Abstract
The location of the electric vehicle charging station is deemed to be a multiple attribute group decision making (MAGDM) issue involving many experts and many conflicting attributes. In practical MAGDM issues, the information of uncertain and fuzzy cognitive decision is well-depicted by linguistic term sets (LTSs). These LTSs could be simply shifted into the probabilistic linguistic sets (PLTSs). In such paper, we design some novel probabilistic linguistic weighted Dice similarity measures (PLWDSM) and the probabilistic linguistic weighted generalized Dice similarity measures (PLWGDSM). Subsequently, the PLWGDSM-based MAGDM methods are presented under PLTSs. In the end, a practical case which concerns about the location planning of electric vehicle charging stations is offered to demonstrate the proposed PLWGDSM’s applicability and advantages.
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1 Introduction
In our everyday lives, decision-making issues are the regular behavior activities (Braglia et al. 2003; Liu et al. 2019a; Tian et al. 2017, 2018). It has been deemed that almost all assessing information is expressed with numerical values (Deng et al. 2000; Bourguignon and Massart 1994; Tsoulfas and Pappis 2008). Due to the human thinking’s fuzziness and vagueness and the objective things’ complexity (Chen et al. 2019a; Chen and Han 2019; Wei et al. 2020a, b; Wang et al. 2020), individuals are willing to express their assessment information by linguistic term sets (LTSs) rather than the form of quantitative in more and more vague decision-making issues. For instance, the DMs may utilize the LTSs like “bad”, “medium” and “good” when a family car’s satisficing degree is assessed. Therefore, more and more studies on diverse linguistic models have brought about extensive attention by more and more scholars. In order to handle these qualitative assessment information easily, Herrera and Martinez (2000a) designed the linguistic term sets (LTSs) for calculating with given words. Herrera and Martinez (2000b) used 2-tuple fuzzy linguistic to combine the linguistic information and numerical. Herrera and Martinez (2001) designed the multi-granular hierarchical linguistic to handle linguistic 2-tuples. Geng et al. (2017) proposed the extension of 2-tuple linguistic DEA for tackling MAGDM issues which considered the attributes’ effect relationships. Furthermore, Rodriguez et al. (2012) brought up the hesitant fuzzy LTSs (HFLTSs) with the aid of HFSs (Torra 2010) and LTSs (Zadeh 1975) which allowed DMs to provide some possible LTSs. However, in most living studies on HFLTSs, all possible values are handled by the DMs by using same weight or importance. Visibly, it is not consistent with our real life. In living situations, the DMs may assign possible linguistic terms so that the furnished information may have different probability distributions. Thus, Pang et al. (2016) raised the probabilistic linguistic term sets (PLTSs) to surmount this defect. Bai et al. (2017) built a comparison method which is more appropriate and proposed a tool which is more effective to manage PLTSs. Zhang et al. (2016) introduced the PLTSs to express the DMs’ preferences information and discussed additive consistency of PLPR from preference relation graph. Lin et al. (2019) put up with the ELECTRE II method to manage PLTSs for edge computing. Liao et al. (2019) raised the novel operations of PLTSs to work out the probabilistic linguistic ELECTRE III method. Liang et al. (2018) developed the probabilistic linguistic grey relational analysis (PL-GRA) for MAGDM based on geometric Bonferroni mean (Wang et al. 2018; Wei et al. 2019; Zhu and Xu 2013). Liao et al. (2017) designed a linear programming algorithm to settle the MADM issues with PLTSs. Chen et al. (2019b) extended MULTIMOORA to the probabilistic linguistic for cloud-based ERP system selection. Feng et al. (Feng et al. 2019) set up the probabilistic linguistic QUALIFLEX method for MAGDM issue. Lu et al. (2019) designed TOPSIS method with entropy weight under the probabilistic linguistic environment for MAGDM issues to select the most appropriable supplier with new agricultural machinery products. Kobina et al. (2017) planned some power operators for MAGDM with PLTSs and classical power aggregation operators (Wei 2019a; Yager 2001; Xu and Yager 2010).
As a major and effective tool, the similarity measure is utilized to depict similarity’s degree between objects (Wang et al. 2019a; Wei 2017, 2018; Wei and Wei 2018). Actually, similarity or dissimilarity’s degree between objects plays a significant role under the currently researches (Li and Cheng 2002; Li 2004; Chen et al. 2016; Peng and Garg 2018; Sharaf 2018; Peng and Li 2019; Xian et al. 2019; Zhang et al. 2019). The measures of Jaccard, cosine and Dice similarity are frequently utilized in different domains (Dice 1945; Jaccard 1901; Salton and McGill 1987). However, Ye (2012a) designed the measures of Jaccard, cosine and Dice similarity between TIFNs for MAGDM issues. Ye (2012b) worked out the MADM approaches by employing the measure of Dice similarity between the expected ITFNs. Ye (2014) designed the Dice measures to tackle the simplified neutrosophic sets. Ye (2016) put up with the generalized Dice measures for MADM issues under simplified neutrosophic setting. Tang et al. (2017) came up with the GDSM to deal with MAGDM issues with intuitionistic information. Mahmood et al. (2016) designed three measures of similarity between simplified neutrosophic HFSs. Mandal and Basu (2016) defined two novel measures of similarity to conquer several limitations of existing all kinds of similarity measures. Wei (2019b) developed the GDSM for MADM issues with HFLTSs. Wang et al. (2019b) designed some new DSM of PFSs and the GDSM of PFSs to tackle MAGDM issues for choosing the most appropriable ERP system. Wei and Gao (2018) worked out some new DSM of picture fuzzy sets for building material recognition.
