An interval-valued intuitionistic fuzzy MABAC approach for material selection with incomplete weight information

Applied Soft Computing 38 (2016) 703–713

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

An interval-valued intuitionistic fuzzy MABAC approach for material selection with incomplete weight information Yi-Xi Xue a , Jian-Xin You a,b , Xiao-Dong Lai c , Hu-Chen Liu a,∗ a b c

School of Management, Shanghai University, Shanghai 200444, PR China School of Economics and Management, Tongji University, Shanghai 200092, PR China School of Tourism and Urban Management, Jiangxi University of Finance and Economics, Nanchang 330013, PR China

a r t i c l e

i n f o

Article history: Received 10 April 2015 Received in revised form 31 August 2015 Accepted 5 October 2015 Available online 23 October 2015 Keywords: Material selection Interval-valued intuitionistic fuzzy sets MABAC method Incomplete weight information

a b s t r a c t In engineering design, selecting the most suitable material for a particular product is a typical multiple criteria decision making (MCDM) problem, which generally involves several feasible alternatives and conflicting criteria. In this paper, we aim to propose a novel approach based on interval-valued intuitionistic fuzzy sets (IVIFSs) and multi-attributive border approximation area comparison (MABAC) for handling material selection problems with incomplete weight information. First, individual evaluations of experts concerning each alternative are aggregated to construct the group interval-valued intuitionistic fuzzy (IVIF) decision matrix. Consider the situation where the criteria weight information is partially known, a linear programming model is established for determining the criteria weights. Then, an extended MABAC method within the IVIF environment is developed to rank and select the best material. Finally, two application examples are provided to demonstrate the applicability and effectiveness of the proposed IVIF-MABAC approach. The results suggest that for the automotive instrument panel, polypropylene is the best, for the hip prosthesis, Co–Cr alloys-wrought alloy is the optimal option. Finally, based on the results, comparisons between the IVIF-MABAC and other relevant representative methods are presented. It is observed that the obtained rankings of the alternative materials are good agreement with those derived by the past researchers. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The selection of the optimum material for a particular product is critical for an enterprise to survive from today’s fierce competitive environment. In most current practices, engineers and designers are forced to select the right material to meet the product’s functional requirements, such as higher product performance, weight saving, and cost reduction [1]. Being the main activity of engineering design process, appropriate selection of materials can significantly reduce manufacturing cost and increase organization competitiveness, customer satisfaction, and profitability [2,3]. But in view of the large number of possible materials and the wide range of manufacturing processes available in the market, material selection for engineering applications has become one of the most challenging issues faced by design engineers. Moreover, material selection is the prerequisite for a chain of other engineering selection problems, which include manufacturing

∗ Corresponding author Tel.: +86 0 21 6613 3703; fax: +86 0 21 6613 4284. E-mail addresses: [email protected], [email protected] (H.-C. Liu). http://dx.doi.org/10.1016/j.asoc.2015.10.010 1568-4946/© 2015 Elsevier B.V. All rights reserved.

process selection, machine selection, tool selection, material handling equipment selection, supplier selection, and so on [4]. Therefore, material selection problem has gathered more and more attention from both academics and practitioners in the past decades [4–6]. Material selection method under uncertain environment can be introduced as a new research area for solving complex material selection problems. Although the existing approaches have had contributions to material selection under uncertainty, most of the related literature described the individual performance of alterative materials with classical fuzzy sets. Because of the complex and unconstructed nature and context of many real world material selection problems, performance information of alternatives usually has to be expressed by the use of more advanced uncertain modelling tools. Recently, several researches have started to propose new methods for material selection of high uncertainty based on, for example, interval 2-tuple linguistic model [7] and uncertain membership linguistic variables [8], which can depict uncertain material performance information more precisely and completely. However, there is few or no researches can be found in the literature concerning material selection by utilizing interval-valued

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Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

intuitionistic fuzzy sets (IVIFSs). The IVIFS theory was introduced by Atanassov and Gargov [9] for dealing with ambiguity in the information and fuzziness in decision makers’ judgments in practical decision making problems. Its basic feature is that both membership and nonmembership functions of an element to a given set are considered and taken on interval values rather than exact numbers. Thus, there is a significantly important need to investigate more effective and suitable mathematical methods by utilizing the IVIFSs in order to better handle material selection problems of high ambiguity and uncertainty. On the other hand, material selection for a specific engineering application can be recognized as a kind of multiple criteria decision making (MCDM) problem [4,10]. It is necessary to take into account several criteria comprehensively when making a decision of material selection, such as performance, price, usability, machinability, recycling, environment, and maintainability [4,6]. Actually, many material selection problems in product design can be taken within the frame of MCDM, and, as reviewed in the literature review section, a lot of MCDM methods have been employed for identifying the most suitable material [11–13]. The multi-attributive border approximation area comparison (MABAC) method is a new MCDM method recently proposed by the research center at the University of Defence in Belgrade [14]. It has a simple computation process, systematic procedure, and a sound logic that represents the rationale of human decision making. Hence, it is an interesting research topic to apply MABAC in the material selection process to rank and determine the best material under the interval-valued intuitionistic fuzzy (IVIF) context. Based on the aforementioned discussions, this paper aims to develop an extended version of the MABAC method for handling material selection problems within the decision environment of IVIFSs. The proposed material selection approach has the ability to reflect both subjective judgments and objective information in realistic applications under the IVIF environment. For some situations where the information about criteria weights is partially known, a linear programming model based on maximum distance measure is further integrated into the proposed approach to determine the weight vector of criteria. Finally, two application examples are examined for the material selection to demonstrate the implementation process of the IVIF-MABAC approach. The results show that the proposed approach can assist engineers and designers to make their efficient decisions for solving intricate material selection problems under uncertainty. The remainder of this paper is organized as follows. Section 2 presents the literature reviews of material selection methods and applications of IVIFSs, and Section 3 introduces some basic concepts, definitions, and operations related to IVIFSs. In Section 4, a new material selection model is proposed by combing IVIFSs and the MABAC method. Furthermore, this section establishes a mathematical programming model for obtaining the criteria weights under limited weight information. In Section 5, two illustrative examples are provided to demonstrate the effectiveness and practicality of the proposed IVIF-MABAC. Finally, concluding remarks and suggestions for future research are given in Section 6.

