Combinatorial design of the MAUT and PAMSSEM II methods for multiple attributes group decision making with probabilistic linguistic information

This paper considers the application of Probabilistic Linguistic Term Sets (PLTS) in Multiple Attribute Group Decision Making (MAGDM) when the weights cannot be determined. Firstly, as an improvement of PROMETHEE, the PAMSSEM method has substantial advantages in handling mixed data and missing data. It can not only make use of the preference threshold as well as the irrelevant threshold to express the preference degree of the decision-maker (DM) but likewise express the compensation limit that DM is willing to accept through the rejection threshold. Simultaneously, the MAUT method cleverly employs the marginal utility function to redistribute the evaluation values under the same attribute within a fixed interval, making the decision matrix standard and reliable. To comprehensively utilize the advantages of these two methods, we integrated the PAMSSEM II and MAUT methods in the probabilistic linguistic environment to form the PL-MAUT-PAMSSEM II method followed by applying it to the problem of inbound tourism destination selection. Secondly, in the process of data processing, an equivalent conversion function will be used to convert the PLTS and the probabilistic hesitant fuzzy set (HPFS) to each other to make the calculation between the matrices feasible and use the entropy method to calculate the weights to ensure the decision-making process. Finally, taking the sample deviation value as the inspection standards, the big data analysis compared with other methods is carried out through a large number of sample examples to scientifically validate the superiority of the newly proposed method.


