An interval type2 fuzzy TOPSIS using the extended vertex method for MAGDM
Abstract
In this article, the technique of order preference by similarity to an ideal solution (TOPSIS) is modified to handle interval type2 fuzzy sets (IT2FSs) using the extended vertex method for distance measure. While the existing TOPSIS techniques for IT2FSs depend on the defuzzification of the average decision matrix or the average weighted decision matrix from the very beginning, the proposed method maintains fuzziness in the preference technique up to the hilt to avoid any information distortion which might lead to false ranking. First, the vertex method for distance measure is extended to encompass IT2FSs. The extended vertex method is an efficient simple formula that requires few computations in contrast to other distance measures based on embedding type 1 fuzzy sets or αcuts that need special algorithms and can be restrictive in applications that require high computations. Second, the fuzzy positive and negative ideal solutions are defined. Then, the relative degree of closeness to the ideal solutions is computed for each alternative using the extended vertex method. As the relative degree of closeness of an alternative increases, its preference increases. Therefore, the preference technique avoids the flaws of the existing techniques and the computations are reduced. Two illustrative examples are given and the results are compared with the results of the existing TOPSIS methods. In light of the results and comparisons, the role of the defined ideal solutions in ranking is clarified.
Keywords
Fuzzy multiattributes group decision making TOPSIS Distance measure Interval type2 fuzzy setsMathematics Subject Classification
90B501 Introduction
In modern decision theory, multiple attributes group decisionmaking (MAGDM) problems play a vital role. In these problems, the best choice is chosen from a set of alternatives counting on experts’ assessments of the alternatives according to their multiple attributes [5]. In many cases, the human evaluations and preferences are often ambiguous and vague. In other cases, some of the evaluation attributes are subjective and qualitative in nature that they cannot be expressed using exact numerical values [17].
The concept of fuzzy sets was introduced to handle vagueness, uncertainty, and imprecision in decisionmaking problems. Type1 fuzzy sets (T1FSs) are the early proposed sets. They are characterized by a crisp membership function in the interval [0, 1]. In many situations it is hard to estimate the exact membership function of fuzzy sets. As a result, T1FSs decrease the flexibility and precision of decisionmaking in an uncertain environment [12]. Recently, MAGDM methods use more elaborate fuzzy sets, e.g. IT2FSs [4, 22]; and intuitionistic fuzzy sets [7, 11, 16, 27, 30].
Type2 fuzzy sets (T2FSs) are an extension of T1FSs. Zadeh [31] proposed T2FSs to express linguistic terms more efficiently than T1FSs. The membership degree of each element in a T2FS is represented by another fuzzy set defined over the interval [0, 1]. Therefore, the membership function of T2FSs is three dimensional. That provides more degrees of freedom in representing uncertainties in recent models [17]. T2FSs have proven to surpass T1FSs in many applications due to their ability to model uncertainties with greater accuracy [20]. However, T2FSs require complicated and massive tiresome operations [17]. This led to the introduction of fuzzy sets that requires simpler and easier computations than T2FSs.
Sambuc [24] introduced intervalvalued fuzzy sets (IVFSs) in which the membership values are expressed by intervals. The exact membership degree lies within the considered interval [28]. Consequently, an IVFS is defined by an upper T1FS membership function and a lower T1FS membership function. Later, Liang and Mendel [19] introduced IT2FSs as a particular case of T2FSs whose secondary membership degree is equal to one. IVFSs can be considered as a special case of IT2FSs in which the membership values are equal in both the upper and lower fuzzy numbers. Although IVFSs and IT2FSs might appear to be similar, they are not totally equivalent. The information capacity of these fuzzy sets is not equal [21]. IT2FSs can represent concepts that cannot be represented by IVFSs. Actually, IT2FSs can be considered as a generalization of IVFSs [28]. For the points of similarity and difference between the two sets, the reader is referred to Niewiadomski [21] and Sola [28].
The technique of order preference by similarity to an ideal solution (TOPSIS) is one of the wellknown methods to solve MAGDM problems. TOPSIS is preferred due to its simplicity, intuitiveness and limited computations [15]. It proved to be a useful practical tool for ranking and selecting among alternatives [23]. A TOPSIS solution is an alternative which is the closest to the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS). First, the weighted ratings are defuzzified into crisp values. Second, their distance from both the positive and negative ideal solutions is calculated. Then, a closeness coefficient is defined to determine the ranking order of the alternatives. Chu and Lin [6] proposed the conversion of the weighted normalized decision matrix to crisp values by defuzzification to change a fuzzy MAGDM problem into a crisp one.
Several modifications have been introduced to fuzzy TOPSIS. These modifications are either in the defuzzification technique or in the preference technique. Defuzzification is simple and easy. Nevertheless, defuzzification loses uncertainty of messages. On the other hand, a fuzzy pairwise comparison is complex and difficult. However, it preserves fuzziness in messages [15]. Ashtiani et al. [1] proposed a triangular intervalvalued fuzzy TOPSIS (IVFTOPSIS) to solve MAGDM problems. Chen and Lee [5] presented an interval type2 fuzzy TOPSIS (IT2FTOPSIS) for fuzzy MAGDM problems. The method relies on the early defuzzification of the average weighted decision matrix. This leads to two drawbacks, it gives an incorrect preferred order of the alternatives in some situations, and the preferred order of the alternatives will change if additional alternatives are added [3]. To overcome these drawbacks, Chen and Hong [3] introduced a new ranking technique for IT2FSs and utilized it in TOPSIS. Still, early defuzzfication is carried out for the attributes’ fuzzy weights and the average fuzzy decision matrix. Rashid et al. [23] extended TOPSIS for generalized intervalvalued trapezoidal fuzzy numbers. Yet, the used heuristic expression to calculate the difference between intervalvalued trapezoidal fuzzy numbers was not justified [8]. Therefore, Dymova et al. [8] introduced an interval type2 fuzzy extension of the TOPSIS method using αcuts to avoid the limitations and drawbacks of the existing methods. Ilieva [15] used the graded mean integration to defuzzify IT2FSs into two crisp values and then working with their average value. Kumar and Garg [18] proposed a TOPSIS method for intervalvalued intuitionistic fuzzy sets based on set pair analysis. Sharaf [25] proposed an IVFTOPSIS using similarity measure based on map distance for preference comparison.
In this article, TOPSIS is modified to handle IT2FSs using the extended vertex method for distance measure. The existing TOPSIS methods for IT2FSs rely on the early defuzzification of the average decision matrix or the average weighted decision matrix. The proposed method maintains fuzziness in the preference technique to avoid the disadvantages of defuzzification which may lead to incorrect ranking. First, the vertex method for distance measure is extended to include IT2FSs. The proposed distance measure is simple and requires few computations. Meanwhile, the other distance measures based on embedding type 1 fuzzy sets or αcuts need special algorithms and can be restrictive in applications that require high computations. The performance of the vertex method is compared versus other distance measures. Second, the fuzzy positive and negative ideal solutions are defined for IT2FSs. Then, the relative degree of closeness to the ideal solutions is computed for each alternative using the proposed distance measure. As the relative degree of closeness of an alternative increases, its preference increases. Therefore, the preference technique depends on a fuzzy basis to avoid the flaws of the existing techniques resulting from the loss of information due to defuzzification and the computations are reduced due to the simple and efficient formula of the extended vertex method for distance measure.
