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Development of a new hesitant fuzzy ranking model for NTMP ranking problem


Nontraditional manufacturing processes (NTMPs) bring the processing capabilities such as machining high strength and hard materials with desired accuracies and surface finish to the manufacturing companies. Therefore, there has been a significant increase in the use and number of NTMPs. Hence, choosing a particular NTMP for a specific application turns out to be a complex decision-making problem, which involves conflicting qualitative and quantitative ranking criteria. In recent NTMP ranking literature, it is noted that fuzzy approaches are better suited for handling uncertainties and incomplete information that exist within the NTMP ranking environment. This paper introduces such a fuzzy approach using the hesitant fuzzy preference selection index (PSI) method for the assessment of the criteria weights and the hesitant fuzzy correlation coefficient principle for ranking and recommending the most appropriate NTMP for a specific application. The proposed methodology and its efficiency in dealing with incomplete information under the fuzzy decision-making environment are explored with a case study. As a result of the study, the proposed model preferred the electron beam machining (EBM) as the most suitable nontraditional manufacturing process. On the other hand, triangular fuzzy TOPSIS methods offered the electrochemical machining (ECM) as the best choice among the alternatives. The differences among the ranking decisions are also analyzed in the paper. It can be concluded from the authors’ various applications of the proposed hesitant fuzzy PSI method that it is extremely effective in representing fuzzy decision-making environments in NTMP ranking decisions.

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Appendix 1: Basic concepts and operational laws for hesitant fuzzy sets

Definition 1.

(Torra and Narukawa 2009; Naz and Akram 2019; Lalotra and Singh 2020; Xia and Xu 2011; Garmendia et al. 2017) A hesitant fuzzy set (HFS) on the reference set \(X\) is defined in terms of a function \(A\) that returns a subset of \(\left[ {0,1} \right]\) when it is applied to \( X\).

$$ A = \left\{ { < {x,h\left( x \right)} > | x \in X} \right\} $$

where \( h\left( x \right)\) is a set of different values in \(\left[ {0,1} \right]\), expressing the possible membership degree of an element \(x \in X\) to \(A\). For convenience, Xu and Xia (2011) called \(h\left( x \right) = \left\{ {\mu_{1} , \mu_{2} , \ldots , \mu_{l} } \right\}\) a hesitant fuzzy element (HFE), which denotes a basic component of the HFS with \(\mu_{i}\) \(\left( {i = 1,2, \ldots , l} \right)\) being the possible membership degrees of an element \(x\) to a given set. The complement of \(A\) is defined as \( A^{c} = \left\{ {\left\langle {x, h^{c} \left( x \right)} \right\rangle |x \in X} \right\}\), where \( h^{c} = \mathop \cup \limits_{\mu \in h} \left\{ {1 - \mu } \right\}\).

Definition 2.

(Torra 2010; Xu and Xia 2011; Liao et al. 2020) Let \( h\), \(h_{1}\) and \(h_{2}\) be three HFEs, then the following relations and operations are defined:

$$ h_{1} \oplus h_{2} = \mathop \bigcup \limits_{{\mu_{1} \in h_{1} , \mu_{2} \in h_{2} }} \left\{ {\mu_{1} + \mu_{2} - \mu_{1} \mu_{2} } \right\} $$
$$ \lambda h = \mathop \bigcup \limits_{\mu \in h} \left\{ {1 - \left( {1 - \mu } \right)^{\lambda } } \right\}, \lambda > 0. $$

In the following works, Liao et al. (2014) gave the generalized form of operation in Eq. (2). Let \(h_{p} \left( {p = 1,2,. . ., r} \right) \) be a collection of HFEs, and then the generalized operation is defined as in Eq. (4).

$$ \oplus_{p = 1}^{r} h_{p} = \mathop \bigcup \limits_{{\mu_{p} \in h_{p} }} \left\{ {1 - \mathop \prod \limits_{p = 1}^{r} \left( {1 - \mu_{p} } \right)} \right\}. $$

It should be noted that the number of values, denoted by \( l_{{h_{p} }}\), may be different from HFE to HFE. Then, based on the above operational law, they also proved the following.

