Interval-valued intuitionistic fuzzy envelopment analysis and preference fusion

Journal Pre-proofs Interval-valued intuitionistic fuzzy envelopment analysis and preference fusion Wei Zhou, Jin Chen, Bingqing Ding, Sun Meng PII: DOI: Reference:

S0360-8352(20)30095-4 https://doi.org/10.1016/j.cie.2020.106361 CAIE 106361

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

23 March 2018 22 November 2019 11 February 2020

Please cite this article as: Zhou, W., Chen, J., Ding, B., Meng, S., Interval-valued intuitionistic fuzzy envelopment analysis and preference fusion, Computers & Industrial Engineering (2020), doi: https://doi.org/ 10.1016/j.cie.2020.106361

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The revised manuscript (No.: CAIE-D-18-00566-R2):

Interval-valued intuitionistic fuzzy envelopment analysis and preference fusion

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Author Information: Title: Interval-valued intuitionistic fuzzy envelopment analysis and preference fusion The name(s) of the author(s): Wei Zhoua,b, Jin Chena, Bingqing Ding c, Sun Meng a The affiliation(s) and address(es) of the author(s): a. School of Finance, Yunnan University of Finance and Economics, Kunming 650221, P. R. China; b. Business School, Sichuan University, Chengdu 610064, P. R. China; c. Business School, East China University of Science and Technology, Shanghai 200237, P. R. China. Full contact details: Corresponding author: Wei Zhou Tel: 08613669736681 E-mail addresses: [email protected] (W. Zhou) Postal address: No.237 Longquan Road, Kunming , China, 650221. Declarations of interest: none Acknowledgments: This work was supported by the Natural Science Foundation of China (Nos. 71561026 and 71840001), Social Science Youth foundation of Ministry of Education of China (No. 18YJC790118) and Applied Basic Research Programs of Science and Technology Commission of Yunnan Province (No. 2017FB102).

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Abstract: The interval-valued intuitionistic fuzzy set (IIFS) is an effective tool to describe qualitative evaluations using two interval values from two perspectives of “good” and “bad”. In this study, a new decision-making method based on the envelopment efficiency of IIFS is proposed, which can be introduced to select the optimal alternative and improve the inefficient ones. To do this, we firstly propose the ratio comparison rules of IIFS, based on which the interval-valued intuitionistic fuzzy envelopment analysis (IIFEA) is demonstrated. After that, we develop the IIFEA-N and IIFEA-M models on the basis of non-membership and membership. Their dual forms are derived so that the IIFEA-N and IIFEA-M models can be transformed into linear programming, which are calculable. With respect to the attributes’ difference, we further construct the preference IIFEA (PIIFEA) model by fusing subjective preference rather than quantitative weights. Therefore, the optimal alternative can be selected based on the decision makers’ preference, and the development approaches also can be given to the inefficient alternatives. Subsequently, a complete decision-making process is provided. At last, an example for the CEO of an international hotel to select a GM of the biggest branch in China is illustrated to show the effectiveness of the proposed methods. Keywords: Envelopment analysis; interval-valued intuitionist fuzzy set; qualitative evaluation; preference fusion; alternative development.

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1. Introduction In some situations, it is difficult for a decision maker (DM) to make decisions due to the lack of sufficient and accurate evaluation data. Therefore, the concept of fuzzy set (FS) was proposed by Zadeh in 1965 to deal with this difficulty. The theory has soon attracted increasing attention so that lots of researches related to FS have been conducted in recent years. For instance, the intuitionistic fuzzy set (IFS) could be the priority to describe some qualitative evaluation based on membership and non-membership which represent “good” and “bad” of an object (Atanassov, 1986). According to IFS, a further study named interval-valued intuitionistic fuzzy set (IIFS) was proposed by Atanassov and Gargov (1989). This study is characterized by the membership and non-membership functions as well, but it concerns more about interval values rather than certain numbers, which makes the results more flexible. In the practical decision-making process, many qualitative evaluations cannot be given in the form of accurate numbers. Hence these assessments are transformed into several ranges and become interval values to make the whole decision-making process more convenient. This paper focuses on the research of IIFS to find the method for comparing different alternatives and selecting the optimal one based on their IIFS efficiencies. Moreover, the DM’s preference and the enhancement suggestions for inefficient alternatives are also taken into account to demonstrate the method’s prominent advantages. To achieve the above aims, the remainder of this paper is organized as follows: The literature review is presented in Section 2. The IIFS is introduced and its enveloped efficiency is defined in Section 3. In Section 4, we propose the interval-valued intuitionistic fuzzy envelopment analysis (IIFEA) models based on non-membership and membership and then derive their dual and computable forms. With respect to the DM’s preference for the attributes, Section 5 develops the IIFEA models by fusing the preference relationship and proposes the preference IIFEA (PIIFEA) models; moreover, a complete decision-making process based on the proposed IIFEA and PIIFEA models is presented. Finally, Section 6 presents an illustrated example, and the advantages and disadvantages of these models are concluded in Section 7.

2. Literature review As aforementioned, the IIFS is an efficient tool to describe qualitative evaluations using two interval values from two perspectives of “good” and “bad.” It has been utilized in different fields. 4

Besides, some extended IIFSs have been further developed, such as the interval-valued versus intuitionistic fuzzy set (Tizhoosh, 2008), the generalized interval-valued intuitionistic fuzzy set (Adak et al., 2012), the interval-valued intuitionistic uncertain linguistic set (Liu, 2013), the interval-valued intuitionistic fuzzy soft set (Zhang et al., 2014), the interval-valued intuitionistic fuzzy rough set (Xu et al., 2016), and the interval-valued complex fuzzy set (Selvachandran et al., 2018). Meanwhile, many decision-making techniques have been proposed based on the IIFS, such as the correlation analysis for IIFS (Bustince and Burillo, 1995), the distance measure for IIFS (Xu, 2010), the TOPSIS method based on nonlinear programming for IIFS (Li, 2010), the Choquet integral-based method for IIFS (Tan, 2011; Meng et al., 2013), the granulation and uncertainty measures for IIFS (Huang et al., 2013), the risk attitudinal ranking method for IIFS (Wu and Chiclana, 2014), the maximizing consensus method for IIFS (Zhang and Xu, 2015), the admissible linear orders for IIFS (Miguel et al., 2016), the aggregation operations and the multiplication operations for IIFS (Liu 2017; Chen and Han, 2018), and the CODAS (Combinative Distance Based Assessment) method for IIFS (Yeni and Özçelik, 2019). Obviously, the IIFS and its decision-making methods have become a popular research topic. We can find that the above decision-making methods can only be used to rank alternatives and select the optimal one under the interval-valued intuitionistic fuzzy environment. However, an important issue is how to enhance the suboptimal alternatives. It cannot be addressed by the above methods. Therefore, in this study, we propose a new decision-making method which mainly includes three functions: (1) it is convenient to utilize IIFS to describe the DMs’ subjective evaluations; (2) the new method can provide a quantitative ranking, which can be used to select the optimal alternative accordingly; and (3) the method shows how to modify the interval-valued membership and non-membership values to enhance the suboptimal and inefficient alternatives. Thus, by using this new decision-making method, the above issue can be addressed. In this study, we introduce the data envelopment analysis (DEA, Charnes et al., 1978) which is a famous technique to measure the efficiency of decision-making unites and has been successfully applied in many fields (Liu et al., 2016). However, we can find the data used in the DEA model are real numbers, which is obviously different from the interval-valued intuitionistic fuzzy number (IIFN) and other fuzzy numbers. In this case, some fuzzy DEA models were developed, such as the ideal-seeking fuzzy DEA model (Hatami-Marbini et al., 2010), the tolerance fuzzy DEA model 5

(Hatami-Marbini et al., 2011), the double front fuzzy DEA model (Ahmady et al., 2013), the cross-efficiency fuzzy DEA model (Dotoli et al., 2015), the flexible cross-efficiency fuzzy DEA model (Hatami-Marbini et al., 2017), and the big-data oriented fuzzy DEA (He et al., 2019). Thus, fuzzy sets and fuzzy numbers can be used to describe the input and output indexes, and then the corresponding fuzzy DEA can be constructed based on the fuzzy inequality constraints (Ghodousian and Khorram, 2008; Ghodousian and Parvari, 2017). Note that the input and output indexes should be pointed out if the DEA model under the interval-valued intuitionistic fuzzy environment is constructed, which is inconsistent with the above methods. Therefore, to address this inconsistency and develop the IIFEA model, this study firstly proposes a ratio comparison rule of IIFS. Another key issue for the interval-valued intuitionistic fuzzy multi-attribute decision making is how to reasonably distribute attributes’ weights. A simple method is to put equal weights on each attribute. However, this is too simple to deal with all interval-valued intuitionistic fuzzy multiattribute decision makings. As a result, some important weighted approaches were proposed, such as the maximum entropy approach (O’Hagan, 1988), the genetic algorithm weighted method (Nettleton and Torra, 2001), the minimum variance weighted method (Fuller, Majlender 2003), the preemptive goal programming weighted method (Wang, Parkan 2007), the deviation entropy weight method (Han 2011), the ordered precise weighted method (Zhou and He, 2014), the PROMETHEE (GAIA brain) method (Bagherikahvarin and Smet, 2017), and the correlation sparse weighted method (Zhou et al., 2019). Even though these general weighted methods can tell how to put accurate weight values on certain attributes, these values could be different and variable. Furthermore, the ranking results and the optimal alternatives could be different because of diverse weighted methods. To avoid this dilemmatic situation, this study only focuses on the DMs’ preference relationship for each attribute rather than the accurate weights. Then, the preference fusion method is developed, based on which we further propose the PIIFEA model. It is noted that the preference fusion method can distinguish different attributes just like the weighted methods. Meanwhile, the given information by this method is more believable and its presentation is more convenient than that of the general weighted methods.

