Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making

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Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making Shu-Ping Wan a, Jun Xu b,∗, Jiu-Ying Dong c,d a

College of Information Technology, Jiangxi University of Finance and Economics, Nanchang, 330013, China College of Modern Economics & Management, Jiangxi University of Finance and Economics, Nanchang, 330013, China School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China d Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China b c

a r t i c l e

i n f o

Article history: Received 3 August 2015 Revised 7 August 2016 Accepted 27 September 2016 Available online xxx Keywords: Multi-attribute group decision making Heterogeneous information Aggregation technique Interval-valued intuitionistic fuzzy numbers Intuitionistic fuzzy programming model

a b s t r a c t Multi-attribute group decision making (MAGDM) has attracted more and more attention in many fields. Correspondingly, a number of usable methods have been proposed for various MAGDM problems, nevertheless, very few research focus on the aggregation techniques of intuitionistic fuzzy information. The aim of this paper is to aggregate decision information into interval-valued intuitionistic fuzzy numbers (IVIFNs) to solve heterogeneous MAGDM problem in which the decision information involves real numbers, interval numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs). There are three issues being addressed in this paper. The first is to propose a new general method to aggregate the attribute value vector into IVIFNs under heterogeneous MAGDM environment utilizing the relative closeness in technique for order preference by similarity to ideal solution (TOPSIS). The second is to construct a multiple objective intuitionistic fuzzy programming model to determine the attribute weights. Borrowing the results of the former two issues, the last is to present a new method to solve heterogeneous MAGDM problem. A comparison analysis with existing method is conducted to demonstrate the advantages of the proposed method. Two examples are provided to verify the practicality and effectiveness of the proposed method. © 2016 Elsevier B.V. All rights reserved.

1. Introduction As a generalization of Zadeh’s fuzzy sets [1], intuitionistic fuzzy (IF) set (IFS) [2] has better agility in expressing the uncertainty and ambiguity since it can be used to describe the characteristics of affirmation, negation and hesitation simultaneously. However, since sufficient information may be unavailable in practice, it is not easy to use the crisp values to express the membership and non-membership degrees of IFS. In such case, an interval may be a more suitable measurement to describe the vagueness. So, Atanassov and Gargov [3] proposed the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) by assigning membership and nonmembership degrees in the form of intervals rather than real numbers. Because of its advantages in describing uncertainty, many researchers have devoted to the theory of IVIFSs and its applications, such as aggregation operators [4–6], entropy measures [7,8], ranking functions [9,10], decision making [11–15], to name a few. ∗

Corresponding address. E-mail address: [email protected] (J. Xu).

In recent years, several scholars have presented some useful and valuable techniques for dealing MAGDM problems by aggregating attribute values into intuitionistic fuzzy number (IFN) and interval-valued intuitionistic fuzzy numbers (IVIFNs). Yue [16] and Yue et al. [17] employed Golden Section idea to aggregate crisp values into IFN. Yue et al. [18] proposed the method based on Minimax Criterion to aggregate crisp values into IFN for MAGDM. Yue [19] developed a new useful and practical method for aggregating crisp values into IFN using the idea of mean value. Later, Xu et al. [20] introduced a general aggregation method, which is a generalization of the method [19]. Yue [21] first defined the concepts of the attribute satisfactory interval and the attribute dissatisfactory interval, respectively, according to attribute values. Then, a method for aggregating the obtained attribute satisfactory interval and attribute dissatisfactory interval into an IVIFN was developed for MAGDM problems. Yue and Jia [22] presented a soft computing model in which it aggregates all individual decisions on an attribute into IVIFN for MAGDM whose attribute values is expressed by real numbers. By the idea of mean and standard deviation in statistics, Yue [23] introduced a straightforward and practical algo-

http://dx.doi.org/10.1016/j.knosys.2016.09.026 0950-7051/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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rithm to aggregate interval numbers into IVIFN. Although these aggregation methods [21–23] have some advantages, they suffer from some limitations: (1) The basic elements of an IVIFN are the interval membership degree, interval non-membership degree and interval hesitation degree. However, those methods [21–23] do not consider interval hesitation degree. (2) Existing aggregation methods [21–23] are just designed to deal with the MAGDM problems with real numbers or interval numbers, but cannot be used to solve heterogeneous MAGDM problems in which the attribute values may be triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs). (3) The weights of attributes are given by DMs in advance in those methods [21–23], which cannot avoid the subjectivity of giving the attribute weights. In fact, MAGDM often involves attribute values which may be described by multiple formats, such as real numbers, intervals, TFNs and TrFNs. Such a MAGDM with multiple types of information is called the heterogeneous MAGDM [24,25]. At present, several methods have been proposed to solve heterogeneous MAGDM [26–32]. Wan and Li [26] and Li and Wan [27] presented the extended fuzzy Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) [33] for solving heterogeneous multi-attribute decision making (MADM) problems. To evaluate emergency risk of community, Zhang et al. [28] developed a fuzzy MAGDM method, which can deal with both objective and subjective data. Wan and Li [29] put forward an intuitionistic fuzzy programming method for heterogeneous MAGDM with Atanassov’s intuitionistic fuzzy truth degrees. Further, considering the interval-valued intuitionistic fuzzy truth degrees of alternatives’ comparison, Wan and Li [30] developed interval-valued intuitionistic fuzzy LINMAP method for heterogeneous MADM; Wan and Dong [31] proposed an interval-valued intuitionistic fuzzy mathematical programming method for hybrid MCGDM. Zhang et al. [32] constructed some deviation models to solve heterogeneous MAGDM. It should be pointed out that the aforesaid methods employed the distance of fuzzy numbers to unify the heterogeneous information. During the process of unifying the heterogeneous information, much useful information may be lost. As mentioned previously, IVIFN has powerful ability to capturing uncertainty and ambiguity. It is more reasonable to convert heterogeneous information into IVIFNs in real-life heterogeneous MAGDM problems. For example, in an actual IT outsourcing service provider evaluation, the attributes might be better to be expressed by using multiple information representations, such as real numbers, interval numbers, TFNs and TrFNs, etc. However, these information representations characterize the fuzziness by membership function only, whereas IVIFSs with interval-valued membership and nonmembership functions are more flexible and abundant in expressing the imprecise or uncertain decision information. For instance, the collective ratings of the attributes product quality and flexibility, with respect to the alternative given by experts, are TFN r12 = (4.66, 6.34, 8.00) and TrFN r14 = (4.18, 5.12, 6.48, 8.02), respectively. However, the ratings for these attributes are divided into two parts: dissatisfaction degree and satisfaction degree, which just are the membership degree and non-membership degree of IVIFN, and there exist some hesitancy when experts evaluate these attributes. So, the attributes product quality and flexibility might be more suitable to be expressed by IVIFNs r12 = ([0.181, 0.283], [0.103, 0.152]) and r14 = ([0.195, 0.241], [0.125, 0.175]), respectively. Therefore, this paper attempts to convert the decision information into interval-valued intuitionistic information. Aggregating heterogeneous decision information into IVIFNs for heterogeneous MAGDM is of great importance for scientific re-

search and actual application. However, there exist two major difficulties and challenges to solving such heterogeneous MAGDM problems. The first is how to develop a new general method for aggregating heterogeneous decision information into IVIFNs. The second is how to determine the attribute weights objectively. In this paper, we develop a new general method for aggregating heterogeneous decision information into IVIFNs. The proposed general aggregating method involves three steps: (i) the extraction of the satisfactory element, dissatisfactory element and uncertain element of each attribute value using the concept of relative closeness of technique for order preference by similarity to ideal solution (TOPSIS) [34]; (ii) the construction of satisfactory interval, dissatisfactory interval and uncertain interval of each attribute according to the statistic method; (iii) the induction of the IVIFN of each attribute by considering membership degree, non-membership degree and hesitation degree. Then, to determine the weights of the attributes objectively, a new multiple objective IF programming model is constructed and transformed into linear program to solve. Thereby, a new method is proposed to solve the heterogeneous MAGDM problems. Compared with existing research, the primary features of the proposed method are illuminated as follows: (1) The proposed general aggregating method can aggregate different types of decision information into IVIFNs, including real numbers, interval numbers, TFNs and TrFNs, while existing methods [21–23] are just suitable to aggregate decision information of single types (i.e., real numbers or interval numbers). (2) The proposed general aggregating method considers all basic elements of IVIFNs in the aggregation process, thus the induced IVIFNs derived by the developed new linear transformation are more persuasive than those derived by the methods [21–23]. (3) A new multiple objective interval-valued IF programming model is constructed to determine objectively the weights of attributes, which avoids the subjective randomness of giving the weights in methods [21–23]. The structure of this paper is organized as follows: In Section 2, we briefly recalls some basic concepts on IVIFSs, interval numbers, TFNs and TrFNs. Section 3 describes the heterogeneous MAGDM problems and presents a general method to aggregate heterogeneous decision information into IVIFNs. In Section 4, the attribute weights are determined objectively by constructing a multiple objective IF programming. Thus, a new method is proposed to solve heterogeneous MAGDM problems. In Section 5, comparison analyses are made with existing methods. Section 6 provides two illustrative examples to verify the effectiveness of the proposed method. Section 7 makes our conclusions. 2. Some basic concepts In the following, we recall some basic notations of IVIFS, interval numbers [35], TFNs [36,37] and TrIFNs [38,39]. 2.1. Interval-valued intuitionistic fuzzy set Definition 1. [3]. Let X be a non-empty set of the universe. An interval-valued intuitionistic fuzzy set A˜ in X is defined as A˜ = {< x, μA˜ (x ), vA˜ (x ) > |x ∈ X }, where μA˜ (x ) = [μl˜ (x ), μh˜ (x )] : X → A

A

[0, 1] and vA˜ (x ) = [vl˜ (x ), vh˜ (x )] : X → [0, 1], such that μh˜ (x ) + A

A

A

vhA˜ (x ) ∈ [0, 1], for any x ∈ X. The interval numbers μA˜ (x ) and vA˜ (x ) indicate, respectively, the membership degree and non-

membership degree of the element x in A˜ . The third parameter πA˜ (x ) = [π l˜ (x ), π h˜ (x )], x ∈ X is called IVIFS index of x in A˜ , A

A

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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where π l˜ (x ) = 1 − μh˜ (x ) − vh˜ (x ) and π h˜ (x ) = 1 − μl˜ (x ) − vl˜ (x ). If A A A A A A μ ˜ (x ) = μl (x ) = μh (x ) and v ˜ (x ) = vl (x ) = vh (x ), then IVIFS A˜ is A

A˜

A

A˜

A˜

A˜

reduced to an intuitionistic fuzzy set [2].

Xu and Chen [4] called ([μl˜ (x ), μh˜ (x )], [vl˜ (x ), vh˜ (x )] ) an A A A A interval-valued intuitionistic fuzzy number (IVIFN). For simplicity, an IVIFN will be denoted by α˜ = ([μl , μh ], [vl , vh ] ), where

[μl , μh ] ⊆ [0, 1], [vl , vh ] ⊆ [0, 1],

μh + v h ≤ 1 .

(1)

Definition 2. [3]. Let α˜ 1 = ([μl1 , μh1 ], [vl1 , vh1 ] ) and ([μl2 , μh2 ], [vl2 , vh2 ] ) be two IVIFNs, the containment is:

α˜ 1 ⊆ α˜ 2 i f f μl1 ≤ μl2 , μh1 ≤ μh2 , vl1 ≥ vl2 and vh1 ≥ vh2 .