Introduction
Decision-making activities are pervasive in all aspects of our lives. The decision-making process includes asking questions, determining alternatives, choosing alternatives, and making decisions (Xu and Zhang 2021). Due to the diversity of decision-making problems and the ambiguity of human thinking, it is not easy for us to use specific numbers to accurately describe the value of each alternative under different standards when making assessments (Saleem and Noor Ul 2014). Therefore, obtaining accurate assessments requires a great price. In the MAGDM problem, the decision-makers (DMs) are required to provide the evaluation value of different alternatives under multiple attributes. Because it is difficult to provide accurate numbers to express preferences, linguistic terms such as ''medium'' or ''good'' are used as evaluation information (Lin et al. 2019;He et al. 2016;Qu et al. 2018;Xi et al. 2018Xi et al. , 2019Bai et al. 2017;Sun et al. 2019). Therefore, more and more scholars are aware of the importance of probability information in decision-making compared with digital information and devote themselves to related research  Based on the above situation, Zadeh (1975) introduced the Linguistic Term Set (LTS), which expresses DM's preferences through one or more linguistic information and uses fuzzy linguistics to model qualitative evaluation information. Since then, scholars have carried out further research based on LTS, including linguistic cloud information (Herrera et al. 2000), two-tuple linguistic model (Xu 2004), uncertain linguistic structure (Herrera et al. 2008), unbalanced linguistic information (Xu 2008), and unbalanced uncertain linguistic information (Wang et al. 2014).
With the enrichment of human practical activities, the difficulty of decision-making is getting bigger, and some models that have been proposed cannot completely and accurately express the preferences of DMs in the actual decision-making problem (Wang et al. 2018a). For example, when a DM evaluates an inbound destination, he cannot be 100% sure that the scheme is ''good,'' or he may think it is ''medium,'' ''good'' or ''very good.'' To help DMs make more reasonable decisions, Rodriguez et al. (2012) proposed the concept of hesitating fuzzy linguistic term set (HFLTS), and Liao et al. (2015) introduced the mathematical form of HFLTS, which allows the DMs to hesitate between several linguistic terms.
However, HFLTS believes that different linguistic terms have the same importance or weight, which is likely to be inconsistent with the true wishes of the DMs. For example, suppose there are 10 DMs evaluating alternatives for inbound tourist attractions, one DM may think it is ''good,'' and the other 9 experts may think it is ''medium.'' Then the evaluation information represented by HFLTS is good; medium f g . The result that does not consider the importance or weight of different linguistic terms may misrepresent the preferences of DMs. To avoid the above problems, Pang et al. (2016) introduced the probabilistic linguistic term set (PLTS) by adding the corresponding probability to the HFLTS, which reduces the information loss in the evaluation process and increases the applicability and accuracy of DMs in expressing its language evaluation. Use PLTS to indicate that the result of the above example is good(0.1), medium(0:9Þ f g . PLTS can reflect the true weights of different attributes corresponding to the alternative and is suitable for group decision-making problems with distributed information (Wang et al. 2018b). Therefore, it is necessary to study PLTS.
The rationality and flexibility of PLTS have attracted the attention of more and more scholars, and scholars have studied PLTS from different angles Mi et al. 2020), including probabilistic linguistic vector item sets promote the application of multi-granular linguistic information (Zhai et al. 2016), MADM problem , the operation of PLTS, the distance measurement of PLTS (Lin et al. 2019;Wang et al. 2018;Lin and Xu 2018;Zhang et al. 2020), the probability formula of PLTS (Bai et al. 2017;Feng et al. 2019;Yu et al. 2019aYu et al. , 2019bXian et al. 2019), probabilistic linguistic preference relations (Gao et al. 2018;Song and Jun 2019;Tian et al. 2019;Zhang et al. 2016Zhang et al. , 2018, etc. At the same time, we also explored to expand the application field of PLTS. Computational intelligence such as the Internet-of-Things (IoT), Cloud computing, and Fog computing have gained a lot of attention from academia over the past few years. Cloud computing, the most widely known field, aims to develop utility programs and provide Internet services (De Donno et al. 2019). With the proliferation of smart devices and appliances, IoT opens up new research perspectives and points out the limitations of Cloud computing. In view of this, Edge computing came into being, the idea is to provide the power of the Cloud at the Edge of the network to solve new challenges that Cloud computing alone cannot handle. In this Edge computing rule, Fog computing stands out, which represents the highest evolution of Edge computing. Fog computing acts as the glue between IoT and Cloud computing by distributing resources horizontally and vertically along the Cloud-to-Thing continuum, assisting and enhancing Cloud-Thing interactions. Therefore, we briefly discussed how to apply PLTS and PL-MAUT-PAMSSEM II for data processing in the information transfer process of IoT, Fog computing and Cloud computing (Shi and Dustdar 2016;Liu et al. 2017;Vaquero and Rodero-Merino 2014).
Many scholars have studied the multi-attribute decisionmaking method and applied it to the probabilistic linguistic environment to solve practical problems. The proposed method has the advantages of simple calculation process and reasonable calculation results, but its ability to deal with missing evaluation needs to be improved, and PL-MAUT-PAMSSEM II can make up for this deficiency. Furthermore, some scholars combine subjective and objective weights to determine criterion weights (Hafezalkotob and Hafezalkotob 2015), however, this weight determination technique has two drawbacks (Wu et al. 2018): (1) they are difficult to express all subjective preferences of DMs in a single linguistic term, and they cannot integrate all evaluations. (2) There are relatively few studies on the determination of the objective weight of attributes in the probabilistic linguistic environment. In order to maintain objectivity, this paper adopts the entropy method to determine the weights when studying the application of MAUT-PAMSSEM II in PLTS. Finally, taking the sample deviation value as the new evaluation criterion, this new method is compared with PL-MAUT, PL-PAMSSEM II, PL-ARAS, PL-PROMETHEE II, PL-TOPSIS, PL-EDAS and PL-MABAC through big data analysis to show its necessity.
In summary, the focus of this article includes the following points: 1. When the attribute weights are unknown, we offer a decision-making method referred to as the PL-MAUT-PAMSSEM II method. It combines MAUT and PAMSSEM II in a probabilistic linguistic setting, providing DM with more method choices while also improving the accuracy of the assessment by taking into account the different importance of linguistic terms. 2. The PL-MAUT-PAMSSEM II method combines PLTS and HPFS using an equivalent transfer function to make calculation between matrices feasible and utilizes the entropy method to calculate the weights to ensure the objectivity of the decision-making process. 3. Apply the newly designed method to solve the problem of inbound tourism destination selection, and use the sample deviation as the new criterion to carry out big data analysis in comparison to other methods through a large number of sample examples in order to obtain more accurate and objective scientific results.
The rest of this article is as follows: Sect. 2 reviews some basic knowledge of PLTS and HPFS. Section 3 introduces the MAUT method and PAMSSEM method, respectively, and includes the entire calculation process of PL-MAUT-PAMSSEM II. Section 4 introduces the entropy method and the whole process of determining attribute weights. Section 5 presents a large sample case analysis of an inbound tourist destination and compares several methods to demonstrate the superiority of the proposed PL-MAUT-PAMSSEM II method. Section 6 is the conclusion of this article.