(Liao et al. 2014) Let \(h_{1}\) and \(h_{2}\) be two HFEs, and \(l_{{h_{1} }}\) and \(l_{{h_{2} }}\) denote the number of values in \(h_{1}\) and \(h_{2}\), respectively. The number of values in the sum of \(h_{1}\) and \( h_{2}\), denoted by \( l_{{h_{1} \oplus h_{2} }}\), is calculated according to Eq. (5).

$$ l_{{h_{1} \oplus h_{2} }} = l_{{h_{1} }} \times l_{{h_{2} }}. $$

When there are \(r\) different HFEs, it still holds. The number of values in the sum of \( r\) HFEs, denoted by \( l_{{ \oplus_{p = 1}^{r} h_{p} }}\), is calculated based on Eq. (6).

$$ l_{{ \oplus_{p = 1}^{r} h_{p} }} = \mathop \prod \limits_{p = 1}^{r} l_{{h_{p} }}. $$

Definition 3.

The existing distance measures are all defined under a strict assumption that the compared HFEs are of equal length. If the two HFEs are not the same length, the one which has fewer elements, should be extended accordingly to either optimistic or pessimistic extension approach. However, the preferred extension technique directly affects the comparison result. Therefore, to overcome this limitation a new distance measure is introduced. Let \(h_{1}\) and \(h_{2}\) be two HFEs, the generalized hesitant fuzzy Hausdorff distance measure (Wang et al. 2016) is given in Eq. (7).

$$ d\left( {h_{1} ,h_{2} } \right) = \left[ {\frac{1}{2}\left( {\mathop {\max }\limits_{{\mu_{{h_{1} }} \in h_{1} }} \mathop {\min }\limits_{{\mu_{{h_{2} }} \in h_{2} }} \left| {\mu_{{h_{1} }} - \mu_{{h_{2} }} } \right|^{\lambda } + \mathop {\max }\limits_{{\mu_{{h_{2} }} \in h_{2} }} \mathop {\min }\limits_{{\mu_{{h_{1} }} \in h_{1} }} \left| {\mu_{{h_{2} }} - \mu_{{h_{1} }} } \right|^{\lambda } } \right)} \right]^{{\frac{1}{\lambda }}}. $$

If \(\lambda = 1\), then Eq. (7) is reduced to the hesitant fuzzy Hamming–Hausdorff distance and if \(\lambda = 2\), then Eq. (7) is reduced to the hesitant fuzzy Euclidean–Hausdorff distance. Throughout the work, Hamming distance and Euclidean distance are referred to measures introduced in above.

Definition 4.

Correlation is an indicator which measures how well two variables move together in a linear fashion, and correlation coefficient is a tool to express this relationship. Therefore, the correlation coefficient has been integrated into different circumstances. Xia and Xu (2011) introduced several correlation coefficients of HFEs, under the assumption that they all are of the same length. However, some of these measures are not suitable when there is only one element in HFEs. Considering the limitations, Chen and Lu (2015) defined the correlation coefficients of HFSs given in Eqs. (8) and (9). Let B and C be two HFSs, then

$$ {\text{Corr}}_{1} \left( {B,C} \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{1}{{l_{{x_{i} }} }}\mathop \sum \nolimits_{j = 1}^{{l_{{x_{i} }} }} \left( {h_{B}^{\tau \left( j \right)} \left( {x_{i} } \right)*h_{C}^{\tau \left( j \right)} \left( {x_{i} } \right)} \right)} \right)}}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{1}{{l_{{x_{i} }} }}\mathop \sum \nolimits_{j = 1}^{{l_{{x_{i} }} }} h_{B}^{\tau \left( j \right)} \left( {x_{i} } \right)^{2} } \right)} \sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{1}{{l_{{x_{i} }} }}\mathop \sum \nolimits_{j = 1}^{{l_{{x_{i} }} }} h_{C}^{\tau \left( j \right)} \left( {x_{i} } \right)^{2} } \right)} }} $$
$$ {\text{Corr}}_{2} \left( {B,C} \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{1}{{l_{{x_{i} }} }}\mathop \sum \nolimits_{j = 1}^{{l_{{x_{i} }} }} \left( {h_{B}^{\tau \left( j \right)} \left( {x_{i} } \right)*h_{C}^{\tau \left( j \right)} \left( {x_{i} } \right)} \right)} \right)}}{{{\text{max}}\left\{ {\left( {\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{1}{{l_{{x_{i} }} }}\mathop \sum \nolimits_{j = 1}^{{l_{{x_{i} }} }} h_{B}^{\tau \left( j \right)} \left( {x_{i} } \right)^{2} } \right)} \right),\left( {\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{1}{{l_{{x_{i} }} }}\mathop \sum \nolimits_{j = 1}^{{l_{{x_{i} }} }} h_{C}^{\tau \left( j \right)} \left( {x_{i} } \right)^{2} } \right)} \right)} \right\}}} $$

