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The Generalized Dice Similarity Measures for Probabilistic Uncertain Linguistic MAGDM and Its Application to Location Planning of Electric Vehicle Charging Stations

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Abstract

The location of the electric vehicle charging station (EVCS) is the important link in the construction of the EVCS. The optimal location of the electric vehicle directly affects the operating efficiency of the charging station and the satisfaction of the electric vehicle user, site selection play an important part throughout whole life cycle, which is deemed to be 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 uncertain linguistic term sets (ULTSs). These ULTSs could be simply shifted into the probabilistic uncertain linguistic sets (PULTSs). In such paper, we design some novel probabilistic uncertain linguistic weighted Dice similarity measures (PULWDSM) and the probabilistic uncertain linguistic weighted generalized Dice similarity measures (PULWGDSM). Subsequently, the PULWGDSM-based MAGDM methods are presented under PULTSs. In the end, a practical case which concerns about the location planning of electric vehicle charging stations is offered to demonstrate the proposed PULWGDSM’s applicability and advantages.

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Appendices

Appendix 1 for Example 1

Let \({\text{PUL}}_{1} \left( p \right) = \left\{ {\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{4}} \right\rangle ,\;\left\langle {\left[ {l_{{2}} ,\;l_{{3}} } \right],\;0.{6}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{0}} ,\;l_{{1}} } \right],\;0.{2}} \right\rangle ,\;\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{8}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{ - 2}} ,\;l_{{ - 1}} } \right],\;0.{2}} \right\rangle ,\;\left\langle {\left[ {l_{{0}} ,\;l_{{1}} } \right],\;0.{8}} \right\rangle } \right\}\) and \({\text{PUL}}_{2} \left( p \right) = \left\{ {\left\langle {\left[ {l_{{ - 3}} ,\;l_{{ - 2}} } \right],\;0.{8}} \right\rangle ,\;\left\langle {\left[ {l_{{ - 2}} ,\;l_{{ - 1}} } \right],\;0.{2}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{6}} \right\rangle ,\;\left\langle {\left[ {l_{{2}} ,\;l_{{3}} } \right],\;0.{4}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{ - 1}} ,\;l_{{1}} } \right],\;0.{7}} \right\rangle ,\;\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{3}} \right\rangle } \right\}\) be two sets of normalized PULTSs, let \(\lambda = 0.3\), then according to the Eqs. (13)–(14), we can obtain:

$$\begin{aligned} & {\text{PULGDSM}}_{{_{{{\text{PULTSs}}}} }}^{1} \left( {{\text{PUL}}_{1} \left( p \right),\;{\text{PUL}}_{2} \left( p \right)} \right) \\ & \quad = \frac{1}{n}\sum\limits_{{j = 1}}^{n} {\frac{{\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right) \cdot \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)}}{{\lambda \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)^{2} + \left( {1 - \lambda } \right)\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)^{2} }}} \\ & \quad = \frac{1}{3} \times \left( \begin{gathered} \frac{{2 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)}}{{0.3 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} }} + \hfill \\ \frac{{2 \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)}}{{0.3 \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} }} + \hfill \\ \frac{{2 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)}}{{0.3 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} }} \hfill \\ \end{gathered} \right) \\ & \quad = 0.7782, \\ \end{aligned} ,$$
$$\begin{aligned} & {\text{PULGDSM}}_{{_{{{\text{PULTSs}}}} }}^{2} \left( {{\text{PUL}}_{1} \left( p \right),\;{\text{PUL}}_{2} \left( p \right)} \right) \\ & \quad = \frac{{\sum\nolimits_{{j = 1}}^{n} {\left( {\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right) \cdot \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)} \right)} }}{{\lambda \sum\nolimits_{{j = 1}}^{n} {\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)^{2} + \left( {1 - \lambda } \right)\sum\nolimits_{{j = 1}}^{n} {\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)} } }} \\ & \quad = \frac{{2 \times \left( \begin{gathered} \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) + \hfill \\ \;\;\;\left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) + \hfill \\ \;\;\;\left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \hfill \\ \end{gathered} \right)}}{{\left( \begin{gathered} \left( {0.3 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)} \right) + \hfill \\ \left( {0.3 \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)} \right) + \hfill \\ \left( {0.3 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)} \right) \hfill \\ \end{gathered} \right)}} \\ = 0.8471. \\ \end{aligned}$$

Appendix 2 for Example 2

Let \({\text{PUL}}_{1} \left( p \right) = \left\{ {\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{4}} \right\rangle ,\;\left\langle {\left[ {l_{{2}} ,\;l_{{3}} } \right],\;0.{6}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{0}} ,\;l_{{1}} } \right],\;0.{2}} \right\rangle ,\;\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{8}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{ - 2}} ,\;l_{{ - 1}} } \right],\;0.{2}} \right\rangle ,\;\left\langle {\left[ {l_{{0}} ,\;l_{{1}} } \right],\;0.{8}} \right\rangle } \right\}\) and \({\text{PUL}}_{2} \left( p \right) = \left\{ {\left\langle {\left[ {l_{{ - 3}} ,\;l_{{ - 2}} } \right],\;0.{8}} \right\rangle ,\;\left\langle {\left[ {l_{{ - 2}} ,\;l_{{ - 1}} } \right],\;0.{2}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{6}} \right\rangle ,\;\left\langle {\left[ {l_{{2}} ,\;l_{{3}} } \right],\;0.{4}} \right\rangle } \right\},\;\left\{ {\left\langle {\left[ {l_{{ - 1}} ,\;l_{{1}} } \right],\;0.{7}} \right\rangle ,\;\left\langle {\left[ {l_{{1}} ,\;l_{{2}} } \right],\;0.{3}} \right\rangle } \right\}\) be two sets of normalized PULTSs, the weight values are: \(\omega = \left( {0.2,\;0.5,\;0.3} \right)^{T}\), \(\lambda = 0.3\) then according to the Eqs. (21)–(22), we can get:

$$\begin{aligned} & {\text{PULWGDSM}}_{{_{{{\text{PULTSs}}}} }}^{1} \left( {{\text{PUL}}_{1} \left( p \right),\;{\text{PUL}}_{2} \left( p \right)} \right) \\ & \quad = \sum\limits_{{j = 1}}^{n} {\omega _{j} \frac{{\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right) \cdot \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)}}{{\lambda \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)^{2} + \left( {1 - \lambda } \right)\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)^{2} }}} \\ & \quad = 0.2 \times \frac{{2 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)}}{{0.3 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} }} + \\ 0.5 \times \frac{{2 \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)}}{{0.3 \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} }} + \\ 0.3 \times \frac{{2 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)}}{{0.3 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} }} \\ & \quad = 0.8450, \\ \end{aligned}$$
$$\begin{aligned} & {\text{PULWGDSM}}_{{_{{{\text{PULTSs}}}} }}^{2} \left( {{\text{PUL}}_{1} \left( p \right),\;{\text{PUL}}_{2} \left( p \right)} \right) \\ & \quad = \frac{{\sum\nolimits_{{j = 1}}^{n} {\omega _{j}^{2} \left( {\left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right) \cdot \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)} \right)} }}{{\lambda \sum\nolimits_{{j = 1}}^{n} {\omega _{j}^{2} \left( {\sum\nolimits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{1j}} ^{{\left( \phi \right)}} g\left( {L_{{1j}} ^{{\left( \phi \right)}} } \right) + p_{{1j}} ^{{\left( \phi \right)}} g\left( {U_{{1j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)^{2} + \left( {1 - \lambda } \right)\sum\nolimits_{{j = 1}}^{n} {\omega _{j}^{2} \left( {\sum\limits_{{\phi = 1}}^{{\# {\text{PUL}}_{j} \left( p \right)}} {\frac{{p_{{2j}} ^{{\left( \phi \right)}} g\left( {L_{{2j}} ^{{\left( \phi \right)}} } \right) + p_{{2j}} ^{{\left( \phi \right)}} g\left( {U_{{2j}} ^{{\left( \phi \right)}} } \right)}}{{2\# {\text{PUL}}_{j} \left( p \right)}}} } \right)} } }} \\ & \quad = \frac{{\;\;\;2 \times \left( \begin{gathered} 0.2^{2} \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) + \hfill \\ 0.5^{2} \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) + \hfill \\ 0.3^{2} \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right) \hfill \\ \end{gathered} \right)}}{{\left( \begin{gathered} 0.2^{2} \times \left( {0.3 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + }}}} \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.6}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + }}}} \hfill \\ {{( - 3 + 3)} \mathord{\left/ {\vphantom {{( - 3 + 3)} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8 + {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} } \right) + \hfill \\ 0.5^{2} \times \left( {0.3 \times \left( {\frac{\begin{gathered} {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} + \hfill \\ {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.6 + {{(3 + 3)} \mathord{\left/ {\vphantom {{(3 + 3)} {6 \times 0.4}}} \right. \kern-\nulldelimiterspace} {6 \times 0.4}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} } \right) + \hfill \\ 0.3^{2} \times \left( {0.3 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(0 + 3)} \mathord{\left/ {\vphantom {{(0 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} + \hfill \\ {{( - 2 + 3)} \mathord{\left/ {\vphantom {{( - 2 + 3)} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.2 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.8}}} \right. \kern-\nulldelimiterspace} {6 \times 0.8}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} + 0.7 \times \left( {\frac{\begin{gathered} {{( - 1 + 3)} \mathord{\left/ {\vphantom {{( - 1 + 3)} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(1 + 3)} \mathord{\left/ {\vphantom {{(1 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}} + }} \hfill \\ {{\;\;(1 + 3)} \mathord{\left/ {\vphantom {{\;\;(1 + 3)} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}}} \right. \kern-\nulldelimiterspace} {6 \times 0.7 + {{(2 + 3)} \mathord{\left/ {\vphantom {{(2 + 3)} {6 \times 0.3}}} \right. \kern-\nulldelimiterspace} {6 \times 0.3}}}} \hfill \\ \end{gathered} }{{2 \times 2}}} \right)^{2} } \right) \hfill \\ \end{gathered} \right)}} \\ & \quad = 0.9204. \\ \end{aligned}$$

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Wei, G., Lin, R., Lu, J. et al. The Generalized Dice Similarity Measures for Probabilistic Uncertain Linguistic MAGDM and Its Application to Location Planning of Electric Vehicle Charging Stations. Int. J. Fuzzy Syst. 24, 933–948 (2022). https://doi.org/10.1007/s40815-021-01084-z

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  • DOI: https://doi.org/10.1007/s40815-021-01084-z

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