α˜ 2 = (2)

Also, the arithmetic operations are expressed as follows [19]:

([μl1

α˜ 1 + α˜ 2 = + vh1 vh2 ] ), (2) λα˜ = ([1 − (1 − μl )λ , 1 − (1 − μh )λ ], [(vl )λ , (vh )λ ] ) λ > 0. (1)

μl2

− μl1 μl2 , μh1

+ μh2

− μh1 μh2 ], [vl1 vl2 ,

Definition 3. [4]. For a set of IVIFNs α˜ j = ([μlj , μhj ], [vlj , vhj ] ) ( j = 1, 2, · · · , n ), an interval-valued intuitionistic fuzzy weighted averaging operator (IVIFWA) is defined as

IV IF WA(a˜1 , a˜2 , · · · , a˜n ) =

1−

n 

  wj

(1 − μhj )

j=1

,

n 



w j a˜ j =

1−

j=1 n 

n 

w

(vlj ) j ,

j=1

 wj

(vhj )

,

dT F N (a, b) =

where w = (w1 , w2 , , wn )T is an associated weight vector satn isfying wj ∈ [0, 1] and j=1 w j = 1. The aggregated value by the operator IVIFWA is also an IVIFN. Definition 4. [4] Let α˜ = ([μl , μh ], [vl , vh ] ) be an IVIFN, the score and accuracy degrees of α˜ can be defined by

S(α˜ ) =

1 l ( μ + μh − v l − v h ) , 2

(4)

H (α˜ ) =

1 l ( μ + μh + v l + v h ) , 2

(5)

(7)

(i) (ii) (iii) (iv)

0 ≤ dTFN (a, b) ≤ 1; dTFN (a, b) = 0 if and only if a = b; dTFN (a, b)=dTFN (b, a); If c = (c1 , c2 , c3 ) is any TFN, then dTFN (a, c) ≤ dTFN (a, b) + dTFN (b, c).

Proof. Obviously, the proposed distance Theorem 1. We need only to prove (iv).

meets

(i)-(iii)

of

It is easy to see that |a1 − c1 | ≤ |a1 − b1 | + |b1 − c1 |, 2|a2 − c2 | ≤ 2|a2 − b2 | + 2|b2 − c2 | and |a3 − c3 | ≤ |a3 − b3 | + |b3 − c3 |. So, we have 1 1 4 ( |a1 − c1 | + 2|a2 − c2 | + |a3 − c3 | ) ≤ 4 ( |a1 − b1 | + 2|a2 − b2 |+ |a3 1 − b3 | ) + 4 (|b1 − c1 | + 2|b2 − c2 | + |b3 − c3 | ). Therefore, dTFN (a, c) ≤ dTFN (a, b) + dTFN (b, c). Namely, the proposed distance satisfies (iv) of Theorem 1.

dIN (a, b) =

1 ( |a1 − b1 | + |a4 − b4 | ). 2

(8)

(2) If a1 = a2 = a3 = a4 and b1 = b2 = b3 = b4 , TrFNs a and b are reduced to real numbers a = a1 and b = b1 , dTrFN (a, b) is reduced to dRN (a, b):

dRN (a, b) = |a1 − b1 |.

(9)

3. Aggregating heterogeneous decision information into IVIFNs This section describes the heterogeneous MAGDM problem and then proposes a general method for aggregating heterogeneous decision information into IVIFNs. 3.1. Presentation of heterogeneous MAGDM problem

where S(α˜ ) ∈ [−1, 1], H (α˜ ) ∈ [0, 1]. and

α˜ 2 =

(1) If S(α˜ 1 ) > S(α˜ 2 ), then α˜ 1 > α˜ 2 . (2) If S(α˜ 1 ) = S(α˜ 2 ), then (i) If H (α˜ 1 ) > H (α˜ 2 ), then α˜ 1 > α˜ 2 . (ii) If H (α˜ 1 ) = H (α˜ 2 ), then α˜ 1 = α˜ 2 .

For convenience, denote M = {1, 2, , m}, N = {1, 2, , n}, and P = {1, 2, , p}. Let {Di |(i ∈ M)} be a group of DMs, Sk (k ∈ P) be a discrete set of alternatives, and Aj (j ∈ N) be a set of attributes. Since there are multiple formats of rating values, the attribute set A = {A1 , A2 , · · · , An } is divided into four subsets Aˆ 1 = {A1 , A2 , · · · , A j1 }, Aˆ 2 = {A j1 +1 , A j1 +2 , · · · , A j2 }, Aˆ 3 = {A j +1 , A j +2 , · · · , A j } and Aˆ 4 = {A j +1 , A j +2 , · · · , An }, where 2

2.2. Distance for TrFN, TFN, interval number and real number Definition 6. [38]. Let a = (a1 , a2 , a3 , a4 ) and b = (b1 , b2 , b3 , b4 ) be two TrFNs, and λ ≥ 0, then (1) a + b = (a1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 ), (2) λa = (λa1 , λa2 , λa3 , λa4 ).

2

1 ( |a1 − b1 | + |a2 − b2 | + |a3 − b3 | + |a4 − b4 | ) 4

(6)

Remark 1. Let dTFN ( •, •), dIN ( •, •) and dRN ( •, •) be the Hamming distances of TFNs, INs and RNs, respectively. Generally, a TrFN is regarded as the generalized form of real number, interval number and TFN. It is worth mentioning that:

3

3

3

1 ≤ j1 ≤ j2 ≤ j3 ≤ n, Aˆt ∩ Aˆ k = ∅ (t, k = 1, 2, 3, 4 ; t = k) and 4 ˆ t=1 At = A, ∅ is the empty set. The rating values in the subsets Aˆ e (e = 1, 2, 3, 4) are in the form of real numbers, interval num-

bers, TFNs and TrFNs, respectively. Denote the subscript sets for subsets Aˆ e (e = 1, 2, 3, 4)by N1 = {1, 2, , j1 }, N2 = {j1 + 1, j1 + 2, , j2 }, N3 = {j2 + 1, j2 + 2, , j3 } and N4 = {j3 + 1, j3 + 2, , n}, respectively. Thus, a group decision matrix of alternative Sk can be expressed as

The Hamming distance between two TrFNs a = (a1 , a2 , a3 , a4 ) and b = (b1 , b2 , b3 , b4 ) is defined as follows [39]:

dT rF N (a, b) =

1 ( |a1 − b1 | + 2|a2 − b2 | + |a4 − b4 | ) 4

Theorem 1. The Hamming distance dTFN (a, b) satisfies the following properties:

(3)

j=1

Definition 5. [4]. Let α˜ 1 = ([μl1 , μh1 ], [vl1 , vh1 ] ) ([μl2 , μh2 ], [vl2 , vh2 ] ) be two IVIFNs, then:

(1) If a2 = a3 and b2 = b3 , TrFNs a and b are reduced to TFNs a = (a1 , a2 , a4 ) and b = (b1 , b2 , b4 ), dTrFN (a, b), is reduced to dTFN (a, b):

(1) If a1 = a2 , a3 = a4 , b1 = b2 and b3 = b4 , TrFNs a and b are reduced to interval numbers a = [a1 , a4 ] and b = [b1 , b4 ], dTrFN (a, b) is reduced to dIN (a, b):

w

(1 − μlj ) j ,

j=1

n 

3

X k = (xki j )m×n

A1 D1 ⎛ k x11 D2 xk21 = . ⎜ .. ⎜ . ⎝ .. Dm xkm1

A2 xk12 xk22 .. . xkm2

··· ··· ··· .. . ···

An ⎞ xk1n xk2n ⎟ .. ⎟ ⎠ . k xmn

(k ∈ P )

(10)

where xki j is attribute value of the alternative Sk on attribute Aj given by DM Di .

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Fig. 1. Process for aggregating IVIFN.

Denote the attribute weight vector by w = (w1 , w2 , , wn ), where wj represents the weight of attrubite Aj such that wj ∈ [0, 1]  (j ∈ N) and nj=1 w j = 1. Practically, DMs may specify some preference relations on weights of attributes according to his/her knowledge, experience and judgment [12]. Thus, the attribute weights are incomplete. Let  be the set of incomplete weights. Generally,  may consist of several basic forms [26–29]:

3.2.1. Compute the Qsd, Qdd and Qud Let xki j be the rating of alternative Sk on attribute Aj given by DM Di . Due to the complexity, fuzziness and uncertainty inherent in the evaluated attributes, identifying the degrees of satisfactory, dissatisfactory and uncertainty implied in xki j is very difficult. Note

A weak ranking: {wi ≥ wj }; A strict ranking: {wi − wj ≥ ε i } (ε i > 0); An interval form: {ε i ≤ wi ≤ ε i + ϕ i } (0 ≤ ε i ≤ ε i + ϕ i ≤ 1); A ranking of differences: {wi − wj ≥ wk − wl }, for j = k = l; A ranking with multiples: {wi ≥ ε i wj } (0 ≤ ε i ≤ 1).

faction; (2) the distance from xki j to Amin is less than the distance j

1) 2) 3) 4) 5)

According to Yue [21–23], we integrate the decision matrices X k = (xki j )m×n (k = 1, 2, · · · , p) into a collective decision matrix with IVIFNs to derive the ranking order of alternatives.

from xki j to Amax , which shows that the DM prefers dissatisfaction; j and (3) the distance from xki j to Amin is equal to the distance from j xki j to Amax , which shows that the DM maintains neutrality. j Based on the above analysis, some notations are introduced to calculate satisfactory degree, dissatisfactory degree and uncertain degree of each element in Akj .

element in Akj . The Qsd, Qdd and Qud of xki j are defined as

For the sake of calculation simplicity, the j-th column vector in is rewritten as

Ak j = (xk1 j , xk2 j , · · · , xkm j ) (k ∈ P, j ∈ N ),

distance from xki j to Amax , which shows that the DM prefers satisj

Definition 11. Let Akj be a benefit attribute vector, xki j be arbitrary

3.2. A general method for aggregating heterogeneous decision information into IVIFNs

Xk

the facts that: (1) the distance from xki j to Amin is greater than the j

(11)

which is the assessment vector of alternative Sk on attribute Aj id and given by all DMs Di (i = 1, 2, …, m). Suppose that Amax , Am j j Amin , respectively, are the largest grade, the middle grade and the j smallest grade employed in rating system (for example, in the 10point scale system for the ratings of real numbers, if attribute Aj id = 5 and Amax = 10; if atis benefit attribute, then Amin = 0, Am j j j id = 5 and Amax = 0). tribute Aj is cost attribute, then Amin = 10, Am j j j

It is easy to see that the elements of Akj are always between Amin j and Amax . j

To integrate the decision matrices X k = (xki j )m×n (k = 1, 2, · · · , p) into a collective IVIF decision matrix, all the elements in vector Akj need to be aggregated into an IVIFN. In what follows, some definitions including the Quasi-satisfactory degree (Qsd), Quasi-dissatisfactory degree (Qdd) and Quasi-uncertain degree (Qud) of xki j are introduced. Then, using the idea of mean and standard deviation, we construct the Quasi-satisfactory interval (Qsi), Quasi-dissatisfactory interval (Qdi) and Quasi-uncertain interval (Qui) of Akj . Finally, the Qsi, Qdi and Qui are used to generate an IVIFN by the proposed normalized method. The aggregating process is depicted in Fig. 1.