2. Literature review 2.1. Material selection methods In the literature, a diversity of material selection methods has been developed to assist designers to select the apt material for a given engineering application and to increase the efficiency in product design and development process. For example, Jahan and Edwards [15] proposed an extended VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method for addressing

material selection problems with simultaneous availability of interval data and target based-criteria. Jeya Girubha and Vinodh [10] applied fuzzy VIKOR and environmental impact analysis for the selection of alternate material for an automotive component. The comprehensive VIKOR method was proposed and utilized by Jahan et al. [2] and Cavallini et al. [16] to ensure the optimal material selection in different engineering applications. Yang and Ju [8] proposed an approach based on the uncertain membership linguistic aggregation operators for material selection problems with uncertain membership linguistic information. Peng and Xiao [17] presented the method of preference ranking organization method for enrichment evaluations (PROMETHEE) combined with analytic network process (ANP) for selecting the most suitable material in a hybrid environment. Yazdani and Payam [5] conducted a comparative study on material selection of microelectromechanical systems electrostatic actuators by using Ashby, VIKOR, and technique for order preference by similarity to ideal solution (TOPSIS) methods. Liu et al. [6] developed a hybrid MCDM model that combines decision making trial and evaluation laboratory (DEMATEL)-based ANP (DANP) and modified VIKOR to solve the material selection problems of multiple dimensions and criteria that are interdependent. Anojkumar, Ilangkumaran, and Sasirekha [11] described the application of four MCDM methods, i.e., FAHP-TOPSIS, FAHP-VIKOR, FAHP-ELECTRE, and FAHP-PROMTHEE, for the selection of suitable materials for sugar industry equipment. In addition, a lot of other methods, such as analytical hierarchy process (AHP) [18], multiobjective optimization on the basis of ratio analysis (MOORA) [3], and grey complex proportional assessment (COPRAS-G) [19], have been utilized by the past researchers in engineering material selections. 2.2. Applications of IVIFSs The IVIFS theory offers an intuitive and computationally feasible way for addressing uncertain and ambiguous properties. In recent years, IVIFSs have received increasing attention from researchers and have been successfully applied in a variety of engineering and management fields. For instance, Chen [20] proposed a prioritized aggregation operator-based approach to handle MCDM problems within the environment of IVIFSs, and Chen [21] developed a outranking method based on likelihood-based preference functions for solving MCDM problems with IVIF information. Zhang and Xu [22] developed a soft computing technique based on maximizing consensus and fuzzy TOPSIS approach to IVIF group decision making from the perspectives of both the ranking and the magnitude of decision data. Qi, Liang, and Zhang [23] introduced a generalized cross-entropy based approach for IVIF group decision making with unknown expert and attribute weights. Zavadskas et al. [24] proposed an extended version of newly developed weighted aggregated sum product assessment (WASPAS) method using IVIFSs for uncertain decision making environment. Hashemi et al. [25] presented a group MCDM model based on the compromise ratio method under the IVIF context and applied it to reservoir flood control operation. Meng, Tan, and Chen [26] developed a method to MCDM with IVIF information based on prospect theory, which considers the decision makers’ attitudes for gains and losses. Chen [27] proposed an extended TOPSIS method involving an inclusion comparison approach for addressing multiple criteria group decision making problems in the framework of IVIFSs. 3. Preliminaries The concept of intuitionistic fuzzy sets (IFSs) was first introduced by Atanassov [28] to generalize fuzzy sets [29]. Its definition can be given as follows:

Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

Definition 1. [28]. Let a set X = {x1 , x2 , ..., xn } be a universe of discourse, an IFS A in X is defined as: A=



  x ∈ X ,

x, A (x) , vA (x)

(1)

where A : X → [0, 1] represents the membership degree and vA : X → [0, 1] represents the nonmembership degree of the element x ∈ X to A, respectively, with the condition that for all x ∈ X, 0 ≤ A (x) + vA (x) ≤ 1. For any IFS A and x ∈ X, A (x) = 1 − A (x) − vA (x) is called the hesitation degree of x to A. For convenience, an intuitionistic fuzzy number (IFN) is denoted as ˛ = (˛ , v˛ ) [30], where ˛ and v˛ are the membership and the nonmembership degrees of the element ˛ ∈ X to A, respectively. Later, Atanassov and Gargov [9] proposed the interval-valued intuitionistic fuzzy set (IVIFS) to better express vague decision information, which is characterized by a membership function and a nonmembership function whose values are intervals rather than crisp numbers.

and Definition 5. Let ˛ ˜ 1 = ([a1 , b1 ] , [c1 , d1 ]) ˛ ˜2 = ([a2 , b2 ] , [c2 , d2 ]) be two IVIFNs, then the order relation between IVIFNs is provided as below [22,25]: 1) If S (˛ ˜ 1 ) > S (˛ ˜ 2 ), then ˛ ˜1 > ˛ ˜ 2; 2) If S (˛ ˜ 1 ) = S (˛ ˜ 2 ), then ˜ 1 ) > H (˛ ˜ 2 ), then ˛ ˜1 > ˛ ˜ 2; a) if H (˛ b) if H (˛ ˜ 1 ) = H (˛ ˜ 2 ), then ˛ ˜1 = ˛ ˜ 2. According to the above definition, it is easily known that ˛ ˜+ = ˜ − = ([0, 0] , [1, 1]) are the largest and the small([1, 1] , [0, 0]) and ˛ est IVIFNs, respectively. Definition 6.

Let ˛ ˜j =

  ˜= x,  ˜ A˜ (x) , v˜ A˜ (x) x ∈ X A    ˜ A˜ (x) = L˜ (x) , U˜ (x) ⊆ [0, 1] where A A L  U

(2)

v˜ A˜ (x) =

and

vA˜ (x) , vA˜ (x) ⊆ [0, 1] are intervals denoting the membership ˜ and the nonmembership degrees of the element x ∈ X to A, with the condition U˜ (x) + vU˜ (x) ≤ 1 for all x ∈ X. Similarly, A A ˜ can be calculated as:  ˜ ˜ (x) = the hesitancy degree of x to A









A

L˜ (x) , U˜ (x) = 1 − U˜ (x) − vU˜ (x) , 1 − L˜ (x) − vL˜ (x) . A

A

A

A

A

A

Specially, for every x ∈ X, if  ˜ A˜ (x) = L˜ (x) = U˜ (x) and v˜ A˜ (x) = A

A

(x) = vU˜ (x), then the IVIFS is degraded to an IFS. Analogously, A ˜ = ( ˜ ˛˜ , v˜ ˛˜ ) is termed as an interval-valued intuitionistic the pair ˛ fuzzy number (IVIFN) [31] and each IVIFN can be simply denoted as ˛ ˜ = ([a, b] , [c, d]), where 0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1 and b + d ≤ 1.

