## 1 Introduction

*et al.*, 2021; Lei

*et al.*, 2021; Liu

*et al.*, 2018; Zhang D.

*et al.*, 2021; Zhang

*et al.*, 2018). In the decision-making process, decision makers are usually experts in their fields. Therefore, decision makers (DMs) would like to use linguistic terms rather than utilize the exact real numbers due to the complication of the socioeconomic setting and fuzziness of human beings’ thinking (Lei

*et al.*, 2021; Wei

*et al.*, 2021a). It means that the linguistic terms given by experts contain uncertainty and preference. To solve the uncertainty of decision-making problems, a lot of effective work has been done. Wang and Garg (2021) proposed new interaction Pythagorean operators and designed an algorithm to solve the MADM issues with Pythagorean fuzzy uncertainties. Yazdi

*et al.*(2020) proposed an integrated method which combined BWM with Weighted Aggregated Sum-Product Assessment (WASPAS) on uncertain decision-making environments with Z-numbers. Xiao

*et al.*(2021) built Taxonomy method for MAGDM based on interval-valued intuitionistic fuzzy information. Zhang H.

*et al.*(2021) defined the CPT-MABAC method for spherical fuzzy MAGDM. Zhang S.

*et al.*(2021) defined the grey relational analysis method based on cumulative prospect theory for intuitionistic fuzzy MAGDM.

*et al.*(2012) defined hesitant fuzzy linguistic term set (HFLTS) to use the hesitancy degree in the linguistic context. Zeng

*et al.*(2019) introduced several weighted operators to aggregate weighted hesitant fuzzy linguistic information. Liu

*et al.*(2019) improved incomplete hesitant fuzzy linguistic preference relations (IHFLPRs). These concepts can describe ambiguity and preference in linguistic term sets, but ignored differences in the importance of evaluation information. Thus, Pang

*et al.*(2016) used probabilistic linguistic term sets (PLTSs) to depict fuzziness and uncertainty with certain probabilities. They proposed some rules of operation and aggregation operators for PLTSs. We can find that PLTSs can more comprehensively and precisely represent the attitude of decision makers. Furthermore, many improvements have been made in decision-making issues on PLTSs. Yue

*et al.*(2020b) put forward the group utility measure, the individual regret measure and the compromise measure under PLTSs. Some studies discussed decision making methods under PLTSs. Wei

*et al.*(2021b) built the DAS method for probabilistic linguistic MAGDM. Chen

*et al.*(2020) combined distillation algorithm with ELECTRE III method on PLTSs. He

*et al.*(2021) modified the FMEA (the failure mode and effect analysis) model on the PLTSs. You

*et al.*(2020) designed PL-VIKOR method and modified the distance measure. Some studies have introduced and defined some new distance formulas. Chang

*et al.*(2021) introduced Hellinger distance measure. Jiang and Liao (2021) defined Kolmogorov-Smirnov distance measure on the PLTSs. Some studies have proposed effective tools to solve decision-making issues under PLTSs. Du and Liu (2021) researched quality function deployment tool under PLTSs. Lin

*et al.*(2021) proposed score C-PLTSs and probability splitting algorithm. And a novel PLTS correlation coefficient was put forward by Luo

*et al.*(2020). Peng and Wang (2020) introduced linguistic scale functions. Shen

*et al.*(2021) came up with a model to reduce limitations of evaluation on the PLTSs. Teng

*et al.*(2021) designed the Choquet integral operator under PLTSs. Wang and Liang (2020) put away a preference degree for g-granularity PLTS. Wang

*et al.*(2021) extended the operational laws of PLTSs. Wang

*et al.*(2020) proposed probabilistic linguistic Z-numbers to describe related information. Xie

*et al.*(2020) defined the dual probabilistic linguistic correlation coefficient. Xu

*et al.*(2020) proposed a method to make probabilistic linguistic more complete in describing evaluation information. Yu

*et al.*(2020) combined stochastic dominance degrees with PLTSs. Yue

*et al.*(2020a) introduced the projection formulas and Qu

*et al.*(2020) introduced new utility functions on the PLTSs. Su

*et al.*(2021a) built PT-TODIM method for probabilistic linguistic MAGDM.

*et al.*(2020) improved customer satisfaction evaluation system on PLTSs. Mo (2020) proposed the D-PLTS method to settle emergency decision-making issues. Pan

*et al.*(2021) designed a probabilistic linguistic data envelopment analysis model. Xu C.

*et al.*(2020) applied probabilistic linguistic preference relations to handle the healthcare insurance audits in China. Gao

*et al.*(2021) proposed the PLTSs to describe information and built the MCGDM framework for the risk assessment. Luo

*et al.*(2021) designed the IDOCRIW-COCOSO model to evaluate tourism attractions on the PLTSs. Ming

*et al.*(2020) structured a medical service evaluation criteria system under PLTSs.

