1 Introduction
MultiAttributive Border Approximation area Comparison (MABAC) developed by Pamučar and Ćirović [1] is one of the recent approaches to multiattribute decision making (MCDM/MADM) problems. MABAC method has attracted researchers due to its inherent characteristics: simple computation and stability in solution. The basic principle used in MABAC is that it divides the performances of each criteria/attribute function into Upper Approximation Area (UAA) containing ideal alternatives and Lower Approximation Area (LAA) containing antiideal alternatives. In other words this method provides a direct relextion of
the relative strength and weakness of an alternative over others’ according to each criteria. In classical MABAC [1] method, the performance rating and criteria weights were represented by crisp/deterministic numerical values. But human judgment, preference values and criteria weights are often imprecise and ambiguous in nature and cannot be represented by crisp numbers in reallife problems. In response to the uncertainty inherent from decision makers’ subjective judgment, Zadeh [2], first, introduced fuzzy set theory (FST). Further extensions of FST done by many authors. Atanassov [3] defined intuitionistic fuzzy set(IFS) as a generalization of fuzzy sets to formulate the nondeterminacy which occurs due to hesitation of decision makers in the system. Further generalization of IFS as intervalvalued intuitionistic fuzzy set (IVIFS) by Atanassov and Gargov [4] where membership and nonmembership are introduced to deal with impreciseness in decision making. Xue [5] used IVIFS in MCDM problems after developing some aggregation operators for IVIFS. In this regard, few extensions of MABAC method could be found in the literature. Recently, Peng and Yang [6] developed Pythagorean Fuzzy Choquet Integral Based MABAC Method for Multiple Attribute Group Decision Making. Xue et al. [7] defined intervalvalued intuitionistic fuzzy MABAC approach and used for material selection with incomplete weight information. Božanić et al. [8] used the fuzzy AHPMABAC hybrid model in ranking potential locations for preparing layingup positions. Roy et al. [9] extended MABAC model using rough approximation and rough numbers and developed a hybrid MADM method for assessing medical tourism sites in India.
Recent researchers are applying Fuzzy Type2 sets having fuzzybased membership function, dealing with benefit and risk under uncertain environment in a better way. Type2 fuzzy sets (T2FS) [10] are extension of ordinary type1 fuzzy sets where traditional Type1 fuzzy sets fails to deal with uncertainties, due to crisp nature of their membership function. Lee and Chen [11] present a new method for handling fuzzy multiple criteria hierarchical group decisionmaking problems based on arithmetic operations and fuzzy preference relations of interval type2 fuzzy sets. Chen and Lee [12] also presented an interval type2 TOPSIS method to handle fuzzy MCDM problem. Chen and Lee [13] suggested a method to handle MCDM problems based on ranking values and arithmetic operation of interval type2 fuzzy set. Wang et al. [14] proposed a method for MAGDM based on interval type2 fuzzy environment. Chen et al. [15] presented a new method to handle fuzzy MAGDM problems based on ranking interval type2 fuzzy sets. Celik et al. [16] proposed an integrated novel type2 fuzzy MCDM method to improve customer satisfaction in public transportation in Istanbul. Chen and Wang [17] applied interval type2 fuzzy set in fuzzy decision making. A linear assignment method for multiplecriteria decision analysis with interval type2 fuzzy sets is discussed in Chen [18]. Kar and Chatterjee [19] applied ranking based interval type2 fuzzy set exploring the risk factors and ranking supplier companies and presented an empirical study. Despite various methods developed based on type2 fuzzy sets, these methods have their own shortcomingslarge number of computations, i.e., computation complexity. But MABAC has simple computation steps and provides stable solutions. With these considerations, the paper extends the MABAC method for IT2TrFNs for making pragmatic and reliable decisions in evaluation and selection of the suitable alternative/candidate for a job.
The remainder of this paper is structured as follows. In Section 2, we first introduce some preliminaries on type2 fuzzy sets and interval type2 trapezoidal fuzzy numbers (IT2TrFNs). In Section 3, we extend MABAC method based on IT2TrFNs. Section 4 presents an illustrative example for validation of our proposed model. A comparison of the proposed MADM method with other two methods is presented in section 5. Finally, we conclude the work in Section 6.
2 Preliminaries
In this section, we briefly review some definitions of type2 fuzzy sets and interval type2 fuzzy sets.
Definition 1
[10] A type2 fuzzy set (T2FS) in the universe of discourse can be represented by a type2 membership function shown as follows:
(1) 
where denotes an interval in . Moreover, the type2 fuzzy set also can be represented as follows:
(2) 
where is the primary membership at , and indicates the second membership at . For discrete space, the symbol is replaced by .
Definition 2
[11] Let be a type2 fuzzy set in the universe of discourse represented by a type2 membership function . If all , then is called an interval type2 fuzzy set (IT2FS). An IT2FS can be regarded as a special case of the T2FS, which is defined as follows:
(3) 
It is obvious that the IT2FSs defined in is fully determined by the primary membership which is called the footprint of uncertainty (FOU), and can be expressed as follows:
(4) 
Because the operations on IT2FS are quite complex, IT2FS is usually considered by studying some simplified versions. Here, we follow the results of Chen [18], who adopted interval type2 trapezoidal fuzzy numbers (IT2TrFNs) for solving MADM problems.