vLA˜

˜ = ([a, b] , [c, d]), ˛ Definition 3. Let ˛ ˜ 1 = ([a1 , b1 ] , [c1 , d1 ]) and ˛ ˜ 2 = ([a2 , b2 ] , [c2 , d2 ]) be three IVIFNs, and  > 0, then the basic operational laws of IVIFNs are defined as follows [31]: ˜1 + ˛ ˜ 2 = ([a1 + a2 − a1 a2 , b1 + b2 − b1 b2 ] , [c1 c2 , d1 d2 ]) ; 1) ˛ ˜1 · ˛ ˜ 2

= ([a1 a2 , b1 b2 ] , [c1 + c2 −c1 c2 , d1 + d2 − d1 d2 ]) ; 2) ˛ 3) ˛ ˜ = 4) ˛ ˜ =



1 − (1 − a) , 1 − (1 − b)







a , b



, c  , d

, 1 − (1 − c) , 1 − (1 − d)







˜2 = ˜ 1, A D A



1  4n n



(j = 1, 2, ..., n) be a col-

IVIFWG (˛ ˜ 1, ˛ ˜ 2 , ..., ˛ ˜ n) =

n 

wj

˛ ˜j

j=1 ⎛⎡ ⎤ ⎡ ⎤⎞ n n n n    



 w w w w = ⎝⎣ a j, b j ⎦ , ⎣1 − 1 − cj j , 1 − 1 − dj j ⎦⎠. j

j

j=1

j=1

j=1

j=1

(5)

T

Especially, if w = 1/n, 1/n, ..., 1/n , then the IVIFWG operator reduces to the interval-valued intuitionistic fuzzy geometric (IVIFG) operator. Definition and 7. Let ˛ ˜ 1 = ([a1 , b1 ] , [c1 , d1 ]) ˛ ˜2 = ([a2 , b2 ] , [c2 , d2 ]) be two IVIFNs, then the Hamming distance for the IVIFNs is calculated as [34]: dH (˛ ˜ 2) = ˜ 1, ˛

    1 |a1 − a2 | + b1 − b2  + |c1 − c2 | + d1 − d2  , 4

(6)

and the Euclidean distance for the IVIFNs is calculated as [34]:



=

;

 1 2 2 (a1 − a2 )2 + (b1 − b2 ) + (c1 − c2 )2 + (d1 − d2 ) . 4

.

˜ = ([a, b] , [c, d]) be an IVIFN, its score function Definition 4. Let ˛ S (˛) ˜ [32], and accuracy function H (˛) ˜ [33] are expressed, respectively, by the following formulas:



 

aj , bj , cj , dj

dE (˛ ˜ 1, ˛ ˜ 2)

The comparison of linguistic information represented by IVIFNs is performed according to the score and accuracy functions defined as follows.





lection of IVIFNs, and w = (w1 , w2 , ..., wn )T be their associated n w = 1, then the intervalweight vector, with wj ∈ [0, 1] and j=1 j valued intuitionistic fuzzy weighted geometric (IVIFWG) operator is defined as [31]:

Definition 2. [9]. Let a set X = {x1 , x2 , ..., xn } be a universe of dis˜ in X is an object having the form: course, an IVIFS A



705

Definition 8.



(2)

(2)

˜1 = Let A



ai , bi

(2)

(2)

, ci , di



 

(1)

(1)

ai , bi 1×n



(1)

(1)

, ci , di

  1×n

(7)

˜2 = and A

be two IVIFSs in the universe X =

˜ 1 and A ˜ 2 can {x1 , x2 , ..., xn }, then the distance measure between A be defined as follows [34]:

           1/  (1) (2)    (1) (2)    (1) (2)    (1) (2)   ,  > 0. ai − ai  + bi − bi  + ci − ci  + di − di 

(8)

i=1

Particularly, if  = 1, then Eq. (8) becomes the Hamming distance: S (˛) ˜ =

1 (2 + a − c + b − d) 4

H (˛) ˜ =a+b−1+

c+d 2

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

(3)

˜ 1, A ˜2 D A



1  4n n

=

     (1) (2)   (1) (2)  ai − ai  + bi − bi 

i=1

(4)

     (1) (2)   (1) (2)  + ci − ci  + di − di  .

(9)

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Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

If  = 2, then Eq. (8) is degenerated to the Euclidean distance:





 ˜2 = ! 1 ˜ 1, A D A 4n n



(1)

ai

(2)

− ai

2



(1)

+ bi

(2)

− bi

2



(1)

+ ci

(2)

− ci

2



(1)

+ di

(2)

− di

2  .

(10)

i=1

4. The IVIF-MABAC approach for material selection To select the optimal material for a given application, we put forward a novel framework based on IVIFSs and the MABAC method for solving material selection problems with incomplete weight information. In short, the proposed approach for material evaluation and selection consists of three main stages: determining the performance of materials, calculating the weights of criteria, and obtaining the ranking orders of alternatives. In the first stage, the performance ratings of alternatives on each criterion provided by decision makers are linguistic terms expressed in IVIFNs. Since the information about criteria weights is usually partially known, in the second stage of the proposed approach, the weight vector of evaluation criteria is calculated by using an optimization model. After obtaining the optimal weights of criteria, the procedure of the MABAC method is employed finally to determine the ranking orders of materials. Fig. 1 delineates the flowchart of the proposed interval-valued intuitionistic fuzzy MABAC (IVIF-MABAC) approach, and its detailed explanations for material selection are given in the following subsections.

with each other by l decision

makers (DMk , k = 1, 2, ..., l) on the basis of n evaluation criteria Cj , j = 1, 2, ..., n . Suppose that Dk =

 dijk

m×n

is a decision matrix, where dijk is the performance or rating

of alternative Ai with respect to criterion Cj given by the decision maker DMk , and x˜ ijk =





akij , bkij , cijk , dijk

is an IVIFN indicating

the range of degrees to which alternative Ai satisfies and dissatisfies criterion Cj , respectively. Since decision makers may come from different departments and have different backgrounds and expertise, each decision maker is given a weight k , k = 1, 2, ..., l (where

l

 = 1) to reflect his/her influence on overall material seleck=1 k tion results. In what follows, the IVIFS theory is adopted to handle the uncertain assessments of alternatives provided by the decision makers. ˜ Step 1: Construct the group IVIF decision matrix X. After obtaining the assessments of decision makers, we need k to aggregate all individual decision matrices   X˜ (k = 1, 2, ..., l) into ˜ the group IVIF decision matrix X = x˜ ij by using the IVIFWG m×n operator:

4.1. Determine the performance of materials Considering a general material selection problem with m alternatives (Ai , i = 1, 2, ..., m), which are evaluated and compared





x˜ ij = IVIFWG x˜ ij1 , x˜ ij2 , ..., x˜ ijl =

Fig. 1. Flow diagram of the proposed material selection approach.

l 

k k

x˜ ij

k=1

,

(11)

Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

x˜ ij =

where



 

aij , bij , cij , dij



, i = 1, 2, ..., m; j = 1, 2, ..., n,

 k

 k

l

by

and

1−

l 

Eq.