The article is organized as follows. Different types of fuzzy sets, distance measures, and the classical TOPSIS are presented in Sect. 2. The extended vertex method is introduced in Sect. 3. The proposed TOPSIS method is introduced in Sect. 4. Two illustrative examples are given to demonstrate the approach; and the results are compared with the results of some existing TOPSIS methods in Sect. 5. Discussion is given in Sect. 6. Finally, the conclusion is given in Sect. 7.
2 Preliminaries
2.1 Interval type2 fuzzy numbers
For the two interval type2 fuzzy numbers \(\tilde{A} = [(a_{1}^{L} , a_{2}^{L} , a_{3}^{L} , a_{4}^{L} ;w_{{1\tilde{A}}}^{L} ,w_{{2\tilde{A}}}^{L} ), (a_{1}^{U} , a_{2}^{U} , a_{3}^{U} , a_{4}^{U} ;w_{{1\tilde{A}}}^{U} ,w_{{2\tilde{A}}}^{U} )]\) and \(\tilde{B} = [(b_{1}^{L} , b_{2}^{L} , b_{3}^{L} , b_{4}^{L} ;w_{{1\tilde{B}}}^{L} ,w_{{2\tilde{B}}}^{L} ), (b_{1}^{U} , b_{2}^{U} , b_{3}^{U} , b_{4}^{U} ;w_{{1\tilde{B}}}^{U} ,w_{{2\tilde{B}}}^{U} )]\), the aggregation operations are defined as follows [5]:
2.2 Distance measure
2.2.1 Concept of distance measures
Distance measures have been extensively studied due to their applications in multiple areas, e.g. risk analysis, data mining, signal processing and pattern recognition [26, 27]. A distance measure depicts the difference between two fuzzy sets. Computing the difference between two fuzzy sets as a crisp number is crucial for ranking and preference. Yet as the distance is computed in an inaccurate domain, due to vagueness, a rational problem arises [13].
Definition 2.2.1.1
 1.
\(d\left( {\tilde{A},\tilde{B}} \right) \ge 0,\) for any two IT2FSs \(\tilde{A}\) and \(\tilde{B}\).
 2.
\(d\left( {\tilde{A},\tilde{B}} \right) = d\left( {\tilde{B},\tilde{A}} \right),\) for any two IT2FSs \(\tilde{A}\) and \(\tilde{B}\).
 3.
\(d\left( {\tilde{A},\tilde{B}} \right) + d\left( {\tilde{B},\tilde{C}} \right) \ge d\left( {\tilde{A},\tilde{C}} \right),\) for any three IT2FSs \(\tilde{A}\), \(\tilde{B}\) and \(\tilde{C}\).
Definition 2.2.1.2
[13] for any three IT2FSs \(\tilde{A}\), \(\tilde{B}\) and \(\tilde{C}\), we can write \(\tilde{A} \prec \tilde{B} \prec\) \(\tilde{C}\) if and only if \(d\left( {\tilde{A},\tilde{B}} \right) < d\left( {\tilde{A},\tilde{C}} \right)\), which means that \(\tilde{B}\) is closer to \(\tilde{A}\) than \(\tilde{C}\).
Most of the distance measures for IT2FSs are generalizations of the distances used in the crisp sets, replacing the characteristic functions by the membership functions, e.g. the normalized Hamming distance, the normalized Euclidean distance and the normalized Hamming distance based on Hausdorff metric. Heidarzade et al. [13] demonstrated that these three distance measures are not appropriate for IT2FSs.
FigueroaGarcia and HernandezPerez [10] proposed a distance measure for triangular IT2FSs, i.e. with triangular lower and upper membership functions, using its decomposition into αcuts. However, the αcuts based distance can be restrictive in applications that require high computational efforts [9]. For this reason, FigueroaGarcia et al. [9] proposed centroid based distance measures for triangular IT2FSs. However, centroids are a sort of defuzzification, regardless the formula used for measuring distance based on them.
Heidarzade et al. [13] proposed a distance measure for IT2FSs. The algorithm assumes n embedded type1 fuzzy numbers within the surface of the footprint of uncertainty (FOU). By increasing n, the surface of FOU is embedded with more type1 fuzzy sets. Therefore, the value of n has a direct impact on the difference between the upper and lower membership functions for the IT2FS.
For details on distance measures for IT2FSs, the reader is referred to Zhang et al. [32], FigueroaGarcia et al. [9] and Heidarzade et al. [13].
2.3 The classical TOPSIS

Step 1. Form the normalized decision matrix.
In this step, the various attribute dimensions are transformed into nondimensional attributes to allow comparing the attributes. 
Step 2. Form the weighted normalized decision matrix.
Since we can’t assume that all the criteria are of equal importance, the decisionmakers assign a set of weights \(w = \left( {w_{1} , w_{2} , \ldots ,w_{m} } \right)\) for the criteria. Each criterion is multiplied by its associated weight  Step 3. Determine the positive and negative ideal solutions.where F_{b} are the benefit criteria, and F_{c} are the cost criteria.$$\begin{aligned} \varvec{v}^{ + } & = \left\{ {\left. {\left( {\mathop {\hbox{max} }\limits_{\varvec{j}} \varvec{v}_{{\varvec{ij}}} j \in F_{b} } \right),\left( {\mathop {\hbox{min} }\limits_{\varvec{j}} \varvec{v}_{{\varvec{ij}}} j \in F_{c} } \right)} \rightj = 1,2, \ldots n} \right\} = \left( {v_{1}^{ + } , v_{2}^{ + } , \ldots v_{m}^{ + } } \right), \\ \varvec{v}^{  } & = \left\{ {\left. {\left( {\mathop {\hbox{min} }\limits_{\varvec{j}} \varvec{v}_{{\varvec{ij}}} j \in F_{b} } \right),\left( {\mathop {\hbox{max} }\limits_{\varvec{j}} \varvec{v}_{{\varvec{ij}}} j \in F_{c} } \right)} \rightj = 1,2, \ldots n} \right\} = \left( {v_{1}^{  } , v_{2}^{  } , \ldots v_{m}^{  } } \right), \\ \end{aligned}$$

Step 4. Calculate the separation measures.
The Euclidean distance is used to compute the separation measures between each alternative and the ideal solutions.$$S_{j}^{ + } = \sqrt {\mathop \sum \limits_{i = 1}^{m} (v_{ij}  v^{ + } )^{2} } ,\;S_{j}^{  } = \sqrt {\mathop \sum \limits_{i = 1}^{m} (v_{ij}  v^{  } )^{2} } ,\;j = 1,2, \ldots n.$$  Step 5. Calculate the relative closeness to the ideal solution.$$R_{j} = \frac{{S_{j}^{  } }}{{S_{j}^{ + } + S_{j}^{  } }}, j = 1,2, \ldots n$$

Step 6. Rank the preference order.