where \(h_{B}^{\tau \left( j \right)} \left( {x_{i} } \right)\) and \(h_{C}^{\tau \left( j \right)} \left( {x_{i} } \right)\) are the \(j^{th}\) largest values in \(h_{B} \left( {x_{i} } \right)\) and \(h_{C} \left( {x_{i} } \right)\).

Appendix 2: Hesitant fuzzy correlation integrated preference selection index method

In this section, the proposed hesitant fuzzy correlation integrated preference selection index (PSI) method is introduced. The algorithm consists of two major steps: evaluation criteria weighting via the hesitant fuzzy PSI and ranking of alternatives depending on the correlation coefficients.

Before introducing the integrated evaluation method, some definitions are given about a random number generation procedure which is used at the very beginning of the method. In addition, a brief review of the triangular fuzzy numbers is provided.

Definition 1.

When there is a lack of measurement; hence, incomplete information, the following approach can be considered with tight intervals to generate near measurement results. Let \(x_{ij}\) be continuous variables which are measurable, and \(\max x_{ij}\) and \(\min x_{ij}\) be the maximum and minimum values that \(x_{ij}\) can attain, respectively. \(M_{ij}\) is defined as a set of numbers being elements of the interval \(I_{ij} = \left[ {\min x_{ij} , \max x_{ij} } \right]\). In order to create an appropriate \(M_{ij}\), \(k\) many numbers are randomly generated between the minimum and the maximum according to uniform distribution.

Example 1.1.

Let \(I_{ij} = \left[ {0.50, 1.30} \right]\) be an interval expressing all possible values that a measurable quantity can take and \(k\) be 2. Then, 2 random numbers are generated respecting the boundary values. The set \(M_{ij}\) is shown as in Eq. (10).

$$ M_{ij} = \left\{ {0.50, 0.79, 0.99, 1.30} \right\}. $$

Definition 2.

\(\tilde{A} = \left( {\min x_{ij} , x_{ij}^{*} , \max x_{ij} } \right)\) is a triangular fuzzy number. The membership function of \(\tilde{A}\) is given in Eq. (11).

$$ \mu_{{\tilde{A}}} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\frac{{x - \min x_{ij} }}{{x_{ij}^{*} - \min x_{ij} }}, \;\min x_{ij} \le x \le x_{ij}^{*} } \\ {\frac{{\max x_{ij} - x}}{{\max x_{ij} - x_{ij}^{*} }}, \;x_{ij}^{*} \le x \le \max x_{ij} } \\ {0, \;\;{\text{otherwise}}} \\ \end{array} } \right.. $$

The membership degrees of the elements of a set \(M_{ij}\) are calculated by using triangular membership function in Eq. (11). The intermediate value of the corresponding triangular fuzzy number (Eq. 10) is estimated to be 0.90 and the membership function is written as in Eq. (12).

$$ \mu_{{\tilde{A}}} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\frac{x - 0.50}{{0.90 - 0.50}}, \;\min x_{ij} \le x \le x_{ij}^{*} } \\ {\frac{1.30 - x}{{1.30 - 0.90}},\; x_{ij}^{*} \le x \le \max x_{ij} } \\ {0, \;{\text{otherwise}}} \\ \end{array} } \right.. $$

The membership degree of each element of \(M_{ij}\) is obtained as \(\mu_{{\tilde{A}}} \left( {0.50} \right) = \mu_{{\tilde{A}}} \left( {1.30} \right) = 0\), \(\mu_{{\tilde{A}}} \left( {0.79} \right) = 0.73\) and \(\mu_{{\tilde{A}}} \left( {0.99} \right) = 0.78\).

Application steps of the hesitant fuzzy model for an MCDM problem with \(m\) alternatives \(A_{i} \left( {i = 1,2, \ldots , m} \right)\) and \(n\) criteria \( C_{j} \left( {j = 1,2, \ldots , n} \right)\) is provided below:

Step 1 Determine the measurement intervals for each alternative.