1 − cijk

k

akij

aij =

(5),

l

,

k=1 l

, dij = 1 −

k=1



1 − dijk

bkij

bij =

k

,

cij =

k=1

.

k=1

If objective criteria are existed in the material selection problem, the performance of each alternative against the objective criteria xij must be normalized and converted into associated IVIFNs. First, the normalized ratings for the alternatives can be calculated as follows: xˆ ij =

xij − min j max j − min j

,

for benefit criteria,

(12)

,

for cost criteria,

(13)

and xˆ ij =

max j − xij max j − min j

where the max j (j ∈ n) and min j (j ∈ n) are the maximum value (or positive-ideal level) and the minimum value (or negative-ideal level) of the jth evaluation criterion, respectively. Other than benefit and cost criteria, the fixation and deviation criteria are also common in material selection problems [4]. The fixation (on xj+ ) criterion is a criterion that the performance value is closer to the fixed number xj+ , the better the alternative; while

H = H1 ∪ H2 ∪ H3 ∪ H4 ∪ H5 . In the sequel, we propose an approach to determine the weights of evaluation criteria by comprehensively utilizing the known weight information acquired in material selection. Step 2: Determine the optimal weights of evaluation criteria. During the material selection process, if the performance values of alternatives have little difference regarding a criterion, it displays that such a criterion plays a less important role when choosing the best material. On the opposite, if all the alternatives have obvious differences on a certain criterion, then this criterion plays a relatively important role in the ranking of alternatives. Hence, if a criterion has similar performance values across alternatives, it should be assessed with a smaller weight; otherwise, the criterion which makes larger deviation should be assigned a larger weight [23]. Particularly, if all alternatives have the same performance values with respect to a given criterion, then this criterion will be unimportant in the material selection process because it does not help in differentiating alternatives [37]. Considering the discrimination among alternative performances, we establish an optimization model based on the IVIF distance measure to determine the weights of evaluation criteria when the information about the criteria weights is incompletely known. First, the distance between the alternative Ai and the other alternatives with respect to the criterion Cj is defined as follows:

the deviation (from xj− ) criterion is a criterion that the performance

value is far away from the fixed number xj− , the better the alternative. To improve generality of the proposed approach, the following two formulas are given for nonmonotonic criteria:

 

xˆ ij =

 

maxx+ − xij − xj+  j

maxx+ − minx+ j

,

for fixation criteria,

j

maxx+ = max

where

i∈m # j "  +  min xj − xij  .and i∈m     xij − xj−  − minxj−

xˆ ij =

maxx− − minx− j

j

where maxx− = max j

i∈m

,

# "  +  xj − xij 

(14)

and

Dij =

1 m−1

 



i = 1, 2, ..., m; j = 1, 2, ..., n.(16)

dH x˜ ij , x˜ gj ,

g=1,g = / i

1  m−1 m

Dj =

m 



dH x˜ ij , x˜ gj , j = 1, 2, ..., n.

(17)

i=1 g=1,g = / i

for deviation criteria,

(15)

# # " "     xij − xj−  and minxj− = min xij − xj−  . i∈m

Then the weighted distance function can be constructed as:

D (w) =

Next, the normalized value for alternative Ai with respect to objective criterion Cj , i.e., xˆ ij , can be written as an IVIFN as x˜ ij =



m 

Next, the overall distance of all the alternatives concerning the criterion Cj is represented as:

minx+ = j

707

Dj wj =

j=1



[25]. As a result, both subjective and xˆ ij , xˆ ij , 1 − xˆ ij , 1 − xˆ ij objective criteria can be considered and handled in order to solve complex material selection problems.

n 

Dij wj

j=1 i=1

1  m−1 n

=

m n  

m

m 



dH x˜ ij , x˜ gj wj .

(18)

j=1 i=1 g=1,g = / i

4.2. Calculate the weights of evaluation criteria Let w = (w1 , w2 , ..., wn )T be the weight vector of the criteria Cj , n j = 1, 2, ..., n, where wj ≥ 0, j = 1, 2, ..., n, j=1 wj = 1, the known weight information on the criteria can be divided into the following five basic ranking forms [22,35,36], for i = / j:

Based on the above analysis, a reasonable weight vector of criteria w = (w1 , w2 , ..., wn )T should be chosen to maximize D (w), and thus, we can reasonably establish the following optimization model:

⎧ m m n

 ⎪ 1   ⎪ ⎪ max D = dH x˜ ij , x˜ gj wj (w) ⎪ m−1 ⎨

  w ≥ wj ;  i   2) A strict ranking: H2 = wi − wj ≥ ˇj ˇj > 0 ;   3) A ranking of differences: H3 = wi − wj ≥ wk − wl (j = / k= / l);    4) A ranking with multiples: H4 = wi ≥ ˇj wj 0 ≤ ˇj ≤ 1 ;   

(M − 1)

To ease exposition, let H denote the set of the known weight information of criteria provided by decision makers and

Via solving model (M-1), we get the optimal solution w∗ , which can be used as the weight vector of the evaluation criteria.

1) A weak ranking: H1 =

5) An interval form: H5 =

ˇi ≤ wi ≤ ˇi + εi

0 ≤ ˇi ≤ ˇi + εi .

j=1 i=1 g=1,g = / i

n  ⎪ ⎪ ⎪ Subject to w ∈ H, wj = 1, wj ≥ 0, j = 1, 2, ..., n. ⎪ ⎩

(19)

j=1

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Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

For other situations where the information regarding criteria weights is completely unknown, we can construct another optimization model for computing the optimal weights of criteria:

(M − 2)

⎧ m m n

 ⎪ 1   ⎪ ⎪ max D (w) = dH x˜ ij , x˜ gj wj ⎪ m−1 ⎨ j=1 i=1 g=1,g = / i

(20)

n  ⎪ ⎪ ⎪ Subject to wj = 1, wj ≥ 0, j = 1, 2, ..., n. ⎪ ⎩ j=1

Model (M-2) can be solved by using Lagrange method, and we can get the criteria weight vector by normalizing the corresponding optimal solutions: m m  

dH x˜ ij , x˜ gj



i=1 g=1,g = / i

wj =

m m n   

dH x˜ ij , x˜ gj



.