However, these DSM do not directly tackle the measures of similarity for PLTSs. Hence, extending the Dice measure to PLTSs to tackling MAGDM issues is essential which can fulfill DMs preference. To do so, the main aims of such paper are: (1) two DSM’ forms within PLTSs are designed; (2) the GDSM and weighted GDSM with PLTSs are proposed; (3) the weighted GDSM are employed to work out the MAGDM issues under PLTSs; (4) within the process of MAGDM, the developed methods’ major merit is more flexible and useful compared with the existing MAGDM issues with PLTSs.
To do so, this essay’s remainder is given in the following. In Sect. 2, some fundamental theories which concerns about PLTSs is proposed. In Sect. 3, several measures of DSM and weighted DSM between PLTSs are put forward. In Sect. 4, the weighted GDSM are used to work out MAGDM issues with PLTSs. A practical case study for site selection of EVCS is offered to validate the developed weighted Dice similarity measure in Sect. 5. The paper with some remarks is concluded in the final section.
2 Preliminaries
Firstly, Xu (2005)worked out the additive linguistic scale and Gou et al. (2017) put up with the corresponding transformation function between the linguistic terms and \( \left[ {0,1} \right] \).
Definition 1
(Xu 2005; Gou et al. 2017) Let \( L = \left\{ {l_{\alpha } \left| {\alpha = - \theta , \ldots , - 2, - 1,0,1,2, \ldots \theta } \right.} \right\} \) be an LTS (Xu 2005), the linguistic terms \( l_{\alpha } \) can express the same information to \( \beta \) which is expressed with transformation mathematical function \( g \) (Gou et al. 2017):
\( \beta \) can also be represented the equal information by linguistic terms \( l_{\alpha } \) which is denoted with the transformation function \( g^{ - 1} \):
Definition 2
(Pang et al. 2016) Given the LTS \( L = \left\{ {l_{\alpha } \left| {\alpha = - \theta , \ldots , - 2, - 1,0,1,2, \ldots \theta } \right.} \right\} \), a PLTS could be designed as:
where \( l^{\left( \phi \right)} \left( {p^{\left( \phi \right)} } \right) \) is the \( \phi {\text{th}} \) linguistic term \( l^{\left( \phi \right)} \) connected with the corresponding probability \( p^{\left( \phi \right)} \), and \( \# PL\left( p \right) \) is the length of \( PL\left( p \right) \). The linguistic term \( l^{\left( \phi \right)} \) in \( PL\left( p \right) \) are listed by ascending order.
In order to easy computation, Pang et al. (Pang et al. 2016) normalized the PLTS \( PL\left( p \right) \) as \( NPL\left( p \right) = \left\{ {l^{\left( \phi \right)} \left( {\tilde{p}^{\left( \phi \right)} } \right)\left| {l^{\left( \phi \right)} \in L,\quad \tilde{p}^{\left( \phi \right)} \ge 0,\quad \phi = 1,2, \ldots ,\# NPL\left( {\tilde{p}} \right),\quad \sum\limits_{\phi = 1}^{{\# NPL\left( {\tilde{p}} \right)}} {\tilde{p}^{\left( \phi \right)} = 1} } \right.} \right\} \), where \( \tilde{p}^{\left( \phi \right)} = {{p^{\left( \phi \right)} } \mathord{\left/ {\vphantom {{p^{\left( \phi \right)} } {\sum\limits_{\phi = 1}^{\# NPL\left( p \right)} {p^{\left( \phi \right)} } }}} \right. \kern-0pt} {\sum\limits_{\phi = 1}^{\# NPL\left( p \right)} {p^{\left( \phi \right)} } }} \) for all \( \phi = 1,2, \ldots ,\# L\left( {\tilde{p}} \right) \).
Definition 3
(Pang et al. 2016) Let \( L = \left\{ {l_{\alpha } \left| {\alpha = - \theta , \ldots , - 1,0,1, \ldots \theta } \right.} \right\} \) be an LTS, \( NPL_{1} \left( p \right) = \left\{ {l_{1}^{\left( \phi \right)} \left( {p_{1}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{1} \left( p \right)} \right.} \right\} \) and \( NPL_{2} \left( p \right) = \left\{ {l_{2}^{\left( \phi \right)} \left( {p_{2}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{2} \left( p \right)} \right.} \right\} \) be two PLTSs, where \( \# NPL_{1} \left( p \right) \) and \( \# NPL_{2} \left( p \right) \) are the length of \( NPL_{1} \left( p \right) \) and \( NPL_{2} \left( p \right) \), respectively. If \( \# NPL_{1} \left( p \right) > \# NPL_{2} \left( p \right) \), then add \( \# NPL_{1} \left( p \right) - \# NPL_{2} \left( p \right) \) linguistic terms to \( NPL_{2} \left( p \right) \). In addition, the newly added linguistic terms should be the smallest linguistic term in \( NPL_{2} \left( p \right) \) and the corresponding probabilities of newly added linguistic terms should be zero.
Definition 4
(Pang et al. 2016) For a PLTS \( NPL\left( p \right) = \left\{ {l^{\left( \phi \right)} \left( {p^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL\left( p \right)} \right.} \right\} \), the expected value \( EV\left( {NPL\left( p \right)} \right) \) and standard deviation \( SD\left( {NPL\left( p \right)} \right) \) of \( NPL\left( p \right) \) is designed in the following:
By using the Eqs. (4)–(5), the order relation between two PLTSs is distinguished as: (1) if \( EV\left( {NPL_{1} \left( p \right)} \right) > EV\left( {NPL_{2} \left( p \right)} \right) \), then \( NPL_{1} \left( p \right) > NPL_{2} \left( p \right) \); (2)if \( EV\left( {NPL_{1} \left( p \right)} \right) = EV\left( {NPL_{2} \left( p \right)} \right) \), then if \( SD\left( {NPL_{1} \left( p \right)} \right) = SD\left( {NPL_{2} \left( p \right)} \right) \), then \( NPL_{1} \left( p \right) = NPL_{2} \left( p \right) \); if \( SD\left( {NPL_{1} \left( p \right)} \right) < SD\left( {NPL_{2} \left( p \right)} \right) \), then, \( NPL_{1} \left( p \right) > NPL_{2} \left( p \right) \).