3. IIFS and its enveloped efficiency In this section, the concepts of IIFS and IIFN are introduced firstly. After that, we propose the comparison rules for IIFS and IIFN. Being different from the original ones which are mainly about 6

subtraction and addition operations, the proposed comparison rules are formed as a ratio and can be applied to calculate the alternatives’ IIFS efficiency. Furthermore, it is also able to illustrate the interval-valued membership and the interval-valued non-membership enhancement schedule and make the inefficient alternatives optimized. As a result, an interval-valued intuitionistic fuzzy decision-making method is developed. 3.1 IIFS, IIFN, and comparison rules As aforementioned, Atanassov and Gargov (1989) developed the IIFS when it was found that making a decision did not merely involve real numbers, the utilization of intervals has made decision-making more flexible and practical. Therefore, the IIFS and IIFN are characterized by membership and non-membership with interval values which enables the DMs to cope with fuzziness and uncertainty. Meanwhile, it is obvious that the IIFS consists of the IFS, which shows its generalization. Therefore, this paper uses the IIFS and IIFN to present the qualitative evaluation information and make decisions. Some basic definitions are introduced as follows: Definition 1 (Atanassov and Gargov, 1989). Let X  ( x1 , x2 ,

, xn ) be a finite set, then an IIFS A

in X can be defined as follows: A  ( xi ,  A ( xi ), v A ( xi )) xi  X  ,

(1)

where  A ( xi )  [0,1] and vA ( xi )  [0,1] are called the interval-valued membership and intervalvalued non-membership of the element xi  X , and sup A ( x)  sup vA ( x)  1 for all xi  X . Definition 2 (Xu, 2009). Let X  ( x1 , x2 ,

, xn ) be a finite set and A  ( xi ,  A ( xi ), v A ( xi ))

xi  X  be an IIFS, the pair (  A ( xi ), vA ( xi )) is called an IIFN which can be presented as ai (  A ( xi ), v A ( xi )) ( [ i , i ],[vi , vi ]) .

To compare the IFNs, Chen and Tan (1994) developed the score function of IFN a , namely, s(a)  a  a . Later, Xu and Yager (2006) proposed an order relation to compare two IFNs.

Similarly, Xu (2009) introduced the score function s (ai ) 

i  i  vi  vi 2

and the accuracy function

h(ai )  ( i  i  vi  vi ) / 2 of IIFN, and provided an order relation between two IIFNs a1 and

a2 : If s(a1 )  s(a2 ) , then a1

a2 ; if s(a1 )=s(a2 ) , then (1) if h(a1 )  h(a2 ) , then a1

a2 ; and (2)

if h(a1 )  h(a2 ) , then a1 ~ a2 . In the next section, we further analyze these comparison rules. 7

3.2 Ratio comparison rule of IIFNs As mentioned before, the subtraction operation shows the distance between interval-valued membership and interval-valued non-membership of IIFN which can be utilized to compare two IIFNs. This comparison rule is direct and reasonable. However, we can also compare two IIFNs by using the ratio instead of the distance between interval-valued membership and interval-valued non-membership of IIFN. In this way, comparing interval-valued membership per unit calculated by the equation

Interval -valued Membership Interval -valued Non -membership

is proposed to be a new comparison rule. It is clear that this

calculation is in the form of a ratio that is similar to general concepts such as income per unit. As can be seen below, Eq. (2) is provided to illustrate this comparison rule. For an IIFN, Eq. (2) can be presented as na  [na , na ]  [  / v  ,   / v  ] , where a  ([  ,   ],[v  , v  ]) is an IIFN and na is its enveloped efficiency. Moreover, this enveloped efficiency can be further obtained based on na  Ana  (1  A)na , where A [0,1] and is a constant. Output Interval  valued Membership ~ . Input Interval  valued Non  membership

(2)

Then, the new comparison rule is derived based on the enveloped efficiency of an IIFN to compare two IIFNs such as a1 and a2 as follows: If na1  na2 , then a1

a2 ; and if na1  na2 ,

then a1 ~ a2 . Additionally, na represents the enveloped efficiency of an IIFN a stated in the form of a ratio. We use Example 1 to show the proposed comparison laws. Example 1. Suppose there are 6 IIFNs namely ([0.35, 0.45],[0.15, 0.25]) , ([0.5, 0.6],[0.3, 0.35]) , ([0.3, 0.4],[0.4, 0.55]) , ([0.55, 0.6],[0.35, 0.4]) , ([0.35, 0.55],[0.3, 0.4]) , and ([0.5, 0.6],[0.35, 0.4])

which are labeled from F1 to F6 at the head of each column in Table 1. Table 1. Calculation and comparison of the six IIFNs.

IIFNs Interval-Valued Membership Interval-Valued Non-Membership Score Value of IIFN Interval-Valued Membership Per Unit Score Value of Interval-Valued Membership Per Unit

F1

F2

F3

F4

F5

F6

[0.35,0.45] [0.50,0.60] [0.30,0.40] [0.55,0.60] [0.35,0.55] [0.50,0.60] [0.15,0.25] [0.30,0.35] [0.40,0.55] [0.35,0.40] [0.30,0.40] [0.35,0.40] 0.200

0.225

-0.125

0.200

0.100

0.175

[1.40,3.00] [1.42,2.00] [0.55,1.00] [1.38,1.71] [0.67,1.14] [0.82,1.83] 2.20

1.71

0.76

1.55

(Here, A is set as 0.5. The explanation can be found in the last paragraph of this section)

8

1.32

1.48

It can be seen from Table 1 that: Based on the cited comparison rules, we have

F2

F4

F1 F 2

F1 F 6

F 5 F 3 . While as for the proposed comparison rules, we have

F4

F5

F6

F 3 . It is clear that there are some differences between the two

comparison laws. The proposed laws focus on the ratio efficiency of IIFN and its meaning is easy to understand.

Figure 1. The efficient frontier of the six IIFNs

To further investigate the proposed comparison rules, we apply the data above to Figure 1 by plotting interval-valued non-membership on the horizontal and interval-valued membership on the vertical axis. The slope of the line connecting each point and the origin corresponds to the interval-valued membership per unit. Meanwhile, the highest slope is attained by the line through the origin and F1, which is named as the efficient frontier of these IIFNs. Even though any two IIFNs can be compared by new comparison rules based on the intervalvalued membership per unit, this direct comparison method is unsuitable to rank the alternatives with some attributes. To further apply the new comparison method to make multi-attribute decisions and improve alternative’s efficiency under the interval-valued intuitionistic fuzzy environment, the enveloped efficiency of an IIFS is defined as follows: Definition 3. If there are K IIFSs, Ak (k  1, 2,

, K ) , are used to evaluate K alternatives

, yK ) with respect to n attributes ( x1 , x2 ,

, xn ) , then any Ak contains n IIFNs. The

( y1 , y2 ,

, K }) is defined as me* where me*  max{me } ,

enveloped efficiency of an IIFS Ae (e {1, 2, me  Ame  (1  A)me , and

p    p    me  1  1e 2 2e q1  1e  q2 2e  

   pn ne  i 1 pi ie = n  ,  qn ne  qi  ie n

i 1

9

(3)

p    p    me  1  1e 2 2e q1  1e  q2 2e  +

   pn ne  i 1 pi ie = , n    qn ne q   i ie n

(4)

i 1





In Eqs. (3) and (4), we have Ae  ([1e , 1e ],[1e , 1e ]),..., ([ ne ,  ne ],[ ne ,  ne ])

is an IIFS,

([ie , ie ],[ ie ,  ie ]) is an IIFN, A [0,1] and is a constant given by the DMs, pi , pi , qi , qi  0

are the weight parameters, ie , ie ,  ie ,  ie [0,1] , 0  ie   ie  1 , e  1,

, K , and i  1, 2,

,n .

Then, based on the optimal operation me*  max{me }  A max{me }  (1  A) max{me } , we can get the optimal enveloped efficiency of an IIFS Ae . For the unknown weight parameters, we have pi , pi , qi , qi  0 (i  1, 2,

, n) . Therefore, Eq.

(5) can be held when these parameters are given suitably.

0  i 1 pi ie / i 1 qi ie  1; n

n

 i 1 pi ie / i 1 qi ie  1. n

n

(5)

Thus, similar to the data envelopment analysis, we can add the constraint conditions me   [0,1] and me +  [0,1] (e {1, 2,

, K }) into Eqs. (3) and (4) to obtain the optimal values.

In the next section, Theorem 1 shows that max{me  }  max{me + } (e {1, 2,

, K }) under the

constraint conditions me   [0,1] , me +  [0,1] and pi , pi , qi , qi  0 . Based on the ratio comparison rules and the enveloped efficiency of IIFS, for any two IIFSs Ag and At , we have: If m*g  mt* , then yg

yt ; and if m*g  mt* , then yg ~ yt .