(21)

j=1 i=1 g=1,g = / i

Fig. 2. Presentation of the G+ , G- , and G approximation areas.

4.3. Obtain the ranking orders of alternatives The MABAC is a new MCDM method proposed by the research center at the University of Defence in Belgrade [14]. Due to its simple computation procedure and the stability (consistency) of solution, the MABAC method is a particularly pragmatic and reliable tool for rational decision making. In this subsection, a modified MABAC method within the IVIF environment is developed to assist decision makers in selecting the optimum material for an engineering design. ˜ Step 3: Calculate the weighted group IVIF decision matrix R. When the weight vector w = (w1 , w2 , ..., wn )T of the criteria Cj (j = 1, 2, ..., n) is determined, the weighted group IVIF decision   ˜ = r˜ij matrix R can be constructed by the following formula: ˜ i x˜ ij = r˜ij = w where r˜ij =



m×n

1 − 1 − aij



w˜ i





, 1 − 1 − bij

a ij , b ij , cij , dij

w˜ i  

, cijw˜ i , dijw˜ i





,

(22)

is the weighted IVIFN, x˜ j are the

˜ elements of the group IVIF decision matrix X. ˜ Step 4: Obtain the border approximation area vector G. The border approximation area for each evaluation criterion is calculated using the IVIFG operator as: g˜ j =

m 

1/m

r˜ij

=



 

a¯ j , b¯ j , c¯ j , d¯ j



, j = 1, 2, ..., n,

(23)

dE r˜ij , g˜ j

˜ and a¯ j = sion matrix R m 

1 − c˜ij

1/m

m  1/m

a ij

i=1 m

, d¯ j = 1 −

i=1



1 − d˜ ij

, b¯ j =

1/m

m  1/m

b ij

, c¯ j = 1 −

i=1

. Based on the values

i=1

g˜ j (j = 1, 2, ..., n) for all the criteria, the border approximation area ˜ can be formed as the following format: vector G ˜ = [˜g1 , g˜ 2 , ..., g˜ n ] . G

(24)

Step 5: Calculate the distance matrix D. Using the IVIF Euclidean distance operator, the distances of the candidate materials from the border approximation area are com  , where puted to construct the distance matrix D = dij

(

dij =





dE r˜ij , g˜ j

−dE r˜ij , g˜ j

m×n

if r˜ij ≥ g˜ j ; if r˜ij < g˜ j ,

(25)

=



1 4

a ij − a¯ j

2

+ b ij − b¯ j

2

+ cij − c¯ j

2

+ dij − d¯ j

2

.

(26)

Especially, alternative Ai will belong to the border approximation area (G) if dij = 0, upper approximation area (G+ ) if dij > 0, and lower approximation area (G− ) if dij < 0. The upper approximation area (G+ ) is the area, which contains the ideal material (A+ ), while the lower approximation area (G− ) is the area which contains the anti-ideal material (A− ) (See Fig. 2). Therefore, in order for alternative Ai to be designated as the best material, it is necessary for it to have as many criteria as possible belonging to the upper approximate area (G+ ). Step 6: Determine the ranking orders of all alternatives. If the distance value dij ∈ G+ , then the alternative Ai is near or equal to the ideal material. If the value dij ∈ G− , it indicates that the alternative Ai is near or equal to the anti-ideal material. Therefore, the values of the criteria functions for the alternative materials can be calculated by adding the distances of the alternatives from the border approximation area vector. That is, the closeness coefficient (CCi ) to the border approximation area for each alternative is defined by calculating the sum of the row elements of the distance matrix D as follows:

i=1

where r˜ij are the elements of the weighted group IVIF deci-



CCi =

n 

dij ,

i = 1, 2, ..., m.

(27)

j=1

Then, all the alternatives are ranked based on the descending order of their closeness coefficients CCi (i = 1, 2, ..., m), and the best material for the given engineering application can be determined accordingly.

5. Illustrative examples In this section, two real material selection examples are cited to demonstrate the implementation process and effectiveness of the proposed IVIF-MABAC approach. The first example is an automotive instrument panel material selection problem that includes subjective data and incomplete weight information. The second example describes a typical biomedical problem, selecting hip prosthesis materials with objective data, target criteria, and unknown weight information.

Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

709

Fig. 3. Decision structure of example 1.

Table 1 Linguistic terms and corresponding IVIFNs for evaluating materials. Linguistic terms

IVIFNs

Very high (VH) High (H) Medium high (MH) Medium (M) Medium low (ML) Low (L) Very low (VL)

([0.9, 1.0], [0.0, 0.0]) ([0.8, 0.8], [0.1, 0.1]) ([0.6, 0.7], [0.2, 0.3]) ([0.5, 0.5], [0.4, 0.5]) ([0.3, 0.4], [0.5, 0.6]) ([0.2, 0.2], [0.7, 0.7]) ([0.0, 0.1], [0.8, 0.9])

and model (M-1) we can construct the following single-objective programming model:

(

max D (w) = 0.554w1 + 0.594w2 + 0.450w3 + 0.431w4 +0.450w5 + 0.547w6 + 0.735w7 + 0.509w8 s.t.w ∈ H

Solving the above model, the criteria weight vector is obtained as follows: w∗ = (0.07, 0.07, 0.15, 0.07, 0.18, 0.145, 0.17, 0.145)T .  ˜ = r˜ij is Then, the weighted group IVIF decision matrix R 4×8

5.1. Application example 1 This case study has been conducted for an automotive component, instrument panel, in an Indian automotive parts factory [10]. For the implementation of sustainable practices, the factory needs to select a right material for the produced instrument panel. The selection of material for the instrumental panel takes eight evaluation criteria into consideration and there are four feasible materials remained for further evaluation after preliminary screening. The decision structure of the problem is depicted in Fig. 3. A team of five experts or decision makers, DM1 , DM2 , DM3 , DM4 , and DM5 , is built to conduct the performance assessment and to determine the most suitable alternative. In view of uncertainty in information and subjectivity in decision makers’ judgments, the candidate materials are evaluated against the eight criteria by applying the linguistic terms as given in Table 1. The obtained linguistic assessments of the four alternatives provided by the five decision makers are tabulated in Table 2. In this example, it is assumed that the five decision makers have equal weights and the information about criteria weights is partially known as follows: H=



established by using Eq. (22) and the border approximation area ˜ is formed through Eq. (23) as shown below: vector G ˜ = [([0.088, 0.102] , [0.865, 0.880]) G ([0.189, 0.214] , [0.723, 0.747]) ([0.223, 0.251] , [0.678, 0.705]) ([0.200, 0.229] , [0.709, 0.739])