The alternatives can be ranked to the descending order of the relative closeness.
3 The extended vertex method
Chen [2] extended the TOPSIS method for group decision making under fuzzy environment. To find the relative degree of closeness, the distance to both the PIS and the NIS must be calculated. Chen [2] proposed the vertex method to calculate the distance between two triangular type1 fuzzy numbers. The vertex method is defined as follows.
Definition 2.3.2.1
Generalizing the concept to handle IT2FSs, the extended vertex method is defined as follows.
Consider the fuzzy numbers \(\tilde{A} = [(a_{1}^{L} , a_{2}^{L} , a_{3}^{L} , a_{4}^{L} ;w_{{1\tilde{A}}}^{L} ,w_{{2\tilde{A}}}^{L} ), (a_{1}^{U} , a_{2}^{U} , a_{3}^{U} , a_{4}^{U} ;w_{{1\tilde{A}}}^{U} ,w_{{2\tilde{A}}}^{U} )]\) and \(\tilde{B} = [(b_{1}^{L} , b_{2}^{L} , b_{3}^{L} , b_{4}^{L} ;w_{{1\tilde{B}}}^{L} ,w_{{2\tilde{B}}}^{L} ), (b_{1}^{U} , b_{2}^{U} , b_{3}^{U} , b_{4}^{U} ;w_{{1\tilde{B}}}^{U} ,w_{{2\tilde{B}}}^{U} )].\)
Definition 2.3.2.2
The properties of the metric distance, \(d\left( {\tilde{A},\tilde{B}} \right) \ge 0, d\left( {\tilde{A},\tilde{B}} \right) = d\left( {\tilde{B},\tilde{A}} \right),\) and \(d\left( {\tilde{A},\tilde{B}} \right) + d\left( {\tilde{B},\tilde{C}} \right) \ge d\left( {\tilde{A},\tilde{C}} \right),\) are trivial from the formula.
Proposition 1
If \(\tilde{A}\) and \(\tilde{B}\) are real numbers, then the distance measure reduces to the Euclidean distance.
Proof
If \(\tilde{A}\) is a real number, then
\(a_{1}^{L} = a_{2}^{L} = a_{3}^{L} = a_{4}^{L} = a_{1}^{U} = a_{2}^{U} = a_{3}^{U} = a_{4}^{U} = a\) and \(w_{{1\tilde{A}}}^{L} = w_{{2\tilde{A}}}^{L} = w_{{1\tilde{A}}}^{U} = w_{{2\tilde{A}}}^{U} = 1\).
Proposition 2
Two IT2FSs \(\tilde{A}\) and \(\tilde{B}\) are identical if and only if \(d\left( {\tilde{A}, \tilde{B}} \right) = 0.\)
Proof
 1.(i) Let \(\tilde{A}\) = \(\tilde{B}\), then$$a_{1}^{L} = b_{1}^{L} , a_{2}^{L} = b_{2}^{L} ,a_{3}^{L} = b_{3, }^{L} a_{4}^{L} = b_{4}^{L} , a_{1}^{U} = b_{1}^{U} ,a_{2}^{U} = b_{2}^{U} , a_{3}^{U} = b_{3}^{U} , a_{4}^{U} = b_{4}^{U} ,w_{{1\tilde{A}}}^{L} = w_{{1 \tilde{B}}}^{L} , w_{2B}^{L} = w_{{2\tilde{A}}}^{L} ,w_{{1\tilde{A}}}^{U} = w_{1B}^{U} ,\, and \,w_{{2\tilde{A}}}^{U} = w_{2B}^{U} .$$
Substituting in the vertex formula gives \(d\left( {\tilde{A}, \tilde{B}} \right) = 0.\)
 2.If \(d\left( {\tilde{A}, \tilde{B}} \right) = 0\), then, which implies that$$\sqrt {\frac{1}{8}\left[ \begin{aligned} & \left( {a_{1}^{L}  b_{1}^{L} } \right)^{2} + \left( {a_{2}^{L}  b_{2}^{L} } \right)^{2} + \left( {a_{3}^{L}  b_{3}^{L} } \right)^{2} + \left( {a_{4}^{L}  b_{4}^{L} } \right)^{2} + \left( {a_{1}^{U}  b_{1}^{U} } \right)^{2} + \left( {a_{2}^{L}  b_{2}^{U} } \right)^{2} + \left( {a_{3}^{U}  b_{3}^{U} } \right)^{2} \\ & \quad + \left( {a_{4}^{U}  b_{4}^{U} } \right)^{2} + \left( {w_{{1\tilde{A}}}^{L}  w_{{1 \tilde{B}}}^{L} } \right)^{2} + \left( {w_{{2\tilde{A}}}^{L}  w_{{2 \tilde{B}}}^{L} } \right)^{2} + \left( {w_{{1\tilde{A}}}^{U}  w_{{1 \tilde{B}}}^{U} } \right)^{2} + \left( {w_{{2\tilde{A}}}^{U}  w_{{2 \tilde{B}}}^{U} } \right)^{2} . \\ \end{aligned} \right]} = 0$$$$a_{1}^{L} = b_{1}^{L} , a_{2}^{L} = b_{2}^{L} ,a_{3}^{L} = b_{3, }^{L} a_{4}^{L} = b_{4}^{L} , a_{1}^{U} = b_{1}^{U} ,a_{2}^{U} = b_{2}^{U} , a_{3}^{U} = b_{3}^{U} , a_{4}^{U} = b_{4}^{U} ,w_{{1\tilde{A}}}^{L} = w_{{1\tilde{B}}}^{L} , w_{2B}^{L} = w_{{2\tilde{A}}}^{L} ,w_{{1\tilde{A}}}^{U} = w_{1B}^{U} , and w_{{2\tilde{A}}}^{U} = w_{2B}^{U} , then \tilde{A}=\tilde{B}$$
Wu and Mendel [29] proposed 32 words for computing with words. They can be grouped into three classes. Class one: smallsounding words (none to very little, teenyweeny, a smidgen, tiny, very small, very little, a bit, little, low amount, small, and somewhat small). Class two: mediumsounding words (some, some to moderate, moderate amount, fair amount, medium, modest amount, and good amount). Class three: largesounding words (sizeable, quite a bit, considerable amount, substantial amount, a lot, high amount, very sizeable, large, humongous amount, huge amount, very high amount, extreme amount, and maximum amount). For the corresponding IT2FSs of these words the reader is referred to Wu and Mendel [29] or Heidarzade et al. [13].