Definition 5
(Lin and Xu 2018) Let \( L = \left\{ {l_{\alpha } \left| {\alpha = { - }\theta , \ldots , - 1,0,1, \cdots \theta } \right.} \right\} \) be a LTS. And let \( NPL_{1} \left( p \right) = \left\{ {l_{1}^{\left( \phi \right)} \left( {p_{1}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{1} \left( p \right)} \right.} \right\} \) and \( NPL_{2} \left( p \right) = \left\{ {l_{2}^{\left( \phi \right)} \left( {p_{2}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{2} \left( p \right)} \right.} \right\} \) be two PLTSs with \( \# NPL_{1} \left( p \right) = \# NPL_{2} \left( p \right){ = }NPL\left( p \right) \), then Hamming distance \( HD\left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \) between \( NPL_{1} \left( p \right) \) and \( NPL_{2} \left( p \right) \) is derived:
3 Some Dice similarity measure for PLTSs
When one vector is zero, the DSM (Dice 1945) can’t induce this undefined setting which conquers the cosine similarity measure’s limitation. Hence, the DSM’s concept is designed in the chapter (Dice 1945).
Definition 6
(Dice 1945) Let \( A{ = }\left( {a_{1} ,a_{2} , \ldots ,a_{n} } \right) \) and \( B{ = }\left( {b_{1} ,b_{2} , \ldots ,b_{n} } \right) \) be two set of positive real numbers. Then the DSM is defined in the following:
where \( A \cdot B = \sum\nolimits_{j = 1}^{n} {a_{j} b_{j} } \) is the inner product between \( A \) and \( B \) and \( \left\| A \right\|_{2} = \sqrt {\sum\nolimits_{j = 1}^{n} {\left( {a_{j} } \right)^{2} } } \) and \( \left\| B \right\|_{2} = \sqrt {\sum\nolimits_{j = 1}^{n} {\left( {b_{j} } \right)^{2} } } \) are the Euclidean norms of \( A \) and \( B \).
The value of DSM belongs to the interval \( \left[ {0,1} \right] \). Thus, if \( a_{j} = b_{j} = 0\left( {j = 1,2, \ldots ,n} \right) \), then \( DSM\left( {A,B} \right){ = }0 \).
3.1 Dice similarity measure for PLTSs
In such section, some DSM and some weighted DSM (WDSM) between PLTs are designed on the basis of the concept of the DSM.
Definition 7
Let \( L = \left\{ {l_{\alpha } \left| {\alpha = { - }\theta , \ldots , - 1,0,1, \cdots \theta } \right.} \right\} \) be an LTS, \( NPL_{1} \left( p \right) = \left\{ {NPL_{1j} \left( p \right)} \right\}\left( {j = 1,2, \ldots ,n} \right) = \left\{ {l_{1j}^{\left( \phi \right)} \left( {p_{1j}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{1j} \left( p \right)} \right.} \right\}\left( {j = 1,2, \ldots ,n} \right) \) and \( NPL_{2} \left( p \right) = \left\{ {NPL_{2j} \left( p \right)} \right\}\left( {j = 1,2, \ldots ,n} \right) = \left\{ {l_{2j}^{\left( \phi \right)} \left( {p_{2j}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{2j} \left( p \right)} \right.} \right\}\left( {j = 1,2, \ldots ,n} \right) \) be two sets of PLTSs, where \( \# NPL_{1j} \left( p \right) \) and \( \# NPL_{2j} \left( p \right) \) are the numbers of PLTS \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \), \( \# NPL_{1j} \left( p \right) = \# NPL_{2j} \left( p \right) = \# NPL_{j} \left( p \right) \) respectively, the probabilistic linguistic DSM (PLDSM) between \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \) is designed as follows:
Example 1
Let \( NPL_{1} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{2} \left( {0.4} \right),l_{3} \left( {0.6} \right)} \right\},\left\{ {l_{1} \left( {0.2} \right),l_{2} \left( {0.8} \right)} \right\},\left\{ {l_{ - 1} \left( {0.2} \right),l_{1} \left( {0.8} \right)} \right\}} \right]\kern-0.15em\right] \) and \( NPL_{2} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{ - 3} \left( {0.8} \right),l_{ - 1} \left( {0.2} \right)} \right\},\left\{ {l_{2} \left( {0.6} \right),l_{3} \left( {0.4} \right)} \right\},\left\{ {l_{ - 2} \left( {0.7} \right),l_{ - 1} \left( {0.3} \right)} \right\}} \right]\kern-0.15em\right] \) be two sets of normalized PLTSs, then in terms of the Eq. (8), we can acquire:
The DSM between \( NPL_{1} \left( p \right) \) and \( NPL_{2} \left( p \right) \) also fulfills the subsequently properties:
-
(1)
\( 0 \le PLDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \le 1 \);
-
(2)
\( PLDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = PLDSM_{{_{PLTSs} }}^{1} \left( {NPL_{2} \left( p \right),NPL_{1} \left( p \right)} \right) \);
-
(3)
\( PLDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = 1, \) if \( \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{1j}^{\left( \phi \right)} g\left( {l_{1j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} = \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{2j}^{\left( \phi \right)} g\left( {l_{2j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} \), \( j = 1,2, \ldots ,n \).