([0.097, 0.116] , [0.849, 0.862]) ([0.091, 0.100] , [0.865, 0.878]) ([0.187, 0.216] , [0.723, 0.746])  ([0.162, 0.181] , [0.767, 0.797])

The distances of the four alternatives from the border approximation area are computed thereafter using Eqs. (25–26) and the distance matrix D is constructed as illustrated in Table 3. Finally, the closeness coefficient to the border approximation area for each alternative is determined via Eq. (27) and the ranking orders of all the materials are derived in accordance with the values of CCi (i = 1, 2, 3, 4), which are shown in the last two columns of Table 3. As we can see, the alternative A3 , that is polypropylene, can be selected as the most appropriate material among the four alternatives. The results acquired by the proposed approach are then compared with the rankings by other material selection methods, such as the fuzzy VIKOR [10], the fuzzy TOPSIS [38], the interval 2-tuple

w1 ≤ 0.07, w2 = w1 , 0.15 ≤ w3 ≤ 0.20, w4 = w2 , 0.12 ≤ w5 ≤ 0.18, 0.10 ≤ w6 ≤ 0.15,

0.12 ≤ w7 ≤ 0.18, w8 = w6 , w5 − w6 ≥ 0.03, w5 − w7 ≥ 0.01, wj ≥ 0, j = 1, 2, ..., 8,

8 

)

wj = 1

.

i=1

In the following, the proposed approach is employed to select the optimum material for the considered engineering application. After converting into corresponding IVIFNs, the linguistic assessare aggregated into the group IVIF decision ments of alternatives   matrix X˜ = x˜ ij by using Eq. (11). Next, by utilizing Eqs. (16–18) 4×8

linguistic VIKOR (ITL-VIKOR) [7], and the uncertain membership linguistic method [8]. The ranking of the alternative materials yielded by the proposed approach is exactly the same with those obtained with the listed methods. Therefore, the proposed IVIFMABAC approach is validated. But, from the case study, we can

710

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Table 2 Linguistic assessments of alternatives by the five decision makers. Alternatives

Decision makers

A1

DM1 DM2 DM3 DM4 DM5 DM1 DM2 DM3 DM4 DM5 DM1 DM2 DM3 DM4 DM5 DM1 DM2 DM3 DM4 DM5

A2

A3

A4

Criteria C1

C2

C3

C4

C5

C6

C7

C8

MH MH MH H MH M MH MH MH H VH H VH H VH H H H H H

H H H MH H MH M MH MH MH H VH VH VH H VH H VH H VH

MH MH MH H MH H MH H M H VH H H H VH H VH H VH H

MH MH H MH H H M H MH MH H VH H VH H H H H H H

H MH MH MH MH H MH H M H H H VH H VH VH VH H H H

MH H MH H H MH M H MH MH VH VH VH H H H H VH H VH

MH MH H MH MH H M MH M M H H H VH VH VH VH H VH H

MH MH MH H MH MH M H MH M VH H H H H H VH H VH M

Table 3 Ranking of the alternatives for example 1.

A1 A2 A3 A4

C1

C2

C3

C4

C5

C6

C7

C8

CCi

Rank

−0.0230 −0.0329 0.0621 0.0187

−0.0118 −0.0527 0.0474 0.0474

−0.0542 −0.0489 0.0664 0.0664

−0.0169 −0.0280 0.0416 0.0172

−0.0618 −0.0555 0.0749 0.0749

−0.0302 −0.0782 0.0907 0.0579

−0.0445 −0.0977 0.0871 0.1231

−0.0210 −0.0538 0.0735 0.0247

−0.263 −0.448 0.544 0.430

3 4 1 2

derive the weights of evaluation criteria when the weight information is incompletely known, which is one advantage of the proposed model. 5.2. Application example 2 The second case study is related to selecting materials for the hip prosthesis [2,39], which comprises three main components: femoral component, acetabular cup, and acetabular interface. The femoral component is a rigid pin implanted into the hollowed out shaft of the femur to substitute the natural femoral head. The hip socket is replaced by an acetabular cup, which is a soft polymer molding fixed to the ilium. The acetabular interface is located between the femoral component and the acetabular cup to reduce wear debris generated by friction. The pin and cup are fixed to the surrounding bone structure by adhesive cement and perform different functions. In this example, selecting material for pin has been considered and the decision structure of this problem is shown in Fig. 4. Table 4 exhibits the performance matrix for the 11 alternative

materials. One may observe that only objective data are existed in this material selection problem. But the selection of the most appropriate material for the engineering application can be taken within the frame of the proposed approach under IVIF context. In what follows, application of the IVIF-MABAC approach in this case is illustrated. Among the nine criteria, C1 , C2 ,. . ., C6 are benefit criteria, C7 and C8 are fixation criteria, and C9 is a cost criterion. Based on nearing to human bone, the most favorable value for C7 and C8 are determined as 14 GPa and 2.1, respectively. Thus, the performance matrix in Table 4 can be normalized by applying Eqs. (12–14) and transformed   into IVIFNs to form the IVIF decision matrix X˜ = x˜ ij as 11×9

shown in Table 5. Then, according to model (M-2) and Eq. (21), the weights of the nine criteria are computed as w∗ = (0.131, 0.125, 0.089, 0.098, 0.133, 0.098, 0.120, 0.121, 0.086)T since the information regarding the criteria weights is completely unknown. Next, the procedure of the MABAC method is implemented to determine the ranking of the alternative materials.

Table 4 Performance matrix for hip joint prosthesis material selection. Alternatives

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11

Criteria C1

C2

C3

C4

C5

C6

C7

C8

C9

10 9 9 9 10 10 8 8 7 7 7

7 7 7 7 9 9 10 10 7 7 7

517 630 610 650 655 896 550 985 680 560 430

350 415 410 430 425 600 315 490 200 170 130

8 10 10 10 2 10 7 7 3 3 3

8 8.5 8 8.4 10 10 8 8.3 7 7.5 7.5

200 200 200 200 238 242 110 124 22 56 29

8 8 7.9 8 8.3 9.1 4.5 4.4 2.1 1.6 1.4

1 1.1 1.1 1.2 3.7 4 1.7 1.9 3 10 5

Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

711

Fig. 4. Decision structure of example 2.