ξikj =

d (xki j , Amin ) j d(

xki j , Amax j

) + d (xki j , Amin ) j

,

(12)

ζikj = 1 − ξikj , ⎧ d (Amax , xki j ) ⎪ j ⎪ ⎪ , ⎪ k k max mid ⎪ ⎪ ⎨ d ( A j , xi j ) + d ( xi j , A j ) d (xki j , Amin ) ηikj = j ⎪ , ⎪ id k min ⎪ d (xi j , A j ) + d (xki j , Am ) ⎪ j ⎪ ⎪ ⎩1,

(13)

if d (xki j , Amax ) < d (xki j , Amin ) j j if d (xki j , Amax ) > d (xki j , Amin ) j j

,

if d (xki j , Amax ) = d (xki j , Amin ) j j (14)

respectively, where d (xki j , Amax ) is the distance between xki j and the j

d ) is the distance belargest grade Amax of attribute A j , d (xki j , Ami j j

d of attribute A , d (xk , Amin ) is tween xki j and the middle grade Ami j j j ij

the distance between xki j and the smallest grade Amin of attribute j Aj . Remark 2. For the benefit attribute vector Ak j = (xk1 j , xk2 j , · · · , xkm j ), the Quasi-satisfactory set (Qss) ξ kj , Quasi-dissatisfactory set (Qds) ζ kj and Quasi-uncertain set (Qus) ηkj of Akj are de-

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S.-P. Wan et al. / Knowledge-Based Systems 000 (2016) 1–16 k }, ζ = {ζ , ζ , , ζ } and η = fined as ξk j = {ξ1kj , ξ2kj , · · · , ξm kj 1j 2j mj kj j

{η1k j , η2k j , · · ·

k }. , ηm j

μlk j = ξkl j /ψk j , μhk j = ξkhj /ψk j , vlk j = ζkl j /ψk j , vhk j =

3.2.2. Calculate Qsi, Qdi and Qui Note that the membership degree and non-membership degree of an IVIFN are intervals rather than real numbers. Moreover, ξikj , ζikj and ηikj are the Qsd, Qdd and Qud of xki j , which is one el-

5

ζkhj /ψk j , (k ∈ P, j ∈ N ),

where ψk j =

ξkl j

+

ξkhj

+ ζkl j

+ ζkhj

(19) +

1 l 2 ( ηk j

+

ηkh j ).

ement in Akj . In such cases, the ranges of ξikj , ζikj and ηikj may be more appropriate measurements of Qsd, Qdd and Qud of Akj , respectively. For these reasons, it is necessary to construct the Qsi, Qdi and Qui of Akj .

Apparently, μlk j , μhk j , vlk j and vhk j satisfy Eq. (1). Let us turn to the aforementioned Example 1. By Eq. (19), A and B can be converted into A’ = ([0.389, 0.425], [0.067, 0.083]) and B’ = ([0.356, 0.436], [0.053, 0.090]). Then, we have s(A’) = 0.332 and s(B’) = 0.324. So, A’ > B’, which is consistent with A > B. Therefore, the new transformations of Eq. (19) can overcome the drawback of Eq. (18).

Definition 12. For the benefit attribute vector Akj , the Qsi ξ˜k j , Qdi ζ˜k j and Qui η˜ k j of alternative Sk on attribute Aj are defined as

3.3. Concrete computation formulas for aggregating heterogeneous information into IVIFNs

ξ˜k j = [ξkl j , ξkhj ] = [max(m(ξk j ) − d (ξk j ), 0 ), m(ξk j ) + d (ξk j )] (k ∈ P, j ∈ N ), (15)

Without loss of generality, suppose that all the attributes are benefit attributes in this subsection.

ζ˜k j = [ζkl j , ζkhj ] = [max(m(ζk j ) − d (ζk j ), 0 ), m(ζk j ) + d (ζk j )] (k ∈ P, j ∈ N ), (16) η˜ k j = [ηkl j , ηkh j ] = [max(m(ηk j ) − d (ηk j ), 0 ), m(ηk j ) + d (ηk j )] (k ∈ P, j ∈ N ), (17)

1 m k respectively, where m ( ξk j ) = m i=1 ξi j , m (ζk j ) = 1 − m (ξk j ),  2 1 m 1 m k k m ( ηk j ) = m i=1 ηi j , d (ξk j ) = i=1 (ξi j − m (ξk j )) , d (ξk j ) = m−1

 

1 m−1

m

i=1

m

2

(ζikj − m(ζk j ))

and

d ( ξk j ) =

are TrFNs). Let Amin = (0, 0, 0, 0 ), Amax = (x+j , x+j , x+j , x+j ) ( j ∈ N4 ) j j where x+j means the largest grade for the attribute A j ∈ Aˆ 4 of TrFNs. Plugging dTrFN ( • ) in Eqs.(12)-(14), it yields that

ξikj =

μlk j = ξkl j /τk j , μhk j = ξkhj /τk j , vlk j = ζkl j /τk j , vhk j ζkhj /τk j , (k ∈ P, j ∈ N ),

(18)

where τk j = ξkl j + ξkhj + ζkl j + ζkhj . The linear transformations of Eq. (18) may cause inconsistency in some situations. Example 1. Let A = ([ξ1l , ξ1h ], [ζ1l , ζ1h ] ) = ([0.75, 0.82], [0.13, 0.16] ) and B = ([ξ2l , ξ2h ], [ζ2l , ζ2h ] ) = ([0.67, 0.82], [0.10, 0.17]), where A and B satisfy the conditions ξ1l + ζ1l ≤ 1 and ξ1h + ζ1h ≤ 1. By Eq. (3), we have s(A) = 0.64 and s(B) = 0.61. Thus, A > B. However, by Eq. (18), A and B can be converted into A’ = ([0.403, 0.441], [0.070, 0.086]) and B’ = ([0.381, 0.466], [0.057, 0.096]). Then, we have s(A’) = 0.344 and s(B’) = 0.347. So, A’ < B’, which is not consistent with A > B. An effective transformation method should consider not only the membership degree and non-membership degree of quasiIVIFN, but also the hesitation degree. However, the hesitation degree is ignored in Eq. (18). Here, some new linear transformations are developed as follows. Definition 13. Let αk j = ([μlk j , μhk j ], [vlk j , vhk j ] ) (k ∈ P, j ∈ N ) be the induced IVIFN by the attribute vector Akj . The values of μlk j , μhk j , vlk j and vhk j can be computed as follows:

aki j + bki j + cikj + dikj 4x+j

,

(20)

ζikj = 1 − ξikj ,

i=1

3.2.3. Induce an IVIFN Let αk j = ([μlk j , μhk j ], [vlk j , vhk j ] ) (k ∈ P, j ∈ N ) be the induced IVIFN by the attribute vector Akj . To satisfy the conditions in Eq. (1), Yue [21] constructed some linear transformations as follows:

=

tor in the form of TrFNs (i.e., xki j = (aki j , bki j , cikj , dikj ) (i = 1, 2, · · · , m )

2

(ηikj − m(ηk j )) represent the mean values and standard deviations of Qss ξ kj , Qds ζ kj and Qus ηkj , respectively. Here we call an ordered pair ([ξkl j , ξkhj ], [ζkl j , ζkhj ] ) a quasi-IVIFN. 1 m−1

3.3.1. For TrFNs If j ∈ N4 , then Ak j = (xk1 j , xk2 j , · · · , xkm j ) is an attribute value vec-

ηikj =

(21)

⎧ 4x+j −aki j −bki j −cikj −dikj ⎪ , 1 + 1 + 1 + 7 + ⎪ | a − x | + | b − ξk j 2 j ξk j 2 x j |+|cξk j − 2 x j |+ 2 x j −aξk j −bξk j −cξk j ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ if aki j + bki j + cikj + dikj > 2x j ⎨ aξ +bξ +cξ +dikj kj kj kj

, ⎪ |bζk j − 12 x+j |+|cζk j − 12 x+j |+|dζk j − 12 x+j |+bζk j +cζk j +dζk j + 12 x+j ⎪ ⎪ ⎪ ⎪ ⎪ if aki j + bki j + cikj + dikj < 2x+j ⎪ ⎪ ⎩ + k k k k 1,

.

(22)

if ai j + bi j + ci j + di j = 2x j

Subsequently, the Qsi, Qdi and Qui of alternative Sk on attribute Aj can be obtained by Eqs. (15–17). Let αk j = ([μlk j , μhk j ], [vlk j , vhk j ] ) (k ∈ P, j ∈ N4 ) be the induced IVIFN by the attribute vector Akj , the μlk j , μhk j , vlk j and vhk j can be computed through Eq. (19).

Example 2. Let Akj = ((2, 3, 4, 5), (2, 5, 6, 8), (1, 3, 4, 7), (4, 5, 7, 8), (2, 3, 6, 8)) be a TrFN vector in the ten-mark system, then Amin = (0, 0, 0, 0 ), Amax = (10, 10, 10, 10 ). According to j j Eqs. (20–22), the Qss, Qds and Qus of Akj are calculated respectively as ξ kj = {0.35, 0.53, 0.38, 0.60, 0.48, }, ζ kj = {0.65, 0.47, 0.62, 0.40, 0.52} and ηkj = {0.70, 0.73, 0.63, 0.73, 0.68}. Subsequently, it follows from Eqs. (15–17) that ξ˜ = [0.36, 0.57], ζ˜ = [0.43, 0.64] kj

kj

and η˜ k j = [0.65, 0.74]. Then, the induced IVIFN α kj = ([0.134, 0.211], [0.160, 0.237]) can be derived by using Eq. (19). 3.3.2. For TFNs, interval numbers, real numbers As for TFNs, interval numbers, real numbers, considering that (1) they are the special cases of TrFNs; (2) their distances can be deduced from the distance of TrFNs; (3) their Qsd, Qdd and Qud mainly rely on the corresponding distance; (4) the calculation for Qsi, Qdi, Qui and final induced IVIFN are based on the Qsd, Qdd and Qud, the computation formulas for aggregating TrFNs into IVIFN can derive these for TFNs, interval numbers, real numbers:

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(1) If j ∈ N3 , then Ak j = (xk1 j , xk2 j , · · · , xkm j ) is an attribute value

vector in the form of TFNs (i.e., xki j = (aki j , bki j , dikj ) (i = 1, 2, , m) are TFNs), Amin = (0, 0, 0 ), Amax = (x+j , x+j , x+j ). According j j to Remark 1, plugging bki j = cikj in Eqs. (20–22), it yields that

ξikj =

aki j + 2bki j + dikj 4x+j

,

ζikj = 1 − ξikj ,

ηikj =

⎧ ⎪ ⎪ ⎪ ⎨ |aξ

4x+j −aξ −2bξ −dξ kj

kj

kj

1 1 7 − x+j |+2|bξ − x+j |+ x+j −aξ −2bξ 2 2 2 kj kj kj kj aξ +bξ +cξ kj

kj

kj

[0.54, 0.82], ζ˜k j = [0.18, 0.46] and η˜ k j = [0.40, 0.61]. Then, the induced IVIFN α kj = ([0.214, 0.329], [0.070, 0.185]) can be derived by using Eq. (19).

(23)

(1) If j ∈ N1 , then Ak j = (xk1 j , xk2 j , · · · , xkm j ) is an attribute value

(24)

, m) are real numbers), Amin = 0, Amax = x+j . According j j

vector in the form of real numbers (i.e., xki j = aki j (i = 1, 2, to Remark 1, plugging aki j = bki j = cikj = dikj in Eqs. (20–22), it yields that

, if aki j + 2bki j + dikj > 2x+j

1 1 1 , if ⎪ 2|bζ − x+ |+|dζ − x+j |+2bζ +dζ + x+j ⎪ 2 2 kj kj kj ⎪ ⎩ kj 2 j kj

0.45}, ζ kj = {0.30, 0.20, 0.35, 0.20, 0.55} and ηkj = {0.50, 0.40, 0.58, 0.40, 0.64}. Subsequently, from Eqs. (15–17) it follows that ξ˜ =

aki j

+

2bki j

+

dikj

<

ξikj =

2x+j .

(25) Likewise, by Eqs. (15–17) and (19), we can get the induced IVIFN. Example 3. Let Akj = ((3, 4, 7), (4, 6, 9), (2, 4, 5), (6, 8, 9), (3, 4, 5)) be a TFN vector in the ten-mark system, then Amin = j (0, 0, 0 ), Amax = ( 10 , 10 , 10 ) . Using Eqs. (23 –25 ), the Qss, Qds and j Qus of Akj are calculated respectively as ξ kj = {0.45, 0.63, 0.38, 0.78, 0.40}, ζ kj = {0.55, 0.37, 0.62, 0.22, 0.60} and ηkj = {0.75, 0.68, 0.75, 0.45, 0.80}. Subsequently, it follows from Eqs. (15–17) that ξ˜ =

x+j

,

(29)

ζikj = 1 − ξikj ,

if aki j + 2bki j + dikj = 2x+j

1,

aki j

ηikj =

(30)

⎧ + x j − aki j ⎪ ⎪ ⎪ , ⎪ 1 + ⎪ ⎪ ⎨ 2xj

if aki j > 12 x+j

ak

ij ⎪ , ⎪ 1 + ⎪ x ⎪ ⎪2 j ⎪ ⎩1,

if aki j < 12 x+j

.