If we take into account the weights \( \omega_{j} \) of \( NPL_{kj} \left( p \right)\left( {k = 1,2} \right) \), then, a probabilistic linguistic weighted DSM (PLWDSM) between \( NPL_{1} \left( p \right) \) and \( NPL_{2} \left( p \right) \) is designed in the following:
where \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) is the weight of \( NPL_{kj} \left( p \right)\left( {k = 1,2} \right) \), with \( \omega_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \). Particularly, if \( \omega = \left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n},{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}, \ldots ,{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right)^{T} \), then the PLWDSM reduces to the PLDSM. Then there is: \( PLWDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = PLDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right). \)
Example 2
Let \( NPL_{1} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{2} \left( {0.4} \right),l_{3} \left( {0.6} \right)} \right\},\left\{ {l_{1} \left( {0.2} \right),l_{2} \left( {0.8} \right)} \right\},\left\{ {l_{ - 1} \left( {0.2} \right),l_{1} \left( {0.8} \right)} \right\}} \right]\kern-0.15em\right] \) and \( NPL_{2} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{ - 3} \left( {0.8} \right),l_{ - 1} \left( {0.2} \right)} \right\},\left\{ {l_{2} \left( {0.6} \right),l_{3} \left( {0.4} \right)} \right\},\left\{ {l_{ - 2} \left( {0.7} \right),l_{ - 1} \left( {0.3} \right)} \right\}} \right]\kern-0.15em\right] \) be two sets of normalized PLTSs, the weight values are: \( \omega = \left( {0.2,0.5,0.3} \right)^{T} \), then in accordance with the Eq. (9), we can acquire:
Obviously, the \( PLWDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \) also fulfills the subsequently properties:
-
(1)
\( 0 \le PLWDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \le 1 \);
-
(2)
\( PLWDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = PLWDSM_{{_{PLTSs} }}^{1} \left( {NPL_{2} \left( p \right),NPL_{1} \left( p \right)} \right) \);
-
(3)
\( PLWDSM_{{_{PLTSs} }}^{1} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = 1, \) if \( \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{1j}^{\left( \phi \right)} g\left( {l_{1j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} = \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{2j}^{\left( \phi \right)} g\left( {l_{2j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} \), \( j = 1,2, \ldots ,n \).
3.2 Another form of the DSM for PLTSs
In such chapter, Dice similarity measure’ another form for PLTSs is designed below:
Definition 8
Let \( L = \left\{ {l_{\alpha } \left| {\alpha = - \theta , \ldots , - 1,0,1, \ldots \theta } \right.} \right\} \) be an LTS, \( NPL_{1} \left( p \right) = \left\{ {NPL_{1j} \left( p \right)} \right\}\left( {j = 1,2, \ldots ,n} \right) = \left\{ {l_{1j}^{\left( \phi \right)} \left( {p_{1j}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{1j} \left( p \right)} \right.} \right\}\left( {j = 1,2, \ldots ,n} \right) \) and \( NPL_{2} \left( p \right) = \left\{ {NPL_{2j} \left( p \right)} \right\}\left( {j = 1,2, \ldots ,n} \right) = \left\{ {l_{2j}^{\left( \phi \right)} \left( {p_{2j}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{2j} \left( p \right)} \right.} \right\}\left( {j = 1,2, \ldots ,n} \right) \) be two sets of PLTSs, where \( \# NPL_{1j} \left( p \right) \) and \( \# NPL_{2j} \left( p \right) \) are the numbers of PLTS \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \), \( \# NPL_{1j} \left( p \right) = \# NPL_{2j} \left( p \right) = \# NPL_{j} \left( p \right) \) respectively, another form of probabilistic linguistic DSM (PLDSM) between \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \) is designed in the following:
Example 3
Let \( NPL_{1} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{2} \left( {0.4} \right),l_{3} \left( {0.6} \right)} \right\},\left\{ {l_{1} \left( {0.2} \right),l_{2} \left( {0.8} \right)} \right\},\left\{ {l_{ - 1} \left( {0.2} \right),l_{1} \left( {0.8} \right)} \right\}} \right]\kern-0.15em\right] \) and \( NPL_{2} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{ - 3} \left( {0.8} \right),l_{ - 1} \left( {0.2} \right)} \right\},\left\{ {l_{2} \left( {0.6} \right),l_{3} \left( {0.4} \right)} \right\},\left\{ {l_{ - 2} \left( {0.7} \right),l_{ - 1} \left( {0.3} \right)} \right\}} \right]\kern-0.15em\right] \) be two sets of normalized PLTSs, then in accordance with the Eq. (10), we can acquire:
Another form of probabilistic linguistic DSM (PLDSM) between \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \) also fulfills the subsequently properties:
-
(1)
\( 0 \le PLDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \le 1 \);
-
(2)
\( PLDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = PLDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \);
-
(3)
\( PLDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = 1,\;if\;NPL_{1} \left( p \right) = NPL_{2} \left( p \right), \) i.e. \( \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{1j}^{\left( \phi \right)} g\left( {l_{1j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} = \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{2j}^{\left( \phi \right)} g\left( {l_{2j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} \), \( j = 1,2, \ldots ,n \).
If we consider the weights \( \omega_{j} \) of \( \tilde{L}_{kj} \left( {\tilde{p}} \right)\;\left( {k = 1,2} \right) \), then, another form of probabilistic linguistic weighted DSM (PLWDSM) between \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \) is developed in the following:
where \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) is the weight vector of \( NPL_{kj} \left( p \right)\;\left( {k = 1,2} \right) \), with \( \omega_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \). Particularly, if \( \omega = \left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n},{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}, \ldots ,{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right)^{T} \), then the PLWDSM reduces to the PLDSM. Then there is: \( PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \).
