,0.3], [0.5,0.7]) [A.sub.2] ([0.2,0.3], [0.1,0.2]) ([0.3,0.4], [0.2,0.3]) [A.sub.3] ([0.3,0.5], [0.2,0.3]) ([0.1,0.3], [0.4,0.6]) [c.sub.3] [c.sub.4] [A.sub.1] ([0.1,0.2], [0.6,0.7]) ([0.0,0.0], [0.7,0.8]) [A.sub.2] ([0.2,0.3], [0.3,0.5]) ([0.1,0.2], [0.4,0.5]) [A.sub.3] ([0.3,0.4], [0.4,0.6]) ([0.2,0.4], [0.3,0.6]) TABLE 2: The detailed comparison of [A.sub.1] and [A.sub.2]. [c.sub.1] [c.sub.2] [A.sub.1] ([0.1, 0.2], [0.0, 0.0]) ([0.1, 0.3], [0.5, 0.7]) [A.sub.2] ([0.2, 0.3], [0.1, 0.2]) ([0.3, 0.4], [0.2, 0.3]) Membership [[a.sub.11], [b.sub.11]] < [[a.sub.12], [b.sub.12]] < degree range [[a.sub.21], [b.sub.21]] [[a.sub.22], [b.sub.22]] Nonmembership [[c.sub.11], [d.sub.11]] < [[c.sub.12], [d.sub.12]] > degree range [[c.sub.21], [d.sub.21]] [[c.sub.22], [d.sub.22]] [c.sub.3] [c.sub.4] [A.sub.1] ([0.1, 0.2], [0.6, 0.7]) ([0.0, 0.0], [0.7, 0.8]) [A.sub.2] ([0.2, 0.3], [0.3, 0.5]) ([0.1, 0.2], [0.4, 0.5]) Membership [[a.sub.13], [b.sub.13]] < [[a.sub.14], [b.sub.14]] < degree range [[a.sub.23], [b.sub.23]] [[a.sub.24], [d.sub.24]] Nonmembership [[c.sub.13], [d.sub.13]] > [[c.sub.14], [d.sub.14]] > degree range [[c.sub.23], [d.sub.23]] [[c.sub.24], [d.sub.24]]

Printer friendly Cite/link Email Feedback | |

Title Annotation: | Research Article |
---|---|

Author: | Yu, Dejian |

Publication: | Journal of Applied Mathematics |

Article Type: | Report |

Date: | Jan 1, 2014 |

Words: | 5928 |

Previous Article: | Time-dependent global attractor for a class of nonclassical parabolic equations. |

Next Article: | Constructing core backbone grid based on the index system of power grid survivability and BBO algorithm. |

Topics: |