Definition 3
[18] Let and be two interval type2 trapezoidal fuzzy numbers (IT2TrFNs), where the height of a generalized fuzzy number is between zero to one. Let and be the heights of and , respectively. A TrIT2FS in the universe of discourse is defined in the form:
(5) 
where and are all real numbers and which satisfy the inequalities ; and . The upper membership function (UMF) and lower membership function (LMF) are defined in the following way (see Fig. 1):
(6) 
and
(7) 
2.1 Geometric Bonferroni Mean operator for IT2TrFNs
Definition 4
[20] Let be a collection of IT2TrFNs and , then, we call:
(8) 
a interval type2 trapezoidal Fuzzy Geometric Bonferroni Mean (TIT2FGBM) operator.
Theorem 2.1
[20] Let , be a collection of IT2TrFNs and , then, the aggregated result by Eq. (14) is also a IT2TrFN, and
(9) 
where
(10) 
and
(11) 
2.2 The rankingbased distance function of IT2TrFNs
Theorem 2.2
[21] Let be a TrIT2FS defined in the universe of discourse . The rankbased distance function between and are defined as follows:
(12) 
Definition 5
[21] Let and be two IT2TrFNs. Then the distance between and is defined as:
(13) 
3 Extended MABAC for group decision making based on IT2TrFNs
The basic steps of MABAC method are: (i) Determine the weights of criteria (ii) Construction of initial decision matrix (iii) Computation of normalized decision matrix and weighted normalized decision matrix (iv) Finding Boarder approximation area(BAA) of each criteria functions for comparison of ideal and antiideal solutions (v) Calculate the relative distance of each alternative’s from ideal solution using BAA (vi) Finally, rank the alternatives based on the overall closeness coefficient of each alternative’s to the ideal solution. Here, we extend MABAC method based on interval type2 trapezoidal fuzzy numbers(IT2TrFNs).
Step 1. Determine the weighting of evaluation criteria
Construct the weighting matrix of the attributes of the th decisionmaker and construct the average weighting matrix , respectively, shown as follows:
(14) 
(15) 
where , is a IT2TrFN, and k denotes the number of decision makers.
Step 2. Construct the IT2TrFNs performance/decision matrix
Construct the decision matrix of the th decisionmaker and construct the average decision matrix , respectively, shown as follows
(16) 
(17) 
where , is a IT2TrFN, and denotes the number of decision makers.
Step 3. Normalization of the elements of the average decision matrix
All relevant candidates’ performance/property values, represented in the form of IT2TrFN, are first normalized and then weighted. For the normalization procedure, distinguishing must be made as to whether a material property is the larger the better (benefit type) or the lower the better (cost type). Let and denote the IT2TrFNs of th criteria of th alternative before and after normalization. The normalization procedure works as follows. For each property, compute the range. That is, where and , .
If attribute is of the benefit type then perform the following normalization operation
(18) 
If attribute is of the cost type then perform the following normalization operation
(19) 
Step 4. Calculation of the weighted matrix
The weighted matrix () is constructed as follows
(20) 
where the elements of the weighted matrix are calculated as follows
(21) 
Step 5. Calculation of Border Approximation Area (BAA)
The border approximation area for each evaluation criterion is calculated using the geometric mean operator for IT2TrFNs [see Eqs.(9)(11)] and defined by
(22) 
After calculating BAA for each criteria function we now form the IT2TrFNs based border approximation area comparison matrix () as defined as
(23) 
Step 6. Calculation of distance matrix
The rankbased distance operator [see Eq.(12)(13)] for IT2TrFNs is used here to measure the rankedbased distances each performance in the weighted matrix are computed to form distance matrix (). Similarly, the rankedbased distance matrix of is calculated. These matrices are described as follows
(24) 
where the elements of the distance matrix are calculated as follows
(25) 
(26) 
where the elements of the distance matrix are calculated as follows
(27) 
Step 7. Determination of the ranking orders of all alternatives
Now, boarder approximation area (BAA) value for each criteria function serves as reference point/benchmark value for criteriawise performance of an alternative . Each individual candidate will belong three different areas namely, the border approximation area (), upper approximation area (), and lower approximation area (). The ideal alternative () can be found in the upper approximation area () whereas the lower approximation area () contains the antiideal alternative ().
(28) 
Therefore, in order to select to be best compromised alternative (suitable candidate for the job), it is necessary for the candidate to be closest to and farthest from . have as many criteria as possible belonging to the upper approximate area (). The closeness coefficient to the border approximation area for each alternative can be computed using the following equation.
(29) 
The alternatives are ranked according to the decreasing value of .