The distance between the first element and the other elements
Distance  Signed distance  Heidarzade et al. method [13]  Extended vertex method 

D (none to very little, teenyweeny)  0.0072  0.0089  0.1879 
D (none to very little, a smidgen)  0.0153  0.0111  0.2374 
D (none to very little, tiny)  0.0261  0.0120  0.2462 
D (none to very little, very small)  0.0443  0.0208  0.3067 
D (none to very little, very little)  0.0441  0.0210  0.3155 
D (none to very little, a bit)  0.3108  0.1321  1.4466 
D (none to very little, little)  0.3748  0.1635  1.7866 
D (none to very little, low amount)  0.3726  0.1640  1.8546 
D (none to very little, small)  0.4171  0.1865  2.1266 
D (none to very little, somewhat small)  0.4791  0.2176  2.3793 
D (none to very little, some)  0.7253  0.3382  3.7383 
D (none to very little, some to moderate)  0.8677  0.4137  4.4649 
D (none to very little, moderate amount)  0.9331  0.4444  4.5810 
D (none to very little, fair amount)  0.9852  0.4723  4.9908 
D (none to very little, medium)  0.9874  0.4733  4.7911 
D (none to very little, modest amount)  1.0363  0.4944  5.0517 
D (none to very little, good amount)  1.2577  0.6060  6.2784 
D (none to very little, sizeable)  1.3947  0.6754  6.9856 
D (none to very little, quite a bit)  1.3947  0.6754  6.9856 
D (none to very little, considerable amount)  1.4189  0.6892  7.1535 
D (none to very little, substantial amount)  1.5543  0.7540  7.7687 
D (none to very little, a lot)  1.5538  0.7542  7.7505 
D (none to very little, high amount)  1.5645  0.7602  7.8420 
D (none to very little, very sizeable)  1.5811  0.7713  8.0113 
D (none to very little, large)  1.5836  0.7716  7.9105 
D (none to very little, very large)  1.8708  0.9818  9.1176 
D (none to very little, humongous amount)  1.8852  0.8982  9.1715 
D (none to very little, huge amount)  1.8991  0.9093  9.2410 
D (none to very little, very high amount)  1.9051  0.9105  9.2567 
D (none to very little, extreme amount)  1.9101  0.9244  9.3083 
D (none to very little, maximum amount)  1.9303  0.9309  9.4422 
From Table 1, it is obvious that the distances obtained by the method of Heidarzade et al. [13] are in an increasing order. As for the distances obtained by the signed distance, they are in an increasing order except for the words very small and very little, little and low amount, and substantial amount and a lot. Regarding the extended vertex method, the distances are also in an increasing order except for the words fair amount and medium, substantial amount and a lot, and very sizeable and large. In any case, the maximum difference was 0.2 on the 0 to 10 scale. This is acceptable in fuzzy sets as these words are quite equivalent when defuzzified, and the order may differ according to the used defuzzification technique. The results indicate the extended vertex method is appropriate for measuring the distances between IT2FSs.
Regarding the computational complexity, the extended vertex method has the least processing time. When implemented using MatLab on a PC (Intel(R) Core(TM) i36100 CPU@2.3 GHz), the processing time was as follows. The signed distance takes 1.8 × 10^{−5} s. The method of Heidarzade et al. [13] takes 3.3 × 10^{−5} s, using the minimum number of embedded T1FSs (only the upper and lower membership functions), in practice large number of embedded T1FSs are used, e.g., 10 or 20. The extended vertex method takes 1.6 × 10^{−5} s. Since calculating the distance between IT2FSs is repeated 2mn times (m criteria and n variables) in IT2FTOPSIS, then the extended vertex method reduces the computations.
4 The proposed TOPSIS
TOPSIS was first introduced by Hwang and Yoon [14] for realvalued data. Later, Chen [2] extended the method to the fuzzy environment using T1FSs. Chen and Lee [5] modified the technique to use IT2FSs. Recent research has been devoted to the fuzzy extension of TOPSIS, but only a few studies handled IT2FSs [8].
In this section, The TOPSIS method is modified. First, the PIS and NIS proposed by Rashid et al. [23] for IVFSs are extended to IT2FSs. Then, the extended vertex method is used to calculate the distance between the alternatives and the ideal solutions.
 Step 1: Constructing the fuzzy decision matrix and the average decision matrixwhere \(\tilde{f}_{{\varvec{ij}}} = \left( {\frac{{\tilde{f}_{ij}^{1} \oplus \tilde{f}_{ij}^{2} \oplus \cdots \oplus \tilde{f}_{ij}^{k} }}{k}} \right)\) is an IT2FS, \(1 \le i \le m, 1 \le j \le n\) and \(1 \le p \le k\).
 Step 2: Constructing the weighting matrix and the average weighting matrix,where \(\tilde{w}_{\varvec{i}} = \left( {\frac{{{\tilde{\text{w}}}_{i}^{1} \oplus {\tilde{\text{w}}}_{i}^{2} \oplus \cdots \oplus {\tilde{\text{w}}}_{i}^{k} }}{k}} \right)\) is an IT2FS, \(1 \le i \le m\) and \(1 \le p \le k\).