Measurement intervals, denoted by \(I_{ij}\), are defined as the interval of reals including possible minimum (\(\min x_{ij}\)) and maximum (\(\max x_{ij} )\) values that a measurable quantity can take. Then, the data is provided in a matrix, as shown in Eq. (13).

$$ I = \left[ {I_{ij} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {I_{11} } & \cdots & {I_{1n} } \\ \vdots & \ddots & \vdots \\ {I_{m1} } & \cdots & {I_{mn} } \\ \end{array} } \right]_{m \times n} $$

where \(I_{ij} = \left[ {\min x_{ij} ,\max x_{ij} } \right].\)

Step 2 Generate random numbers to construct the set of possible measurement values for each alternative.

According to the sensitivity of the variability of the criterion value, the decision maker determines how many values should be generated within the range. In other words, the \(k\) value is declared by the decision maker. Hence, \(k\) many random numbers are generated based on the uniform distribution, and the set \(M_{ij}\) is constituted for each alternative under each criterion.

Step 3. Construct the hesitant fuzzy decision matrix.

Considering each of the \(k_{j}\) many values in the set \(M_{ij}\), the set of possible membership degrees, \(h_{ij}\) is calculated using the membership function given in Eq. (12). \(h_{ij} = \left\{ {\mu_{ij}^{1} , \mu_{ij}^{2} , \ldots , \mu_{ij}^{{k_{j} }} } \right\}\) is an HFE, expressing \(k_{j}\) different membership degrees assigned to the \(i{th}\) alternative for the \(j{th}\) criterion. The hesitant fuzzy decision matrix, shown in Eq. (14) also, denoted by \( \tilde{R}\), is constructed by HFEs,\(h_{ij} \left( {i = 1, 2, ..., m;j = 1, 2, . .., n} \right)\)

$$ \tilde{R} = \left[ {h_{ij} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {h_{11} } & \cdots & {h_{1n} } \\ \vdots & \ddots & \vdots \\ {h_{m1} } & \cdots & {h_{mn} } \\ \end{array} } \right]_{m \times n}. $$

Step 4 Construct the normalized hesitant fuzzy decision matrix.

The normalized hesitant fuzzy decision matrix, denoted by \( \tilde{R}_{N}\), is shown in Eq. (15).

$$ \tilde{R}_{N} = \left[ {{h_{ij}}_{N} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {{h_{11}}_{N} } & \cdots & {{h_{1n}}_{N} } \\ \vdots & \ddots & \vdots \\ {{h_{m1}}_{N} } & \cdots & {{h_{mn}}_{N} } \\ \end{array} } \right]_{m \times n} $$

where \({h_{ij}}_{N} = \left\{ {\begin{array}{*{20}c} {h_{ij} , \;{\text{if}} j {\text{is a}} \,{\text{benefit{-}type criterion}}} \\ {h_{ij}^{c} ,\; {\text{if}} \,j \,{\text{is a cost{-}type criterion}}} \\ \end{array} } \right.\).

\(h_{ij}^{c}\) indicates the complement of the corresponding HFE and is calculated according to Definition 1 in Section 2. It should be noted that it is assumed that a criterion is either a benefit criterion or a cost criterion and none of the criteria excludes the classification.

Step 5 Determine the preference variation value for each criterion.

Preference variation value, \(PV_{j}\), for each criterion is determined by using Eq. (16) or Eq. (17). These equations are derived from distance measures introduced in Definition 3, in Section 2.

$$ \left( {PV_{j} } \right)_{Hamming} = \frac{1}{m}\mathop \sum \limits_{i = 1}^{m} d_{Hamming} ({{h_{ij}}_N} , \overline{{h_{j} }} ), j = 1, 2, \ldots , n $$
$$ \left( {PV_{j} } \right)_{{{\text{Euclidean}}}} = \frac{1}{m}\mathop \sum \limits_{i = 1}^{m} d_{{{\text{Euclidean}}}} ({h_{ij}}_{N} , \overline{{h_{j} }} ), j = 1, 2, \ldots , n. $$

In Eqs. (16) and (17), \(\overline{{h_{j} }}\) is the average HFE which is calculated by using the summation and the product operators introduced in Eqs. (2) and (3), respectively.

$$ \overline{{h_{j} }} = \frac{1}{{\left( {k_{j} } \right)^{m} }} \oplus_{i = 1}^{m} {h_{ij}}_{N} , j = 1, 2, \ldots , n. $$

\(\left( {k_{j} } \right)^{m}\) indicates the length of the HFE found by the summation of \(m\) many HFEs, \(\oplus_{i = 1}^{m} {h_{ij}}_{N}\) and it is calculated according to Eq. (6) in Section 2.