*et al.*(2019) used the MABAC algorithm to select the optimal green supplier. To select the optimal university, Gong

*et al.*(2020) designed a new UTAE (undergraduate teaching audit and evaluation) approach combined with the MABAC method. Biswas (2020) selected the MABAC method to prepare a comparative analysis of supply chain performances. In order to make better use of the MABAC method, experts put it in different linguistic environments. Verma (2021) applied IFS (intuitionistic fuzzy set) with the MABAC algorithm. Liang

*et al.*(2019) came up with the MABAC approach based on TFN to evaluate the risk of rock-burst. Hu

*et al.*(2019) combined the MABAC method with the similarity of interval type-2 fuzzy numbers (IT2FNs). Sun

*et al.*(2018) extended the MABAC method to HFLTSs (hesitant fuzzy linguistic term sets) for patients’ prioritization. Aydin (2021) applied the MABAC method with Fermantean fuzzy sets into decision-making process. Liu and Zhang (2021) integrated the MABAC model with prospect theory (PT) on a normal wiggly hesitant fuzzy set (NWHFS). Additionally, many studies combined MABAC with another algorithm to solve MADM or MAGDM problems. Pamucar

*et al.*(2018) defined the IR-AHP-MABAC (interval rough analytic hierarchy process-MABAC) model to assess the quality of websites. Jiang

*et al.*(2022) built the picture fuzzy MABAC method based on prospect theory for MAGDM.

*et al.*(2018) built a new model based on CPT to tackle portfolio selection. Zhao

*et al.*(2021b) combined CPT with TODIM method under several linguistic environments, such as pythagorean fuzzy sets (2021), the 2-tuple linguistic pythagorean fuzzy sets (Zhao

*et al.*, 2021c). Additionally, Zhao

*et al.*(2021a) introduced the intuitionistic fuzzy MABAC method based on CPT. Furthermore, picture fuzzy sets (Jiang

*et al.*, 2021a, 2021b) were dealt with CPT. Su

*et al.*(2021b) built the probabilistic uncertain linguistic EDAS method based on prospect theory for MAGDM.

## 2 Preliminaries

### 2.1 PLTSs

*et al.*(2016) came up with PLTS, which have different weights and probabilities.

##### Definition 1 (Gou *et al.*, 2017)*.*

*χ*by transformation function

*κ*, and the formula is as follows:

##### (1)

\[ \begin{aligned}{}& \kappa :[{\zeta _{-ƛ}},{\zeta _{ƛ}}]\to [0,1],\\ {} & \kappa ({\zeta _{ƛ}})=\frac{ƛ+\partial }{2\partial }=\chi .\end{aligned}\]*χ*can also be translated into linguistic terms ${\zeta _{ƛ}}$ by the shifting function ${\kappa ^{-1}}$:

##### Definition 2 (Pang *et al.*, 2016)*.*

##### (3)

\[ L(p)=\Bigg\{{\zeta ^{(\gamma )}}\big({p^{(\gamma )}}\big)\Big|{\zeta ^{(\gamma )}}\in L,{p^{(\gamma )}}\geqslant 0,\gamma =1,2,\dots ,\mathrm{\# }L(p),{\sum \limits_{\gamma =1}^{\mathrm{\# }L(p)}}{p^{(\gamma )}}\leqslant 1\Bigg\}.\]*γ*th linguistic term ${\zeta ^{(\gamma )}}$ and its corresponding probability value ${p^{(\gamma )}}$. The linguistic terms ${\zeta ^{(\gamma )}}$ are arranged in ascending order in the set $L(p)$. $\mathrm{\# }L(p)$ represents the linguistic terms’ length in $L(p)$.

*et al.*(2016) normalized the PLTS $L(p)$ as $\textit{NL}(\tilde{p})=\big\{{\zeta ^{(\gamma )}}({\tilde{p}^{(\gamma )}})\big|{\zeta ^{(\gamma )}}\in L,{\tilde{p}^{(\gamma )}}\geqslant 0,\gamma =1,2,\dots ,\mathrm{\# }L(\tilde{p}),{\textstyle\sum _{\gamma =1}^{\mathrm{\# }L(p)}}{\tilde{p}^{(\gamma )}}=1\big\}$, where ${\tilde{p}^{(\gamma )}}={p^{(\gamma )}}/{\textstyle\sum _{\gamma =1}^{\mathrm{\# }L(p)}}{p^{(\gamma )}}$ for all $\gamma =1,2,\dots ,\mathrm{\# }L(\tilde{p})$.

##### Definition 3 (Pang *et al.*, 2016)*.*

##### Definition 4 (Pang *et al.*, 2016)*.*

##### (4)

\[\begin{aligned}{}& \textit{SF}\big(\textit{NL}(\tilde{p})\big)={\sum \limits_{\gamma =1}^{\mathrm{\# }\textit{NL}(\tilde{p})}}\kappa \big(\textit{NL}(\tilde{p})\big){\tilde{p}^{(\gamma )}}\Big/{\sum \limits_{\gamma =1}^{\mathrm{\# }\textit{NL}(\tilde{p})}}{\tilde{p}^{(\gamma )}},\end{aligned}\]##### (5)