Multiplying by corresponding   criteria weights, the weighted group ˜ = r˜ij IVIF decision matrix R is obtained using Eq. (22) and the 11×9 ˜ is established by employing border approximation area vector G

 

Eqs. (23–24). Then the distance matrix D = dij

11×9

and the

closeness coefficients of the alternatives are calculated by Eqs. (25–27). The closeness coefficients CCi (i = 1, 2, ..., 11) obtained are: CC1 = 2.328, CC2 = 1.691, CC3 = 1.659, CC4 = 1.656, CC5 = 2.381, CC6 = 4.368, CC7 = 1.704, CC8 = 2.754, CC9 = 2.206, CC10 = 0.541, CC11 = 0.683. Afterward, the ranking of the materials are determined in line with the descending order of their closeness coefficients as: A6 A8 A5 A1 A9 A7 A2 A3 A4 A11 A10 . Therefore, the material A6 , i.e., Co–Cr alloys-wrought alloy, is the top choice and can be proposed as the optimal material. The above material selection problem was also solved by Jahan et al. [2] using a comprehensive VIKOR method and by Jahan and Edwards [39] based on the TOPSIS and VIKOR methods. Hence, a comparison with the results yielded by the methods of [2,39]

is carried out to validate the results of the proposed IVIF-MABAC approach. In additon, we used the same case study to analyze the IVIF-TOPSIS [22] and the IVIF-ELECTRE [40] methods to further illustrate the effectiveness of the proposed appraoch. Fig. 5 displays the ranking orders of all the alternative materials as determined using these five approaches. The results show that the first choice of material for the hip prosthesis remains the same, i.e., A6 (Co–Cr alloys-wrought alloy), by the proposed approach and the methods given by Jahan et al. [2] and Jahan and Edwards [39]. Moreover, the Spearman’s rank correlation coefficients between the rank orders derived by the proposed approach and the methods of [2,39] are 0.855 and 0.836, respectively. These support the exactness of the IVIF-MABAC approach presented in this paper. But according to the IVIF-TOPSIS and the IVIF-ELECTRE methods, alternative A8 has a higher priority in comparison with alternative A6 , and is the best option for the considered materal selection probem. There are also great differences between the ranking order

Fig. 5. Rankings of the alternatives for example 2.

([0.556, 0.556], [0.444, 0.444])

([0.845, 0.845], [0.155, 0.155]) ([0.968, 0.968], [0.032, 0.032]) ([0.167, 0.167], [0.833, 0.833]) ([0.167, 0.167], [0.833, 0.833])

([0.929, 0.929], [0.071, 0.071]) ([0.900, 0.900], [0.100, 0.100])

([0.600,0.600], [0.400, 0.400]) ([0.536, 0.536], [0.464,0.464]) ([1,1], [0,0])

([0.625,0.625], [0.375, 0.375]) ([0.625, 0.625], [0.375,0.375]) ([0.125,0.125], [0.875,0.875]) ([0.125, 0.125], [0.875, 0.875]) ([0.125, 0.125], [0.875, 0.875])

([0,0], [1,1])

([0.667, 0.667], [0.333, 0.333]) ([0.667, 0.667], [0.333,0.333]) ([1,1], [0,0])

([1,1], [0,0])

([0,0], [1,1])

([0,0], [1,1])

([0,0], [1,1])

([0.667, 0.667], [0.333, 0.333])

([1,1], [0,0])

([1,1], [0,0])

([0.333, 0.333], [0.667, 0.667])

([0.333, 0.333],[0.667,0.667])

([0,0], [1,1])

([0,0], [1,1])

([0,0], [1,1])

A4

A5

A6

A7

A8

A9

A10

A11

([0.450,0.450], [0.550,0.550]) ([0.234, 0.234], [0.766, 0.766]) ([0,0], [1,1])

([1,1], [0,0])

([0,0], [1,1])

([1,1], [0,0])

([1,1], [0,0]) ([0,0], [1,1]) ([0.667, 0.667], [0.333, 0.333]) A3

([0,0], [1,1]) ([0.667, 0.667], [0.333, 0.333]) A2

([0.157,0.157], [0.843, 0.843]) ([0.360, 0.360], [0.640, 0.640]) ([0.324, 0.324], [0.676, 0.676]) ([0.396, 0.396], [0.604, 0.604]) ([0.405, 0.405], [0.595, 0.595]) ([0.840, 0.840], [0.160,0.160]) ([0.216,0.216], [0.784, 0.784]) ([1,1], [0,0]) ([0,0], [1,1])

([0.394,0.394], [0.606, 0.606]) ([0.766, 0.766], [0.234,0.234]) ([0.149,0.149], [0.851,0.851]) ([0.085, 0.085], [0.915, 0.915]) ([0,0], [1,1])

([1,1], [0,0])

([0.333,0.333], [0.667, 0.667]) ([0.433, 0.433], [0.567,0.567]) ([0,0], [1,1])

([0.657,0.657], [0.343, 0.343]) ([0.671, 0.671], [0.329,0.329]) ([1,1], [0,0])

([0.989, 0.989], [0.011, 0.011]) ([0.989, 0.989], [0.011, 0.011]) ([0.978, 0.978], [0.022, 0.022]) ([0.700, 0.700], [0.300, 0.300]) ([0.667, 0.667], [0.333, 0.333]) ([0.922, 0.922], [0.078, 0.078]) ([0.900, 0.900], [0.100,0.100]) ([0.778, 0.778], [0.222, 0.222]) ([0,0], [1,1])

([0.157, 0.157], [0.843, 0.843]) ([0.157, 0.157], [0.843, 0.843]) ([0.171, 0.171], [0.829, 0.829]) ([0.157, 0.157], [0.843, 0.843]) ([0.114, 0.114], [0.886, 0.886]) ([0,0], [1,1]) ([0.191, 0.191], [0.809, 0.809]) ([0.191, 0.191], [0.809, 0.809]) ([0.191, 0.191], [0.809, 0.809]) ([0.191, 0.191], [0.809, 0.809]) ([0.018, 0.018], [0.982, 0.982]) ([0,0], [1,1]) ([0.333, 0.333], [0.667, 0.667]) ([0.500, 0.500], [0.500, 0.500]) ([0.333,0.333], [0.667, 0.667]) ([0.467, 0.467], [0.533, 0.533]) ([1,1], [0,0])

C8 C7 C6 C5

([0.750, 0.750], [0.250, 0.250]) ([1,1], [0,0])

C4

([0.468, 0.468], [0.532, 0.532]) ([0.606, 0.606], [0.394, 0.394]) ([0.596, 0.596], [0.404, 0.404]) ([0.638, 0.638], [0.362, 0.362]) ([0.628, 0.628], [0.372, 0.372]) ([1,1], [0,0])

C3 C2

([1,1], [0,0])

C1

A1

Table 5 IVIF decision matrix for example 2.