Step 6 Determine the overall weight of each criterion.

The overall weight, \(w_{j}\), is calculated according to Eq. (19) or Eq. (20) depending on selected distance measure. Weights should satisfy the following two: \( w_{j} \left[ {0,1} \right], j = 1, 2, ..., n\), and \(\mathop \sum \nolimits_{j = 1}^{n} w_{j} = 1\).

$$ \left( {w_{j} } \right)_{{{\text{Hamming}}}} = \frac{{(\varphi_{j} )_{{{\text{Hamming}}}} }}{{\mathop \sum \nolimits_{j = 1}^{n} (\varphi_{j} )_{{{\text{Hamming}}}} }} , j = 1, 2, \ldots , n $$
$$ \left( {w_{j} } \right)_{{{\text{Euclidean}}}} = \frac{{(\varphi_{j} )_{{{\text{Euclidean}}}} }}{{\mathop \sum \nolimits_{j = 1}^{n} (\varphi_{j} )_{{{\text{Euclidean}}}} }} , j = 1, 2, \ldots , n $$

\((\varphi_{j} )_{{{\text{Hamming}}}}\) and \((\varphi_{j} )_{{{\text{Euclidean}}}}\) are the deviation in preference value of each criterion and are calculated with Eqs. (21) and (22), respectively.

$$ (\varphi_{j} )_{{{\text{Hamming}}}} = 1 - \left( {PV_{j} } \right)_{{{\text{Hamming}}}} $$
$$ (\varphi_{j} )_{{{\text{Euclidean}}}} = 1 - \left( {PV_{j} } \right)_{{{\text{Euclidean}}}} $$

Step 7 Construct the weighted hesitant fuzzy decision matrix.

The weighted hesitant fuzzy decision matrix is denoted by \(\tilde{R}_{w}\) and depicted in Eq. (23). Each element of \(\tilde{R}_{w}\) is calculated by using Eq. (3).

$$ \tilde{R}_{w} = \left[ {h_{ij}^{w} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {h_{11}^{w} } & \cdots & {h_{1n}^{w} } \\ \vdots & \ddots & \vdots \\ {h_{m1}^{w} } & \cdots & {h_{mn}^{w} } \\ \end{array} } \right]_{m \times n} , \;\;{\text{where}}\;\; h_{ij}^{w} = w_{j} h_{ij}. $$

Step 8 Calculate the correlation coefficient between each alternative and the ideal alternative.

The correlation coefficient between each alternative \(A_{i}\) and the ideal alternative \(A^{*}\), \(Corr_{i}\), is calculated by using measures in Eqs. (8) and (9). \(A^{*}\) is selected to be as in Eq. (24) since the maximum value of a membership degree can attain is 1.

$$ A^{*} = \left\{ {h_{1} ,h_{2} , \ldots ,h_{n} } \right\} = \left\{ {\underbrace {{\left\{ {1,1, \ldots ,1} \right\}}}_{{k_{1} }},\,\underbrace {{\left\{ {1,1, \ldots ,1} \right\}}}_{{k_{2} }}, \ldots \underbrace {{\left\{ {1,1, \ldots ,1} \right\}}}_{{k_{n} }}\;} \right\}. $$

Step 9 Rank the alternatives in accordance with the values of correlation coefficients.

All alternatives are sorted in decreasing order according to correlation coefficient values, and hence a complete ranking is obtained. The larger the value of correlation coefficient, the higher the priority of alternative is \( A_{i}\).

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Atalay, K.D., İç, Y.T., Keçeci, B. et al. Development of a new hesitant fuzzy ranking model for NTMP ranking problem. Soft Comput 25, 14537–14548 (2021).

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  • Nontraditional manufacturing process (NTMP)
  • Hesitant fuzzy sets
  • Hesitant fuzzy PSI
  • Hesitant fuzzy correlation coefficient
  • Hesitant fuzzy NTMP ranking model