 Step 3: Constructing the normalized weighted decision matrixwhere \(\tilde{v}_{ij} = {\tilde{\text{w}}}_{i} \otimes \tilde{f}_{ij}\), \(1 \le i \le m\) and \(1 \le j \le n.\)
 Step 4: Defining the fuzzy PIS \(\left( {{\tilde{\mathbf{V}}}^{ + } } \right)\) and the fuzzy NIS solution (\({\tilde{\mathbf{V}}}^{  }\)) where,\({\mathbf{V}}^{ + } = \left[ {\begin{array}{*{20}c} {\tilde{v}_{1}^{ + } } & {\tilde{v}_{2}^{ + } } \\ \end{array} \ldots \tilde{v}_{m}^{ + } } \right]\).$$\tilde{v}_{i}^{ + } \left\{ {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {\left( {\mathop {\hbox{max} }\limits_{j} v_{1ij}^{L} ,\mathop {\hbox{max} }\limits_{j} v_{2ij}^{L} ,\mathop {\hbox{max} }\limits_{j} v_{3ij}^{L} ,\mathop {\hbox{max} }\limits_{j} v_{4ij}^{L} ;\mathop {\hbox{max} }\limits_{j} w_{1ij}^{L} ,\mathop {\hbox{max} }\limits_{j} w_{2ij}^{L} } \right),} \hfill \\ {\left( {\mathop {\hbox{max} }\limits_{j} v_{1ij}^{U} ,\mathop {\hbox{max} }\limits_{j} v_{2ij}^{U} ,\mathop {\hbox{max} }\limits_{j} v_{3ij}^{U} ,\mathop {\hbox{max} }\limits_{j} v_{4ij}^{U} ;\mathop {\hbox{max} }\limits_{j} w_{1ij}^{U} ,\mathop {\hbox{max} }\limits_{j} w_{2ij}^{U} } \right)} \hfill \\ \end{array} } \right],\quad {\text{if}}\quad f_{i} \in F_{b} } \hfill \\ {\left[ {\begin{array}{*{20}l} {\left( {\mathop {\hbox{min} }\limits_{j} v_{1ij}^{L} ,\mathop {\hbox{min} }\limits_{j} v_{2ij}^{L} ,\mathop {\hbox{min} }\limits_{j} v_{3ij}^{L} ,\mathop {\hbox{min} }\limits_{j} v_{4ij}^{L} ;\mathop {\hbox{min} }\limits_{j} w_{1ij}^{L} ,\mathop {\hbox{min} }\limits_{j} w_{2ij}^{L} } \right),} \hfill \\ {\left( {\mathop {\hbox{min} }\limits_{j} v_{1ij}^{U} ,\mathop {\hbox{min} }\limits_{j} v_{2ij}^{U} ,\mathop {\hbox{min} }\limits_{j} v_{3ij}^{U} ,\mathop {\hbox{min} }\limits_{j} v_{4ij}^{U} ;\mathop {\hbox{min} }\limits_{j} w_{1ij}^{U} ,\mathop {\hbox{min} }\limits_{j} w_{2ij}^{U} } \right)} \hfill \\ \end{array} } \right],\quad {\text{if}}\quad f_{i} \in F_{c} ,} \hfill \\ \end{array} } \right.$$\({\mathbf{V}}^{  } = \left[ {\begin{array}{*{20}c} {\tilde{v}_{1}^{  } } & {\tilde{v}_{2}^{  } } \\ \end{array} \ldots \tilde{v}_{m}^{  } } \right]\).$$\tilde{v}_{i}^{  } \left\{ {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {\left( {\mathop {\hbox{min} }\limits_{j} v_{1ij}^{L} ,\mathop {\hbox{min} }\limits_{j} v_{2ij}^{L} ,\mathop {\hbox{min} }\limits_{j} v_{3ij}^{L} ,\mathop {\hbox{min} }\limits_{j} v_{4ij}^{L} ;\mathop {\hbox{min} }\limits_{j} w_{1ij}^{L} ,\mathop {\hbox{min} }\limits_{j} w_{2ij}^{L} } \right),} \hfill \\ {\left( {\mathop {\hbox{min} }\limits_{j} v_{1ij}^{U} ,\mathop {\hbox{min} }\limits_{j} v_{2ij}^{U} ,\mathop {\hbox{min} }\limits_{j} v_{3ij}^{U} ,\mathop {\hbox{min} }\limits_{j} v_{4ij}^{U} ;\mathop {\hbox{min} }\limits_{j} w_{1ij}^{U} ,\mathop {\hbox{min} }\limits_{j} w_{2ij}^{U} } \right)} \hfill \\ \end{array} } \right],\quad {\text{if}}\quad f_{i} \in F_{b} } \hfill \\ {\left[ {\begin{array}{*{20}l} {\left( {\mathop {\hbox{max} }\limits_{j} v_{1ij}^{L} ,\mathop {\hbox{max} }\limits_{j} v_{2ij}^{L} ,\mathop {\hbox{max} }\limits_{j} v_{3ij}^{L} ,\mathop {\hbox{max} }\limits_{j} v_{4ij}^{L} ;\mathop {\hbox{max} }\limits_{j} w_{1ij}^{L} ,\mathop {\hbox{max} }\limits_{j} w_{2ij}^{L} } \right),} \hfill \\ {\left( {\mathop {\hbox{max} }\limits_{j} v_{1ij}^{U} ,\mathop {\hbox{max} }\limits_{j} v_{2ij}^{U} ,\mathop {\hbox{max} }\limits_{j} v_{3ij}^{U} ,\mathop {\hbox{max} }\limits_{j} v_{4ij}^{U} ;\mathop {\hbox{max} }\limits_{j} w_{1ij}^{U} ,\mathop {\hbox{max} }\limits_{j} w_{2ij}^{U} } \right)} \hfill \\ \end{array} } \right],\quad {\text{if}}\quad f_{i} \in F_{c} ,} \hfill \\ \end{array} } \right.$$
 Step 5: Constructing the ideal separation matrix \({\mathbf{S}}^{ + }\) and the antiideal separation matrix \({\mathbf{S}}^{  }\) using the extended vertex method for distance measure.
 Step 6: Calculating the relative degree of closeness of each alternative to the ideal solution and ranking.$$R\left( {x_{j} } \right) = \frac{{S^{  } \left( {x_{j} } \right)}}{{S^{ + } \left( {x_{j} } \right) + S^{  } \left( {x_{j} } \right)}}$$
5 Examples
In this section, two examples are solved. The first example is due to Rashid et al. [23], the second example is due Chen and Hong [3]. The results of the proposed IT2FTOPSIS are compared with their results.
Example 1
 Step 1: a) Constructing the decision matrices, and the average decision matrix
 Step 2: a) Constructing the weighting matrices, and the average weighting matrix
 Step 3: Constructing the weighted normalized decision matrix$$\begin{aligned} \tilde{v}_{11} & = \left[ {\left( {0.5408,0.6094,0.7157,0.7516;0.8,0.8} \right)\left( {0.4464,0.5577,0.7727,0.8577;1,1} \right)} \right], \\ \tilde{v}_{12} & = \left[ {\left( {0.5408,0.6094,0.7157,0.7516;0.8,0.8} \right)\left( {0.4464,0.5577,0.7727,0.8577;1,1} \right)} \right], \\ \tilde{v}_{13} & = \left[ {\left( {0.7264,0.7981,0.8818,0.9080;0.8,0.8} \right)\left( {0.6446,0.7564,0.9287,0.9900;1,1} \right)} \right], \\ \tilde{v}_{21} & = \left[ {\left( {0.4965,0.5469,0.6216,0.6484;0.8,0.8} \right)\left( {0.4285,0.5104,0.6619,0.7233;1,1} \right)} \right], \\ \tilde{v}_{22} & = \left[ {\left( {0.8046,0.8705,0.9244,0.9390;0.8,0.8} \right)\left( {0.7470,0.8426,0.9569,0.9900;1,1} \right)} \right], \\ \tilde{v}_{23} & = \left[ {\left( {0.6408,0.7040,0.7778,0.8010;0.8,0.8} \right)\left( {0.5704,0.6673,0.8193,0.8733;1,1} \right)} \right], \\ \tilde{v}_{31} & = \left[ {\left( {0.1622,0.2019,0.2772,0.3082;0.8,0.8} \right)\left( {0.1059,0.1686,0.3192,0.3978;1,1} \right)} \right], \\ \tilde{v}_{32} & = \left[ {\left( {0.2180,0.2644,0.3416,0.3727;0.8,0.8} \right)\left( {0.1534,0.2286,0.3837,0.4590;1,1} \right)} \right], \\ \tilde{v}_{33} & = \left[ {\left( {0.2399,0.2870,0.3755,0.4108;0.8,0.8} \right)\left( {0.1682,0.2465,0.4236,0.5100;1,1} \right)} \right], \\ \tilde{v}_{41} & = \left[ {\left( {0.3649,0.4170,0.4967,0.5262;0.8,0.8} \right)\left( {0.2897,0.3737,0.5378,0.6048;1,1} \right)} \right], \\ \tilde{v}_{42} & = \left[ {\left( {0.3798,0.4331,0.5181,0.5510;0.8,0.8} \right)\left( {0.2977,0.3841,0.5648,0.6358;1,1} \right)} \right], \\ \tilde{v}_{43} & = \left[ {\left( {0.3880,0.4393,0.5295,0.5632;0.8,0.8} \right)\left( {0.3049,0.3924,0.5774,0.6500;1,1} \right)} \right], \\ \tilde{v}_{51} & = \left[ {\left( {0.8324,0.8747,0.9198,0.9363;0.8,0.8} \right)\left( {0.7938,0.8474,0.9546,0.9900;1,1} \right)} \right], \\ \tilde{v}_{52} & = \left[ {\left( {0.7551,0.7944,0.8278,0.8436;0.8,0.8} \right)\left( {0.7194,0.7693,0.8610,0.8852;1,1} \right)} \right], \\ \tilde{v}_{53} & = \left[ {\left( {0.7466,0.7855,0.8278,0.8436;0.8,0.8} \right)\left( {0.7111,0.7605,0.8797,0.9043;1,1} \right)} \right], \\ \tilde{v}_{61} & = \left[ {\left( {0.6203,0.6705,0.7673,0.8137;0.8,0.8} \right)\left( {0.5563,0.6047,0.8608,0.9800;1,1} \right)} \right], \\ \tilde{v}_{62} & = \left[ {\left( {0.4926,0.5282,0.5865,0.6238;0.8,0.8} \right)\left( {0.4270,0.4703,0.6528,0.7000;1,1} \right)} \right], \\ \tilde{v}_{63} & = \left[ {\left( {0.4785,0.5127,0.5580,0.5776;0.8,0.8} \right)\left( {0.4157,0.4576,05916,0.6322;1,1} \right)} \right] \\ \end{aligned}$$
 Step 4: Define the fuzzy positive ideal solution \({\mathbf{V}}^{ + } = \left[ {\begin{array}{*{20}c} {\tilde{v}_{1}^{ + } } & {\tilde{v}_{2}^{ + } } & {\tilde{v}_{3}^{ + } } & {\tilde{v}_{4}^{ + } } & {\tilde{v}_{5}^{ + } } & {\tilde{v}_{6}^{ + } } \\ \end{array} } \right]\) and the fuzzy negative ideal solution \({\mathbf{V}}^{  } = \left[ {\begin{array}{*{20}c} {\tilde{v}_{1}^{  } } & {\tilde{v}_{2}^{  } } & {\tilde{v}_{3}^{  } } & {\tilde{v}_{4}^{  } } & {\tilde{v}_{5}^{  } } & {\tilde{v}_{6}^{  } } \\ \end{array} } \right]\).$$\begin{aligned} \tilde{v}_{1}^{ + } & = \left[ {\left( {0.7264,0.7981,0.8818,0.9080;0.8,0.8} \right)\left( {0.6466,0.7564,0.9287,0.9900;1,1} \right)} \right], \\ \tilde{v}_{2}^{ + } & = \left[ {\left( {0.8046,0.8705,0.9244,0.9390;0.8,0.8} \right)\left( {0.7470,0.8426,0.9569,0.9900;1,1} \right)} \right], \\ \tilde{v}_{3}^{ + } & = \left[ {\left( {0.2399,0.2870,0.3755,0.4108;0.8,0.8} \right)\left( {0.1682,0.2465,0.4236,0.5100;1,1} \right)} \right], \\ \tilde{v}_{4}^{ + } & = \left[ {\left( {0.3649,0.4170,0.4967,0.5262;0.8,0.8} \right)\left( {0.2897,0.3737,0.5378,0.6048;1,1} \right)} \right], \\ \tilde{v}_{5}^{ + } & = \left[ {\left( {0.8324,0.8747,0.9198,0.9363;0.8,0.8} \right)\left( {0.7938,0.8474,0.9546,0.9900;1,1} \right)} \right], \\ \tilde{v}_{6}^{ + } & = \left[ {\left( {0.4785,0.5127,0.5580,0.5776;0.8,0.8} \right)\left( {0.4157,0.4576,0.5916,0.6322;1,1} \right)} \right], \\ \tilde{v}_{1}^{  } & = \left[ {\left( {0.5408,0.6094,0.7157,0.7516;0.8,0.8} \right)\left( {0.4464,0.5577,0.7727,0.8577;1,1} \right)} \right], \\ \tilde{v}_{2}^{  } & = \left[ {\left( {0.4965,0.5469,0.6216,0.6484;0.8,0.8} \right)\left( {0.4285,0.5104,0.6619,0.7233;1,1} \right)} \right], \\ \tilde{v}_{3}^{  } & = \left[ {\left( {0.1622,0.2019,0.2772,0.3082;0.8,0.8} \right)\left( {0.1059,0.1686,0.3192,0.3978;1,1} \right)} \right], \\ \tilde{v}_{4}^{  } & = \left[ {\left( {0.3880,0.4393,0.5295,0.5632;0.8,0.8} \right)\left( {0.3049,0.3924,0.5774,0.6500;1,1} \right)} \right], \\ \tilde{v}_{5}^{  } & = \left[ {\left( {0.7466,0.7855,0.8275,0.8436;0.8,0.8} \right)\left( {0.7111,0.7605,0.8610,0.8852;1,1} \right)} \right], \\ \tilde{v}_{6}^{  } & = \left[ {\left( {0.6203,0.6705,0.7673,0.8137;0.8,0.8} \right)\left( {0.5563,0.6047,0.8606,0.9800;1,1} \right)} \right]. \\ \end{aligned}$$
 Step 5: Constructing the ideal and the antiideal separation matrices using the extended vertex method for distance measure.ThenSimilarly$$\begin{aligned} \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i1} ,v_{i}^{ + } ) & = 0.1745 + 0.3053 + 0.0915 + 0 + 0 + 0.2178 = 0.7891, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i2} ,v_{i}^{ + } ) & = 0.1745 + 0 + 0.0323 + 0.0203 + .0872 + .0388 = 0.3531, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i3} ,v_{i}^{ + } ) & = 0 + 0.1539 + 0 + 0.0306 +0.0864 + 0 =0.2709. \\ \end{aligned}$$Then$$\begin{aligned} \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i1} ,v_{i}^{  } ) & = 0 + 0 + 0 + 0.0306 +0.0864 + 0 = 0.1170, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i2} ,v_{i}^{  } ) & = 0 + 0.3053 +0.0602 +0.0104 +0.0112 +0.1809 =0.5680, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i3} ,v_{i}^{  } ) & = 0.1745 +0.1522 +0.0915 + 0 + 0 + 0.2178 = 0.6360. \\ \end{aligned}$$
 Step 6: Calculating the relative degree of closeness of each alternative to the fuzzy ideal solutions.From the results,\(R({\text{x}}_{3} ) > R({\text{x}}_{2} ) > R({\text{x}}_{1} )\), the ranking is \({\text{x}}_{3} > {\text{x}}_{2} > {\text{x}}_{1}\). Then, x_{3} is the best alternative. The result coincides with the results of Rashid et al. [23].$$R\left( {{\text{x}}_{1} } \right) = \frac{0.1170}{0.1170 +0.7891} = 0.1291, \quad R\left( {{\text{x}}_{2} } \right) = \frac{0.5680}{0.5680 + 0.3531} = 0.6167, \quad R\left( {{\text{x}}_{3} } \right) = \frac{0.6360}{0.6360 + 0.2709} = 0.7013.$$
Example 2