Example 4
Let \( \tilde{L}_{1} \left( {\tilde{p}} \right) = \left[\kern-0.15em\left[ {\left\{ {l_{2} \left( {0.4} \right),l_{3} \left( {0.6} \right)} \right\},\left\{ {l_{1} \left( {0.2} \right),l_{2} \left( {0.8} \right)} \right\},\left\{ {l_{ - 1} \left( {0.2} \right),l_{1} \left( {0.8} \right)} \right\}} \right]\kern-0.15em\right] \) and \( \tilde{L}_{2} \left( {\tilde{p}} \right) = \left[\kern-0.15em\left[ {\left\{ {l_{ - 3} \left( {0.8} \right),l_{ - 1} \left( {0.2} \right)} \right\},\left\{ {l_{2} \left( {0.6} \right),l_{3} \left( {0.4} \right)} \right\},\left\{ {l_{ - 2} \left( {0.7} \right),l_{ - 1} \left( {0.3} \right)} \right\}} \right]\kern-0.15em\right] \) be two sets of normalized PLTSs, the weight values are: \( \omega = \left( {0.2,0.5,0.3} \right)^{T} \), then according to the Eq. (9), we can obtain:
Obviously, the \( PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \) also meets the following properties:
-
(1)
\( 0 \le PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \le 1 \);
-
(2)
\( PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) \);
-
(3)
\( PLWDSM_{{_{PLTSs} }}^{2} \left( {NPL_{1} \left( p \right),NPL_{2} \left( p \right)} \right) = 1, \) if \( \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{1j}^{\left( \phi \right)} g\left( {l_{1j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} = \sum\nolimits_{\phi = 1}^{{\# NPL_{j} \left( p \right)}} {\frac{{p_{2j}^{\left( \phi \right)} g\left( {l_{2j}^{\left( \phi \right)} } \right)}}{{\# NPL_{j} \left( p \right)}}} \), \( j = 1,2, \ldots ,n \).
3.3 The generalized Dice similarity measure for PLTSs
In such chapter, as the Dice similarity measure’s generalization, the probabilistic linguistic generalized Dice similarity measure (PLGDSM) between \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \) are designed below.
Definition 9
Let \( L = \left\{ {l_{\alpha } \left| {\alpha = - \theta , \ldots , - 1,0,1, \ldots \theta } \right.} \right\} \) be an LTS, \( NPL_{1} \left( p \right) = \left\{ {NPL_{1j} \left( p \right)} \right\}\left( {j = 1,2, \ldots ,n} \right) = \left\{ {l_{1j}^{\left( \phi \right)} \left( {p_{1j}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{1j} \left( p \right)} \right.} \right\}\left( {j = 1,2, \ldots ,n} \right) \) and \( NPL_{2} \left( p \right) = \left\{ {NPL_{2j} \left( p \right)} \right\}\left( {j = 1,2, \ldots ,n} \right) = \left\{ {l_{2j}^{\left( \phi \right)} \left( {p_{2j}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{2j} \left( p \right)} \right.} \right\}\left( {j = 1,2, \ldots ,n} \right) \) be two sets of PLTSs, where \( \# NPL_{1j} \left( p \right) \) and \( \# NPL_{2j} \left( p \right) \) are the numbers of PLTS \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \), \( \# NPL_{1j} \left( p \right) = \# NPL_{2j} \left( p \right) = \# NPL_{j} \left( p \right) \) respectively, the PLGDSM between \( NPL_{1j} \left( p \right) \) and \( NPL_{2j} \left( p \right) \) is proposed in the following:
where \( \lambda \) is a positive parameter for \( 0 \le \lambda \le 1 \).
Example 5
Let \( NPL_{1} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{2} \left( {0.4} \right),l_{3} \left( {0.6} \right)} \right\},\left\{ {l_{1} \left( {0.2} \right),l_{2} \left( {0.8} \right)} \right\},\left\{ {l_{ - 1} \left( {0.2} \right),l_{1} \left( {0.8} \right)} \right\}} \right]\kern-0.15em\right] \) and \( NPL_{2} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{ - 3} \left( {0.8} \right),l_{ - 1} \left( {0.2} \right)} \right\},\left\{ {l_{2} \left( {0.6} \right),l_{3} \left( {0.4} \right)} \right\},\left\{ {l_{ - 2} \left( {0.7} \right),l_{ - 1} \left( {0.3} \right)} \right\}} \right]\kern-0.15em\right] \) be two sets of normalized PLTSs, let \( \lambda = 0.3 \), then according to the Eqs. (12–13), we can obtain:
Then, the PLGDSM involves some special situations by modifying the parameter value \( \lambda \).
If \( \lambda = 0.5 \), the two PLGDSM (12) and (13) reduced to PLDSM (14) and (15):
If \( \lambda = 0,1 \), the two PLGDSM reduced to the subsequently measures of asymmetric similarity respectively:
In terms of the above analysis, it could be found that these four measures of asymmetric similarity are the corresponding extension of the relative projection measure of the PLTSs.