(31)

if aki j = 12 x+j

Subsequently, by Eqs. (15–17) and (19), the induced IVIFN can be obtained.

kj

[0.35, 0.70], duced IVIFN applying Eq.

ζ˜k j = [0.30, 0.65] and η˜ k j = [0.55, 0.83]. Then, the inα kj = ([0.132, 0.259], [0.113, 0.240]) can be derived by (19).

(1) If j ∈ N2 , then Ak j = (xk1 j , xk2 j , · · · , xkm j ) is an attribute value vector in the form of interval numbers (i.e., xki j = [aki j , dikj ] (i = 1, 2, · · · , m ) are interval numbers), Amin = j [0, 0], Amax = [x+j , x+j ]. According to Remark 1, j bki j and cikj = dikj in Eqs. (20–22), it yields that

ξikj =

aki j + dikj 2x+j

plugging

X = (rk j ) p×n

ηikj =

=

kj

kj

and η˜ k j = [0.46, 0.88]. Then, the induced IVIFN α kj = ([0.157, 0.292], [0.082, 0.217]) can be derived by using Eq. (19). 4. A novel approach for heterogeneous MAGDM problems

,

(26)

ζikj = 1 − ξikj , S1 = S2 .. . Sp

aki j

Example 5. Let Akj = (3.4, 5.4, 7.2, 8.1, 5.9) be a real number vector in the ten-mark system, then Amin = 0, Amax = 10. According to j j Eqs. (29–31), the Qss, Qds and Qus of Akj are calculated respectively as ξ kj = {0.34, 0.54, 0.72, 0.81, 0.59}, ζ kj = {0.66, 0.46, 0.28, 0.19, 0.41} and ηkj = {0.68, 0.92, 0.56, 0.38, 0.82}. Subsequently, it follows from Eqs. (15–17) that ξ˜ = [0.42, 0.78], ζ˜ = [0.22, 0.58]

(27)

⎛

A1

([μl11 , μh11 ], [vl11 , vh11 ] ) ( ⎜ [μl21 , μh21 ], [vl21 , vh21 ] ) ⎜ .. ⎝ .

([μlp1 , μhp1 ], [vlp1 , vhp1 ] )

⎧ 2x+j − aki j − dikj ⎪ ⎪ ⎪ , ⎪ 3 1 + + k k ⎪ ⎪ ⎨ 2 x j + | 2 x j − ai j | − ai j ak + d k

ij ij ⎪ , ⎪ ⎪ dikj + 12 x+j + |dikj − 12 x+j | ⎪ ⎪ ⎪ ⎩1,

By the general aggregation method proposed in Section 3, we can aggregate the decision matrices X k = (xki j )m×n (k ∈ P) into a collective decision matrix with the induced IVIFNs, i.e.,

···

A2

([μl12 , μh12 ], [vl12 , vh12 ] ) ([μl22 , μh22 ], [vl22 , vh22 ] )

.. . ([μlp2 , μhp2 ], [vlp2 , vhp2 ] )

A

n ⎞ ([μl1n , μh1n ], [vl1n , vh1n ] ) l h l h ( [ μ2 n , μ2 n ] , [ v 2 n , v 2 n ] ) ⎟ , ⎟ .. ⎠ .

··· ··· .. . ···

(32)

([μlpn , μhpn ], [vlpn , vhpn ] )

where rk j = ([μlk j , μhk j ], [vlk j , vhk j ] ) (k ∈ P, j ∈ N) is an induced IVIFN

if aki j + dikj > x+j if aki j + dikj < x+j

.

(28)

if aki j + dikj = x+j

Subsequently, by Eqs. (15–17) and (19), the induced IVIFN can be obtained. Example 4. Let Akj = ([4, 10], [7, 9], [4, 9], [6, 10], [2, 7]) be a interval number vector in the ten-mark system, then Amin = [0, 0], j Amax = [10, 10]. According to Eqs. (26–28), the Qss, Qds and Qus j of Akj are calculated respectively as ξ kj = {0.70, 0.80, 0.65, 0.80,

aggregated by the attribute vector Akj in Xk . Once the attribute weight vector w = (w1 , w2 , , wn ) is completely known, the overall evaluation rk = ([μlk , μhk ], [vlk , vhk ] ) of the alternative Sk is computed using Eq. (3), where

μlk = 1 −

n 

w

(1 − μlk j ) j , μhk = 1 −

j=1

vlk =

n  j=1

n 

w

(1 − μhk j ) j ,

j=1 wj

(vlk j ) , vhk =

n 

wj

(vhk j ) .

(33)

j=1

Since the attribute weight vector w = (w1 , w2 , , wn ) is incompletely known, we develop a new IF programming model to determine the weights of attribute in the sequel.

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4.1. Construct an intuitionistic fuzzy programming model to determine the attribute weights A reasonable attribute weight vector w should make the collective overall rating rk = ([μlk , μhk ], [vlk , vhk ] ) of alternative Sk (k ∈ P) as large as possible. To this end, a multiple objective IF mathematical optimization model can be constructed to determine the attribute weights:

max{rk }(k ∈ P ) . s.t.w ∈ 

(34)

According to Definition 2, the bigger [μlk , μhk ] and the smaller

There are several methods to solve a multiple objective programming. Here, we apply membership function-based linear sum max and Z min (t = 1, 2, approach to solving Eq. (37). Suppose that Ztk tk 3, 4; k = 1, 2, , p) are the maximum and minimum values of the single objective function Ztk (t = 1, 2, 3, 4; k = 1, 2, …, p) ignoring the other objectives in Eq. (37). The corresponding linear membership function of the objective function Ztk is defined as follows:

⎧ ⎨1, μtk (w ) = (Z max − Ztk )/(Z max − Z min ), tk tk ⎩ tk 0,

[vlk , vhk ], the larger rk is. It is natural to maximize [μlk , μhk ] and

μlk

μhk

mized, while and are maximized. To solve the Eq. (34), it can be transformed into a multiple objective optimization model:

max{μlk }

(k ∈ P ) h max{μk } (k ∈ P ) min{vlk } (k ∈ P ) . min{vhk } (k ∈ P ) s.t.w ∈ 



max 1 −



n  j=1 n 

(35)

 wj

(1 − ulk j )



(k ∈ P )

(36)

j=1

s.t. w ∈  is worth mentioning that the objective max{1 −  w w (1 − μlk j ) j } may be equivalent to min{ nj=1 (1 − μlk j ) j }, n which is equivalent to min{ j=1 w j ln(1 − μlk j )} since 0 ≤ μlk j ≤ 1 and logarithm (ln) is an increasing function. LikeIt

j=1

 w wise, the objective max{1 − nj=1 (1 − μhk j ) j } is equivalent to   w min{ nj=1 w j ln(1 − μhk j )}. The objective min{ nj=1 (vlk j ) j } may n be equivalent to min{ j=1 w j ln(vlk j )} since 0 ≤ vlk j ≤ 1 and logarithm (ln) is an increasing function. Similarly, the objective   w min{ nj=1 (vhk j ) j } is equivalent to min{ nj=1 w j ln(vhk j )}. Thus, Eq. (36) can be converted into a multiple objective linear programming model:



min Z1k =

 min Z2k =

 min Z3k =

 min Z4k =

n 



w j ln(1 − μlk j )

j=1 n 

j=1 n 

s.t.w ∈ ,

(39)

where δ tk is the weighs of objective Ztk such that δ tk ∈ [0, 1] (t = 1, p 2, 3, 4; k = 1, 2, , p) and k=1 (δ1k + δ2k + δ3k + δ4k ) = 1. If the importance of different objectives is equal, the objective weights can take equal values, i.e., δtk = 41p (t = 1, 2, 3, 4; k = 1, 2, …, p). It is easily seen that Eq. (39) is a linear programming. Using the simplex method to solve Eq. (39), the attribute weight vector w can be derived. Remark 3. We establish a multi-objective IVIF programming model that maximizes the collective overall IVIF ratings of all alternatives to determine optimal weight of attributes. Moreover, this model is transformed into the equivalent linear programming model which is solved by using membership function-based linear sum approach. However, Jin et al [8] constructed a programming model that minimizes the entropy values of alternatives, Zhang and Xu [41] presented a multi-choice linear programming model that maximizing the closeness indices. Wan et al. [14] established a multi-objective interval programming model based on the comprehensive values of alternatives, Based on the above analysis, the major difference is that the proposed model is a multi-objective IVIF programming model in which the objective is an IVIFN, which is well-suited to express the hesitancy degree inherent in DMs’ judgments, while the objectives in methods [8,14,41] are crisp values or interval numbers which cannot reflect the hesitancy degrees inherent in DMs’ judgments. Thus, the proposed model in this paper may lead to much less information loss than that of methods [8,14,41] in the determination of the weights.

(k ∈ P )

w j ln(1 − μ )

(k ∈ P )

4.2. Algorithm for solving heterogeneous MAGDM

 w j ln(v ) l kj

(k ∈ P )

 w j ln(vhk j )

(k ∈ P )

j=1

s.t.w ∈ .

[δ1k μ1k (w ) + δ2k μ2k (w ) + δ3k μ3k (w ) + δ4k μ4k (w )]

 h kj

j=1 n 

p 

w

j=1

n

(38)

k=1

(1 − uhk j ) j (k ∈ P )   n .  w min (vlk j ) j (k ∈ P )  j=1  n  l wj min ( vk j ) (k ∈ P ) max 1 −

min max if Ztk ≤ Ztk ≤ Ztk max if Ztk > Ztk

Then, Eq. (37) can be solved by the following linear programming model:

max

Plugging Eq. (33) into Eq. (35), Eq. (35) can be further rewritten as

min if Ztk < Ztk

(t = 1, 2, 3, 4; k = 1, 2, · · · , p).

minimize [vlk , vhk ] for maximizing rk . Let a = [al , au ] and b = [bl , bu ] be two nonnegative interval numbers, then a ≤ b if al ≤ bl and au ≤ bu [40]. Thus, rk will be maximized if vlk and vhk are mini-

7

(37)

On the basis of the above analysis, an algorithm for solving heterogeneous MAGDM problems can be described as follows: Step 1. Collect the decision matrix X k = (x˜ki j )m×n of the alternatives by Eq. (10). Step 2. Identify the smallest grade and the largest grade of the attribute vector, and transform the cost attribute value to benefit attribute value using the following equation:

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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x˜ikj =

⎧ + x − ak , ⎪ ⎪ ⎪akj , i j ⎪ ⎪ ⎪ ⎪[xi j+ − dk , x+ − ak ], ⎪ ⎪ ij ⎪ ⎨[akj , dk ]i,j j ij

ij

(x+j − dikj , x+j − bki j , x+j − aki j ), ⎪ ⎪ ⎪ ⎪ (aki j , bki j , dikj ), ⎪ ⎪ ⎪ ⎪ (x+ − dk , x+ − ck , x+ − bki j , x+j − aki j ), ⎪ ⎪ ⎩(akj , bk ,i jck ,jdk ),i j j ij ij ij ij

if if if if if if if if

j ∈ N1c j ∈ N1b j ∈ N2c j ∈ N2b , j ∈ N3c j ∈ N3b j ∈ N4c j ∈ N4b

(40)

where x+j is the largest grade of the attribute Aj , Nic and Nib (i = 1, 2, 3, 4) denote the subscript sets of cost and benefit attributes in subset Aˆ i , respectively. Step 3. Calculate the Qsd ξikj , Qdd ζikj and Qud ηikj of each attribute value xi kj as follows: (1) If j ∈ 31); (2) If j ∈ 28); (3) If j ∈ 25); (4) If j ∈ 22);

N1 , then ξikj , ζikj and ηikj can be computed by Eqs. (29–

[21–23] distinguish the satisfaction and dissatisfaction through 0.5, which results in the methods [21–23] can only handle real number or interval number attribute vectors. Hence, the proposed method has wider application scope than the methods [21–23]. (3) The proposed method considers the satisfaction, dissatisfaction and hesitation index of each attribute value simultaneously, whereas the methods [21–23] just consider the satisfaction and dissatisfaction index. So, the proposed method induces an IVIFN that takes into account as much as possible all the information in the aggregation process. (4) The proposed method objectively determines the attribute weights by constructing a multiple objective IF mathematical programming model, whereas the attribute weights of the methods [21–23] are given in advance. It is very difficult to avoid subjective randomness while giving attribute weights a priori.