Preliminaries
In this section, we will give some basic concepts and formulas of PLTS, HPFS, and the equivalent transfer functions.

The probabilistic linguistic term set
Definition 1 (Pang et al. 2016) The probabilistic linguistic term set (PLTS) on LTS L can be defined as: 1gwhere l ðkÞ ðp ðkÞ Þ is made up of linguistic term l ðkÞ and its associated probability p ðkÞ , and the cardinality of PLðpÞ is #PLðpÞ. If k ¼ p ðkÞ ¼ 1, then the PLTS reduces to a linguistic term.
The size of PLTS can be compared by the following scoring function and variance function: is the membership degree of the linguistic term l k ð Þ in PLTS PLðpÞ, then the score function of PLðpÞ is defined as: Definition 3 Suppose that r l k ð Þ À Á is the membership degree of the linguistic term l k ð Þ in PLTS PLðpÞ, and SðPLðpÞÞ is the score value of the PLTS PLðpÞ.Then the variance function of PLðpÞ is defined as:

The hesitant probabilistic fuzzy set
Definition 4 (Torra 2010; Torra and Narukawa 2009) Let X be a reference set. Accordingly, an HFS A on X is defined in terms of a function h A x ð Þ that when applied to X returns a finite subset of [0, 1]: where h A x ð Þ is a set of multiple different values on [0, 1], which represents the membership degree of the element x in X on A, and h A x ð Þ represents HFE (Xia and Xu 2011).
Example 1 Let X¼ x 1 ; x 2 f g be a reference set, and h A x 1 ð Þ ¼ 0:1; 0:3 f g and h A x 2 ð Þ ¼ 0:2; 0:3; 0:5 f g be two HFEs of x i ¼ i ¼ 1; 2 ð Þ to a set A. Then, A can be expressed as an HFS as follows: A ¼ x 1 ; 0:1; 0:3 f g h i ; x 2 ; 0:2; 0:3; 0:5 f g h i f g Definition 5 (Xu and Zhou 2016) Let X be a reference set, then an HPFS A p on X can be expressed by as: Combinatorial design of the MAUT and PAMSSEM II methods for multiple attributes group decision… 2095 where hðp x Þ is a set of multiple different values on [0, 1], which described by a probability distribution p x . Among them, p x represents the membership degree of the elements x on the set X on A p . hðp x Þ can be expressed as an HPFE as follows: Among them, p i represents probability of the possible value c i , and it satisfies P k i¼1 p i ¼ 1, k represents the number of c i ðp i Þ in hðp x Þ.
gbe an HPFE. Its variance can be defined as follows: Definition 8 (Xu and Zhou 2016) Let h i ði ¼ 1; 2; . . .; IÞ be a collection of HPFS, x ¼ ðx 1 ; x 2 ; . . .; x i Þ T be the weight vector of h i with x i 2 ½0; 1 and P I i¼1 x i ¼ 1, p i be the probability of c i in the HPFE h i , c rðiÞ be the i th largest of h i , p rðiÞ be the probability of c rðiÞ in the HPFE h i , and x rðiÞ be the i th largest of x, then the aggregation operator can be defined as follows:

The equivalent transformation functions
Definition 9 (Gou and Xu 2016; Zhang and Qi 2013) Let h L ¼ l t t 2 Àd; d ½ j f g be a HFLE on LTS L = l t t ¼ Àd; . . .; À1; 0; 1; . . .; d j f g , and H ¼ e e 2 0; 1 ½ j f g be a HFS. Then the linguistic variable l t and membership degree e can be transformed into each other by the function r and r À1 given as: (2) PL 1 ðpÞ PL 2 ðpÞ ¼ S  (4) PL 1 ðpÞøPL 2 ðpÞ ¼ S 3 Combinatorial of the MAUT and PAMSSEM II methods

The PAMSSEM method
The PAMSSEM method was introduced by Martel et al (2008) and applied to cloud resource integration management (Yazir et al. 2012;Góreck D 2012). The PAMSSEM method includes phase I and phase II. The first stage gives a partial ranking of the alternatives by calculating the entering and leaving flow, and the second stage gives the full ranking of the alternatives by calculating the net flow.
The purpose of PAMSSEM is to establish preference relationships (Bélanger et al. 2000a;Bélanger et al. 2000b;Bélanger and Martel 2012). After the DMs are given the decision matrixes, the indifference threshold (q), the preference threshold (p), and the veto threshold (v), the alternatives are ranked by establishing a local preference model (Alinezhad and Khalili 2019). The PROMETHEE method is proposed to deal with incomparable situations when comparing targets, and the method is simple, clear, and stable (Brans et al. 1986). As an improvement of this method, PAMSSEM also has the above advantages. In addition, the advantages of the PAMSSEM method include the ability to handle missing evaluations. The discrimination threshold allows A i to be better than A i 0 while A i 0 is better than A i , and the rejection threshold prevents A i 0 from being better than A i . Among them, when the threshold value is biased, it may have an impact on the ranking results, but this does not conflict with the stability of the method. Changes in parameters will inevitably lead to a change in the ranking, which just shows that the method considers the particularity of each scheme.
In mathematics, the probability density function of a continuous random variable is a function that describes the probability that the output value of this random variable is near a certain value point. The probability that the value of a random variable falls within a certain region is integral to the probability density function in this region. The decision matrix given by DMs represents the distribution of random variables (or statistical variables) X ij with probability density or quality function f ij ðx ij Þ. A clear assessment can be seen as the degradation of the distribution, the distribution is reduced to only a point ðpðX ij ¼ e ij Þ ¼ 1Þ, and the probability distribution function is as follows (Guitouni et al. 1999):

The MAUT method
The Muti-Attribute Utility Theory (MAUT) method was introduced by Keeney and Raiffa in 1976 (Emovon et al. 2016). The method assumes that each DM tries to optimize, either consciously or implicitly, an aggregate function that integrates the opinions of all DMs. This function may be unknown before the decision starts, so DMs need to construct it first (Nemery and Ishizaka 2013; Wallenius et al. 2008a, b). The utility function in economics is used to express the quantitative relationship between the utility obtained by consumers in consumption and the combination of commodities consumed and is used to measure the degree of satisfaction that consumers obtain from consuming a given combination of commodities. Compared with the previous method, this method is more convenient to solve the MAGDM problems, which is one of its advantages. Furthermore, this method gives DMs more initiative, making the evaluation results more real and accurate, and more reflective of the DM's wishes.