\[\begin{aligned}{}& \textit{DF}\big(\textit{NL}(\tilde{p})\big)=\sqrt{{\sum \limits_{\gamma =1}^{\mathrm{\# }\textit{NL}(\tilde{p})}}{\big(\kappa \big(\textit{NL}(\tilde{p})\big){\tilde{p}^{(\gamma )}}-\textit{SF}\big(\textit{NL}(\tilde{p})\big)\big)^{2}}}\Big/{\sum \limits_{\gamma =1}^{\mathrm{\# }\textit{NL}(\tilde{p})}}{\tilde{p}^{(\gamma )}}.\end{aligned}\]- (1) If $\textit{SF}({\textit{NL}_{1}}(\tilde{p}))>\textit{SF}({\textit{NL}_{2}}(\tilde{p}))$, then ${\textit{NL}_{1}}(\tilde{p})>{\textit{NL}_{2}}(\tilde{p})$;
- (2) If $\textit{SF}({\textit{NL}_{1}}(\tilde{p}))=\textit{SF}({\textit{NL}_{2}}(\tilde{p}))$, then $\textit{DF}({\textit{NL}_{1}}(\tilde{p}))=\textit{DF}({\textit{NL}_{2}}(\tilde{p}))$ and ${\textit{NL}_{1}}(\tilde{p})={\textit{NL}_{2}}(\tilde{p})$; if $\textit{DF}({\textit{NL}_{1}}(\tilde{p}))<\textit{DF}({\textit{NL}_{2}}(\tilde{p}))$, then ${\textit{NL}_{1}}(\tilde{p})>{\textit{NL}_{2}}(\tilde{p})$.

##### Definition 5 (Lin *et al.*, 2019)*.*

##### (6)

\[ d\big({\textit{NL}_{1}}(\tilde{p}),{\textit{NL}_{2}}(\tilde{p})\big)=\frac{{\textstyle\textstyle\sum _{\gamma =1}^{\mathrm{\# }{\textit{NL}_{1}}(\tilde{p})}}\big|{\tilde{p}_{1}^{(\gamma )}}\kappa ({\zeta _{1}^{(\gamma )}})-{\tilde{p}_{2}^{(\gamma )}}\kappa ({\zeta _{2}^{(\gamma )}})\big|}{\mathrm{\# }\textit{NL}(\tilde{p})}.\]### 2.2 Cumulative Prospect Theory

*m*is the number of attributes for alternatives;

*j*expresses the

*j*th attribute; $\lambda ({x_{j}})$ reflects the DMs’ subjective feeling value according to the actual gains or the losses. The formula of $\lambda ({x_{j}})$ is defined as follows (Tversky and Kahneman, 1992):

*j*th attribute. $\Delta >0$ signifies the gain, $\Delta x<0$ signifies the loss and $\Delta x=0$ signifies no gain or loss.

*τ*and

*ς*represent the parameters of risk attitudes. These depict the sensitivity of decision makers to gains and losses.

*θ*depicts the coefficient of loss aversion, and $\theta >1$. The higher the value of

*θ*, the more risk averse the decision maker is. Tversky and the others obtained the parameters of the value function in CPT with the method of linear regression when the parameters were $\tau =\varsigma =0.88$, $\theta =2.25$. It was more consistent with the empirical data. And it is worth mentioning that we take the probabilistic linguistic border approximation area as the reference point to gains and losses in this paper.

## 3 CPT-PL-MABAC Model for MAGDM Issues

*n*qualitative attributes $\mathrm{\Im }=\{{\mathrm{\Im }_{1}},{\mathrm{\Im }_{2}},\dots ,{\mathrm{\Im }_{n}}\}$. Experts will evaluate every attribute and use linguistics ${\zeta _{ij}^{k}}$ $(i=1,2,\dots ,m,j=1,2,\dots ,n,k=1,2,\dots ,q)$ to express the value of evaluation. $\varpi =({\varpi _{1}},{\varpi _{2}},\dots ,{\varpi _{n}})$ represents the attribute weight vector, where ${\varpi _{j}}\in [0,1]$, ${\textstyle\sum _{j=1}^{n}}{\varpi _{j}}=1$ and $\mathrm{\aleph }=\{{\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{q}}\}$ is a collection of

*q*experts.

### 3.2 The CPT-PL-MABAC Calculating Procedure

**Step 1.**Transform cost attributes into beneficial ones.

**Step 2.**Shift the value of evaluation information $L=\{{\zeta _{ij}^{k}}\big|k=1,2,\dots ,q,i=1,2,\dots ,m,j=1,2,\dots ,n\}$ into PLTSs $\textit{PL}=\{{\zeta _{ij}^{(\gamma )}}({p_{ij}^{(\gamma )}})\big|\gamma =1,2,\dots ,\mathrm{\# }{L_{ij}}(p)\}$. Build the evaluation matrix $M={({\textit{PL}_{ij}}(p))_{m\times n}}$, ${\textit{PL}_{ij}}(p)=\big\{{\zeta _{ij}^{(\gamma )}}({p_{ij}^{(\gamma )}})\big|\gamma =1,2,\dots ,\mathrm{\# }{L_{ij}}(p)\big\}$ $(i=1,2,\dots ,m,j=1,2,\dots ,n)$.

**Step 3.**Obtain the normalized decision matrix $\textit{NM}={({\textit{NPL}_{ij}}(\tilde{p}))_{m\times n}}$ with PLTSs, ${\textit{NPL}_{ij}}(\tilde{p})=\{{\zeta _{ij}^{(\gamma )}}({\tilde{p}_{ij}^{(\gamma )}})\big|\gamma =1,2,\dots ,\mathrm{\# }{L_{ij}}(\tilde{p}),{\textstyle\sum _{\gamma =1}^{\mathrm{\# }{L_{ij}}(\tilde{p})}}{\tilde{p}_{ij}^{(\gamma )}}=1\}$ $(i=1,2,\dots ,m,j=1,2,\dots ,n)$.