Here, the parameter A in this paper presents the DM’s risk preference and A [0,1] . Even though A is a variable, it is a constant in the enveloped efficiency calculation process. Therefore, the above equations hold true. Moreover, in our opinion, the bigger A , the more significance is put on the minimum membership degree which makes the DMs become risk averters; the smaller A , the more significance is put on the maximum membership degree and they tend to be risk seekers. Generally, we can set A as 0.5; thus, the DM can be considered to be risk-neutral. In this paper, we set A as 0.5. Of course, if A’s value is different, the calculation process is the same.

4. Envelopment analysis of IIFNs Although a decision can be made by comparing the IIFSs using the rules above, an existing 10

problem is how to obtain the enveloped efficiency me* , which is equivalent to the calculation of the weight parameters pi , pi , qi , and qi (i  1, 2,

, n) . In the following, we introduce the IIFEA

model for IIFS to address this issue according to the DEA principle. 4.1 IIFEA models and their dual forms The interval-valued intuitionistic fuzzy decision-making scenario is set as Definition 3. The enveloped efficiency of IIFS is calculated by me* . Then, we can construct the IIFEA model as follows:

  n p     n p     i 1 i ie   i 1 i ie  m  A max{m }  (1  A) max{m }  A max  n   (1  A) max  n        i 1 qi  ie    i 1 qi  ie   n p    /  n q    1, k  1, 2, , K i 1 i ik  i 1 i ik n n    p  / q    1, k  1, 2, , K   i ik i 1 i ik s.t.  i 1  pi , pi , qi , qi  0, i  1, 2, , n  0  A  1.  e

* e

 e

(6)

In Model (6), if me *  max{me } and me *  max{me } are obtained by substituting the optimal values of the IIFEA model, then we have me*  max{me }  me*  me* . Moreover, pi , pi , qi , qi are the decision variables in the above model. Furthermore, we can derive Theorem 1. Theorem 1. In the IIFEA model, we have me *  me +* . Proof. Based on the IIFEA model and Eqs. (3) and (4), we have

me  i 1 pi ie / i 1 qi ie and me  i 1 pi ie / i 1 qi ie , n

n

n

n

where me  [0,1] , me  [0,1] , ie  ie , and  ie   ie . Let me *  max{me }= i 1 pi*ie /  i 1 qi*ie , we have n

n

me *   i 1 pi*ie /  i 1 qi*ie   i 1 pi*ie /  i 1 qi*ie . n

If



n



n

n

pi*ie /  i 1 qi*ie  1 , then we have n

i 1

me +*  max{me }  max

If

n



p *ie /  i 1 qi*ie  1 , then i 1 i n

n



p  ie /  i 1 qiie   i 1 pi*ie /  i 1 qi*ie me * . i 1 i n

n

max



n

n



p  ie /  i 1 qiie  1 when i 1 i n

n

pi*  pi*

qi*  qi* . However, for pi* , qi*  0 and me [0,1] , we have me +*  1 . Then, me *  me +* .

11

and

Therefore, we have me *  me +* in the IIFEA model. Thus, we complete the proof of Theorem 1.

■

However, it is hard to obtain the optimal alternative by this non-linear IIFEA model. Then, the IIFEA model is further investigated as follows: For



q   0 and

n   i 1 i ik

 If w1 



q   i 1 i ie n



n i 1



q   0 , Eq. (5) can be transformed into Eq. (7).

n   i 1 i ik

pi ik  i 1 qi ik  0;

1



n

, w2 



q

  i 1 i ie n



then Model (6) can be presented as below:

1



n i 1

pi ik  i 1 qi ik  0. n

,  i1  w2 pi ,  i2  w1 pi ,  i1  w1qi , and  i2  w2 qi ,

me*  max(me )  max A i 1  1i ie  (1  A) i 1  i2 ie n

n



 n  1    n  2   0, k  1, 2, , K i 1 i ik  i 1 i ik  n  2    n  1   0, k  1, 2, , K   i1 i ik i ik s.t.  i 1 n n  i2 ie  1;  i1 ie  1 i 1 i 1  1  A  0;  1i ,  i2 , i1 , i2  0; i  1, 2, , n Furthermore, we can derive the dual model of Model (8) as follows:  e*  min( 1e   e2 )  K  1 +   1 + ;  K  2    2  , i  1, 2, , n e ie e ie k 1 k ik  k 1 k ik K 1    i  1, 2, , n   k ik  Aie , s.t.  k 1   K  k2 ik  (1  A)ie , i  1, 2, , n  k 1 1 2  k ,  k  0; 1  A  0; k  1, 2, , K . If  e  min{ e } and  e 1*

1

2*

(7)

(8)

(9)

 min{ e 2 } are the optimal values obtained based on Model (9),

2* 1 2 1 2 1 2 then we have  e*   1* e   e . Moreover,  i ,  i ,  i ,  i , k ,  k are the decision variables in Models

2* (8) and (9). In Model (9),  e*   1* represents the enveloped efficiency of IIFS Ae and e  e

 e* [0,1] . If  e*  1 , then the alternative ye is efficient; and if  e*  1 , then the alternative ye is relatively inefficient and its interval-valued membership and non-membership values should be modified. Thus, we can select the optimal alternative based on the proposed ratio operation and IIFEA model. Moreover, this IIFEA model is also called the IIFEA-N model as its principle is to modify the interval-valued non-membership values of IIFNs. Being similar to Model (8), Model (6) can be changed into Model (10) which can be applied to analyze the enveloped efficiency of all the alternatives on the condition of changing the interval-valued membership values. Therefore, the 12

following IIFEA model is called the IIFEA-M model. If v1 



n i 1

pi ie



1

, v2 



n i 1

pi ie



1

,  i1  v 2 pi ,  i2  v1 pi ,  i1  v1qi , and  i2  v 2 qi

then Model (6) can be presented as follows:



1/ me*  min A i 1 i1 ie  (1  A) i 1 i2 ie n

n



 n  1    n  2   0, k  1, 2, , K i 1 i ik  i 1 i ik n n 2  1    i 1  i ik   i 1 i  ik  0, k  1, 2, , K s.t.  n n   1i ie  1;   i2 ie  1 i 1  i 1 1 2 1 2  i ,  i i , i  0; 1  A  0; i  1, 2, , n. Furthermore, we can derive the dual model of Model (10) as follows:

(10)

 e*  max( 1e   e2 )  K  2   A  ;  K  1   (1  A)  , i  1, 2, , n ie ie k 1 k ik  k 1 k ik K 1  1   i  1, 2, , n   k ik   e ie , s.t.  k 1  K  k2 ik   e2 ie , i  1, 2, , n  k 1  1k ,  k2  0; 1  A  0; k  1, 2, , K .

(11)

If  e1*  max{ 1e } and  e 2*  max{ e2 } are optimal values obtained based on Model (11), then 2* 1 2 1 2 1 2 we have  e*   1* e   e . In Models (10) and (11),  i ,  i ,  i ,  i , k ,  k are the decision variables.

Moreover, the IFEA-M model and the weight parameters pi , pi , qi , and qi (i  1, 2,

, n)

can be calculated. In Model (11), we have  e*  1 . Then, we can use 1 /  e* to represent the enveloped efficiency of IIFS in Ae . If  e*  1 , then the alternative ye is efficient; and if  e*  1 , then the alternative

ye is relatively inefficient and its interval-valued membership and

non-membership values should be modified. Meanwhile, we can rank the alternatives and select the optimal one under the interval-valued intuitionistic fuzzy environment. Based on Models (9) and (11), we can derive the following theories: Theorem 2. If  e* and  e* are the optimal solutions of Models (9) and (11) respectively, then

 e*  0 and  e*  0 . Proof. If  e*  0 , then



K k 1

k1 ik+  0 ,



K k 1



K k 1

k1 ik+   e1 ie+ =0 and



k2 ik =0 , and k1  k2  0 (k  1, 2,



Furthermore, we have



K k 1

k1ik  Aie and



K k 1

K k 1

k1ik+ =0 and



K k 1

K k 1

k2 ik   e2 ie =0 . Thus, we have

, K).

k2 ik =0 which is contradictory with

k2 ik  (1  A)ie , in Model (9). Therefore, we have  e*  0 .

13

Similarly, we can get  e*  0 . Thus, we complete the proof of Theorem 2.

■

Theorem 3. If  e* and  e* are the optimal solutions of Models (9) and (11) respectively, then

 e*   e*  1

(12)

Proof. According to Model (9), we have

 e*  min( 1e   e2 )  K ( 1 /  ) +  ( 1 /  ) + e e ie  k 1 k e ik  K ( 2 /  )   ( 2 /  )  , e e ie  k 1 k e ik  K s.t.  k 1 ( 1k /  e )ik  A(ie /  e ),  K  k 1 ( 1k /  e )ik  (1  A)(ie /  e ),  1 2  k ,  k  0; 1  A  0; k  1, 2, , K . 

i  1, 2,

,n

i  1, 2,

,n

i  1, 2,

,n

i  1, 2,

,n

(13)

2 * 2* Furthermore, Model (13) can be transformed into Model(14) when ( 1k /  e* )   1* k , ( k /  e )   k

and 1/  e   e . Here,  1k and  k2 are the decision variables.

 e*  max( 1e   e2 )  K  2*   A  ;  K  1*   (1  A)  i  1, 2, , n ie ie k 1 k ik  k 1 k ik K 1*  1   i  1, 2, , n  k 1  k ik   e ie s.t.   K  k2*ik   e2 ie i  1, 2, , n  k 1  1k ,  k2  0; 1  A  0; k  1, 2, , K .