The PL-MAUT-PAMSSEM II method
Based on the advantages of the MAUT method and the PAMSSEM method proposed above, we combine these two methods and apply them to PLTS. PAMSSEM is a method to rank schemes based on preference. According to the 2001-2019 inbound tourism data provided by the official website of the Bureau of Statistics, we invited multiple DMs to evaluate multiple popular inbound tourism destinations and asked the DMs to set specific thresholds p, q, and v. Some attribute values are removed by thresholds p, q, v, and the attribute values closest to the threshold p can reflect the preferences of DMs. MAUT is a method for ranking alternatives based on a utility function. Utility functions are also tools to reflect the preferences or intentions of DMs. Therefore, this paper combines the two methods to optimize the decision-making process, make the decision-making results more real and accurate, and provide decision-makers with better travel plans. The decision-making processes based on PL-MAUT-PAMSSEM II method are as follows: For a MAGDM problem in PLTS, let A ¼ fA 1 ; A 2 ; . . .; A m g be a finite set of alternatives, C ¼ fC 1 ; C 2 ; . . .; C n g be the set of attributes and x¼ x 1 ; x 2 ; . . .; x n ð Þ T be the weight vector of attributes C j ðj ¼ 1; 2; . . .; nÞ, with x j 2 ½0; 1, j ¼ 1; 2; . . .; n and P n j¼1 ij Þ t ¼ 1; 2; . . .; #LSðpÞ ij n o is a PLTS, which is an evaluation value of alternative A i about attribute C j . Then the goal is to rank the alternatives.
Step 2 Calculate the decision matrixes R ¼ ½LSðPÞ ij mÂn after aggregation by formulas (8) and (9) Step 3 Normalized the decision matrixes Step 4 Use formula (1) to get the score matrix of the alternatives under different attributes S: To calculate the utility function, the DM needs to first standardize the score value matrix to readjust the original performance between 0 and 1. The rescaling or normalization step is usually based on the minimum and maximum performance of the alternatives on each attribute: When maximizing the attribute, or When minimizing the attribute.
Step 5 Calculate the weight of each attribute x j by formula (28)-(32).
Step 6 Calculate the marginal utility score as follows: Step 7 Calculate the local outranking index d j ðA i ; A i 0 Þ by the following expression: f j A i 0 ð Þ and f j A i ð Þ are, respectively, the probability density functions (discrete) which are assumed to be equal one. d j ðA i ; A i 0 Þ is index computed according to the following formula: where D j ¼ u ij À u i 0 j , which indicates the difference between the level, and threshold q, p is the values determined by the DMs for each attribute. When the number of ordinal levels is greater than 3, after modification, we can obtain a function equivalent to the ordinal evaluation local ranking index: Step 8 Calculate the concordance index CðA i ; A i 0 Þ for each pair of alternatives as follows: where x j represent the weight of each attribute and P n j¼1 x j ¼ 1 Step 9 Calculate the local discordance index DðA i ; A i 0 Þ as follows: f j A i 0 ð Þ and f j A i ð Þ are, respectively, the probability density functions (discrete) which are both assumed to be equal to 1. D j ðA i ; A i 0 Þ is index computed according to the following formula: where the threshold v is determined by DMs. In addition, D j ðA i ; A i 0 Þ for the cardinal attributes is determined by following formula: where a j is the number of measurement scale levels of the j-th attribute (a j [ 3) and nðx j Þ is a non-decreasing function of the relative importance of the j-th criterion. This function can be calculated by following formula: Step 10 Calculate the outranking degree uðA i ; A i 0 Þ as follows: Step 11 Calculate the entering and leaving flows of each alternative: Step 12 Calculate the net flow of each alternative: Step 13 Rank all alternatives according to the value of uðA i Þ.

The entropy method
Definition 11 (He et al. 2016;Yang and Cui 2011;Wang et al. 2015;Zou et al. 2006;Xu et al. 2021) The entropy method is an objective method to calculate the weight. It determines the weight of each attribute according to the observed value of each attribute. This paper analyzes the big data of inbound tourism destination selection through the weight and decision information obtained by objective calculation and gives the deviation values of all algorithms mentioned in this paper. According to m alternatives and n attributes, the DMs give the decision matrixes, and the steps of the entropy method to determine the weight are as follows: Step 1 Calculate the proportion of the i-th alternative under the j-th attribute as follows: . . .; m; j ¼ 1; 2; . . .; n ð Þ X m i¼1 q ij ¼ 1: Step 2 Calculate the entropy value of the j-th index by following formula: where k [ 0, ln is the natural logarithm, and e ij ! 0.Generally let k ¼ 1 ln m , then 0 e 1. When q ij ¼ 0, define lim q ij !0 ðq ij ln q ij Þ ¼ 0.
Step 3 Calculate the difference coefficient of the j-th index by following formula: Step 4 Calculate the weights by following formula: Step 5 The final weight vector X is:

Case study: choice of inbound tourism destinations
In this section, we will use the PL-MAUT-PAMSSEM II method proposed above to conduct a purpose-competitive evaluation of the alternatives to provide decision-makers with a choice reference. Among them, the single sample calculation example aims to show the specific operation process of the method, and the inspection standards still maintain the original simple sorting criteria. However, high accuracy inspection standards are needed in the analysis of large sample computing study. Therefore, this paper designs a new sample deviation value evaluation criterion to make the research results more objective and accurate.

Data sources and processing
After long-term and continuous research on the influencing factors of regional inbound tourism development, a relatively complete attribute index system has been formed (Zhao 2008;Wu et al. 2020). This paper selects the economic level, degree of openness, hotel supply, transportation facilities, and tourism resources as the attribute indicators of the competitiveness evaluation of inbound tourism destinations, the number of inbound tourists is selected as the actual reference value for the competitiveness evaluation of inbound tourism destinations. Among them, the economic level is measured by GDP, the degree of openness is measured by import and export volume, and the number of inbound tourists is measured by the number of inbound overnight tourists. Through the comprehensive calculation of one-star, two-star, three-star, four-star and five-star hotels, the hotel supply is given 1, 3, 5, 7, and 9 points respectively. Transportation facilities are given 1, 3, 5, and 7 points, respectively, according to the comprehensive measurement of inland river network density, highway network density, railway network density and air transportation employment density. Tourism resources are given 1, 2, 3, 7, 9, and 11 points, respectively, through the comprehensive calculation of 1A scenic spot, 2A scenic spot, 3A scenic spot, 4A scenic spot, 5A scenic spot (including 5A scenic spot, historical and cultural name, world natural heritage and world cultural heritage) and world double heritage. The data required for the article comes China's inbound tourism has experienced an everchanging historical period, and external factors closely related to it are also changing one after another. The data sources of economic level and openness are measured separately by GDP and import and export volume, but the historical data of the two variable indicators of GDP (calculated at the price of the current year) and import and export volume are not comparable. To ensure the comparability of the historical data of the research attribute indicators, we need to eliminate the direct price effect caused by the exchange rate of RMB against the US dollar and in the domestic consumption level according to the corresponding scientific method (Wu and Zhang 2013). The main reason is that the import and export volume is officially counted in US dollars. Official statistics will be affected by the direct price effect of changes in the exchange rate of RMB against the US dollar and changes in domestic consumption levels. Furthermore, official statistics of GDP (calculated at the price of the year) in RMB will be affected by direct price effects caused by changes in domestic consumption levels.

Example procedure for a single sample
According to the data from 2001 to 2019 after the above processing, we invited 10 DMs to score the five tourist destinations A1 (Guangdong), A2 (Yunnan), A3 (Shaanxi), A4 (Sichuan) and A5 (Inner Mongolia) under the five standards of C1 (economic level), C2 (openness), C3 (hotel supply), C4 (transportation facilities), and C5 (Tourism Resources), then we can obtain decision-making information and form probabilistic linguistic decision matrixes ðPL ij ðpÞÞ 5Â5 . The weight vector of criteria is To obtain the decision matrixes, we collected the scores of 10 DMs in 2019, 2015, 2011, and 2007 and converted them to PLTS through the language scale. We then list the final evaluation results of these tourist destinations in Tables 1,  2, 3, and 4.

The decision steps
Step 1 Split the matrixes, take attribute C 1 as an example, and we can get Table 5.
Step 2 Calculate the hesitant probabilistic fuzzy decision matrixes by formula (9), take attribute C 1 as an example, we can get Table 6.
Step 3 Calculate the aggregated hesitant probabilistic fuzzy decision matrix by formula (8), and we can get Table 7.  Step 4 Calculate the probabilistic linguistic decision matrix by formula (9), and we can get Table 8.