N2 , then ξikj , ζikj and ηikj can be computed by Eqs. (26–

Table 1 presents a comprehensive comparison between the proposed method and existing methods.

N3 , then ξikj , ζikj and ηikj can be computed by Eqs. (23–

6. Illustrative examples

N4 , then ξikj , ζikj and ηikj can be computed by Eqs. (20–

Step 4. Calculate the Qsi ξ˜k j , Qdi ζ˜k j and Qui η˜ k j of attribute vector Akj by Eqs. (15–17); Step 5. Induce the IVIFN corresponding to the attribute vector Akj through Eq. (19) and form a collective IVIF matrix M = (rkj )p × n (i.e., Eq. (32)). Step 6. Construct a multiple objective IF programming model (i.e., Eq. (34)) and convert it into a linear programming model (i.e., Eq. (39)). Step 7. Determine the attribute weights by solving the linear programming model Eq. (39). Step 8. Derive the comprehensive evaluation value rk of each alternative Sk by Eq. (3). Step 9. Compute the score s(rk ) and accuracy degree h(rk ) of comprehensive assessment rk by using Eqs. (4) and (5). Step 10. Rank the alternatives according to the Definition 5 and select the best one. The schematic structure of the algorithm is depicted in Fig. 2. 5. Comparison analysis with existing methods In this section, we compare the proposed method with existing methods [21–23]. The method [22] aggregated real numbers into IVIFN. The methods [21,23] aggregated interval numbers into IVIFN. Their objective and idea are the same, that is to aggregate the decision information into IVIFN for GDM problems. Notably, in contrast with existing methods [21–23], the proposed method here can be highlighted from the following aspects: (1) The attribute values in the method [22] are real numbers, and the attribute values in the methods [21, 23] are interval numbers, namely, they only considered the assessment information of attribute as single types (i.e., real numbers or interval numbers); whereas the attribute values in the proposed method may be multiple types (i.e., real numbers, interval numbers, TFNs and TrFNs). Therefore, the proposed method can be applied to solve heterogeneous MAGDM problems while the methods [21–23] cannot. (2) In aggregation stage, the proposed method distinguishes the satisfaction and dissatisfaction by calculating the relative closeness between each attribute value and the ideal satisfaction and dissatisfaction, whereas the methods

In this section, an IT outsourcing service provider evaluation example and a supplier selection example are used to illustrate the effectiveness of the proposed method. 6.1. An IT outsourcing service provider evaluation example China CNR Corporation Limited (hereinafter “CNR) is a world class enterprise in the industry of railway transport equipment. CNR is mainly engaged in design, manufacture, refurbishment, service and lease of railway transportation equipment. Headquartered in Beijing, CNR has 88,700 employees divided among over 20 subsidiaries, among which are the leading manufacturers in diesel coaches, wagons and locomotives in China. The annual output of CNR is 1100 urban railway vehicles, 460 diesel locomotives, 26,0 0 0 freight wagons and 2300 passenger coaches, which share over half of the home market, and are exported to more than 60 countries and regions. To cope with today’s fast growing IT environment, CNR is engaged in decision about an IT outsourcing service project which provides a cost-effective alternative to customer’s IT needs. CNR plans to select the best service providers from four candidates, whose key competencies are evaluated by the following six attributes. Now there are five experts are invited to form a DM team. In the evaluation stage, the DM team considers six attributes for each provider, which are described in Li [27,28]. These attributes include the research and development capability (A1 ), the product quality (A2 ), the technological level (A3 ), flexibility (A4 ), delivery time (A5 ) and price (A6 ), in which the former four ones are benefit attributes, the latter two ones are cost attributes. Due to the imprecision and subjectivity in human opinions, the research and development capability (A1 ), product quality (A2 ) and technological level (A3 ) are suitably represented by TFNs. The assessments of A4 are represented by TrFNs. The assessments of delivery time (A5 ) are represented by interval numbers to consider the uncertainties in the product process. The assessments of the providers on A6 can be represented by real numbers. Experts adopt the ten-mark system to evaluate the four candidates IT outsourcing providers based on the six attributes. The decision information can be expressed by the matrices, which are listed in Table 2. The DM team gives the known information on the attributes’ importance as follows:



=



w1 ≥ 0.1; w1 − w4 ≤ 0.1; w2 ≤ 0.2; w2 − w6 ≥ 0.05; w1 + w3 + w4 ≤ 0.6; w4 − w5 ≥ 0.05; w5 ≥ 0.05; w6 ≥ 0.05.

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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Fig. 2. Framework of proposed heterogeneous MAGDM approach.

6.1.1. Decision process using the proposed method Obviously, the decision problem mentioned above is a heterogeneous MAGDM problem involving four different data formats: real numbers, interval numbers, TFNs and TrFNs. To solve this issue, we apply the proposed method to the selection of the IT outsourcing providers below. Step 1. The decision matrix of providers listed in Table 2. Step 2. Owing to the attribute values given on the basis of the ten-mark system, we have

Amin j

⎧ ⎪ ⎨0,

[0, 0], = ⎪ ⎩ ( 0, 0, 0 ), ( 0, 0, 0, 0 ),

if if if if

j ∈ N1 j ∈ N2 , j ∈ N3 j ∈ N4

Amax j

⎧ ⎪ ⎨10,

[10, 10], = ⎪ ⎩(10, 10, 10 ), (10, 10, 10, 10 ),

if if if if

j j j j

∈ N1 ∈ N2 . ∈ N3 ∈ N4

(41)

We use Eq. (41) to transform the cost attribute value to benefit attribute value. The transformation results are shown in Table 3. Step 3. According to Step 3, we can obtain the Qsd, Qdd and Qud of each attribute value. The results are summed up in Table 4. Step 4. By Eqs. (15–17), the Qsi ξ˜k j , Qdi ζ˜k j and Qui η˜ k j of Akj are presented in Table 5. Step 5. By Eq. (19), the induced IVIFNs of Akj are also presented in Table 5. So, the collective IVIF matrix M = (rkj )p × n is constructed by Eq. (32).

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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Table 1 Comparisons between the proposed method and existing methods. Characteristics

Method [22]

Method [21]

Method [23]

The proposed method

Attribute values

Real number

Interval number

Interval number

Solve problem

MAGDM with real numbers

Attribute weight

Given in advance

MAGDM with interval numbers Given in advance

MAGDM with interval numbers Given in advance

TFN, TrFN, interval and real number, heterogeneous MAGDM

Quasi-satisfactory degree (Qsd)

Derived from [0,0.5]

Derived from [0,0.5]

Derived from [0,0.5]

Quasi-dissatisfactory degree (Qdd)

Derived from [0.5,1]

Derived from [0.5,1]

Derived from [0.5,1]

Quasi-uncertain degree (Qud)

None

None

None

Quasi-satisfactory set (Qss)

None

None

None

Quasi-dissatisfactory set (Qds) Quasi-uncertain set (Qus)

None

None

None

None

None

None

Satisfactory interval

Based on Minimax elements in [0,0.5]

Based on Minimax elements in [0,0.5]

Dissatisfactory interval

Based on Minimax elements in [0.5,1]

Based on Minimax elements in [0.5,1]

Hesitancy interval

None

None

Based on mean and standard deviation of all elements in [0,0.5] Based on mean and standard deviation of all elements in [0.5,1] None

τ kj in linear transformation

ξkl j + ξkhj + ζkl j + ζkhj

ξkl j + ξkhj + ζkl j + ζkhj

max{ξkl j + ξkhj + ζkl j + ζkhj }

Determined by multi-objective IF programming Derived from relative closeness between elements and the ideal satisfaction Derived from relative closeness between elements and the ideal dissatisfaction Derived from relative closeness between elements and the ideal median Comprised of Qsd of each element Comprised of Qdd of each element Comprised of Qud of each element Based on mean and standard deviation of Qss Based on mean and standard deviation of Qds Based on mean and standard deviation of Qus ξkl j + ξkhj + ζkl j + ζkhj + 1 (ηkl j + ηkh j ) 2

Table 2 Decision matrix of four alternatives. Providers

DMs

A1

A2

A3

A4

A5

A6

S1

D1 D2 D3 D4 D5

(5.3,6.5,8.9) (6.5,7.6,8.4) (4.7,6.8,8.2) (5.3,6.1,7.9) (4.1,5.3,7.7)

(3.3,5.0,6.7) (5.0,6.7,8.3) (5.0,6.7,8.3) (3.3,5.0,6.7) (6.7,8.3,10)

(4.5,7.1,8.3) (3.3,4.5,6.7) (5.5,6.1,8.3) (5.5,6.4,8.5) (3.8,6.5,6.7)

(4.2,5.3,6.5,7.9) (4.4,4.5,6.6,7.8) (4.1,4.4,5.5,8.8) (5.2,6.3,7.5,8.9) (3.0,5.1,6.3,6.7)

[0.7,4.4] [1.1,3.3] [1.1,2.6] [0.0,2.4] [2.3,3.8]

5.6 3.9 2.8 1.8 4.1

S2

D1 D2 D3 D4 D5

(6.6,7.6,8.8) (5.4,6.4,7.9) (5.2,6.4,8.5) (4.6,6.8,7.8) (4.1,5.2,6.5)

(6.7,8.3,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (1.7,3.3,5.0)

(4.1,6.7,8.3) (4.5,6.4,7.3) (6.7,8.1,10) (3.3,5.0,7.7) (6.9,8.1,10)

(4.2,5.3,5.4,6.7) (5.3,6.4,7.4,7.9) (4.2,5.2,6.4,6.5) (5.5,5.6,6.8,6.8) (3.1,4.1,4.2,5.5)

[1.3,4.6] [1.2,3.5] [0.4,1.7] [0.6,3.3] [1.6,3.7]

4.2 2.7 3.8 2.3 3.1

S3

D1 D2 D3 D4 D5

(6.6,7.6,8.8) (5.5,5.7,8.8) (4.3,5.5,7.9) (4.4,6.0,7.3) (4.7,5.4,6.5)

(5.0,6.7,8.3) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (5.0,6.7,8.3)

(5.1,6.6,8.0) (3.9,6.5,6.7) (6.1,6.7,8.3) (5.7,7,3,10) (4.3,5.0,6.7)

(4.5,4.6,5.6,7.8) (5.0,5.5,7.7,8.8) (3.0,4.0,5.0,9.0) (5.4,5.7,6.6,7.8) (5.3,6.8,7.4,8.5)

[2.0,4.4] [1.4,2.4] [0.0,3.3] [0.3,2.5] [1.4,2.7]

2.9 3.5 3.8 2.3 3.1

S4

D1 D2 D3 D4 D5

(5.3,6.4,7.5) (6.3,7.2,8.4) (5.1,6.2,6.7) (4.5,5.7,6.8) (4.3,5.6,7.4)

(8.3,10,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (3.3,5.0,6.7)

(3.3,4.7,6.7) (5.9,6.5,6.7) (4.6,5.5,6.7) (4.3,5.0,6.7) (5.7,7.3,10)

(4.2,4.3,5.4,7.5) (3.2,5.3,6.4,6.7) (3.1,6.1,6.2,7.7) (2.4,4.5,4.7,5.8) (4.2,4.3,6.6,8.6)

[1.6,3.7] [0.5,2.6] [1.0,2.9] [1.6,3.4] [2.1,4.6]