**Step 4.**Figure up the combined weight of attributes.

- 2. Let ${\mathrm{\wp }_{j}}$ be the entropy of the
*j*th attribute, and calculate it by using Eq. (11):##### (11)

\[\begin{aligned}{}{\mathrm{\wp }_{j}}& =-\frac{1}{\ln m}{\sum \limits_{i=1}^{m}}\\ {} & \hspace{1em}\times \bigg[\frac{d({\textit{NL}_{ij}}({\tilde{p}_{j}}),{\textit{ML}_{j}}({\bar{p}_{j}}))}{{\textstyle\textstyle\sum _{i=1}^{m}}d({\textit{NL}_{ij}}({\tilde{p}_{j}}),{\textit{ML}_{j}}({\bar{p}_{j}}))}\ln \bigg(\frac{d({\textit{NL}_{ij}}({\tilde{p}_{j}}),{\textit{ML}_{j}}({\bar{p}_{j}}))}{{\textstyle\textstyle\sum _{i=1}^{m}}d({\textit{NL}_{ij}}({\tilde{p}_{j}}),{\textit{ML}_{j}}({\bar{p}_{j}}))}\bigg)\bigg].\end{aligned}\] - 3. Compute the objective weights of the
*j*th attribute by using Eq. (12):##### (12)

\[ {w_{oj}}=\frac{1-{\mathrm{\wp }_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}(1-{\mathrm{\wp }_{j}})},\hspace{1em}j=1,2\dots ,n,\]

##### (13)

\[ {w_{cj}}=\frac{{w_{oj}}\ast {w_{sj}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{w_{oj}}\ast {w_{sj}}},\]**Step 5.**Figure out the probabilistic linguistic border approximation area (PLBAA) matrix $\mathrm{PLBAA}={({\textit{PLBAA}_{j}})_{1\times n}}$. The PLBAA could be obtained according to Eqs. (14)–(16).

##### (15)

\[\begin{aligned}{}& {\textit{PLBAA}_{j}}=\big\{{\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}({\stackrel{\leftrightarrow }{p}_{j}^{\hspace{0.1667em}(\gamma )}})\big|\gamma =1,2,\dots ,\mathrm{\# }{\textit{NL}_{ij}}(\stackrel{\leftrightarrow }{p})\big\},\end{aligned}\]##### (16)

\[\begin{aligned}{}& {\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big({\stackrel{\leftrightarrow }{p}_{j}^{\hspace{0.1667em}(\gamma )}}\big)={\kappa ^{-1}}\Bigg(\sqrt[m]{{\prod \limits_{i=1}^{m}}\kappa \big({\zeta _{ij}^{(\gamma )}}\big)}\hspace{0.1667em}\Bigg)\Bigg(\sqrt[m]{{\prod \limits_{i=1}^{m}}{p_{ij}^{(\gamma )}}}\hspace{0.1667em}\Bigg).\end{aligned}\]**Step 6.**Figure up the Hamming distance from PLBAA by Eq. (17) and the cumulative prospect distance matrix by using Eq. (18).

##### (17)

\[\begin{aligned}{}& d({\textit{NL}_{ij}}(\tilde{p}),{\textit{PLBAA}_{j}})=\Bigg({\sum \limits_{\gamma =1}^{\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})}}\big|\kappa \big({\zeta _{ij}^{(\gamma )}}\big)\big({p_{ij}^{(\gamma )}}\big)-\kappa \big({\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big)\big({\stackrel{\leftrightarrow }{p}_{j}^{(\gamma )}}\big)\big|\Bigg)\Big/\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p}),\end{aligned}\]##### (18)

\[\begin{aligned}{}& {\Lambda _{ij}}=\left\{\begin{array}{l}{[d({\textit{NL}_{ij}}(\tilde{p}),{\textit{PLBAA}_{j}})]^{\tau }}{w_{j}},\\ {} \hspace{1em}\text{if}\hspace{2.5pt}\big({\textstyle\textstyle\sum _{\gamma =1}^{\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})}}\kappa \big({\zeta _{ij}^{(\gamma )}}\big)\big({p_{ij}^{(\gamma )}}\big)-\kappa \big({\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big)\big({\stackrel{\leftrightarrow }{p}_{j}^{(\gamma )}}\big)\big)\big/\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})>0,\\ {} 0,\\ {} \hspace{1em}\text{if}\hspace{2.5pt}\big({\textstyle\textstyle\sum _{\gamma =1}^{\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})}}\kappa \big({\zeta _{ij}^{(\gamma )}}\big)\big({p_{ij}^{(\gamma )}}\big)-\kappa \big({\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big)\big({\stackrel{\leftrightarrow }{p}_{j}^{(\gamma )}}\big)\big)\big/\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})=0,\\ {} -\theta {[d({\textit{NL}_{ij}}(\tilde{p}),{\textit{PLBAA}_{j}})]^{\varsigma }}{w_{j}},\\ {} \hspace{1em}\text{if}\hspace{2.5pt}\big({\textstyle\textstyle\sum _{\gamma =1}^{\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})}}\kappa \big({\zeta _{ij}^{(\gamma )}}\big)\big({p_{ij}^{(\gamma )}}\big)-\kappa \big({\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big)\big({\stackrel{\leftrightarrow }{p}_{j}^{(\gamma )}}\big)\big)\big/\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})<0.\end{array}\right.\end{aligned}\]**Step 7.**Calculate the probabilistic linguistic total prospect value.