([1,1], [0,0])

Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

C9

712

obtained by the proposed aprooah and those produced by the IVIFTOPSIS and the IVIF-ELECTRE. The main reasons that brought the inconsistencies mainly lie in the characteristics of the IVIF-TOPSIS and the IVIF-ELECTRE methods. The IVIF-TOPSIS is based on the principle that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest from the negative ideal solution. It introduces two reference points, but it does not consider the relative importance of the distances from these points. Besides, with further comparison between the IVIFTOPSIS index and IVIF-MABAC index, it can be concluded that the resolution of IVIF-MABAC index is very bigger than that of IVIFTOPSIS index, explicating in a sense the method of IVIF-MABAC is superior to the IVIF-TOPSIS. The IVIF-ELECTRE is based on pairwise comparisons of alternatives for each criterion and the computation of concordance and discordance indices. Using the IVIF-ELECTRE, the outranked alternatives can be identified, but we might be unable to differentiate the priority orders of certain alternatives. For example, the priority orders of alternatives A3 , A4 , and A7 cannot be differentiated with the IVIF-ELECTRE method. By contrast, the proposed IVIF-MABAC appraoch can obtain the distinct ranking results of the alternatives and renders the ranking of A7 A3 A4 . Moreover, we also notice that the computation process is more complex and cumbersome by using the IVIF-ELECTRE method, which makes employing it in practical applications challenging. • The two examples presented above have demonstrated the applicability and effectiveness of the proposed approach for selecting the most suitable material in the engineering design processes. However, in comparison to the extant material selection methods, the IVIF-MABAC framework proposed in this paper has the following desirable advantages: • The uncertainty and fuzziness of decision making information can be well reflected and modeled using IVIFSs. Moreover, both subjective assessments and objective data can be considered by the proposed approach during the material selection process. • Based on the maximum distance measure, the proposed model can be used for managing material selection problems in which the information about criteria weights is incompletely known or even completely unknown. • All three categories of criteria in engineering design, including benefit, cost, and fixation criteria, can be taken into account in the material selection. The proposed approach is a general method and can consider any number of evaluation criteria. • By using the modified MABAC method, a more reasonable and credible ranking result of alternative materials can be achieved, which makes the decision results certain and facilitates material selection assistance and judgment. 6. Conclusions In this paper, we proposed a novel approach based on an extended MABAC method within the IVIF environment to solve material selection problems with incomplete weight information. The performance or rating of each alternative on every of the criteria is estimated based on IVIFSs, and a new MCDM method, the MABAC, is used to rank and select the most desirable candidate material. Additionally, incomplete weight information is more realistic in many practical engineering design problems, especially in complex and uncertain environments. To this end, a linear programming model based on the IVIF distance measure is defined to obtain the criteria weights when the information about criteria weights is partially known or completely unknown a priori. Two material selection examples were presented to demonstrate the feasibility and applicability of the proposed approach. In order to inspect effectiveness of the suggested model, comparative

Y.-X. Xue et al. / Applied Soft Computing 38 (2016) 703–713

analyses with some relevant representative methods have also been carried out. The experimental results show that the ranking results produced by the IVIF-MABAC approach are consistent with the previous material selection methods; and that the proposed procedure is intelligible to the material selection decision making process under uncertainty. The needed computations of the proposed approach are straightforward and easy-to-use in real-life engineering applications. Also, it assists engineers and designers in making critical decisions during the selection of the best alternative for multifaceted material selection problems. For future research, extending the IVIF-MABAC model by adding other characteristics, such as prioritized aggregation operator and Choquet integral, in the decision making process is recommended. In addition, the developed new approach in this paper can be utilized for solving other material selection problems to further show its robustness and efficiency. Acknowledgments The authors are very grateful to the respected editor and the anonymous referees for their constructive comments and suggestions, which helped to improve the overall quality of the paper. This work was partially supported by the National Natural Science Foundation of China (No. 71402090), the Project funded by China Postdoctoral Science Foundation (Nos. 2014M560356, 2015T80456), the Program for Young of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. QD2015019), and the Chen Guang project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation. References [1] M. I˙ pek, I˙ .H. Selvi, F. Findik, O. Torkul, I.H. Cedimo˘glu, An expert system based material selection approach to manufacturing, Mater. Design 47 (2013) 331–340. [2] A. Jahan, F. Mustapha, M.Y. Ismail, S.M. Sapuan, M. Bahraminasab, A comprehensive VIKOR method for material selection, Mater. Design 32 (3) (2011) 1215–1221. [3] P. Karande, S. Chakraborty, Application of multi-objective optimization on the basis of ratio analysis (MOORA) method for materials selection, Mater. Design 37 (2012) 317–324. [4] A. Jahan, K.L. Edwards, A state-of-the-art survey on the influence of normalization techniques in ranking: improving the materials selection process in engineering design, Mater. Design 65 (2015) 335–342. [5] M. Yazdani, A.F. Payam, A comparative study on material selection of microelectromechanical systems electrostatic actuators using Ashby, VIKOR and TOPSIS, Mater. Design 65 (2015) 328–334. [6] H.C. Liu, J.X. You, L. Zhen, X.J. Fan, A novel hybrid multiple criteria decision making model for material selection with target-based criteria, Mater. Design 60 (2014) 380–390. [7] H.C. Liu, L. Liu, J. Wu, Material selection using an interval 2-tuple linguistic VIKOR method considering subjective and objective weights, Mater. Design 52 (2013) 158–167. [8] S.H. Yang, Y.B. Ju, A novel multiple attribute material selection approach with uncertain membership linguistic information, Mater. Design 63 (2014) 664–671. [9] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 31 (3) (1989) 343–349. [10] R. Jeya Girubha, S. Vinodh, Application of fuzzy VIKOR and environmental impact analysis for material selection of an automotive component, Mater. Design 37 (2012) 478–486. [11] L. Anojkumar, M. Ilangkumaran, V. Sasirekha, Comparative analysis of MCDM methods for pipe material selection in sugar industry, Expert Syst. Appl. 41 (6) (2014) 2964–2980. [12] P. Chatterjee, S. Chakraborty, Material selection using preferential ranking methods, Mater. Design 35 (2012) 384–393. [13] H.C. Liu, L.X. Mao, Z.Y. Zhang, P. Li, Induced aggregation operators in the VIKOR method and its application in material selection, Appl. Math. Model. 37 (9) (2013) 6325–6338.

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