Example results for a large number of samples
To verify the accuracy of the newly proposed method in competitiveness evaluation, we will extend the above examples. First, all provinces in China (A total of 31 provinces except Hong Kong, Macao and Taiwan) are scored under five criteria: C1 (economic level), C2 (openness), C3 (hotel supply), C4 (transportation facilities), and C5 (tourism resources). In order to obtain an objective decision matrix, we calculated the ranking of 31 provinces under the above five indicators from 2001 to 2019 according to the collected actual data, and scored the 31 provinces through the equal difference sequence, and the scoring results were just from l 3 f g to l À3 f g. Then, we   ( take four years of data as a group for calculation, including one-year interval (2019,2017,2015,2013), 2 years interval (2019,2016,2013,2010), 3 years interval (2019,2015,2011,2007), 4 years interval (2019,2014,2009,2004) and 5 years interval (2019,2013,2007,2001). Finally, to calculate the competitiveness score value of any five provinces under the above five combinations, we will use Java (computer programming) to perform big data analysis on these 849,555 ¼ 5 Â C 5 31 À Á data samples. First, calculate the average of the inbound tourist numbers in any five provinces in the 31 provinces under the five combinations, normalize it to x t1 ðt ¼ 1; 2; . . .; 31Þ and then normalize the competitiveness scores of the five provinces after sorting. Converted to y t2 ðt ¼ 1; 2; . . .; 5Þ, and finally, the advantages and disadvantages of the PL-MAUT-PAMSSEM II method are compared by calculating the sample deviation value x t 1 À y t 2 j j ð t 1 ¼ 1; 2; . . .; 31; t 2 ¼ 1; 2; . . .; 5Þ, and the calculation results are listed in Table 12.
(Due to the large sample size of the data, the differences between samples are small, but these differences cannot be ignored, so we will modify the values of p, q, v and present this operation in Table 10.) The same parameter values and weights are applied to the PL-MAUT (Wallenius et al. 2008a, b), PL-PAMSSEM II (Martel et al. 2008), PL-ARAS (Zavadskas and Turskis 2010), PL-PROMETHEE II (Liu and Li 2018), PL-TOP-SIS (Lin et al. 2021), PL-EDAS (He et al. 2021) and PL-MABAC (Wei et al. 2020) methods, respectively, but each method obtains different output values. Therefore, when different decision methods are applied to MAGDM problems, they will always get different calculation results. We respectively present the calculation results of a single sample and a large number of sample examples in Table 11 and Table 12. Comparing the sample deviation values of various methods after big data analysis, we can find that the error between the calculation results of PL-MAUT-PAMSSEM II method and the actual situation is the smallest and most accurate. Therefore, the comparison of various methods can better highlight the objectivity, rationality, and superiority of the PL-MAUT-PAMSSEM II method proposed in this paper in solving practical problems. Meanwhile, by calculating 169,911 data samples under each year combination, we get the running times of all algorithms and present them in Table 13. The running time of the new method proposed in this paper is not optimal or the worst, but the gap with the shortest running time can always be controlled within one second, which means that the time gap of each sample data is within 1/169911 s. The new method not only has a small gap in running time, but also has the highest computational accuracy. Therefore, the PL-MAUT-PAMSSEM II method, which comprehensively considers the calculation accuracy and running time, has great advantages in practical decision-making. In addition, the method can be used for big data analysis in the tourism field, and it can be applied to other fields with the same effect. The perception layer of the IoT obtains a large number of preference data, effectively eliminates abnormal data and missing data by the preference threshold, calculates the objective weight Table 11 Ranking of eight methods in a single sample study

Methods
Ranking according to the entropy method, and integrates various influence factors through Fog computing, which can improve the objectivity of decision-making. After the network layer transmission, the preference value is characterized by the utility function, and the effective data are processed by high-performance Cloud computing, and finally transformed into useful information for end users, which is conducive to the effective evaluation of warehouse management points or transportation points. Compared with the traditional evaluation process, the mixed application of PL-MAUT-PAMSSEM II method, Fog computing and Cloud computing can improve the stability and accuracy of the calculation results.