(14)

We can find that Model (14) is the same as Model (11). Then, we have  e*   e*  1 . Thus, we complete the proof of Theorem 3.

■

Therefore, we can only use Model (9) or (11) to calculate the enveloped efficiency of IIFS and make a decision under the interval-valued intuitionistic fuzzy environment. 4.2 Alternative development based on the IIFEA models As aforementioned, Models (9) and (11) can be applied to solve the IIFEA models. Here, we take Model (9) to demonstrate the calculation process of the IIFEA models. Obviously, the slack variables si1 , si2  si1 , and si2  should be added into Model (9). To further obtain the accurate slack variables, the Archimedes dimensionless  should also be introduced. Thus, we can obtain 14

the solution of the IIFEA-N model by solving Model (15). It is pointed out that the dimensionless

 can be set as 0.0001 in the practical application.



min  1e   e2    i 1 ( si1  si2  si1  si2 ) n



 K  1 +  s1 = 1 + , i  1, 2, , n e ie  k 1 k ik i  K  2   s 2 = 2  , i  1, 2, , n e ie  k 1 k ik i  K  1    s1 =A  , i  1, 2, , n ie s.t.  k 1 k ik i  K  2    s 2 =(1  A)  , i  1, 2, , n ie  k 1 k ik i  s1 , s 2 , s1 , s 2  0; i  1, 2, , n  i1 i 2 i i  k ,  k  0;1  A  0; k  1, 2, , K .

(15)

If  e1*  min{ e1} and  e 2*  min{ e 2 } are optimal values obtained based on Model (15), then 2* 1 2 1 2 1 we also have  e*   1* e   e . In this model,  k and  k are the decision variables and si , si , si ,

and si2  are slack variables. Moreover, we further define the comparison relationship as follows: (1) If  e*  1 and si1  si2  si1  si2  0 , then the alternative ye responds to Ae is efficient; (2) if

 e*  1 , si1  0 , and si2  0 , or  e  1 , si1  0 , and si2  0 , then the alternative ye responds to Ae is weakly efficient; and (3) if  e*  1 , then the alternative ye responds to Ae is inefficient. Based on these efficiency definitions, the comparison rules of alternatives under the interval-valued intuitionistic fuzzy environment can be given as follows: (1) If  e*   k* , then ye

yk (e, k {1, 2,

, K }) .

(2) If ye is weakly efficient, yk is efficient, and  e*   k* , then ye (3) If ye is efficient and yk is efficient, then ye

yk (e, k {1, 2,

yk (e, k {1, 2,

, K })

, K })

According to the three comparison rules above, we can distinguish and rank most of the alternatives under the interval-valued intuitionistic fuzzy environment. A more precise method can be built through using the IIFEA model again for the alternatives with the same efficiency values. Moreover, the alternative development methods, namely, Model (16) or (17), are proposed to explain how to optimize the inefficient alternatives by adjusting the interval-valued membership and the interval-valued non-membership and change them into efficient ones. Thus, the proposed IIFEA models can be used to calculate and rank the efficiency values of the alternatives, among which, the optimal object could be selected. In addition, this method illustrates how to improve the inefficient alternatives by modifying the interval-valued membership and interval-valued 15

non-membership values, which makes it different from other interval-valued intuitionistic fuzzy decision-making methods.

ie*  Aie  si1 , ie+*  (1  A) ie  si2  ie* = e1* ie  si1 ,  ie* = e2* ie  si2 .

(16)

 ie* =A ie  si1  ie+* =(1  A) ie+  si 2 , ie*   e1* ie  si1 , ie+*   e2* ie  si 2 .

(17)

where  k1* ,  k2* , si1 , si2  , si1 , and si2  are obtained based on Model (15),  k1* ,  k2* , si1 , si 2 , si1 , and si 2 are calculated by introducing the slack variables si1 , si 2 , si1 , and si2  into Model (11). Then, if the alternative ye responds to the IIFS Ae is inefficient, the interval-valued non-membership and interval-valued membership values of this alternative, [ ie ,  ie ] and [ ie , ie ] , should be transferred into [ ie* ,  ie* ] and [ ie* , ie* ] respectively. Therefore, the

modified alternative can be changed into a relatively efficient one. Note that the IIFEA-N model is applied in this paper. Therefore, Model (16) is the corresponding interval-valued membership modification formula. 5. Preference fusion, alternative enhancement, and modeling process in the IIFS environment 5.1 Preference fusion and alternative enhancement Basically, DMs have intuitive perception toward the important degrees of various attributes. As a result, the weight of an attribute is significant as it shows the different important degrees of this attribute. Thus, many weighted methods have been developed in recent years. However, these methods that calculate weights could be different, and the corresponding optimal alternatives may be contradictory. In this study, we focus on the attributes’ preference relationship rather than the accurate weights and propose the following preference interval-valued intuitionistic fuzzy envelopment analysis (PIIFEA) model. Above all, we form the intuitionistic decision scenario: if a DM gives K IIFSs Ak (k  1, 2,

( x1 , x2 ,

, K ) , to evaluate K

alternatives ( y1 , y2 ,

, yK ) with respect to n attributes

, xn ) . Thus, Ak  {([ 1k , 1k ],[ 1k , 1k ]),..., ([ nk , nk ],[ nk , nk ])} where k  1, 2,

Besides, the DM provides the preference relationship about three attributes xg

xl

,K .

xm according

to his/her subjective assessment. This DM wants to choose the best object based on the efficiency measurement and enhance the inefficiency alternatives. It is pointed out that the complex preference relationship can be introduced and modeled 16

similarly. Meanwhile, the decision-making process remains unchanged. To show the above DM’s preference relationship, namely xg

xl

xm , in Model (6), the

following relationship conditions can be used.

pg  pl  pm , pg  pl  pm , qg  ql  qm , and qg  ql  qm . These relationship conditions are utilized to form the PIIFEA model, which means that the preference relationship is presented by the weight parameters of the attributes x g , xl , and xm . Thus, the PIIFEA model is constructed as follows: n   n p  p  ie    i ie * i 1 i 1 i me =max  A n  (1  A) n      q    i 1 qi  ie  i1 i ie   n p    /  n q     1, k  1, 2, i 1 i ik  i 1 i ik  n p    / n q    1, k  1, 2,  i 1 i ik  i 1 i ik  p  p  p l m  g   s.t.  pg  pl  pm     qg  ql  qm q   q   q  l m  g     p , p , q , q   0; i  1, 2, , n.  i i i i

,K ,K (18)

In this model, pi , pi , qi , and qi are the decision variables. If w1  (i 1 qi ie ) 1 , n

w2  (i 1 qi ie ) 1 ,  i1  w2 pi ,  i2  w1 pi ,  i1  w1qi , and  i2  w2 qi , then we have: n



me*  max A i 1  1i ie  (1  A) i 1  i2 ie

s.t.

n

n



 n  1    n  2   0, k  1, 2, , K i 1 i ik  i 1 i ik n n   i2 ik   i 1 i1 ik  0, k  1, 2, , K  i  1  n  n  2   1;  1   1   i ie i 1 i ie  i 1  1   1  0;  2   2  0 l m l  m  1m  l1  0;  m2  l2  0  1 1 2 2  l   g  0;  l   g  0  1 1 2 2 l   g  0; l   g  0  1 2 1 2  i ,  i , i , i  0;1  A  0, i  1, 2, , n

(19)

Model(20) is the dual model of Model(19). In Models(19) to (21), 1i , i2 , i1 , i2 , 1k , k2 , 1ml , 1lg ,

17

2 2 and lg2 are the decision variables. By introducing the slack variables ml , lg2 , 1ml , lg1 ,  ml

si1 , si2 , si1 , si2 and the Archimedes dimensionless  , Model (20) is transformed into Model (21).

It can calculate the envelopment efficiency of IIFS with the DM’s preference relationship.

 e*  min( 1e   e2 )  K  1 +   1 + , i  1, 2, , n; i  g , m, l e ie  k 1 k ik  K  2    2  , i  1, 2, , n; i  g , m, l e ie  k 1 k ik  K  1    A  , i  1, 2, , n; i  g , m, l ie  k 1 k ik  K 2    k 1  k ik  (1  A)ie , i  1, 2, , n; i  g , m, l  K 1 + K 1 1 +  k 1  k2 mk  ml   e2 ie i  1, 2, , n  k 1  k  mk  ml   e ie  K 1 + K    1lg   1e ie+   k 1 k2 gk  lg   e2ie i  1, 2, , n s.t.  k 1 k gk K 2    2   K  1 +  1  1   1 + ,n  k mk ml lg e ie  k 1  k  mk  ml  lg   e  ie i  1, 2, k  1  K   K  1     1  A  ;  2     ml2  (1  A) me   k mk ml me k  1 k 1 k mk  K  K 1  1   k 1 k2 gk+  lg2  (1  A) ge+  k 1  k  gk   lg  A ge ; K  K 1  1 1  2 + 2 2 +  k 1  k  mk   ml   lg  Ale ;  k 1  k  mk   ml   lg  (1  A)le  1 2  k ,  k  0; k  1, 2, , K 1 , 1 ,  2 ,  2 ,  1 ,  1 ,  2 ,  2  0; 1  A  0.  ml lg ml lg ml lg ml lg