**Step 8.**Rank the value of ${\Lambda _{i}^{\ast }}$ $(i=1,2,\dots ,m)$ to obtain the best alternative.

## 4 An Example Analysis and Comparative Analysis

### 4.1 An Example Analysis

##### Table 1

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | SA | SI | A | M |

${\mathrm{\Re }_{2}}$ | I | DI | SI | I |

${\mathrm{\Re }_{3}}$ | M | I | A | DI |

${\mathrm{\Re }_{4}}$ | I | DI | I | I |

${\mathrm{\Re }_{5}}$ | M | I | M | M |

##### Table 2

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | SI | I | DA | DA |

${\mathrm{\Re }_{2}}$ | A | I | DI | M |

${\mathrm{\Re }_{3}}$ | SA | M | SA | A |

${\mathrm{\Re }_{4}}$ | DI | M | DI | M |

${\mathrm{\Re }_{5}}$ | A | A | SI | I |

##### Table 3

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | A | SI | SA | SA |

${\mathrm{\Re }_{2}}$ | I | I | SI | I |

${\mathrm{\Re }_{3}}$ | M | SI | A | DI |

${\mathrm{\Re }_{4}}$ | SI | DA | SA | I |

${\mathrm{\Re }_{5}}$ | M | DA | M | A |

##### Table 4

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | SA | I | DA | DA |

${\mathrm{\Re }_{2}}$ | A | DI | A | M |

${\mathrm{\Re }_{3}}$ | M | I | A | A |

${\mathrm{\Re }_{4}}$ | SI | DA | DI | A |

${\mathrm{\Re }_{5}}$ | M | DA | SI | I |

##### Table 5

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | SA | M | DA | DA |

${\mathrm{\Re }_{2}}$ | SA | SI | A | SI |

${\mathrm{\Re }_{3}}$ | I | I | A | SI |

${\mathrm{\Re }_{4}}$ | DI | SA | DI | A |

${\mathrm{\Re }_{5}}$ | A | DA | A | I |

##### Table 6

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{2}}$ | ${\zeta _{-2}}$ | ${\zeta _{1}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{-1}}$ | ${\zeta _{-3}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{0}}$ | ${\zeta _{-1}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-1}}$ | ${\zeta _{-3}}$ | ${\zeta _{-1}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{0}}$ | ${\zeta _{-1}}$ | ${\zeta _{0}}$ | ${\zeta _{0}}$ |

##### Table 7

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ | ${\zeta _{3}}$ | ${\zeta _{3}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{1}}$ | ${\zeta _{-1}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{2}}$ | ${\zeta _{0}}$ | ${\zeta _{2}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

##### Table 8

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ | ${\zeta _{2}}$ | ${\zeta _{2}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{-1}}$ | ${\zeta _{-1}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{0}}$ | ${\zeta _{-2}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-2}}$ | ${\zeta _{3}}$ | ${\zeta _{2}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{0}}$ | ${\zeta _{3}}$ | ${\zeta _{0}}$ | ${\zeta _{1}}$ |

##### Table 9

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{2}}$ | ${\zeta _{-1}}$ | ${\zeta _{3}}$ | ${\zeta _{3}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ | ${\zeta _{1}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{0}}$ | ${\zeta _{-1}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-2}}$ | ${\zeta _{3}}$ | ${\zeta _{-3}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{0}}$ | ${\zeta _{3}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

##### Table 10

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{2}}$ | ${\zeta _{0}}$ | ${\zeta _{3}}$ | ${\zeta _{3}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{2}}$ | ${\zeta _{-2}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{-1}}$ | ${\zeta _{-1}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-3}}$ | ${\zeta _{2}}$ | ${\zeta _{-3}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{1}}$ | ${\zeta _{3}}$ | ${\zeta _{1}}$ | ${\zeta _{-1}}$ |

**Step 1.**Transform the cost attribute ${\mathrm{\Im }_{2}}$ into a beneficial one. If the cost attribute value is ${\zeta _{\lambda }}$, then transform it into beneficial attribute value ${\zeta _{-\lambda }}$ (see Tables 11–15).