Conclusions
The basic information of the destination is the information content that potential tourists in China need most. Since the choices, as well as preferences of inbound tourists' travel destinations, are continuously evolving, grasping the preferences and changing trends is critical for each region to formulate the development and marketing strategies of the source market. At present, there are still gaps in the research on the choice of inbound tourism destinations. Some scholars focus on the research on tourist behavior. However, individual behavior has great uncertainty, which is not equal to group behavior and macroscopic laws. In addition, traditional evaluation methods are basically carried out in the real-number environment, making it impossible to quantify the degree to which these factors influence tourists' choice of tourism locations. At the same time, considering that people are more accustomed to using qualitative information when it comes to group decisionmaking to expressing preferences, PLTS can solve this problem very well. PLTS consists of several possible linguistic terms and their relative distributions and is an important means of expressing DM's wishes or evaluations in a linguistic environment. PLTS is suitable for expressing qualitative evaluation information in the actual decision-making process, but owing to its complex structure, PLTS is limited in the calculation, which HPFS can compensate for. Therefore, we combined PLTS and HPFS with the help of equivalent transfer functions in the stage of constructing a decision matrix and aggregating decision information. Both the PAMSSEM method and the MAUT method have their advantages. The PAMSSEM method offers considerable advantages in handling mixed data (cardinal and ordinal evaluation values) and missing data (irrelevant evaluation values or relevant but difficult to obtain evaluation values), not only calculating the consistency index of the scheme but also presenting the rejection threshold for calculating the inconsistency index of the scheme, which improves the overall consistency. Since the MAUT method uses the marginal utility function to redistribute the attribute values of the alternatives in the interval, the decision matrix after this processing is more conducive to expressing the satisfaction of the DM. However, when the two methods are, respectively, applied to the calculation of inbound tourism destination selection, the calculation results show that their stability is relatively weak, with a large gap with the expected ranking target. In order to take better advantage of these two methods, we consider a combined design of MAUT and PAMSSEM II methods in a probabilistic linguistic setting.
In this paper, the PL-MAUT-PAMSSEM II method normalizes the evaluation values and calculates the marginal utility score from the marginal utility function. The marginal utility scores of the alternatives are then compared in pairs based on different criteria. Finally, by establishing the priority relationship, constructing the local preference model, and calculating the consistency index and non-consistency index under the pre-set threshold, and calculating the net advantage flow, the score value of the scheme is obtained. In the big data analysis, using the sample deviation value as the inspection standards, a large number of sample examples were used to carry out a scientific comparison with other methods, and it is found that compared with other methods, the PL-MAUT-PAMSSEM II method has the smallest error between the competitiveness evaluation and the actual situation, and the result is the most accurate. Although the running time of the new method proposed in this paper is not optimal or the worst, but the gap between the running time and the shortest running time is very small. Hence, the combined application of MAUT and PAMSSEM II methods in the context of probabilistic linguistic is of great significance. In future work, we will broaden the application fields of the PL-MAUT-PAMSSEM II method, such as applying it to Artificial Intelligence, the Neural Networks, Autonomous Systems, etc. Meanwhile, we consider combining MAUT-PAMSSEM II with some information aggregation operators in the probabilistic linguistic environment so as to develop a new method with better performance.
Author contributions QK and LW conceived and worked together to achieve this work, QK compiled the computing program by Java and analyzed the data, QK and LW wrote the paper. Finally, all the authors have read and approved the final manuscript.
Funding This work is supported by the Visual Computing and Virtual Reality Key Laboratory of Sichuan Province Item (Grant No. SCVCVR2019.04VS).
Data availability The authors confirm that the data supporting the findings of this study are available within the article or its supplementary materials.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.