18

(20)



2 2 min  1e   e2   ( i 1 ( si1  si2  si1  si2 )  1ml  1lg  ml  lg2   1ml   lg1   ml   lg2 ) n

 K  1 +  s1   1 + , i  1, 2, , n; i  g , m, l e ie  k 1 k ik i  K  2   s 2   2  , i  1, 2, , n; i  g , m, l e ie  k 1 k ik i  K  1    s1  A  , i  1, 2, , n; i  g , m, l ie  k 1 k ik i  K 2  2   k 1  k ik  si  (1  A)ie , i  1, 2, , n; i  g , m, l  K 1 + K 1 1 1 + 2  2 2 2   k 1  k  mk  ml  sm   e ie ;  k 1  k  mk  ml  sm   e  ie i  1, 2,  K 1 + K 1 1 1 + 2  2 2 2   k 1  k  gk  lg  sg   e ie ;  k 1  k  gk  lg  sg   e  ie i  1, 2,  K  1 +  1ml  1lg  sl1   1e ie+ i  1, 2, , n  s.t.  k 1 k mk  K  2    2   2  s 2   2  i  1, 2, , n ml lg l e ie  k 1 k mk K K 1  1 1         s  A  ;  2     ml2  sm2  (1  A)me   k mk ml m me k 1 k mk  k 1 K  K 1  1 1  2  2 2   k 1  k  gk   lg  sg  A ne ;  k 1  k tk   lg  sg  (1  A) ne  K 1  1 1 1   k 1  k  mk   ml   ln  sl  Ale  K 2 + 2 2 2 +  k 1  k  mk   ml   lg  sl  (1  A)le  1 2 1 2 1 2  k ,  k , si , si , si , si  0; i  1, 2, , n; k  1, 2, , K 1 , 1 ,  2 ,  2 ,  1 ,  1 ,  2 ,  2  0; 1  A  0.  ml lg ml lg ml lg ml lg

,n



(21)

,n

As a result, Model (21) can provide the efficiency values of K alternatives ( y1 , y2 ,

, yK ) on

the aspects of the DM’s preferences under the interval-valued intuitionistic fuzzy environment. Also, we can use the above comparison rules to select the optimal alternative and find the inefficient ones. More importantly, this model shows how to modify the attributes’ interval-valued non-membership values of inefficient alternatives to optimize them. Therefore, the above PIIFEA model is named as the PIIFEA-N model. Note that Model (20) focuses on the interval-valued non-membership of attributes and calculates the optimal interval-valued non-membership values under the condition of keeping interval-valued membership values fixed. To investigate the approach to modify the interval-valued membership and optimize the inefficient alternative, we introduce the PIIFEA model which focuses on the interval-valued membership and is called the PIIFEA-M model. It is found that these two PIIFEA models are not conflicting. Both of them are used to show how to change interval-valued non-membership and interval-valued membership to optimize the inefficient alternatives. In the following, the PIIFEA-M model is constructed. 19

Let

v1 



n i 1

pi ie



1

,

v2 



n i 1

pi ie



1

,  i1  v 2 pi ,  i2  v1 pi ,

 i1  v1qi , and

 i2  v 2 qi , we can get the linear PIIFEA model according to Model (19), namely Model (22). Furthermore, we can derive the dual form of Model (22), and then obtain the computable linear programming of PIIFEA-M model by introducing the slack variables si1 , si2 , si1 , si2 and the Archimedes dimensionless  . Here,  1i ,  i2 , i1 , i2 are the decision variables. The dual form of Model (22) is similar to Model (20), and the computable linear programming of the PIIFEA-M model is similar to Model (21). To avoid repetition, we don’t present the above PIIFEA-M models.



(1/ me* )  min A i 1 i1 ie  (1  A) i 1 i2 ie

s.t.

n

n



 n  1    n  2   0, k  1, 2, , K i 1 i ik  i 1 i ik n n   i2 ik   i 1 i1 ik  0, k  1, 2, , K  i  1  n  n  1   1;  2   1   i ie i 1 i ie  i 1  1   1  0;  2   2  0;  1   1  0;  2   2  0 l m l m l m l  m 1 1 2 2 1 1 2  l   g  0;  l   g  0; l   g  0; l   g2  0  1 2 1 2  i ,  i , i , i  0; 1  A  0; i  1, 2, , n.

(22)

Thus, the efficiency values of alternatives with respect to the DM’s preferences can be calculated by applying the PIIFEA-N and PIIFEA-M models under the interval-valued intuitionistic fuzzy environment. Similarly, the above alternative development method, Model (16) or (17), could also be utilized to enhance inefficient alternatives by modifying the interval-valued membership and non-membership values. Therefore, we can rank the alternatives based on the efficiency values calculated by the PIIFEA-N or PIIFEA-M model, and optimize the inefficient alternatives through using Model (16) or (17). 5.2 Decision-making process based on the IIFEA and PIIFEA models The optimal alternative can be selected by comparing the enveloped efficiency values of all the alternatives under the interval-valued intuitionistic fuzzy environment, which are calculated by the IIFEA model. Meanwhile, as for the DM’s preference relationship, we further propose the PIIFEA model to calculate the enveloped efficiency values of alternatives. It can be found that the envelopment analysis model proposed above can demonstrate the DM’s subjective preference, which is different from other interval-valued intuitionistic fuzzy multi-attribute decision-making approaches. In addition, the IIFEA and PIIFEA models are able to provide quantitative results to 20

optimize inefficient alternatives. In the following, the decision-making process based on intervalvalued intuitionistic fuzzy enveloped efficiency calculated by the IIFEA and PIIFEA models are demonstrated as follows: Step 1. There are D DMs who give their qualitative evaluation using the IIFSs for K alternatives ( y1 , y2 ,

, yK ) with respect to n attributes ( x1 , x2 ,

intuitionistic fuzzy matrices U d  ([ ik d , ik d ], [ ik d , ik d ]) nK 1, 2,

, xn ) , and D interval-valued (i  1, 2,

, n; k  1, 2,

, K; d 

, D) are obtained.

Step 2. Utilize the general operations to aggregate U d  [ik d , ik d ], [ ik d , ik d ] i  1, 2,

, n; d  1, 2,

n K

(k  1, 2,

, K;

, D) and obtain the aggregated interval-valued intuitionistic fuzzy matrix

U  [ik , ik ], [ ik , ik ]

n K

(i  1, 2,

, n; k  1, 2,

, K) .

In this step, the general aggregation operations can be achieved by the interval-valued intuitionistic fuzzy averaging, geometric, weighted averaging, or weighted geometric aggregation operators (Xu and Yager, 2006; Xu, 2007). With respective the difficult to obtain accurate weight values, the first two operators are more suitable for the proposed models. Step 3. Use the IIFEA-N model to calculate the enveloped efficiency values of K alternatives ( y1 , y2 ,

, yK ) based on the U  [ik , ik ], [ ik , ik ]

slack variables are si1 , si2  , si1 , and si2  (i  1, 2,

n K

(i  1, 2,

, n; k  1, 2,

, K ) and the

, n) .

Step 4. Rank all the alternatives according to the enveloped efficiency values  e* (e {1, 2,

, K }) .

Step 5. Utilize the alternative development methods (Model (16)) and the IIFEA-N model to calculate the optimal interval-valued non-membership and interval-valued membership values, and provide the suggestions to optimize the inefficient alternative. It is pointed out that: (1) As aforementioned in Subsection 2.2, the parameter A in objective function of the IIFEA and PIIFEA models is set as 0.5; (2) the PIIFEA-N model should be used in Steps 3 if the DM can give his/her subjective preference or intuitionistic comparison about some attributes; (3) if there is only one DM, then Step 1 should be omitted and the decision-making process starts from Step 2; (4) If the DM focuses on the interval-valued membership modification to enhance the inefficient alternatives, then the IIFEA-M and PIIFEA-M models should be used. It is pointed the calculation process is the same as the above steps. 6. Illustrative example 21

In this section, we apply the above definitions and methods to a situational efficiency decisionmaking case. The example is about an international hotel’s promotion plan, which demonstrates the feasibility of the proposed methods under the interval-valued intuitionistic fuzzy environment. After that, further analyses are provided to show the effectiveness of the above methods. 6.1. Example and calculations Example 2. Suppose that a CEO of an international hotel who wants to promote one of his subordinates to be the general manager (GM) of their biggest branch in China in order to reinforce the organizational culture, improve the interactions between the branch and the headquarter and increase the market occupancy eventually. After discussing with the board, there are six candidates ( y1 , y2 , y3 , y4 , y5 , y6 ) who have different leadership talents. However, the CEO still needs to select

one subordinate among the six candidates based on knowing his/her competitiveness and shortages. Also, knowing how to enhance all the candidates to be better is in the scope of the CEO’s consideration. In this case, the CEO decides to introduce the qualitative information presented by the IIFNs to accurately describe the candidates’ competitiveness from the aspects of “good” and “bad” and calculate their competitiveness values and find the shortages. It is easy to find that the proposed IIFEA and PIIFEA models can be used to address this issue. To elicit and get the credible evaluation information presented by the IIFNs from experts and DMs, the following three important pre-modeling steps have been performed which can be similarly applied to repeat the whole decision-making process: (1) Select professional and experienced experts or DMs. In this case, the CEO invites three experts including the GM of their headquarter d1 , the director of human resources d 2 , and the marketing director d 3 . (2) Provide a questionnaire to show the IIFN evaluation information with respect to some suitable attributes. In this case, the four competitiveness attributes are used. They are personal morality x1 , environmental adaptability x2 , decision-making skill x3 , and personal development potential on the basis of working performance x4 . (3) Organize a survey for the selected experts and DMs. If the responses to the questionnaire are credible, then we can go to the given modeling process. Otherwise, the questionnaire survey should be done again. Thus, we can get the IIFN evaluation information. In this case, after the above pre-modeling steps, there are three interval-valued intuitionistic fuzzy 22

matrices U d  ([ ik d , ik d ], [ ik d , ik d ]) 46

(i  1, 2,3, 4; k  1, 2,

, 6; d  1, 2,3) provided by the

above three DMs. More details about the matrices could be seen in Table 2. The CEO believes that personal development potential x4 is the most important, following with environmental adaptability x2 , decision-making skills x3 and personal morality x1 . Hence, we have a preference relationship x4

x2

x1 .