##### Table 11

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{2}}$ | ${\zeta _{2}}$ | ${\zeta _{1}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{-1}}$ | ${\zeta _{3}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{0}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-1}}$ | ${\zeta _{3}}$ | ${\zeta _{-1}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{0}}$ | ${\zeta _{1}}$ | ${\zeta _{0}}$ | ${\zeta _{0}}$ |

##### Table 12

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{-2}}$ | ${\zeta _{1}}$ | ${\zeta _{3}}$ | ${\zeta _{3}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{2}}$ | ${\zeta _{0}}$ | ${\zeta _{2}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{1}}$ | ${\zeta _{-1}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

##### Table 13

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{1}}$ | ${\zeta _{2}}$ | ${\zeta _{2}}$ | ${\zeta _{2}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{-1}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{0}}$ | ${\zeta _{2}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-2}}$ | ${\zeta _{-3}}$ | ${\zeta _{2}}$ | ${\zeta _{-1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{0}}$ | ${\zeta _{-3}}$ | ${\zeta _{0}}$ | ${\zeta _{1}}$ |

##### Table 14

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{2}}$ | ${\zeta _{1}}$ | ${\zeta _{3}}$ | ${\zeta _{3}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{1}}$ | ${\zeta _{3}}$ | ${\zeta _{1}}$ | ${\zeta _{0}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{0}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-2}}$ | ${\zeta _{-3}}$ | ${\zeta _{-3}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{0}}$ | ${\zeta _{-3}}$ | ${\zeta _{-2}}$ | ${\zeta _{-1}}$ |

**Step 2.**Shift the evaluation information with LTSs into a decision matrix $M={({\textit{PL}_{ij}}(p))_{m\times n}}$ with PLTSs (see Table 16).

##### Table 15

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | ${\zeta _{2}}$ | ${\zeta _{0}}$ | ${\zeta _{3}}$ | ${\zeta _{3}}$ |

${\mathrm{\Re }_{2}}$ | ${\zeta _{2}}$ | ${\zeta _{2}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ |

${\mathrm{\Re }_{3}}$ | ${\zeta _{-1}}$ | ${\zeta _{1}}$ | ${\zeta _{1}}$ | ${\zeta _{-2}}$ |

${\mathrm{\Re }_{4}}$ | ${\zeta _{-3}}$ | ${\zeta _{-2}}$ | ${\zeta _{-3}}$ | ${\zeta _{1}}$ |

${\mathrm{\Re }_{5}}$ | ${\zeta _{1}}$ | ${\zeta _{-3}}$ | ${\zeta _{1}}$ | ${\zeta _{-1}}$ |

**Step 3.**Normalize the decision matrix with PLTSs. Transform the decision matrix $M={({\textit{PL}_{ij}}(p))_{m\times n}}$ into the normalized decision matrix $\textit{NM}={(\textit{NPL}{i_{ij}}(\tilde{p}))}_{m\times n}$, ${\textit{NPL}_{ij}}(\tilde{p})=\big\{{\zeta _{ij}^{(\gamma )}}({\tilde{p}_{ij}^{(\gamma )}})\big|\gamma =1,2,\dots ,\mathrm{\# }{L_{ij}}(\tilde{p}),{\textstyle\sum _{\gamma =1}^{\mathrm{\# }{L_{ij}}(\tilde{p})}}{\tilde{p}_{ij}^{(\gamma )}}=1\big\}$ $(i=1,2,\dots ,m,j=1,2,\dots ,n)$ (see Table 17).

##### Table 16

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ |

${\mathrm{\Re }_{1}}$ | $\{{\zeta _{-2}}(0.2),{\zeta _{1}}(0.2),{\zeta _{2}}(0.6)\}$ | $\{{\zeta _{-2}}(0.4),{\zeta _{-1}}(0.4),{\zeta _{0}}(0.2)\}$ |

${\mathrm{\Re }_{2}}$ | $\{{\zeta _{-1}}(0.4),{\zeta _{1}}(0.4),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{-3}}(0.4),{\zeta _{-2}}(0.2),{\zeta _{-1}}(0.4)\}$ |

${\mathrm{\Re }_{3}}$ | $\{{\zeta _{-1}}(0.2),{\zeta _{0}}(0.6),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{-2}}(0.2),{\zeta _{-1}}(0.6),{\zeta _{0}}(0.2)\}$ |

${\mathrm{\Re }_{4}}$ | $\{{\zeta _{-3}}(0.4),{\zeta _{-2}}(0.4),{\zeta _{-1}}(0.2)\}$ | $\{{\zeta _{0}}(0.2),{\zeta _{2}}(0.2),{\zeta _{3}}(0.6)\}$ |

${\mathrm{\Re }_{5}}$ | $\{{\zeta _{0}}(0.0),{\zeta _{0}}(0.6),{\zeta _{1}}(0.4)\}$ | $\{{\zeta _{-1}}(0.2),{\zeta _{1}}(0.6),{\zeta _{3}}(0.2)\}$ |

Alternatives | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | $\{{\zeta _{1}}(0.2),{\zeta _{2}}(0.2),{\zeta _{3}}(0.6)\}$ | $\{{\zeta _{0}}(0.2),{\zeta _{2}}(0.2),{\zeta _{3}}(0.6)\}$ |

${\mathrm{\Re }_{2}}$ | $\{{\zeta _{-3}}(0.2),{\zeta _{-2}}(0.4),{\zeta _{1}}(0.4)\}$ | $\{{\zeta _{-2}}(0.2),{\zeta _{-1}}(0.4),{\zeta _{0}}(0.4)\}$ |

${\mathrm{\Re }_{3}}$ | $\{{\zeta _{1}}(0.0),{\zeta _{1}}(0.8),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{-3}}(0.4),{\zeta _{-2}}(0.2),{\zeta _{1}}(0.4)\}$ |