A company wants to hire a system analyst. Three decision makers D_{1}, D_{2}, and D_{3} will count on two attributes to rate the applicants: emotional steadiness (f1) and oral communication skills (f2). The decision makers use two sets of linguistic terms: the weighting set \(W = \{ {\text{Very Low}}\left( {\text{VL}} \right), {\text{Low}}\left( {\text{L}} \right), {\text{Medium}}  {\text{low}}\left( {\text{ML}} \right),{\text{Medium}}\left( {\text{M}} \right),\)
 Step 1: Constructing the decision matrices and the average decision matrixwhere$$\begin{aligned} \tilde{f}_{11} & = \left[ {\left( {0,0.03,0.03,0.17;1,1} \right)\left( {0,0.03,0.03,0.17;1,1} \right)} \right], \\ \tilde{f}_{12} & = \left[ {\left( {0.07,0.23,0.23,0.43;1,1} \right)\left( {0.07,0.23,0.23,0.43;1,1} \right)} \right], \\ \tilde{f}_{13} & = \left[ {\left( {0,0.07,0.07,0.1;1,1} \right)\left( {0,0.07,0.07,0.1;1,1} \right)} \right], \\ \tilde{f}_{14} & = \left[ {\left( {0,0.07,0.07,0.1;1,1} \right)\left( {0,0.07,0.07,0.1;1,1} \right)} \right], \\ \tilde{f}_{21} & = \left[ {\left( {0,0.03,0.03,0.17;1,1} \right)\left( {0,0.03,0.03,0.17;1,1} \right)} \right], \\ \tilde{f}_{22} & = \left[ {\left( {0,0.07,0.07,0.1;1,1} \right)\left( {0,0.07,0.07,0.1;1,1} \right)} \right], \\ \tilde{f}_{23} & = \left[ {\left( {0.07,0.23,0.23,0.43;1,1} \right)\left( {0.07,0.23,0.23,0.43;1,1} \right)} \right], \\ \tilde{f}_{24} & = \left[ {\left( {0.3,0.5,0.5,0.7;1,1} \right)\left( {0.3,0.5,0.5,0.7;1,1} \right)} \right]. \\ \end{aligned}$$
 Step 2: a) Constructing the weighting matrices and the average weighting matrix.where$$\begin{aligned} \tilde{w}_{1} & = \left[ {\left( {0.63,0.83,0.83,0.97;1,1} \right)\left( {0.63,0.83,0.83,0.97;1,1} \right)} \right], \\ \tilde{w}_{2} & = \left[ {\left( {0.63,0.83,0.83,0.97;1,1} \right)\left( {0.63,0.83,0.83,0.97;1,1} \right)} \right]. \\ \end{aligned}$$
 Step 3: Constructing the average weighted decision matrix.where,$$\begin{aligned} \tilde{v}_{11} & = \left[ {\left( {0,0.0249,0.0249,0.1649;1,1} \right)\left( {0,0.0249,0.0249,0.1649;1,1} \right)} \right], \\ \tilde{v}_{12} & = \left[ {\left( {0.0441,0.1909,0.1909,0.4171;1,1} \right)\left( {0.0441,0.1909,0.1909,0.4171;1,1} \right)} \right], \\ \tilde{v}_{13} & = \left[ {\left( {0,0.0581,0.0581,0.0970;1,1} \right)\left( {0,0.0581,0.0581,0.0970;1,1} \right)} \right], \\ \tilde{v}_{14} & = \left[ {\left( {0,0.0581,0.0581,0.0970;1,1} \right)\left( {0,0.0581,0.0581,0.0970;1,1} \right)} \right], \\ \tilde{v}_{21} & = \left[ {\left( {0,0.0249,0.0249,0.1649;1,1} \right)\left( {0,0.0249,0.0249,0.1649;1,1} \right)} \right], \\ \tilde{v}_{22} & = \left[ {\left( {0,0.0581,0.0581,0.0970;1,1} \right)\left( {0,0.0581,0.0581,0.0970;1,1} \right)} \right], \\ \tilde{v}_{23} & = \left[ {\left( {0.0441,0.1909,0.1909,0.4171;1,1} \right)\left( {0.0441,0.1909,0.1909,0.4171;1,1} \right)} \right], \\ \tilde{v}_{24} & = \left[ {\left( {0.1890,0.4150,0.4150,0.6790;1,1} \right)\left( {0.1890,0.4150,0.4150,0.6790;1,1} \right)} \right]. \\ \end{aligned}$$
 Step 4: Define the fuzzy positive ideal solution \({\mathbf{V}}^{ + } = \left[ {\begin{array}{*{20}c} {\tilde{v}_{1}^{ + } } & {\tilde{v}_{2}^{ + } } \\ \end{array} } \right]\) and the fuzzy negative ideal solution \({\mathbf{V}}^{  } = \left[ {\begin{array}{*{20}c} {\tilde{v}_{1}^{  } } & {\tilde{v}_{2}^{  } } \\ \end{array} } \right].\)$$\begin{aligned} \tilde{v}_{1}^{ + } & = \left[ {\left( {0.0441,0.1909,0.1909,0.4171;1,1} \right)\left( {0.0441,0.1909,0.1909,0.4171;1,1} \right)} \right], \\ \tilde{v}_{2}^{ + } & = \left[ {\left( {0.1890,0.4150,0.4150,0.6790;1,1} \right)\left( {0.1890,0.4150,0.4150,0.6790;1,1} \right)} \right]. \\ \tilde{v}_{1}^{  } & = \left[ {\left( {0,0.0249,0.0249,0.0970;1,1} \right)\left( {0,0.0249,0.0249,0.0970;1,1} \right)} \right], \\ \tilde{v}_{2}^{  } & = \left[ {\left( {0,0.0249,0.0249,0.0970;1,1} \right)\left( {0,0.0249,0.0249,0.0970;1,1} \right)} \right]. \\ \end{aligned}$$
 Step 5: Constructing the ideal and the antiideal separation matrices.Then$$\begin{aligned} \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i1} ,v_{i}^{ + } ) & = 0.1737 + 0.3887 = 0.5624, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i2} ,v_{i}^{ + } ) & = 0 + 0.3966 = 0.3966, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i3} ,v_{i}^{ + } ) & = 0.1869 + 0.2180 = 0.4049, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i4} ,v_{i}^{ + } ) & = 0.1869 + 0 = 0.1869. \\ \end{aligned}$$Then$$\begin{aligned} \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i1} ,v_{i}^{  } ) & = 0.0339 + 0.0339 = 0.0678, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i2} ,v_{i}^{  } ) & = 0.1997 + 0.0235 = 0.2232, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i3} ,v_{i}^{  } ) & = 0.0235 + 0.1997 = 0.2232, \\ \mathop \sum \limits_{i = 1}^{m} d(\tilde{v}_{i4} ,v_{i}^{  } ) & = 0.0235 + 0.4119 = 0.4354. \\ \end{aligned}$$
 Step 6: Calculating the relative degree of closeness of each alternative to the ideal solution.$$\begin{aligned} & R\left( {{\text{x}}_{1} } \right) = \frac{0.0678}{0.0678 + 0.5624} = 0.1076,\quad R\left( {{\text{x}}_{2} } \right) = \frac{0.2232}{0.2232 + 0.3966} = 0.3601, \\ & R\left( {{\text{x}}_{3} } \right) = \frac{0.2232}{0.2332 +0.4049} = 0.3554, \quad R\left( {{\text{x}}_{4} } \right) = \frac{0.4354}{0.4354 + 0.1809} = 0.6997. \\ \end{aligned}$$