x3

In order to find the ideal person who is the most qualified to be the GM on the basis of knowing his/her competitiveness and shortages, the decision-making process based on enveloped efficiency of IIFS is introduced as follows: Step 1. According to the above pre-modeling steps, we get three interval-valued intuitionistic fuzzy matrices which are shown in Table 2. Table 2. The interval-valued intuitionistic fuzzy matrices in Example 2. Matrices

Alternatives Attributes

x1 x2 U1

x3 x4

x1 x2 U2

x3

x4 x1

x2 U3

x3 x4

y1

y2

y3

y4

y5

y6

([0.30,0.35],

([0.25,0.35],

([0.35,0.40],

([0.35,0.40],

([0.25,0.30],

([0.10,0.15],

[0.55,0.60])

[0.60,0.65])

[0.55,0.60])

[0.55,0.60])

[0.65,0.70])

[0.70,0.85])

([0.40,0.45],

([0.35,0.40]

([0.30,0.35],

([0.45,0.50],

([0.60,0.65],

([0.20,0.30],

[0.50,0.55])

[0.50,0.55])

[0.55, 0.60])

[0.45, 0.50])

[0.30,0.35])

[0.60,0.65])

([0.40,0.45],

([0.10,0.15],

([0.25,0.30],

([0.55,0.60],

([0.45,0.50],

([0.20,0.25],

[0.50,0.55])

[0.70,0.80])

[0.65, 0.70])

[0.35, 0.40])

[0.45,0.50])

[0.40, 0.50])

([0.20,0.35],

([0.10,0.15],

([0.50,0.60],

([0.10,0.15],

([0.20,0.35],

([0.25,0.30],

[0.40,0.55])

[0.70,0.75])

[0.35, 0.40])

[0.70, 0.80])

[0.40,0.55])

[0.65,0.70])

([0.60,0.55],

([0.40,0.45],

([0.55,0.60],

([0.20,0.25],

([0.65,0.70],

([0.20,0.25],

[0.40,0.45])

[0.50,0.55])

[0.35,0.40])

[0.60, 0.65])

[0.25,0.30])

[0.65,0.70])

([0.45,0.50],

([0.25,0.35],

([0.40,0.45],

([0.40,0.45],

([0.35,0.40],

([0.15,0.25],

[0.45,0.50])

[0.45,0.55])

[0.45,0.50])

[0.50, 0.55])

[0.55,0.60])

[0.70,0.75])

([0.45,0.50],

([0.20,0.25],

([0.30,0.35],

([0.35,0.40],

([0.55,0.60],

([0.15,0.20],

[0.45,0.50])

[0.65, 0.70])

[0.60,0.65])

[0.55,0.60])

[0.35,0.40])

[0.55, 0.60])

([0.25,0.30],

([0.20,0.25],

([0.20,0.30],

([0.20,0.25],

([0.25,0.30],

([0.30,0.35],

[0.45,0.50])

[0.60, 0.70])

[0.50,0.60])

[0.65, 0.70])

[0.45,0.50])

[0.60, 0.65])

([0.30,0.35],

([0.35,0.40],

([0.35,0.40],

([0.30,0.40],

([0.55,0.60],

([0.15,0.25],

[0.60,0.65])

[0.55,0.60])

[0.55,0.60 ])

[0.55, 0.60])

[0.30,0.35])

[0.45,0.60])

([0.55,0.60],

([0.50,0.55],

([0.40,0.45],

([0.25,0.30],

([0.45,0.50],

([0.40,0.50],

[0.35,0.40])

[0.40,0.45])

[0.50, 0.55])

[0.60, 0.70)]

[0.45,0.50])

[0.35,0.40])

([0.40,0.45],

([0.30,0.40], ([0.55,0.60]), ([0.25,0.35],

([0.40,0.45],

([0.20,0.25],

[0.50,0.55])

[0.50, 0.60])

[0.35,0.40])

[0.55, 0.60])

[0.50,0.55])

[0.65,0.75])

([0.20,0.25],

([0.30,0.40],

([0.35,0.40],

([0.30,0.40],

([0.20,0.25],

([0.55,0.60],

[0.60,0.65])

[0.40,0.50])

[0.45, 0.60])

[0.50,0.60])

[0.60,0.65])

[0.35, 0.40])

Step 2. If the questionnaire is credible, then we aggregate the above matrices U 1 , U 2 , and U 3 by 23

using the interval-valued intuitionistic fuzzy averaging aggregation operator. Thus, we get the aggregated interval-valued intuitionistic fuzzy matrix shown in Table 3. Note that the intervalvalued intuitionistic fuzzy weighted operators are suitable if the accurate weights are available. Table 3. The aggregated interval-valued intuitionistic fuzzy matrix in Example 2. Alternatives

y1

Attributes

y2

y3

y4 ([0.636,0.73], [0.182,0.234])

y5

y6

x1

([0.804,0.810], ([0.708,0.786], ([0.810,0.856], [0.132,0.190]) [0.165,0.215]) [0.106,0.144])

([0.882,0.916], ([0.388,0.522], [0.049,0.074]) [0.205,0.357])

x2

([0.852,0.89], ([0.756,0.825], ([0.748,0.803], ([0.7520,0.808], ([0.857,0.895], ([0.592,0.738], [0.079,0.11]) [0.09,0.136]) [0.124,0.165]) [0.135,0.192]) [0.074,0.105]) [0.147,0.195])

x3

([0.802,0.849], ([0.496,0.618], ([0.764,0.818], [0.113,0.151]) [0.228,0.336]) [0.137,0.182])

([0.781,0.844], [0.106,0.144])

([0.852,0.89], ([0.456,0.550], [0.078,0.11]) [0.143,0.225])

x4

([0.52,0.659], ([0.496,0.618], ([0.740,0.832], [0.108, 0.179]) [0.168, 0.263]) [0.079,0.144])

([0.496,0.618], [0.228,0.336])

([0.520,0.659], ([0.764,0.818], [0.108,0.179]) [0.137,0.182])

In the following, we apply the IIFEA-N and PIIFEA-N models to separately calculate the enveloped efficiency values of the candidates to show their competitiveness and development schemes. It is pointed out that the PIIFEA-N model is constructed by fusing the preference relationship x4

x2

x3

x1 . Thus, Step 3 is the calculation process using the IIFEA-N model,

and Step 3º is the calculation process using the PIIFEA-N model. Step 3. Based on U  [ik , ik ], [ ik , ik ]

46

, we construct the IIFEA-N model as Model (23)

where  1k and  k2 are the decision variables and si1 , si2 , si1 and si2  are the slack variables. If

 e1* and  e 2* are the optimal values of  e1 and  e 2 obtained by solving Model (23), then we 2* have  e*   1* e   e . Furthermore, by solving Model (23), we obtain the efficiency values and the

slack variables shown in Table 4.

24



min  e1   e2  0.0001  i 1 ( si1  si2  si1  si2  ) 4



0.19011  0.21521  0.14431  0.23441  0.07451  0.35761  s11   e1 1e  1 1 1 1 1 1 1 1  0.1101  0.1362  0.1653  0.1924  0.1055  0.1956  s2   e 2 e 0.151 1  0.336 1  0.182 1  0.144 1  0.110 1  0.225 1  s1   1  1 2 3 4 5 6 3 e 3e  0.17911  0.26321  0.14431  0.33641  0.17951  0.18261  s14   e1 4e  0.13212  0.16522  0.10632  0.18242  0.04952  0.20562  s12   e2 1e  2 2 2 2 2 2 2 2  0.0791  0.0902  0.1243  0.1354  0.0745  0.1476  s2   e 2 e 0.113 2  0.228 2  0.137 2  0.106 2  0.078 2  0.143 2  s 2   2  1 2 3 4 5 6 3 e 3e  2 2 2 2 2 2 2 0.1081  0.1682  0.0793  0.2284  0.1085  0.1376  s4   e2 4e  1 1 1 1 1 1 1  0.8041  0.7082  0.8103  0.6364  0.8825  0.3886  s1  A1e s.t.  1 1 1 1 1 1 1  0.8521  0.7562  0.7483  0.7524  0.8575  0.5926  s2  A 2 e  1 1 1 1 1 1 1  0.8021  0.4962  0.7643  0.7814  0.8525  0.4566  s3  A3e 0.520 1  0.496 1  0.740 1  0.496 1  0.520 1  0.764 1  s1  A  1 2 3 4 5 6 4 4e  2 2 2 2 2 2 2 0.8101  0.7862  0.8563  0.7304  0.9165  0.5226  s1  (1  A) 1e  2 2 2 2 2 2 2  0.8901  0.8252  0.8033  0.8084  0.8955  0.7386  s2  (1  A) 2 e 0.849 2  0.618 2  0.818 2  0.844 2  0.890 2  0.550 2  s 2  (1  A)   1 2 3 4 5 6 3 3e  2 2 2 2 2 2 2  0.6591  0.6182  0.8323  0.6184  0.6595  0.8186  s4  (1  A) 4e  1 2 1 2 1 2 k , k , si , si , si , si  0; k  1, 2,3, 4,5, 6; i  1, 2,3, 4 1  A  0; e  1, 2,3, 4,5, 6. 