2.9 3.5 4.7 1.7 3.2

Step 6. By Eq. (34), a multiple objective IF program is formed as follows:

max{r1 = ([0.221, 0.273], [0.102, 0.155] )w1 + ([0.181, 0.283], [0.083, 0.185] )w2 + ([0.195, 0.257], [0.113, 0.175] )w3 + ([0.195, 0.241], [0.125, 0.172] )w4 + ([0.293, 0.350], [0.060, 0.118] )w5 + ([0.184, 0.291], [0.082, 0.189] )w6 } max{r2 = ([0.201, 0.245], [0.119, 0.164] )w1 + ([0.183, 0.344], [0.051, 0.212] )w2 + ([0.215, 0.318], [0.070, 0.173] )w3

+ ([0.168, 0.237], [0.123, 0.192] )w4 + ([0.290, 0.351], [0.060, 0.120] )w5 + ([0.227, 0.286], [0.092, 0.151] )w6 } max{r3 = ([0.199, 0.261], [0.106, 0.168] )w1 + ([0.254, 0.325], [0.070, 0.141] )w2 + ([0.207, 0.274], [0.099, 0.166] )w3 + ([0.202, 0.257], [0.113, 0.168] )w4 + ([0.259, 0.336], [0.063, 0.140] )w5 + ([0.217, 0.299], [0.080, 0.162] )w6 } max{r4 = ([0.204, 0.252], [0.114, 0.161] )w1 + ([0.216, 0.353], [0.039, 0.177] )w2 + ([0.176, 0.255], [0.108, 0.187] )w3

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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Table 3 Benefited decision matrix of four alternatives. Providers

DMs

A1

A2

A3

A4

A5

A6

S1

D1 D2 D3 D4 D5

(5.3,6.5,8.9) (6.5,7.6,8.4) (4.7,6.8,8.2) (5.3,6.1,7.9) (4.1,5.3,7.7)

(3.3,5.0,6.7) (5.0,6.7,8.3) (5.0,6.7,8.3) (3.3,5.0,6.7) (6.7,8.3,10)

(4.5,7.1,8.3) (3.3,4.5,6.7) (5.5,6.1,8.3) (5.5,6.4,8.5) (3.8,6.5,6.7)

(4.2,5.3,6.5,7.9) (4.4,4.5,6.6,7.8) (4.1,4.4,5.5,8.8) (5.2,6.3,7.5,8.9) (3.0,5.1,6.3,6.7)

[5.6,9.3] [6.7,8.9] [7.4,8.9] [7.6,10] [6.2,7.7]

4.4 6.1 7.2 8.2 5.9

S2

D1 D2 D3 D4 D5

(6.6,7.6,8.8) (5.4,6.4,7.9) (5.2,6.4,8.5) (4.6,6.8,7.8) (4.1,5.2,6.5)

(6.7,8.3,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (1.7,3.3,5.0)

(4.1,6.7,8.3) (4.5,6.4,7.3) (6.7,8.1,10) (3.3,5.0,7.7) (6.9,8.1,10)

(4.2,5.3,5.4,6.7) (5.3,6.4,7.4,7.9) (4.2,5.2,6.4,6.5) (5.5,5.6,6.8,6.8) (3.1,4.1,4.2,5.5)

[5.4,8.7] [6.5,8.8] [8.3,9.6] [6.7,9.4] [6.3,8.4]

5.8 7.3 6.2 7.7 6.9

S3

D1 D2 D3 D4 D5

(6.6,7.6,8.8) (5.5,5.7,8.8) (4.3,5.5,7.9) (4.4,6.0,7.3) (4.7,5.4,6.5)

(5.0,6.7,8.3) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (5.0,6.7,8.3)

(5.1,6.6,8.0) (3.9,6.5,6.7) (6.1,6.7,8.3) (5.7,7,3,10) (4.3,5.0,6.7)

(4.5,4.6,5.6,7.8) (5.0,5.5,7.7,8.8) (3.0,4.0,5.0,9.0) (5.4,5.7,6.6,7.8) (5.3,6.8,7.4,8.5)

[5.6,8.0] [7.6,8.6] [6.7,10] [7.5,9.7] [7.3,8.6]

7.1 6.5 5.3 8.3 6.8

S4

D1 D2 D3 D4 D5

(5.3,6.4,7.5) (6.3,7.2,8.4) (5.1,6.2,6.7) (4.5,5.7,6.8) (4.3,5.6,7.4)

(8.3,10,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (3.3,5.0,6.7)

(3.3,4.7,6.7) (5.9,6.5,6.7) (4.6,5.5,6.7) (4.3,5.0,6.7) (5.7,7.3,10)

(4.2,4.3,5.4,7.5) (3.2,5.3,6.4,6.7) (3.1,6.1,6.2,7.7) (2.4,4.5,4.7,5.8) (4.2,4.3,6.6,8.6)

[6.3,8.4] [7.4,9.5] [7.1,9.0] [6.6,8.4] [5.4,7.9]

4.9 8.1 3.9 5.8 7.0

Table 4 Qsd, Qdd and Qud of each attribute value. Providers

DM

A1

A2

A3

A4

A5

A6

S1

D1 D2 D3 D4 D5

0.69,0.31,0.62 0.75,0.25,0.50 0.66,0.34,0.66 0.64,0.36,0.71 0.57,0.43,0.77

0.50,0.50,1.00 0.67,0.33,0.67 0.67,0.33,0.67 0.50,0.50,1.00 0.83,0.17,0.33

0.66,0.34,0.63 0.48,0.52,0.79 0.66,0.34,0.67 0.68,0.32,0.64 0.57,0.43,0.75

0.60,0.40,0.75 0.58,0.42,0.75 0.57,0.43,0.75 0.70,0.30,0.61 0.53,0.47,0.79

0.74,0.26,0.51 0.78,0.22,0.44 0.82,0.18,0.37 0.88,0.12,0.24 0.69,0.31,0.61

0.44,0.56,0.88 0.61,0.39,0.78 0.72,0.28,0.56 0.82,0.18,0.36 0.59,0.41,0.82

S2

D1 D2 D3 D4 D5

0.58,0.42,0.77 0.66,0.34,0.69 0.67,0.33,0.66 0.64,0.36,0.68 0.53,0.47,0.85

0.83,0.17,0.33 0.67,0.33,0.67 0.83,0.17,0.33 0.67,0.33,0.67 0.33,0.67,0.67

0.64,0.36,0.65 0.61,0.39,0.74 0.83,0.17,0.35 0.53,0.47,0.76 0.83,0.17,0.33

0.54,0.46,0.85 0.68,0.32,0.65 0.56,0.44,0.82 0.62,0.38,0.77 0.42,0.58,0.81

0.71,0.30,0.59 0.76,0.24,0.47 0.89,0.11,0.21 0.81,0.19,0.39 0.78,0.22,0.44

0.58,0.42,0.84 0.73,0.27,0.54 0.62,0.38,0.76 0.77,0.23,0.46 0.69,0.31,0.62

S3

D1 D2 D3 D4 D5

0.77,0.23,0.47 0.67,0.33,0.67 0.59,0.41,0.75 0.59,0.41,0.76 0.55,0.45,0.76

0.67,0.33,0.67 0.83,0.17,0.33 0.83,0.17,0.33 0.67,0.33,0.67 0.67,0.33,0.67

0.66,0.34,0.69 0.57,0.43,0.75 0.70,0.30,0.59 0.77,0.23,0.47 0.53,0.47,0.85

0.56,0.44,0.80 0.68,0.32,0.65 0.53,0.47,0.73 0.64,0.36,0.73 0.70,0.30,0.60

0.68,0.32,0.64 0.81,0.19,0.38 0.84,0.16,0.33 0.86,0.14,0.28 0.79,0.21,0.41

0.71,0.29,0.58 0.65,0.35,0.70 0.53.0.47,0.94 0.83,0.17,0.34 0.68,0.32,0.64

S4

D1 D2 D3 D4 D5

0.64,0.36,0.72 0.73,0.27,0.54 0.60,0.40,0.80 0.57,0.43,0.81 0.58,0.42,0.73

0.94,0.06,0.11 0.67,0.33,0.67 0.83,0.17,0.33 0.67,0.33,0.67 0.50,0.50,1.00

0,49,0.51,0.80 0.64,0.36,0.73 0.56,0.44,0.84 0.53,0.47,0.85 0.77,0.23,0.47

0.53,0.47,0.81 0.54,0.46,0.78 0.58,0.42,0.71 0.43,0.57,0.81 0.59,0.41,0.71

0.73,0.27,0.53 0.84,0.16,0.31 0.81,0.19,0.39 0.75,0.25,0.50 0.67,0.33,0.67

0.49,0.51,0.98 0.81,0.19,0.38 0.39,0.61,0.78 0.58,0.42,0.84 0.70,0.30,0.60

+ ([0.172, 0.216], [0.146, 0.190] )w4 + ([0.279, 0.334],

−0.432w5 − 0.337w6 }

[0.069, 0.125] )w5 + ([0.158, 0.280], [0.088, 0.211] )w6 }

min{−0.302w1 − 0.393w2 − 0.321w3 − 0.297w4

s.t.w ∈ .

(42)

According to Eq. (37), Eq. (42) can be transformed into the following multiple objective linear programming model:

min{−0.249w1 − 0.199w2 − 0.217w3 − 0.217w4 −0.347w5 − 0.203w6 } min{−0.224w1 − 0.202w2 − 0.242w3 − 0.184w4 −0.343w5 − 0.257w6 } min{−0.222w1 − 0.292w2 − 0.232w3 − 0.226w4 −0.300w5 − 0.244w6 } min{−0.229w1 − 0.243w2 − 0.194w3 − 0.188w4 −0.327w5 − 0.171w6 } min{−0.319w1 − 0.332w2 − 0.297w3 − 0.276w4 −0.432w5 − 0.344w6 } min{−0.282w1 − 0.421w2 − 0.383w3 − 0.270w4

−0.409w5 − 0.355w6 } min{−0.290w1 − 0.435w2 − 0.294w3 − 0.244w4 −0.407w5 − 0.328w6 } min{−2.282w1 − 2.487w2 − 2.184w3 − 2.076w4 −2.807w5 − 2.497w6 } min{−2.128w1 − 2.974w2 − 2.665w3 − 2.092w4 −2.820w5 − 2.383w6 } min{−2.241w1 − 2.666w2 − 2.316w3 − 2.179w4 −2.768w5 − 2.523w6 } min{−2.171w1 − 3.241w2 − 2.221w3 − 1.926w4 −2.674w5 − 2.427w6 } min{−1.867w1 − 1.687w2 − 1.745w3 − 1.763w4 −2.138w5 − 1.665w6 } min{−1.810w1 − 1.550w2 − 1.754w3 − 1.653w4

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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S.-P. Wan et al. / Knowledge-Based Systems 000 (2016) 1–16 Table 5 Qsi ξ˜k j , Qdi ζ˜k j , Qui η˜ k j and aggregated IVIFNs of attribute vectors. Providers

Attributes

ξ˜k j

ζ˜k j

η˜ k j

IVIFNs

S1

A1 A2 A3 A4 A5 A6

[0.588,0.728] [0.494,0.773] [0.528,0.695] [0.532,0.658] [0.713,0.853] [0.493,0.779]

[0.272,0.412] [0.227,0.506] [0.305,0.472] [0.342,0.468] [0.147,0.287] [0.221,0.507]

[0.549,0.782] [0.455,1.011] [0.632,0.779] [0.657,0.798] [0.294,0.574] [0.464,0.896]

([0.221,0.273],[0.102,0.155]) ([0.181,0.283],[0.083,0.185]) ([0.195,0.257],[0.113,0.175]) ([0.195,0.241],[0.125,0.172]) ([0.293,0.350],[0.060,0.118]) ([0.184,0.291],[0.082,0.189])

S2

A1 A2 A3 A4 A5 A6

[0.551,0.673] [0.462,0.871] [0.553,0.821] [0.468,0.657] [0.707,0.855] [0.600,0.756]