From the results,\(R({\text{x}}_{4} ) > R({\text{x}}_{2} ) > R({\text{x}}_{3} ) > R({\text{x}}_{1} ).\) Then, the ranking is \({\text{x}}_{4} > {\text{x}}_{2} > {\text{x}}_{3} > {\text{x}}_{1}\). This ranking coincides with the ranking of Chen and Lee [5]. The ranking of Chen and Hong [3] is \({\text{x}}_{4} > {\text{x}}_{2} = {\text{x}}_{3} > {\text{x}}_{1}\). Therefore, the best and the worst alternative are the same as their result. However, the second alternative is not equivalent to the third. The second alternative is better than the third with a slight difference.
6 Discussion
According to Chen and Hong [3], the second and third alternatives should be equally preferred, and this was supported by the results of their proposed IT2FTOPSIS. As justified by Chen and Hong [3], this is due to the decision makers’ similar evaluations of the alternatives with respect to the given attributes together with the equivalence in weights. The decision makers’ evaluations of the second attribute with respect to the emotional steadiness (f1) are \(\left\{ {{\text{MP}}, {\text{MP}}, {\text{P}}} \right\}\), and for the oral communication skills (f2) are \(\left\{ {{\text{P}}, {\text{VP}}, {\text{P}}} \right\}\). While the evaluations of the third attribute with respect to the emotional steadiness are \(\left\{ {{\text{P}}, {\text{VP}}, {\text{P}}} \right\},\) and for the oral communication skills are \(\left\{ {{\text{MP}}, {\text{MP}}, {\text{P}}} \right\}\). Both are given equal weights \(\left\{ {{\text{MH}}, {\text{H}}, {\text{MH}}} \right\}\). This is obvious in the average weighted decision matrix where \(\tilde{v}_{12} = \tilde{v}_{23}\) and \(\tilde{v}_{13} = \tilde{v}_{22}\).
On the other hand, the influence of the ideal solutions was not taken into consideration. In this example, the positive ideal solution for the emotional steadiness is different from that of the oral communication skills, i.e.\(\tilde{v}_{1}^{ + } \ne \tilde{v}_{2}^{ + }\). Consequently, the distance between \(\tilde{v}_{12}\) and \(\tilde{v}_{1}^{ + }\) is different from the distance between \(\tilde{v}_{23}\) and \(\tilde{v}_{2}^{ + }\). Similarly, the distance between \(\tilde{v}_{13}\) and \(\tilde{v}_{1}^{ + }\) is different from the distance between \(\tilde{v}_{22}\) and \(\tilde{v}_{2}^{ + }\). Therefore, this difference has an impact on the ranking and lead to the preference of the second alternative over the third one. This asserts that early defuzzification may affect the results and give an incorrect preference. In addition, it emphasizes on the role of the defined ideal solutions in ranking.
When proposing the PIS and the NIS, some researchers use the absolute ideal solutions \(\left\{ {v^{ + } = \left( {1,1,1,1;1,1} \right)\; {\text{and}}\; v^{  } = \left( {0,0,0,0;1,1} \right)} \right\}\), which represent the perfect PIS and NIS that can be attained. Other researchers use the relative ideal solutions, as in the proposed IT2FTOPSIS, which are related to the performance of the available alternatives with respect to the selected attributes. Resolving the same example with the absolute ideal solutions \(\left\{ {\tilde{v}_{1}^{ + } = \tilde{v}_{2}^{ + } = \left( {1,1,1,1;1,1} \right) {\text{and }}\tilde{v}_{1}^{  } = \tilde{v}_{2}^{  } = \left( {0,0,0,0;1,1} \right)} \right\}\), the distances from the ideal solutions are the same for the second and third alternatives and they are equally preferred. Subsequently, it can be concluded that the relative ideal solutions are more discriminating than the absolute ideal solutions in IT2FTOPSIS.
7 Conclusion
In this article, an IT2FTOPSIS was proposed using the extended vertex method for distance measure. While the existing TOPSIS techniques for IT2FSs depend on the defuzzification of the average decision matrix or the average weighted decision matrix in the early steps, the proposed method maintains fuzziness in the preference technique to avoid any information distortion which might lead to incorrect results. First, the vertex method is extended to include IT2FSs. This distance measure is a simple formula that requires few computations. Meanwhile, the other distance measures are either inappropriate or requires extensive computations. The performance of the extended vertex method was examined using the 32 words for computing with words proposed by Wu and Mendel [29]. The results indicate that the method is efficient in measuring the distance between IT2FSs. In addition, it has the least processing time compared to other distance measures for trapezoidal IT2FSs. Second, the fuzzy positive and negative ideal solutions are defined. Then, the relative degree of closeness to the ideal solutions is computed for each alternative. As the relative degree of closeness of an alternative increases, its preference increases. Two illustrative examples were solved using the proposed IT2FTOPSIS. Regarding the first example, the ranking coincide with the ranking of the IVFTOPSIS proposed by Rashid et al. [23]. As for the second example, the ranking coincides with the ranking of the IT2FTOPSIS proposed by Chen and Lee [5]. On the other hand, when compared to the ranking of Chen and Hong [3], the second alternative was not equivalent to the third one, it proved to be better. This can be attributed to maintaining fuzziness throughout the solution steps. It was also found that the defined ideal solutions have an impact on the ranking. The relative ideal solutions are more discriminating than the absolute ideal solutions in IT2FTOPSIS.
Notes
Compliance with ethical standards
Conflict of interest
The author declares that there is no conflict of interest.
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