(23)

Table 4. The calculated results of the IFEA-N model.

 e*

s11

s12

y1 y2 y3 y4

0.9864

0.0548

0.0000

0.0174 0.0000 0.0383 0.0000 0.0144 0.0000 0.0377 0.0000 0.0231 0.0073 0.0516 0.0000 0.0187 0.0066

0.7537

0.0439

0.0000

0.0713 0.0115 0.0406 0.0000 0.0513 0.0141 0.0666 0.0307 0.1583 0.0000 0.0365 0.0072 0.1083 0.0000

1.0000

0.0000

0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.7132

0.0500

0.0199

0.0000 0.0370 0.0403 0.0120 0.0000 0.0283 0.1026 0.0327 0.0158 0.0000 0.0693 0.0204 0.0000 0.0035

y5 y6

1.0000

0.0000

0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.8309

0.0820

0.0000

0.0046 0.0000 0.0414 0.0058 0.0000 0.0000 0.2425 0.1096 0.1849 0.0000 0.2417 0.1148 0.2104 0.0000

IIFEA-N

s31

s14

s12 

s22 

Step 3º. Based on U  [ik , ik ], [ ik , ik ]

s32 

46

s42 

and x4

s11

x2

s12

x3

s31

s14

s12 

s22 

s32 

x1 , we construct the PIIFEA-N

model as Model (24) where  1k ,  k2 , 1 ,  2 ,  1 and  2 are the decision variables and si1 , si2 , si1 and si2  are the slack variables. If  e1* and  e 2* are the optimal values of  e1 and  e 2 obtained 2* by solving Model (24), then we have  e*   1* e   e . Furthermore, by solving Model (24), we

obtain the efficiency values and the slack variables shown in Table 5. 25

s42 

1 1    4 ( si1  si2  si1  si2 )  131  32   24  132    1 2  min   e   e  0.0001  i 1   2   2   1 + 1 + 1 + 2 + 2 + 2    32 24 13 32 24 13 32 24   1 0.19011  0.21521  0.14431  0.23441  0.07451  0.35761  13  s11   e1 12e  1 1 1 1 1 1 1 1 1 2 0.1101  0.1362  0.1653  0.1924  0.1055  0.1956  32  s2   e 2 e 0.151 1  0.336 1  0.182 1  0.144 1  0.110 1  0.225 1   1  s1   1 2 1 2 3 4 5 6 24 3 e 3e  1 1 1 0.17911  0.26321  0.14431  0.33641  0.17951  0.18261  13  32  24  s14   e1 42e  (24) 0.13212  0.16522  0.10632  0.18242  0.04952  0.20562  132  s12    e2 11e  2 2 2 2 2 2 2 2 2 1 0.0791  0.0902  0.1243  0.1354  0.0745  0.1476  32  s2   e 2 e 0.113 2  0.228 2  0.137 2  0.106 2  0.078 2  0.143 2   2  s 2   2 1 1 2 3 4 5 6 24 3 e 3e  0.10812  0.16822  0.07932  0.22842  0.10852  0.13762  132  322  242  s42   e2 41e  1 1 1 1 1 1 1 1 1 0.8041  0.7082  0.8103  0.6364  0.8825  0.3886  13  s1  A1e s.t.  1 1 1 1 1 1 1 1 1 0.8521  0.7562  0.7483  0.7524  0.8575  0.5926   32  s2  A 2 e 0.802 1  0.496 1  0.764 1  0.781 1  0.852 1  0.456 1   1  s1  A 1 1 2 3 4 5 6 24 3 3e  1 1 1 1 1 1 1 1 1 0.5201  0.4962  0.7403  0.4964  0.5205  0.7646  13  32   24  s14  A41e  2 2 2 2 2 2 2 2 2 0.8101  0.7862  0.8563  0.7304  0.9165  0.5226  13  s1  (1  A) 1e  2 2 2 2 2 2 2 2 2 0.8901  0.8252  0.8033  0.8084  0.8955  0.7386  32  s2  (1  A) 2 e 0.849 2  0.618 2  0.818 2  0.844 2  0.890 2  0.550 2   2  s 2  (1  A)  2 1 2 3 4 5 6 24 3 3e  2 2 2 2 2 2 2 2 2 2 0.6591  0.6182  0.8323  0.6184  0.6595  0.8186  13   32   24  s4  (1  A)  42e  1 2 1 2 1 2 k , k , si , si , si , si  0; i  1, 2,3, 4; k  1, 2,3, 4,5, 6  1 1 1 2 2 2 1 1 1 2 2 2 13 ,32 , 24 ,13 ,32 , 24 ,13 ,32 ,  24 , 13 , 32 ,  24  0; 1  A  0; e  1, 2,3, 4,5, 6.

Table 5. The calculated results of the PIIFEA-N model. IIFEA-N

 e*

s11

s12

y1

0.9746

0.0562

0.0000

0.0191 0.0000 0.0396 0.0000 0.0159 0.0000 0.0374 0.0000 0.0235 0.0000 0.0515 0.0000 0.0191 0.0000

y2 y3 y4 y5 y6

0.726

0.0418

0.0000

0.0789 0.0000 0.0402 0.0000 0.0648 0.0000 0.0470 0.0000 0.1393 0.0000 0.0323 0.0000 0.1042 0.0000

0.9681

0.0315

0.0000

0.0309 0.0000 0.0242 0.0000 0.0236 0.0000 0.0715 0.0000 0.0783 0.0000 0.0539 0.0000 0.0592 0.0000

0.5243

0.0288

0.0000

0.0000 0.0000 0.0241 0.0000 0.0000 0.0000 0.0834 0.0000 0.0000 0.0000 0.0604 0.0000 0.0000 0.0000

1.0000

0.0000

0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.6916

0.0960

0.0000

0.0293 0.0000 0.0412 0.0000 0.0068 0.0000 0.2403 0.0000 0.1915 0.0000 0.1975 0.0000 0.1706 0.0000

131

1  32

1  24

y1 y2 y3

0.0000

0.0014

0.0000

0.0000

0.0000

0.0000

y4 y5

IIFEA-N

s31

132

s14

 322

s12 

 242

s22 

131

s32 

321

s42 

1  24

s11

132

s12

322

s31

 242

s14

s12 

s22 

s32 

s42 

/

/

/

/

/

0.0000 0.0014 0.0000 0.0000 0.0009 0.0000 0.0000 0.0011 0.0000

/

/

/

/

/

0.0109

0.0000 0.0000 0.0140 0.0000 0.0116 0.0000 0.0000 0.0030 0.0000

/

/

/

/

/

0.0252

0.0000

0.0000 0.0196 0.0000 0.0000 0.0890 0.0000 0.0000 0.0693 0.0000

/

/

/

/

/

0.0000

0.0034

0.0116

0.0000 0.0004 0.0089 0.0000 0.0141 0.0027 0.0000 0.0116 0.0087

/

/

/

/

/

0.0000

0.0000

0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

/

/

/

/

/

26

y6

0.0000

0.0206

0.0000

0.0000 0.0101 0.0000 0.0000 0.1260 0.0000 0.0000 0.0791 0.0000

/

/

/

/

Table 6. The candidate development schedules based on the IIFEA-N model. Attributes Alternatives

x1

x2

x3

x4

Membership

Nonmembership

Membership

Nonmembership

Membership

Nonmembership

Membership

Nonmembership

[0.8417 ,0.8616] [0.7746, 0.8225],

[0.0772, 0.1001] [0.1211, 0.1369]

[0.8520, 0.8900] [0.7856, 0.8615]

[0.0790, 0.1100] [0.090, 0.1360]

[0.8251, 0.8677] [0.6543, 0.7263]

[0.0956, 0.1366] [0.1567, 0.2224]

[0.5273, 0.6656] [0.4960, 0.6180]

[0.1080, 0.1790] [0.1565, 0.2489]

[0.8100, 0.8560] [0.7386, 0.7993]

[0.1060, 0.1440] [0.1320, 0.1937]

[0.7480, 0.8030] [0.7847, 0.8284],

[0.1240, 0.1650] [0.1151, 0.1632]

[0.7640, 0.8180] [0.7968, 0.8440]

[0.1370, 0.1820] [0.1060, 0.1440]

[0.7400, 0.8320] [0.4960, 0.6215]

[0.0790, 0.1440] [0.1910, 0.3077]

[0.8820 [0.0490, [0.8570, 0.9160] 0.0740] 0.8950] [0.6305, [0.1230, [0.7016, y6 0.7637] 0.1949] 0.8528], (Note: The bold and italic numbers are unchanged)

[0.0740, 0.1050] [0.1414, 0.1470]