[0.327,0.449] [0.129,0.538] [0.179,0.447] [0.343,0.532] [0.145,0.293] [0.244,0.400]

[0.654,0.833] [0.352,0.714] [0.362,0.796] [0.700,0.856] [0.291,0.585] [0.488,0.799]

([0.201,0.245],[0.119,0.164]) ([0.183,0.344],[0.051,0.212]) ([0.215,0.318],[0.070,0.173]) ([0.168,0.237],[0.123,0.192]) ([0.290,0.351],[0.060,0.120] ([0.227,0.286],[0.092,0.151])

S3

A1 A2 A3 A4 A5 A6

[0.541,0.711] [0.643,0.824] [0.551,0.741] [0.546,0.694] [0.727,0.865] [0.572,0.788]

[0.289,0.459] [0.176,0.357] [0.259,0.449] [0.306,0.454] [0.135,0.273] [0.212,0.428]

[0.570,0.875] [0.352,0.714] [0.522,0.818] [0.623,0.780] [0.269,0.547] [0.424,0.856]

([0.199,0.261],[0.106,0.168]) ([0.254,0.325],[0.070,0.141] ([0.207,0.274],[0.099,0.166]) ([0.202,0.257],[0.113,0.168]) ([0.259,0.336],[0.063,0.140]) ([0.217,0.298],[0.080,0.162])

S4

A1 A2 A3 A4 A5 A6

[0.559,0.689] [0.550,0.900] [0.489,0.706] [0.474,0.598] [0.691,0.829] [0.428,0.760]

[0.312,0.441] [0.100,0.450] [0.294,0.511] [0.402,0.526] [0.171,0.309] [0.239,0.572]

[0.622,0.846] [0.200,0.901] [0.578,0.895] [0.713,0.812] [0.342,0.618] [0.484,0.948]

([0.204,0.252],[0.114,0.161]) ([0.216,0.353],[0.039,0.177]) ([0.176,0.255],[0.108,0.187]) ([0.172,0.216],[0.146,0.190]) ([0.279,0.334],[0.069,0.125]) ([0.158,0.280],[0.088,0.211])

−2.120w5 − 1.889w6 }

+0.409w5 + 0.355w6 ) − 0.334 )/0.025

min{−1.781w1 − 1.960w2 − 1.795w3 − 1.783w4

+δ8 ((0.290w1 + 0.435w2 + 0.294w3 + 0.244w4

−1.968w5 − 1.819w6 }

+0.407w5 + 0.328w6 ) − 0.307 )/0.029

min{−1.824w1 − 1.734w2 − 1.677w3 − 1.660w4

+δ9 ((2.282w1 + 2.487w2 + 2.184w3 + 2.076w4

−2.083w5 − 1.557w6 } s.t.w ∈ .

+2.807w5 + 2.497w6 ) − 2.283 )/0.117 (43)

By using Eq. (43), we obtain the maximum and minimax = mum values corresponding to each objective as follows: Z11 min max min max = −0.221, Z11 = −0.254, Z21 = −0.212, Z21 = −0.246, Z31 min max min max = −0.244, Z31 = −0.258, Z41 = −0.208, Z41 = −0.237, Z12 min max min max = −0.310, Z12 = −0.336, Z22 = −0.323, Z22 = −0.365, Z32 min max min max = −0.332, Z32 = −0.350, Z42 = −0.309, Z42 = −0.337, Z13 min max min max = −2.284, Z13 = −2.402, Z23 = −2.364, Z23 = −2.586, Z33 min = −2.467, Z max = −2.331, Z min = −2.469, Z max = −2.370, Z33 43 43 14 min = −1.862, Z max = −1.708, Z min = −1.814, Z max = −1.754, Z14 24 24 34 min = −1.866, Z max = −1.697, Z min = −1.799. Based on −1.834, Z34 44 44 these ideal values, we construct some membership functions corresponding to the objectives by Eq. (38). Combined with these membership functions, the linear programming model is constructed by Eq. (39) as follows:

min =

δ1 ((0.249w1 + 0.199w2 + 0.217w3 + 0.217w4 + 0.347w5 +0.203w6 ) − 0.222 )/0.033 +δ2 ((0.224w1 + 0.202w2 + 0.242w3 + 0.184w4 +0.343w5 + 0.257w6 ) − 0.212 )/0.034 +δ3 ((0.222w1 + 0.292w2 + 0.232w3 + 0.226w4 +0.300w5 + 0.244w6 ) − 0.248 )/0.025 +δ4 ((0.229w1 + 0.243w2 + 0.194w3 + 0.188w4 +0.327w5 + 0.171w6 ) − 0.208 )/0.029 +δ5 ((0.319w1 + 0.332w2 + 0.297w3 + 0.276w4 +0.432w5 + 0.344w6 ) − 0.310 )/0.025 +δ6 ((0.282w1 + 0.421w2 + 0.383w3 + 0.270w4 +0.432w5 + 0.337w6 ) − 0.323 )/0.043 +δ7 ((0.302w1 + 0.393w2 + 0.321w3 + 0.297w4

+δ10 ((2.128w1 + 2.974w2 + 2.665w3 + 2.092w4 +2.820w5 + 2.383w6 ) − 2.369 )/0.217 +δ11 ((2.241w1 + 2.666w2 + 2.316w3 + 2.179w4 +2.768w5 + 2.523w6 ) − 2.380 )/0.110 +δ12 ((2.171w1 + 3.241w2 + 2.221w3 + 1.926w4 +2.674w5 + 2.427w6 ) − 2.319 )/0.136 +δ13 ((1.867w1 + 1.687w2 + 1.745w3 + 1.763w4 +2.138w5 + 1.665w6 ) − 1.756 )/0.108 +δ14 ((1.810w1 + 1.550w2 + 1.754w3 + 1.653w4 +2.120w5 + 1.889w6 ) − 1.709 )/0.104 +δ15 ((1.781w1 + 1.960w2 + 1.795w3 + 1.783w4 +1.968w5 + 1.819w6 ) − 1.844 )/0.070 +δ16 ((1.824w1 + 1.734w2 + 1.677w3 + 1.660w4 +2.083w5 + 1.557w6 ) − 1.698 )/0.101 s.t.w ∈ .

(44)

Step 7. By using Lingo 11.0 to solve Eq. (44), we can get the weight vector of attribute w = (0.10, 0.200, 0.12, 0.29, 0.24, 0.05) 1 1 1 for δ = ( 16 , 16 , · · · , 16 ). Step 8. Use Eq. (3) to calculate the comprehensive evaluation value of each provider Sk (k ∈ P), which is shown in Table 6. Step 9. Use Eq. (4) to obtain the scores of the comprehensive evaluation value of each provider, which are also shown in Table 6. Step 10. According to Definition 5, the ranking order of the providers is S3 S2 S1 S4 and the best provider is S3 . 6.1.2. Comparison with extended TOPSIS method Hwang and Yoon [34] firstly proposed the TOPSIS method on the ground of the fact that the chosen alternative should have the

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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S.-P. Wan et al. / Knowledge-Based Systems 000 (2016) 1–16 Table 8 D+ , D− , RCC and ranking order of each provider.

Table 6 Comprehensive evaluation values, scores and ranking of the four providers. Providers

Comprehensive assessments

Scores

Ranking

S1 S2 S3 S4

([0.219,0.285],[0.092,0.159]) ([0.214,0.300],[0.080,0.168]) ([0.228,0.295],[0.086,0.155]) ([0.210,0.285],[0.086,0.167])

0.127 0.133 0.141 0.121

3 2 1 4

shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). Lots of work has been done about heterogeneous MAGDM problems with extended TOPSIS methods [42–44]. For well comparison, we extend TOPSIS method to manage appropriately the IT outsourcing service provider evaluation example, and the steps can be summarized as follows. Step 1 and 2. Refer to Step 1and 2 in Section 4.2. Step 3. Form the benefit decision matrices X k = (x˜i kj )m×n , where

x˜ikj =

⎧ k xi j , ⎪ ⎪ ⎨[xk , x¯k ],

if j ∈ N1 if j ∈ N2

⎪ (ak , bk , ck ), ⎪ ⎩ ikj ikj ikj k ( ai j , bi j , ci j , di j ),

if j ∈ N3 if j ∈ N4

−i j

ij

.

(45)

Step 4. Determine the ideal decision of group decision of alternatives. We denoted the positive ideal decision of group decision of alternatives by X + = (x˜i + ) , where j m×n

x˜i+j =

⎧ xki j , ⎪ ⎪max k∈P ⎪ ⎪ ⎨[max xk , max x¯ki j ], ( ⎪ ⎪ ⎪ ⎪ ⎩(

k∈P

if j ∈ N1

k∈P max aki j , max bki j , max cikj , k∈P k∈P k∈P max aki j , max bki j , max cikj , max dikj k∈P k∈P k∈P k∈P −i j

if j ∈ N2

)

(46)

if j ∈ N3

),

if j ∈ N4

and the negative ideal decision of group decision of alternatives by X − = (x˜i− ) , where j m×n

x˜i−j =

⎧ min xk , ⎪ ⎪ ⎪ k∈P i jk ⎪ ⎨[min x , min x¯ki j ], ( ⎪ ⎪ ⎪ ⎪ ⎩(

k∈P

if j ∈ N1

k∈P min aki j , min bki j , min cikj , k∈P k∈P k∈P min aki j , min bki j , min cikj , min dikj k∈P k∈P k∈P k∈P −i j

if j ∈ N2

)

if j ∈ N3

),

.

D− = k

n  m 

D+

D−

RCC

Ranking

S1 S2 S3 S4

5.068 4.905 3.674 5.691

5.230 5.392 6.323 4.606

0.508 0.524 0.632 0.447

3 2 1 4

w j dis(x˜ikj , x˜i−j ),

(49)

respectively, where wj denote weight attribute Aj . Step 6. Calculate the relative closeness (RC) of the alternative Sk by Eq. (50) as follows

RCk =

D+ k

D− k

+ D− k

.

(50)

Step 7. Rank the alternatives according to the RCk of the alternative Sk . In the following, the extend method mentioned above is applied to the same example in Section 6.1. The ideal decisions of group decision of alternatives are calculated by Step 4 as shown in Table 7. By Step 5 and Step 6, taking the attribute weight vector as w = (0.10, 0.200, 0.12, 0.29, 0.24, 0.05)obtained by Step 7 in Section 6.1.1, the separations and RC of alternatives are calculated as shown in Table 8. By Step 7, the rankings of alternatives are obtained as shown in Table 8. From Table 8, the ranking of the providers is S3 S2 S1 S4 , which seems to be in line with the proposed method in Table 6 in Section 6.1.1. Therefore, the proposed method here is reliable. Based on the comparison analysis between the extended TOPSIS method and the proposed method mentioned above, it is not hard to see that our proposed method has the following desirable advantages: (1) The latter utilizes the multi-objective IF programming model to determine the attribute weights, which is more objective and reasonable, whereas the former needs the DM to provide the attribute weights in advance, which is inevitably subjective. (2) The collective overall ratings of alternatives in the latter are presented by IVIFNs, which is well-suited to express the hesitancy degree inherent in DMs’ judgments, while the collective overall rating in the former is a relative closeness which is a crisp value and cannot reflect the hesitancy degree inherent in DMs’ judgments.