In various situations, the weight \( \omega_{j} \left( {j = 1,2, \ldots ,n} \right) \) of the elements \( \tilde{L}_{kj} \left( {\tilde{p}} \right)\left( {k = 1,2} \right) \) could be taken into consideration. For instance, in the process of MADM, there exist different importance for the considered attributes, thus different weights should be considered to assign. However, the subsequently two probabilistic linguistic weighted GDSM (PLWGDSM) for PLTSs are further to proposed, respectively, as follows:
where \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) is the weight of \( NPL_{kj} \left( p \right)\left( {k = 1,2} \right) \), with \( \omega_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \). Particularly, if \( \omega = \left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n},{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}, \ldots ,{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right)^{T} \), then the PLWGDSM reduces to the PLGDSM. Then there is \( PLWGDSM_{{_{PLTSs} }}^{k} \left( {\tilde{L}_{1} \left( {\tilde{p}} \right),\tilde{L}_{2} \left( {\tilde{p}} \right)} \right) = \) \( PLGDSM_{{_{PLTSs} }}^{k} \left( {\tilde{L}_{1} \left( {\tilde{p}} \right),\tilde{L}_{2} \left( {\tilde{p}} \right)} \right)\left( {k = 1,2} \right) \).
Example 6
Let \( NPL_{1} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{2} \left( {0.4} \right),l_{3} \left( {0.6} \right)} \right\},\left\{ {l_{1} \left( {0.2} \right),l_{2} \left( {0.8} \right)} \right\},\left\{ {l_{ - 1} \left( {0.2} \right),l_{1} \left( {0.8} \right)} \right\}} \right]\kern-0.15em\right] \) and \( NPL_{2} \left( p \right) = \left[\kern-0.15em\left[ {\left\{ {l_{ - 3} \left( {0.8} \right),l_{ - 1} \left( {0.2} \right)} \right\},\left\{ {l_{2} \left( {0.6} \right),l_{3} \left( {0.4} \right)} \right\},\left\{ {l_{ - 2} \left( {0.7} \right),l_{ - 1} \left( {0.3} \right)} \right\}} \right]\kern-0.15em\right] \) be two sets of normalized PLTSs, the weight values are: \( \omega = \left( {0.2,0.5,0.3} \right)^{T} \), \( \lambda = 0.3 \) then according to the Eq. (20–21), we can get:
After that, the PLWGDSM involves some special cases by modifying the parameter value \( \lambda \).
If \( \lambda = 0.5 \), the two weighted GDSM (20) and (21) reduced to weighted DSM (22) and (23):
If \( \lambda = 0,1 \), the two PLWGDSM reduces to the subsequent asymmetric weighted DSM, respectively:
In terms of the above analysis, it could be found that these four measures of asymmetric weighted similarity are the corresponding extension of the relative weighted projection measure of PLTSs.
4 The weighted GDSM for probabilistic linguistic MAGDM with entropy weight
In this chapter, we put forward a novel probabilistic linguistic weighted GDSM (PLWGDSM) method for MAGDM issues with unknown weight. The subsequently mathematical notations are made use of solving the probabilistic linguistic MAGDM issues. Let \( A = \left\{ {A_{1} ,A_{2} , \ldots ,A_{m} } \right\} \) be a discrete collection of alternatives, and \( G = \left\{ {G_{1} ,G_{2} , \ldots ,G_{n} } \right\} \) with weight vector \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \), where \( \omega_{j} \in \left[ {0,1} \right] \), \( j = 1,2, \ldots ,n \), \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), and a collection of experts \( E = \left\{ {E_{1} ,E_{2} , \ldots ,E_{q} } \right\} \). Suppose that there are \( n \) qualitative attribute \( A = \left\{ {A_{1} ,A_{2} , \ldots ,A_{m} } \right\} \) and their values are evaluated by qualified experts and denoted as linguistic expressions information \( l_{ij}^{k} \left( {i = 1,2, \ldots ,m,\;j = 1,2, \ldots ,n,\;k = 1,2, \ldots ,q} \right) \).
Then, PLWGDSM method is designed to solve the MAGDM problems with entropy weight. The elaborated calculating procedures are given in the following:
Step 1 Shift cost attribute into beneficial attribute. If the cost attribute value is \( \;l_{\tau } \), then the corresponding beneficial attribute value is \( l_{ - \tau } \left( {\tau = - 3, - 2, - 1,0,1,2,3} \right) \).
Step 2 Convert the linguistic information \( l_{ij}^{k} \left( {i = 1,2, \ldots ,m,\;j = 1,2, \ldots ,n,\;k = 1,2, \ldots ,q} \right) \) into PLTs \( l_{ij}^{\left( \phi \right)} \left( {p_{ij}^{\left( \phi \right)} } \right),\;\phi = 1,2, \ldots ,\;\# L_{ij} \left( p \right) \) and construct the probabilistic linguistic assessing matrix \( PL = \left( {PL_{ij}^{{}} \left( p \right)} \right)_{m \times n} \), \( PL_{ij}^{{}} \left( p \right) = \left\{ {l_{ij}^{\left( \phi \right)} \left( {p_{ij}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# L_{ij}^{{}} \left( p \right)} \right.} \right\} \) \( \left( {i = 1,2, \ldots ,m,j = 1,2, \ldots ,n} \right) \).
Step 3 Derive the normalized probabilistic linguistic matrix \( NPL = \left( {NPL_{ij}^{{}} \left( p \right)} \right)_{m \times n} \), \( NPL_{ij}^{{}} \left( p \right) = \left\{ {l_{ij}^{\left( \phi \right)} \left( {p_{ij}^{\left( \phi \right)} } \right)\left| {\;\phi = 1,2, \ldots ,\# NPL_{ij}^{{}} \left( p \right)} \right.} \right\} \)\( \left( {i = 1,2, \ldots ,m,\;j = 1,2, \ldots ,n} \right) \). Thus, probabilistic linguistic information for given alternative \( A_{i} \in A \) with regard to all the attribute \( G \) can be depicted as: \( PLA_{i} = \left( {l_{i1}^{\left( \phi \right)} \left( {\tilde{p}_{i1}^{\left( \phi \right)} } \right),l_{i2}^{\left( \phi \right)} \left( {\tilde{p}_{i2}^{\left( \phi \right)} } \right), \ldots ,l_{in}^{\left( \phi \right)} \left( {\tilde{p}_{in}^{\left( \phi \right)} } \right)} \right) \), \( \phi = 1,2, \ldots ,\# L_{ij}^{{}} \left( {\tilde{p}} \right) \).