${\mathrm{\Re }_{4}}$ | $\{{\zeta _{-3}}(0.6),{\zeta _{-1}}(0.2),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{-1}}(0.4),{\zeta _{0}}(0.2),{\zeta _{1}}(0.4)\}$ |

${\mathrm{\Re }_{5}}$ | $\{{\zeta _{-2}}(0.4),{\zeta _{0}}(0.4),{\zeta _{1}}(0.2)\}$ | $\{{\zeta _{-1}}(0.6),{\zeta _{0}}(0.2),{\zeta _{1}}(0.2)\}$ |

**Step 4.**Figure out the combined weight.

##### Table 17

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ |

${\mathrm{\Re }_{1}}$ | $\{{\zeta _{-2}}(0.2),{\zeta _{1}}(0.2),{\zeta _{2}}(0.6)\}$ | $\{{\zeta _{0}}(0.2),{\zeta _{1}}(0.4),{\zeta _{2}}(0.4)\}$ |

${\mathrm{\Re }_{2}}$ | $\{{\zeta _{-1}}(0.4),{\zeta _{1}}(0.4),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{1}}(0.4),{\zeta _{2}}(0.2),{\zeta _{3}}(0.4)\}$ |

${\mathrm{\Re }_{3}}$ | $\{{\zeta _{-1}}(0.2),{\zeta _{0}}(0.6),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{0}}(0.2),{\zeta _{1}}(0.6),{\zeta _{2}}(0.2)\}$ |

${\mathrm{\Re }_{4}}$ | $\{{\zeta _{-3}}(0.4),{\zeta _{-2}}(0.4),{\zeta _{-1}}(0.2)\}$ | $\{{\zeta _{-1}}(0.6),{\zeta _{1}}(0.2),{\zeta _{2}}(0.2)\}$ |

${\mathrm{\Re }_{5}}$ | $\{{\zeta _{0}}(0.0),{\zeta _{0}}(0.6),{\zeta _{1}}(0.4)\}$ | $\{{\zeta _{-3}}(0.2),{\zeta _{-1}}(0.6),{\zeta _{1}}(0.2)\}$ |

Alternatives | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | $\{{\zeta _{1}}(0.2),{\zeta _{2}}(0.2),{\zeta _{3}}(0.6)\}$ | $\{{\zeta _{0}}(0.2),{\zeta _{2}}(0.2),{\zeta _{3}}(0.6)\}$ |

${\mathrm{\Re }_{2}}$ | $\{{\zeta _{-3}}(0.2),{\zeta _{-2}}(0.4),{\zeta _{1}}(0.4)\}$ | $\{{\zeta _{-2}}(0.2),{\zeta _{-1}}(0.4),{\zeta _{0}}(0.4)\}$ |

${\mathrm{\Re }_{3}}$ | $\{{\zeta _{1}}(0.0),{\zeta _{1}}(0.8),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{-3}}(0.4),{\zeta _{-2}}(0.2),{\zeta _{1}}(0.4)\}$ |

${\mathrm{\Re }_{4}}$ | $\{{\zeta _{-3}}(0.6),{\zeta _{-1}}(0.2),{\zeta _{2}}(0.2)\}$ | $\{{\zeta _{-1}}(0.6),{\zeta _{1}}(0.2),{\zeta _{2}}(0.2)\}$ |

${\mathrm{\Re }_{5}}$ | $\{{\zeta _{-2}}(0.4),{\zeta _{0}}(0.4),{\zeta _{1}}(0.2)\}$ | $\{{\zeta _{-1}}(0.6),{\zeta _{0}}(0.2),{\zeta _{1}}(0.2)\}$ |

- (3) The objective weight of the
*j*th attribute is computed by Eq. (12), and the results are as follows: ${w_{o1}}=0.515$, ${w_{o2}}=0.136$, ${w_{o3}}=0.215$, ${w_{o4}}=0.134$.

##### Table 18

The mean value | |

${_{{\mathrm{\Im }_{1}}}}$ | $\{{\zeta _{0.2700}}(0.2400),{\zeta _{0.5000}}(0.5200),{\zeta _{0.7000}}(0.2400)\}$ |

${_{{\mathrm{\Im }_{2}}}}$ | $\{{\zeta _{0.3300}}(0.3200),{\zeta _{0.5300}}(0.4000),{\zeta _{0.7700}}(0.2800)\}$ |

${_{{\mathrm{\Im }_{3}}}}$ | $\{{\zeta _{0.3000}}(0.2800),{\zeta _{0.5000}}(0.4000),{\zeta _{0.8000}}(0.3200)\}$ |

${_{{\mathrm{\Im }_{4}}}}$ | $\{{\zeta _{0.2700}}(0.4000),{\zeta _{0.5000}}(0.2400),{\zeta _{0.7300}}(0.3600)\}$ |

##### Table 19

Alternatives | ${_{{\mathrm{\Im }_{1}}}}$ | ${_{{\mathrm{\Im }_{2}}}}$ | ${_{{\mathrm{\Im }_{3}}}}$ | ${_{{\mathrm{\Im }_{4}}}}$ |

${\mathrm{\wp }_{j}}$ | 0.8689 | 0.9653 | 0.9453 | 0.9657 |

**Step 5.**According to Eqs. (14)–(16), the $\textit{PLBAA}={({\textit{PLBAA}_{j}})_{1\times 4}}$ for all attributes can be obtained (see Table 20).