4 Illustrative example
In this section, we adopt one example from Chen and Lee [12] to illustrate the fuzzy multiple attributes group decisionmaking process of our proposed method. Assume that there are three decisionmakers , and of a software company to hire a system analysis engineer and assume that there are three alternatives/candidates for this job, say, , and . The selection would be done based on five attributes namely, ‘Emotional Steadiness’, ‘Oral Communication Skill’, ‘Personality’, ‘Past Experience’, ‘SelfConfidence’. Let be the set of alternatives, where , and let be the set of attributes, where Emotional Steadiness; Oral Communication Skill; Personality; Past Experience; SelfConfidence.
Assume that the three decision makers , and use the linguistic terms shown in Table 1 to represent the weights of the five attributes, respectively, as shown in Table 3. In Table 3, five benefit type (maximizing criteria) attributes are considered, including ‘Emotional Steadiness’ (denoted by ), ‘Oral Communication Skill’ (denoted by ), ‘Personality’ (denoted by ), ‘Past Experience’ (denoted by ) and ‘SelfConfidence’ (denoted by ). Assume that the three decisionmakers , and use the linguistic terms shown in Table 2 to represent the evaluating values of the candidates with respect to different attributes, respectively, as shown in Table 5. The linguistic terms shown in Table 3 and 5 can be represented by the IT2TrFN ratings as shown in Tables 1 and 2 respectively.
Step 1: Based on Table 3 and Eq. (1415), we can get the aggregated IT2TrFN weights for the decision attributes which are shown in Table 4.
Table 3 and 4 about here
Step 2: Based on Table 5 and Eq. (1617), we can construct the average initial decision matrix (Table 6).
Table 5 and 6 about here
Step 3: For normalization it should be noted that in Table 2, the universe of discourse of the linguistic terms for the ratings is [0, 10]. Therefore, we normalize each entries in Table 6 according to Eq.(1819), since all the attributes are maximizing type in this example. Data normalization is essential for all kinds of decisionmaking problems, to ensure we obtain dimensionless units, from heterogeneous data measurements, which can be aggregated for rating and ranking decision alternatives.
Step 4: Based on Eq.(2021), we can get the IT2TrFNbased weighted decision matrix , shown as follows:
Table 7 about here
Step 5. Using the geometric aggregation operation for interval valued numbers, the IT2TrFNbased border approximation area (BAA) matrix (Table 8) for each evaluation criterion is computed according to Eq.(22).
Table 8 about here
Step 6. Rankbased distance matrices (Table 9 and 10) of weighted matrix and BAA are calculated using Eq.(1213) in order to compare the attributewise performances of each candidate.
Table 9 and 10 about here
Step 7. Finally, we can verify each candidate’s strength and weakness in every attribute by subtracting from as defined above in Eq.(24) and (26). The final score values according to Eq.(29) of the candidates and final rank of the candidates is given in Table 11.
Table 11 about here
5 Comparison and Discussion
In order to verify the validity of our proposed method, we perform a comparison of our proposed method with two other previous methods including TOPSIS by Chen and Lee [12] and Gong et al. [20] which also deal with IT2TrFNs. The results are shown as follows:
Table 12 about here
Now, let us observe few things from Table 11. Candidate has better ‘Past Experience’ than and where as performed best in all others attributes. So, it is clear that no single candidate is best in all the attributes. So, we proceed for a compromised solution. It is also clear from Table 12 that the three methods have the similar results. Note that is best compromised alternative/candidate according to all three models under fixed preference of attribute weights. This shows the method we proposed in this paper is reasonable.
The main reason of using MABAC method is the simple computation procedure and the stability (consistency) of solution (Pamučar and Ćirović [1]; Xue et al. [7]). The MABAC method is a particularly pragmatic and reliable tool for rational decision making. One more benefit of this method is that it enables us to visualize of performance and assessment of individual candidates as per each criteria and vice versa (Table 11). From the distance matrix Table 11, one can directly conclude whether an alternative performs better than the ideal person or not. In other words, this particular method enables us to understand easily the strength and weakness of a candidate compared to others. But TOPSIS [20] and VIKOR [18] methods do not produce such a direct observation.
6 Conclusions
This study proposes an extended MABAC for multiattribute decision making model based on IT2TrFNs to facilitate a more precise analysis of the alternatives, considering several criteria in fuzzy environment. Different relative weights of attributes is more realistic in many practical MADM problems, especially in complex and uncertain environments. MABAC possess simple computation procedure and the stability (consistency) of solution. Here, we utilize IT2TrFNbased MABAC to evaluate and select the most suitable candidate for a software company which is heading to hire a system analysis engineer based on few attributes. Finally, the proposed method is validated comparing with other two methods from literature. The proposed method could be applied to other MADM problems such as supplier selection, project portfolio selection etc. In future applications of the IT2TrFNbased MABAC method it would be interesting to combine with various methods for determining the weights of attributes (e.g., Entropy, AHP, DEMATEL, ANP etc).
Acknowledgment.
The first author would like to thank Department of Science and Technology, New Delhi, India, for their help and supports in this research work under INSPIRE program with sanction order number: DST/INSPIRE Fellowship/2013/544.
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