Step 4 Compute the weight values with entropy.
The attributes’ weight is very significant in decision making issues. Entropy (Shannon 1948) is a conventional term from information theory which is also used to determine weight of attributes. Firstly, the normalized decision matrix \( NL_{ij}^{{}} \left( p \right) \) is derived as follows:
Then, the information of Shannon entropy \( E = \left( {E_{1} ,E_{2} , \ldots ,E_{n} } \right) \) is calculated in the following:
and \( NL_{ij}^{{}} \left( p \right)\ln NL_{ij}^{{}} \left( p \right) \) is defined as 0, if \( NL_{ij}^{{}} \left( p \right) = 0 \).
Finally, the attribute weights \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \) is computed:
Step 5 Decide the probabilistic linguistic positive ideal solution (PLPIS):
Step 6 Calculate the PLWGDSM between \( PLA_{i} \left( {i = 1,2, \ldots ,m} \right) \) and \( PLPIS \) as follows:
or
Step 7 All the given alternatives \( A_{i} \left( {i = 1,2, \ldots ,m} \right) \) can be ranked and the optimal one(s) could be selected by \( PLWGDSM_{{_{PLTSs} }}^{1} \left( {PLA_{i} ,PLPIS} \right) \) or \( PLWGDSM_{{_{PLTSs} }}^{2} \left( {PLA_{i} ,PLPIS} \right) \) \( \left( {i = 1,2, \ldots ,m} \right) \). If any alternative has the highestvalues \( PLWGDSM_{{_{PLTSs} }}^{1} \left( {PLA_{i} ,PLPIS} \right) \) or \( PLWGDSM_{{_{PLTSs} }}^{2} \left( {PLA_{i} ,PLPIS} \right) \), then, it is the optimal alternative.
5 A case study and comparative analysis
5.1 A case study for site selection of EVCS
With the rapid development of economy, resource shortage and environmental pollution become more and more serious, thus, people pay more and more attention to their health and living environment. At present, the huge car market is aggravating the cost of resources and adding more pressure to the urban environment. And electric vehicle because of its energy saving and environmental protection characteristics is becoming the main development direction of the automobile industry. Liu et al. (2019b) proposed the integrated MCDM method by a grey decision-making trial and evaluation laboratory (DEMATEL) and uncertain linguistic multi-objective optimization by ratio analysis plus full multiplicative form (UL-MULTIMOORA) for obtaining the most suitable EVCS site in terms of multiple interrelated criteria. Wu et al. (2017) defined the hesitant fuzzy integrated MCDM method for quality function deployment with a case study in electric vehicle. Liu et al. (2017) explored the critical factors influencing the diffusion of electric vehicles in China form the multi-stakeholder perspective. As a supporting infrastructure for electric vehicles, charging stations must be planned and constructed first. The site selection of EVCS is deemed as a kind of MAGDM issue (Wu et al. 2019a, b; Deng and Gao 2019; Li and Lu 2019; Lu and Wei 2019; Wang et al. 2019c). Thus, in this section a numerical case is designed for site selection of EVCS. There are five possible EVCS sites \( A_{i} \left( {i = 1,2,3,4,5} \right) \) to be assessed. The invited experts select four attributes to assess five underlying EVCS sites: ③ G1 the traffic convenience; ④ G2 the service capability; ① G3is waste discharge; ② G4 is construction cost. The construction cost (G4) is not beneficial attribute, others are beneficial attribute. The five underlying EVCS sites \( A_{i} \left( {i = 1,2,3,4,5} \right) \) are to be assessed by utilizing the linguistic variables
by five DMs, as listed in the Tables 1, 2, 3, 4 and 5.
Whereafter, we employ the PLWGDSM method developed for selecting the optimal EVCS sites.
Step 1 Shift cost attribute G4 into beneficial attribute. If the cost attribute value is \( \;l_{\tau } \), then the corresponding beneficial attribute value is \( l_{ - \tau } \left( {\tau = - 3, - 2, - 1,0,1,2,3} \right) \) (See Tables 6, 7, 8, 9 and 10)
Step 2 Shift the linguistic information into probabilistic linguistic assessing matrix (Table 11).
Step 3 Calculate the normalized assessing matrix with PTSs (Table 12).
Step 4 Compute the weight values for attributes from Eqs. (28)–(30): \( w_{1} = 0.1416,w_{2} = 0.4892,w_{3} = 0.1021,w_{4} = 0.2671 \).
Step 5 Determine the PLPIS by Eqs. (31)–(32) (Table 13):
Step 6 Calculating the PLWGDSM between \( PLA_{i} \left( {i = 1,2, \ldots ,5} \right) \) and \( PLPIS \) (Tables 14, 15):
Step 7 All the given alternatives \( A_{i} \left( {i = 1,2, \ldots ,m} \right) \) can be ranked and the optimal one(s) can be selected by \( PLWGDSM_{{_{PLTSs} }}^{1} \left( {PLA_{i} ,PLPIS} \right) \) or \( PLWGDSM_{{_{PLTSs} }}^{2} \left( {PLA_{i} ,PLPIS} \right) \) \( \left( {i = 1,2, \ldots ,5} \right) \) (Tables 16, 17).
From the Tables 16 and 17, taking different \( \lambda \) and different PLWGDSM, the ranking orders can be different. Then \( A_{1} \), \( A_{5} \) and \( A_{3} \) should be the optimal EVCS sites in accordance with the principle of the maximum PLWGDSM.