[0.8520, 0.8900], [0.6049, 0.7604]

[0.0780, 0.1100] [0.1384, 0.2250]

[0.5200, 0.6590] [0.7640, 0.8180]

[0.1080, 0.1790] [0.1370, 0.1820]

y1 y2 y3 y4 y5

Table 7. The candidate development schedules based on the PIFEA-N model. Attributes Alternatives

x1

x2

x3

x4

Membership

Nonmembership

Membership

Nonmembership

Membership

Nonmembership

Membership

Nonmembership

[0.8414, 0.8615] [0.7550, 0.8183] [0.8815, 0.9099] [0.7194, 0.7904] [0.8820, 0.9160] [0.6283, 0.7195]

[0.0758, 0.0989] [0.1232, 0.1748] [0.0659, 0.0745] [0.1532, 0.2096] [0.0490, 0.0740] [0.1090, 0.2393]

[0.8520, 0.8900] [0.7560, 0.8250] [0.7480, 0.8030] [0.7520, 0.8080], [0.8570, 0.8950] [0.5920, 0.7380],

[0.0790, 0.1100] [0.0900, 0.1360] [0.1240, 0.1650] [0.1350, 0.1920] [0.0740, 0.1050] [0.1470, 0.1950]

[0.8255, 0.8681] [0.6353, 0.7220] [0.8423, 0.8772] [0.7810, 0.8440], [0.8520, 0.8900], [0.6475, 0.7206]

[0.0939, 0.1160] [0.1491, 0.2712] [0.1061, 0.1324] [0.1060, 0.1440] [0.0780, 0.1100] [0.1137, 0.2182]

[0.5200, 0.6590] [0.4960, 0.6180] [0.7400, 0.8320] [0.4960, 0.6180] [0.5200, 0.6590] [0.7640, 0.8180]

[0.1080, 0.1790] [0.1680, 0.2630] [0.0790, 0.1440] [0.2280, 0.3360] [0.1080, 0.1790] [0.1370, 0.1820]

y1 y2 y3 y4 y5 y6

(Note: The bold and italic numbers are unchanged)

Step 4. According to Table 4, we have y3 ~ y5

y1

y6

y2

y4 . Thus, the candidate y3 and

y5 are the best among these six candidates. While it can be found in Table 5, we have y5

y1

y3

y2

y6

y4 and the enveloped efficiency value of the candidate y5 is the highest.

Step 5. Based on Tables 6 and 7 and Model (16), we can modify the interval-valued membership and non-membership values to enhance the “inefficient” candidates as follows: (1) If the four attributes are equal and the IIFEA-N model is used, then the alternative development schemes of the candidates are shown in Table 6. Here, y3 and y5 are “efficient” candidates and can be unmodified. 27

/

(2) If the preference relationship x4

x2

x3

x1 is considered and the PIIFEA-N model is

used, then the alternative development schemes of the candidates are shown in Table 7. Here, only y5 is an “efficient” candidate and can be unmodified.

It is clear that the modified membership and non-membership values are the advised results obtained by the models above. We can address the two issues given by the CEO: (1) Despite the preference relationship, both the candidate 3 and 5 are qualified to be the GM because they have the capability to do better than others on the aspects of personal morality and decision-making skills. They can also adapt to the new environment as soon as possible and have lots of potential for personal development. (2) If the four attributes’ preference relationship is considered, the candidate 5 is even better than the candidate 3. Therefore, the candidate 5 can be selected as GM. For other candidates, their development schemes can be seen in table 7. Take the candidate 3 as an example, he or she should reduce his non-membership values of the four attributes to [0.0659,0.0745], [0.1240,0.1650],[0.1061,0.1324], [0.0790,0.1440], respectively. 6.2. Result analysis and comparison There are some conclusions according to the comparison between Tables 4 and 5:  The efficiency values calculated by the IIFEA-N model are larger than which calculated by the PIIFEA-N model. The reason is clear to see. The PIIFEA-N model considers more elements than IIFEA-N model does, which not merely requires alternatives to be the most efficient, but under the subjective preference environment as well.  The slack variables calculated by the IIFEA-N and PIIFEA-N models are different, which is mainly about the differences between the alternative development methods for the inefficient candidates. Specifically, unlike the PIIFEA-N model, the objective of the IIFEA-N model is to calculate the optimal enveloped efficiencies of all the alternatives. While for the PIIFEA-N model, the objective is to measure the optimal enveloped efficiencies of all the alternatives with satisfying the preference relationship provided by DMs. By comparing Tables 6 and 7, we can derive the following conclusions:  Compared with other decision-making approaches, the proposed method not only shows the optimal alternatives, but it is also able to calculate attributes’ modified interval-valued membership and non-membership values to help optimize the inefficient alternatives. This 28

prominent advantage cannot be achieved by other decision-making approaches.  Although the alternatives’ enveloped efficiency values are calculated differently by the IIFEA and PIIFEA models, the latter is more reliable for DMs to utilize in real-life decision-making because the enveloped efficiency values are calculated according to preference relationships. In real life, DMs may have their own consideration about different importance degrees of each attribute as they have to make decisions based on their experience, the actual environment and specific goals that are difficult to be measured in quantitative ways. Hence, the PIIFEA model is more subjective and quite flexible for DMs who have certain targets. That is also the second advantage compared with other similar decision-making methods.  It is summarized that the qualitative efficiency of an alternative can be improved by increasing

the

interval-valued

membership

values

and

decreasing

the

interval-valued

non-membership values. However, the enveloped efficiency value of the efficient alternative cannot be increased using this approach. The example’s final results are calculated by the IIFEA-N and PIIFEA-N models which focus on the interval-valued non-membership values. When a leader considers assigning his subordinates to the right positions which are suitable for them according to their specific capabilities, the proposed method could be used to help the leader make the decision. More importantly, if the candidates are not qualified yet and need to enhance themselves in certain aspects, the efficiency enhancement approaches are also of great significance to help the candidates to be better.

7. Conclusions In this study, the IIFN is considered to be an effective tool in the qualitative decision-making process. Even though there are some prior researches about the envelopment analysis which utilize real numbers instead of interval values, and have been proved to be effective in decision-making, it is believed that IIFN is more convenient for DMs to give their evaluations when they feel it is hard to assess an alternative with an accurate IFN or real number from two perspectives of “good” and “bad”. Think about the case in the last section, the 6 candidates’ performance may be changeable as time goes by, which also leads the DMs’ evaluations for them become floating relatively. Thus, using an accurate IFN or real number seems unfair to the candidates. Due to this reason, an IIFN allows a DM to give his/her assessment for one candidate in a range from two perspectives of “good” 29

and “bad” based on this candidate’s performance at a given time. Therefore, it is necessary to take the above situations into account. Thus, in this paper, the main contribution is the proposal of the IIFEA and PIIFEA models. Compared with the similar methods, they have three desired advantages: (1) the methods can be applied to make decisions in the interval-valued intuitionistic fuzzy environment by calculating the enveloped efficiency values of all the alternatives, which is more reasonable than the corresponding aggregation decision-making methods; (2) instead of accurate attribute weights, the DM’s qualitative preferences can be directly used in the proposed models; (3) not only the method is helpful for DMs to select the optimal alternative, it also provides the development schemes for the inefficient alternatives by using the models proposed above. We believe that the IIFEA and PIIFEA models could be applied in many decision-making fields under the interval-valued intuitionistic fuzzy environment. However, it is admitted that the proposed methods are hampered by certain limitations, such as (1) In the enveloped efficiency of IIFS, A is a vital constant; however, it is subjectively determined by the DM, and (2) the methods are proposed based on the small sample, which cannot deal with the large scale decision-making issues under the interval-valued intuitionistic fuzzy environment. Moreover, it is also pointed out that these proposed models can be further improved by generalizing the IIFS and introducing the fuzzy inequality constraints (Ghodousian and Parvari, 2017). Although there are still some aspects waiting to be optimized, further studies in the future about the proposed method could be worthy of improving decision-making effectiveness. Declarations of interest: none References [1] Adak, A.K., Bhowmik, M., & Pal, M. (2012). Interval cut-set of generalized interval-valued intuitionistic fuzzy sets. International Journal of Fuzzy System Applications, 2(3), 35–50. [2] Ahmady, N., Azadi, M., Sadeghi, S.A.H., et al. (2013). A novel fuzzy data envelopment analysis model with double frontiers for supplier selection. International Journal of Logistics Research and Applications, 16, 87–98. [3] Atanassov, K. & Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets & Systems, 31(3), 343–349. [4] Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets & Systems, 20(1), 87–96. [5] Bagherikahvarin, M., & Smet, Y. D. (2017). Determining new possible weight values in promethee: a procedure based on data envelopment analysis. Journal of the Operational Research 30

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Highlights:

The interval-valued intuitionistic fuzzy envelopment analysis (IIFEA) is proposed. 33

Two IIFEA models are proposed from the perspectives of non-membership and membership. The IIFEA model is further developed by fusing the DM’s preference for attributes. The alterative development method is designed to enhance the “bad” alternatives. These methods are shown by an example of selecting and enhancing the alternatives.

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Author Contributions: Wei Zhou: Conceptualization, Methodology, Data analysis. Jin Chen: Investigation, Writing-Original draft preparation. Bingqing Ding: Visualization, Validation, Software. Sun Meng: Writing- Reviewing and Editing.

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