**Step 6.**The Hamming distance can be calculated by using Eq. (17) and the cumulative prospect Hamming distance can be calculated by using Eq. (18) (see Tables 21–22).

##### Table 20

PLBAA | |

${_{{\mathrm{\Im }_{1}}}}$ | $\{{\zeta _{0.0000}}(0.0000),{\zeta _{0.4503}}(0.5102),{\zeta _{0.6635}}(0.2297)\}$ |

${_{{\mathrm{\Im }_{2}}}}$ | $\{{\zeta _{0.0000}}(0.2862),{\zeta _{0.4599}}(0.3565),{\zeta _{0.7463}}(0.2639)\}$ |

${_{{\mathrm{\Im }_{3}}}}$ | $\{{\zeta _{0.0000}}(0.0000),{\zeta _{0.4342}}(0.3482),{\zeta _{0.7905}}(0.2862)\}$ |

${_{{\mathrm{\Im }_{4}}}}$ | $\{{\zeta _{0.0000}}(0.3565),{\zeta _{0.4342}}(0.2297),{\zeta _{0.7137}}(0.3288)\}$ |

##### Table 21

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | 0.0726 | 0.1130 | 0.2779 | 0.1774 |

${\mathrm{\Re }_{2}}$ | 0.0615 | 0.1575 | −0.0417 | 0.0338 |

${\mathrm{\Re }_{3}}$ | 0.0504 | 0.1221 | 0.1472 | −0.0328 |

${\mathrm{\Re }_{4}}$ | −0.0830 | −0.0759 | −0.0480 | 0.1005 |

${\mathrm{\Re }_{5}}$ | 0.0615 | −0.0332 | 0.0695 | 0.1005 |

**Step 7.**Figure up the probabilistic linguistic total prospect value, which is computed by using Eq. (19) (see Table 23).

##### Table 22

Alternatives | ${\mathrm{\Im }_{1}}$ | ${\mathrm{\Im }_{2}}$ | ${\mathrm{\Im }_{3}}$ | ${\mathrm{\Im }_{4}}$ |

${\mathrm{\Re }_{1}}$ | 0.0236 | 0.0419 | 0.0724 | 0.0307 |

${\mathrm{\Re }_{2}}$ | 0.0204 | 0.0561 | −0.0462 | 0.0072 |

${\mathrm{\Re }_{3}}$ | 0.0171 | 0.0448 | 0.0624 | −0.0157 |

${\mathrm{\Re }_{4}}$ | −0.0596 | −0.0664 | −0.0524 | 0.0186 |

${\mathrm{\Re }_{5}}$ | 0.0204 | −0.0321 | 0.0322 | 0.0186 |

**Step 8.**According to the above calculation, the rank of alternatives is ${\mathrm{\Re }_{1}}>{\mathrm{\Re }_{3}}>{\mathrm{\Re }_{5}}>{\mathrm{\Re }_{2}}>{\mathrm{\Re }_{4}}$, and the optimal location is ${N_{1}}$.

### 4.2 Comparative Analysis

*et al.*, 2016), the PL-TOPSIS method (Pang

*et al.*, 2016) and the PL-GRA method (Liang

*et al.*, 2018) (let $\rho =0.5$) (see Table 24).

##### Table 24

Methods | Order | Optimal alternative | Bad alternative |

PLWA operator (Pang et al., 2016) | ${\mathrm{\Re }_{1}}>{\mathrm{\Re }_{3}}>{\mathrm{\Re }_{2}}>{\mathrm{\Re }_{5}}>{\mathrm{\Re }_{4}}$ | ${\mathrm{\Re }_{1}}$ | ${\mathrm{\Re }_{4}}$ |

PL-TOPSIS method (Pang et al., 2016) | ${\mathrm{\Re }_{1}}>{\mathrm{\Re }_{3}}>{\mathrm{\Re }_{2}}>{\mathrm{\Re }_{5}}>{\mathrm{\Re }_{4}}$ | ${\mathrm{\Re }_{1}}$ | ${\mathrm{\Re }_{4}}$ |

PL-GRA method (Liang et al., 2018) | ${\mathrm{\Re }_{1}}>{\mathrm{\Re }_{2}}>{\mathrm{\Re }_{3}}>{\mathrm{\Re }_{5}}>{\mathrm{\Re }_{4}}$ | ${\mathrm{\Re }_{1}}$ | ${\mathrm{\Re }_{4}}$ |

PL-MABAC | ${\mathrm{\Re }_{1}}>{\mathrm{\Re }_{2}}>{\mathrm{\Re }_{3}}>{\mathrm{\Re }_{5}}>{\mathrm{\Re }_{4}}$ | ${\mathrm{\Re }_{1}}$ | ${\mathrm{\Re }_{4}}$ |

CPT-PL-MABAC method | ${\mathrm{\Re }_{1}}>{\mathrm{\Re }_{3}}>{\mathrm{\Re }_{5}}>{\mathrm{\Re }_{2}}>{\mathrm{\Re }_{4}}$ | ${\mathrm{\Re }_{1}}$ | ${\mathrm{\Re }_{4}}$ |