(47)

if j ∈ N4

w j dis(x˜ikj , x˜i+j ),

n  m 

Providers

j=1 i=1

Step 5. Calculate D+ and D− , i.e., the separations between the k k group decision of each alternative and X + as well as X − , respectively. They can be given by the following equations:

D+ = k

13

(48)

j=1 i=1

Table 7 Ideal decision of group decision of alternatives. Provider +

X−

X

A1

A2

A3

A4

A5

A6

DM1 DM2 DM3 DM4 DM5

(6.6,7.6,8.9) (6.5,7.5,8.8) (5.2,6.8,8.5) (5.3,6.8,8.5) (4.7,5.6,7.7)

(8.3,10,10) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (6.7,8.3,10)

(5.1,7.1,8.3) (5.9,6.5,7.3) (6.7,8.1,10) (5.7,7.3,10) (6.9,8.1,10)

(4.5,5.3,6.5,7.9) (5.3,6.4,7.7,8.8) (4.2,6.1,6.4,9.0) (5.5,6.3,7.5,8.9) (5.3,6.8,7.4,8.6)

[6.3,9.3] [7.6,9.5] [8.3,10] [7.6,10] [7.3,8.6]

7.1 8.1 7.2 8.3 7.0

DM1 DM2 DM3 DM4 DM5

(4.3,5.4,7.5) (5.4,5.7,7.9) (4.3,5.5,6.7) (4.4,5.7,6.8) (4.1,5.2,6.5)

(3.3,5.0,6.7) (5.0,6.7,8.3) (5.0,6.7,8.3) (3.3,5.0,6.7) (1.7,3.3,5.0)

(3.3,4.7,6.7) (3.3,4.5,6.7) (4.6,5.5,6.7) (3.3,5.0,6.7) (3.8,5.0,6.7)

(4.2,4.3,5.4,6.7) (3.2,4.5,6.4,6.7) (3.0,4.0,5.0,6.5) (2.4,4.5,4.7,5.8) (3.0,4.1,4.2,5.5)

[5.4,8.0] [6.5,8.6] [6.7,8.9] [6.3,8.4] [5.4,7.7]

4.4 6.1 3.9 5.8 5.9

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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S.-P. Wan et al. / Knowledge-Based Systems 000 (2016) 1–16 Table 9 Three evaluation matrices provided by nine DMs. Suppliers

DMs

Product quality (A1 )

Delivery capability (A2 )

Cost reduction performance (A3 )

Post-sales service (A4 )

S1

The production manager (D1 ) The quality manager (D2 ) The material manager (D3 ) The maintenance manager (D4 ) The development manager (D5 ) The planning manager (D6 ) The purchasing manager (D7 ) The finance manager (D8 ) A group leader (D9 )

97 98 98 99 99 100 87 90 92

27 28 28 29 29 30 30 40 45

55 63 68 45 62 60 56 63 57

89 98 90 91 83 70 86 88 79

S2

The production manager (D1 ) The quality manager (D2 ) The material manager (D3 ) The maintenance manager (D4 ) The development manager (D5 ) The planning manager (D6 ) The purchasing manager (D7 ) The finance manager (D8 ) A group leader (D9 )

77 93 79 78 76 77 78 80 78

36 43 33 60 32 34 35 38 51

52 55 51 69 41 53 55 52 53

90 76 83 81 71 79 77 78 80

S3

The production manager (D1 ) The quality manager (D2 ) The material manager (D3 ) The maintenance manager (D4 ) The development manager (D5 ) The planning manager (D6 ) The purchasing manager (D7 ) The finance manager (D8 ) A group leader (D9 )

85 89 62 83 87 73 89 77 90

90 89 89 90 72 89 88 91 93

80 81 85 88 87 97 77 89 90

33 41 52 44 49 32 55 46 53

Table 10 Qsd, Qdd and Qud of each attribute value. Suppliers

DMs

A1

A2

A3

A4

S1

D1 D2 D3 D4 D5 D6 D7 D8 D9

(0.97,0.03,0.06) (0.98,0.02,0.04) (0.98,0.02,0.04) (0.99,0.01,0.02) (0.99,0.01,0.02) (1.0 0,0.0 0,0.0 0) (0.87,0.13,0.26) (0.90,0.10,0.20) (0.92,0.08,0.16)

(0.27,0.73,0.54) (0.28,0.72,0.56) (0.28,0.72,0.56) (0.29,0.71,0.58) (0.29,0.71,0.58) (0.30,0.70,0.60) (0.30,0.70,0.60) (0.40,0.60,0.80) (0.45,0.55,0.90)

(0.55,0.45,0.90) (0.63,0.37,0.74) (0.68,0.32,0.64) (0.45,0.55,0.90) (0.62,0.38,0.76) (0.60,0.40,0.80) (0.56,0.44,0.88) (0.63,0.37,0.74) (0.57,0.43,0.86)

(0.89,0.11,0.22) (0.98,0.02,0.04) (0.90,0.10,0.20) (0.91,0.09,0.18) (0.83,0.17,0.34) (0.70,0.30,0.60) (0.86,0.14,0.28) (0.88,0.12,0.24) (0.79,0.21,0.42)

S2

D1 D2 D3 D4 D5 D6 D7 D8 D9

(0.77,0.23,0.46) (0.93,0.07,0.14) (0.79,0.21,0.42) (0.78,0.22,0.44) (0.76,0.24,0.48) (0.770.23,0.46) (0.78,0.22,0.44) (0.80,0.20,0.40) (0.78,0.22,0.44)

(0.36,0.64,0.72) (0.43,0.57,0.86) (0.33,0.67,0.66) (0.60,0.40,0.80) (0.32,0.68,0.64) (0.34,0.66,0.68) (0.35,0.65,0.70) (0.38,0.62,0.76) (0.51,0.49,0.98)

(0.52,0.48,0.96) (0.55,0.45,0.90) (0.51,0.49,0.98) (0.69,0.31,0.62) (0.41,0.59,0.82) (0.53,0.47,0.94) (0.55,0.45,0.90) (0.52,0.48,0.96) (0.53,0.47,0.94)

(0.90,0.10,0.20) (0.76,0.24,0.48) (0.83,0.17,0.34) (0.81,0.19,0.38) (0.71,0.29,0.58) (0.79,0.21,0.42) (0.77,0.23,0.46) (0.78,0.22,0.44) (0.80,0.20,0.40)

S3

D1 D2 D3 D4 D5 D6 D7 D8 D9

(0.85,0.15,0.30) (0.89,0.11,0.22) (0.62,0.38,0.76) (0.83,0.17,0.34) (0.87,0.13,0.26) (0.73,0.27,0.54) (0.89,0.11,0.22) (0.77,0.23,0.46) (0.90,0.10,0.20)

(0.90,0.10,0.20) (0.89,0.11,0.22) (0.89,0.11,0.22) (0.90,0.10,0.20) (0.72,0.28,0.56) (0.89,0.11,0.22) (0.88,0.12,0.24) (0.91,0.09,0.18) (0.93,0.07,0.14)

(0.80,0.20,0.40) (0.81,0.19,0.38) (0.85,0.15,0.30) (0.88,0.12,0.24) (0.87,0.13,0.26) (0.97,0.03,0.06) (0.77,0.23,0.46) (0.89,0.11,0.22) (0.90,0.1,0.20)

(0.33,0.67,0.66) (0.41,0.59,0.82) (0.52,0.48,0.96) (0.44,0.56,0.88) (0.49,0.51,0.98) (0.32,0.68,0.64) (0.55,0.45,0.90) (0.46,0.54,0.92) (0.53,0.47,0.94)

6.2. A supplier selection example This section will apply another example (adapted from [22]) involving the assessments for three potential suppliers to illustrate the proposed method. A manufacturing company intends to select an appropriate supplier to increase its customer base. After detailed survey, data eliciting and statistical treatment, we get tree potential suppliers S1 , S2 and S3 as alternatives for further evaluation, with the specifications

listed in Table 9. This is a GMADM problem with real numbers. All attributes are benefit criteria. To address this issue, we utilize the proposed method to the selection of the supplier below. First, owing to the assessments given with the hundred-mark system, it is easy to see that Amin = 0 and Amax = 100. Accordj j ing to Step 3 in Section 4.2, the Qsd, Qdd and Qud of each attribute value can be calculated and their results are summed up in Table 10. Then, by Step 4 in Section 4.2, the Qsi ξ˜k j , Qdi ζ˜k j and Qui η˜ k j of Akj are presented in Table 11. By Step 5 in Section 4.2,

Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026

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S.-P. Wan et al. / Knowledge-Based Systems 000 (2016) 1–16

15

Table 11 Qsi ξ˜k j , Qdi ζ˜k j , Qui η˜ k j and aggregated IVIFNs of attribute vectors. Suppliers

Attributes

ξ˜k j

ζ˜k j

η˜ k j

IVIFNs

S1

A1 A2 A3 A4

[0.909,1.002] [0.255,0.381] [0.522,0.654] [0.780,0.940]

[0.0 0 0,0.091] [0.619,0.745] [0.346,0.478] [0.060,0.220]

[0.0 0 0,0.182] [0.510,0.761] [0.713,0.892] [0.120,0.440]

([0.434,0.479],[0.0 0 0,0.044]) ([0.097,0.144],[0.235,0.283]) ([0.186,0.233],[0.124,0.171]) ([0.342,0.412],[0.026,0.096])

S2

A1 A2 A3 A4

[0.744,0.847] [0.307,0.497] [0.463,0.606] [0.742,0.847]

[0.153,0.256] [0.503,0.693] [0.394,0.537] [0.153,0.258]

[0.305,0.512] [0.646,0.865] [0.779,1.003] [0.307,0.516]

([0.309,0.352],[0.063,0.106]) ([0.111,0.180],[0.182,0.251]) ([0.160,0.210],[0.136,0.186]) ([0.308,0.351],[0.064,0.107])

S3

A1 A2 A3 A4

[0.723,0.910] [0.818,0.940] [0.800,0.921] [0.366,0.534]

[0.090,0.277] [0.060,0.182] [0.079,0.201] [0.466,0.634]

[0.179,0.554] [0.120,0.365] [0.159,0.401] [0.730,0.981]

([0.305,0.385],[0.038,0.117]) ([0.365,0.419],[0.027,0.081]) ([0.351,0.404],[0.035,0.088]) ([0.128,0.187],[0.163,0.222])

Table 12 Comprehensive evaluation values, scores and ranking of the three suppliers. Suppliers

Comprehensive assessments

Scores

Ranking

S1 S2 S3

([0.301,0.355],[0.0 0 0,0.106]) ([0.244,0.293],[0.091,0.141]) ([0.279,0.343],[0.054,0.125])

0.275 0.152 0.222

1 3 2

the induced IVIFNs of Akj are also presented in Table 11. In method [22] the weight vector w = (0.3, 0.2, 0.2, 0.3) of attribute is given in advance. By Step 8 in Section 4.2, the comprehensive evaluation value of each supplier Sk is obtained as shown in Table 12. By Step 9 and 10 in Section 4.2, the scores and suppliers’ ranking are also shown in Table 12. The ranking order obtained by the proposed in this paper is S1 S3 S2 , which is in accordance with that obtained by method [22]. Thus, the proposed method is reliable. 7. Conclusions This paper puts forward a new method for aggregating the attribute information into IVIFNs and applies it to solve heterogeneous MAGDM problem. The primary contributions of the proposed method are summarized as follows: (1) A new general method is developed to aggregate heterogeneous information into IVIFNs. This method can aggregate different types of information (including real numbers, interval numbers, TFNs and TrFNs) into IVIFNs. (2) The attribute weights are determined objectively by constructing a multiple objective IF programming which is transformed to a linear programming model. (3) Combining the proposed aggregating heterogeneous information method with the obtained attribute weights, a new method is presented to solve heterogeneous MAGDM problems. Although the proposed method is developed for solving heterogeneous MAGDM problems, it is also appropriate for the complex multi-attribute large-group decision-making problems. Future research will extend the developed method to heterogeneous MAGDM including IFNs [45–47], hesitant fuzzy elements [48], linguistic terms [49] and hesitant fuzzy linguistic terms [50,51]. Acknowledgment The authors are very grateful to the anonymous referees for their comments and suggestions. This research was supported by the National Natural Science Foundation of China (Nos. 71061006, 61263018 and 61602219), Young scientists Training object of

Jiangxi province (No. 20151442040081), the Science and Technology Project of Jiangxi province educational department of China (Nos. GJJ151601, GJJ150466 and GJJ150464).

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Please cite this article as: S.-P. Wan et al., Aggregating decision information into interval-valued intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Knowledge-Based Systems (2016), http://dx.doi.org/10.1016/j.knosys.2016.09.026