Furthermore, for the two PLWGDSM’s special situations, we acquire the subsequent results:
-
When \( \lambda = 0 \), the two PLWGDSM reduced to the corresponding weighted projection measures of \( A_{i} \;(i = 1,2,3,4,5) \) on PLPIS. Thus, \( A_{1} \) should be the optimal EVCS site in accordance with the maximum PLWGDSM. In such situation, we could obtain the same optimal EVCS site as the above mentioned three methods. Thus our proposed method is effective.
-
When \( \lambda = 0.5 \), the two PLWGDSM reduced to the PLWDSM on PLPIS. Thus, \( A_{1} \) should be the optimal EVCS site in accordance with maximum PLWDSM between PLTSs. For such case, we could also obtain the same optimal EVCS site as the above mentioned three methods. Thus our proposed method is effective.
-
When \( \lambda = 1 \), the two PLWGDSM reduced to the corresponding weighted projection measures of PLPIS on \( A_{i} \;(i = 1,2,3,4,5) \). Thus, \( A_{3} \) should be the optimal EVCS site in accordance with the principle of the maximum degree of PLWDSM between PLTSs.
However, in accordance with different PLWGDSM and different \( \lambda \), there exists slightly different with ranking orders. Therefore some value of \( \lambda \) and some measure can assign to the presented PLWGDSM methods to fulfill the requirements of DMs preference and flexible decision making issues.
5.2 Comparative analysis
Firstly, PL-GRA method (Liang et al. 2018) (let \( \rho = 0.5 \)) is used to compare with our proposed PLWGDSM method, then we can get the calculating results: \( \varepsilon_{1}^{ + } = 0.6865,\varepsilon_{2}^{ + } = 0.4094,\varepsilon_{3}^{ + } = 0.3848,\varepsilon_{4}^{ + } = 0.6220,\varepsilon_{5}^{ + } = 0.4451. \) Furthermore, we can derive the ranking order: \( A_{1} > A_{4} > A_{5} > A_{2} > A_{3} \). Thus, we also have the same optimal EVCS site \( A_{1} \).
Secondly, probabilistic linguistic weighted average (PLWA) operator (Pang et al. 2016) is used to compare with our proposed PLWGDSM method. If the attribute weights are completely known, the calculating results is: \( E\left( {Z_{1} \left( w \right)} \right) = s\begin{array}{*{20}r} \hfill {_{0.2735} } \\ \end{array} ,E\left( {Z_{2} \left( w \right)} \right) = \begin{array}{*{20}r} \hfill {s_{ - 0.2052} } \\ \end{array} , \) \( E\left( {Z_{3} \left( w \right)} \right) = s_{ - 0.6567} , \) \( E\left( {Z_{4} \left( w \right)} \right) = s_{ - 0.0132} , \) \( E\left( {Z_{5} \left( w \right)} \right) = s_{ - 0.0920} \) and we can obtain the sorting order: \( A_{1} > A_{4} > A_{5} > A_{2} > A_{3} \), thus, we have the same optimal EVCS site \( A_{1} \).
Finally, PL-TOPSIS method (Pang et al. 2016) is employed to compare with PLWGDSM method, then we can acquire the calculating results and sorting results (Table 18).
In terms of the above analysis, it can be found that these above mentioned methods have the same optimal EVCS site \( A_{1} \), and there are slightly different in the three methods’ ranking results from our presented PLWGDSM methods, which can confirm the PLWGDSM methods we presented are more flexible and fulfill the requirements of DMs’ preference. All these methods have their good advantages: (1) PL-GRA method emphasis the shape similarity degree from the positive ideal solution; (2) PLWA operator emphasis group influences; (3) PL-TOPSIS method emphasis the distance similarity degree from the positive and negative ideal solution with incomplete weight information. (4) Some value of \( \lambda \) and some measure can assign to the presented PLWGDSM methods to fulfill the requirements of DMs’ preference and flexible decision making. Evidently, on the basis of the Dice measures and the projection measures, the MAGDM methods are the special situations of the presented MAGDM methods based on PLWGDSM. Thus, in the process of MAGDM, the MAGDM methods put forward in such paper are more useful and more flexible compared with existing MAGDM issues under PLTSs.
6 Conclusion
In this paper, we design some novel DSM of PLTSs and the GDSM of PLTSs and indicate that the DSM and PLTSs’ asymmetric measures are special situations of the PLWGDSM with different parameter values. Then, we propose the PLWGDSM-based MAGDM methods with PLTSs. In the end, a demonstrative case study for location planning of electric vehicle charging stations is offered to illustrate the PLWGDSM’s efficiency. Thus, the main contributions of such paper are: (1) two DSM’ forms within PLTSs are designed; (2) the GDSM and weighted GDSM with PLTSs are defined; (3) the weighted GDSM are utilized to tackle the MAGDM issues under PLTSs; (4) within the process of MAGDM, the developed methods’ major merit is more flexible and useful compared with the existing MAGDM issues with PLTSs. In the future, the proposed PLWGDSM of PLTSs can be widely applied and investigated in dynamic and intricate MADM or MAGDM issues and various unpredictable environments. The designed methods could also tackle other issues, such as environmental sustainability competency analysis, intelligent sustainable supplier selection and comprehensive assessment for water pollution.
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Wei, G., Wei, C., Wu, J. et al. Probabilistic linguistic multiple attribute group decision making for location planning of electric vehicle charging stations based on the generalized Dice similarity measures. Artif Intell Rev 54, 4137–4167 (2021). https://doi.org/10.1007/s10462-020-09950-2
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DOI: https://doi.org/10.1007/s10462-020-09950-2