T-spherical uncertain linguistic MARCOS method based on generalized distance and Heronian mean for multi-attribute group decision-making with unknown weight information

The T-spherical uncertain linguistic (TSUL) sets (TSULSs) integrated by T-spherical fuzzy sets and uncertain linguistic variables are introduced in this article. This new concept is not only a generalized form but also can integrate decision-makers’ quantitative evaluation ideas and qualitative evaluation information. The TSULSs serve as a reliable and comprehensive tool for describing complex and uncertain decision information. This paper focuses on an extended MARCOS (Measurement of Alternatives and Ranking according to the Compromise Solution) method to handle the TSUL multi-attribute group decision-making problems where the weight information is completely unknown. First, we define, respectively, the operation rules and generalized distance measure of T-spherical uncertain linguistic numbers (TSULNs). Then, we develop two kinds of aggregation operators of TSULNs, one kind of operator with independent attributes is T-spherical uncertain linguistic weighted averaging and geometric (TSULWA and TSULWG) operators, and the other is T-spherical uncertain linguistic Heronian mean aggregation operators (TSULHM and TSULWHM) considering attributes interrelationship. Their related properties are discussed and a series of reduced forms are presented. Subsequently, a new TSUL-MARCOS-based multi-attribute group decision-making model combining the proposed aggregation operators and generalized distance is constructed. Finally, a real case of investment decision for a community group-buying platform is presented for illustration. We further test the rationality and superiorities of the proposed method through sensitivity analysis and comparative study.


Introduction
Multiple decision agents, including experts or stakeholders in different fields, select the optimal option(s) from several alternatives, this process is named multi-attribute group decision-making (MAGDM) [1,2]. On the one hand, group B Haolun Wang hlwang71162@nchu.edu.cn 1 some scholars extend and propose various linguistic information models from specific requirements, such as hesitant fuzzy linguistic term sets [5], intuitionistic linguistic sets [6], 2-tuple fuzzy linguistic representation model [7], etc. Xu [8] believed that the uncertain information of evaluation can be expressed more accurately through linguistic terms interval, while a single linguistic variable cannot be realized. Therefore, Xu [8] proposed the concept of uncertain linguistic variables. Since then, the uncertain linguistic sets have received extensive attention, and various extended studies have emerged.
The intuitionistic uncertain linguistic sets (IULSs) [9] have the advantages of both uncertain linguistic variables and intuitionistic fuzzy sets. The intuitionistic uncertain linguistic number consists of an uncertain linguistic part and an intuitionistic fuzzy part. The former represents the qualitative evaluation value of a certain attribute by decision-makers, and the latter indicates the level of subordination and nonsubordination to the uncertain linguistic part [10][11][12], the intuitionistic fuzzy part meets the sum of membership degree (MD) and non-membership degree (ND) is not greater than one [13]. The IULSs cannot be adopted once the sum of MD and ND is greater than one, many scholars explored the combination of an uncertain linguistic variable and Pythagorean fuzzy set [14], then introduced Pythagorean uncertain linguistic sets (PyULSs) [15][16][17][18][19], so the modeling ability of uncertain information is improved to a certain extent. However, the PyULSs cannot be applied if the sum of squares of MD and ND of Pythagorean fuzzy numbers is more than one. Scholars proposed q-rung orthopair uncertain linguistic sets (q-ROULSs) [21][22][23][24][25][26] by pooling the strengths of q-rung orthopair fuzzy sets [20] and uncertain linguistic variables, in which q-rung orthopair fuzzy part satisfies the q-power sum of MD and ND degree less than or equal to one. The q value can be adjusted by decision-makers to expand the expression range of acceptable q-rung orthopair uncertain linguistic information and further improve the modeling ability of uncertain information. The IULSs and PyULSs are particular forms of the q-ROULSs (when q 1 and q 2). Therefore, the q-rung orthopair uncertain linguistic sets are more flexible and comprehensive than intuitionistic uncertain linguistic sets and Pythagorean uncertain linguistic sets in describing qualitative evaluation information. In addition, Cuong [27] found that picture fuzzy sets theory can manage awkward and complicated information. This concept makes the description and evaluation information more reliable and widely proficient by adding an abstinence degree (AD) [28,29]. Wei et al. [30,31] proposed picture uncertain linguistic sets (PULSs) by combining picture fuzzy sets with uncertain linguistic sets, in which the sum of MD, AD, and ND in the picture fuzzy part is less than or equal to one. Furthermore, Naeem et al. [32] extended interval-valued PULSs to solve the problem of supplier selection. Some weighted and ordered weighted generalized Hamacher aggregation operators (AOs) based on interval-valued PULSs are developed by Garg et al. [33] In 2019, Mahmood et al. [34] proposed the concept of Tspherical fuzzy set (T-SFS), which not only has the same structural framework of MD, AD and ND as PFS, but also has a larger and more flexible decision space, that is, it has the sum of the q-powers of MD, AD and ND less than or equal to one. Obviously, the T-SFS is reduced to the spherical fuzzy set, PFS, q-ROFS, PyFS and IFS under certain conditions. Therefore, the T-SFS can better represent the uncertainty of evaluation information. Since the generalized feature of T-SFS with no limitations, it has widely concerned by many scholars. Current studies on T-SFS mainly focus on the following three aspects: (1) Some information measures of T-spherical fuzzy number have been studied by scholars, such as entropy measure [35], similarity measure [35][36][37], divergence measure [38], correlation coefficient [39], etc.
For MAGDM problems, the MARCOS method is a relatively new and stable technique for selecting the best candidate(s) for decision problems. In 2020, Stevič et al. [52] first proposed the MARCOS method, whose basic idea is to determine the utility function of the alternative by determining the connection between the alternative and the reference points and to achieve compromise ordering related to ideal solutions and anti-ideal solutions. Different from existing methods (TOPSIS [53], VIKOR [54], EDAS [55], TODIM [46], etc.), which have disadvantages such as complex calculation, pre-setting of parameters, and ignoring the relative importance of distance, the MARCOS method has below merits: (1) The reference points are determined at the beginning of the model; (2) The utility degree is more accurately determined from ideal and anti-ideal perspectives; (3) A new method to determine utility functions is proposed, which is different from the TOPSIS's distance approximation and VIKOR's compromise mechanism; (4) Ability to handle a large number of conflicting attributes and alternatives [56].
So far, the MARCOS method has been extended and applied in different decision environments to deal with various real MAGDM problems, as shown in Table 1. We can find that there are still some spaces for expansion and improvement of MARCOS method from Table 1. This study aims to modify the MARCOS method for MAGDM problems with interrelationship among decision attributes in T-spherical uncertain linguistic environment. The motivation of this article as follows: (1) As a generalized form for expressing uncertain and vague information, the TSULSs can effectively characterize and identify decision-makers' preference information in a decision-making environment with increasing complexity. The existing MARCOS methods have been extended and applied in different MAGDM environments, like IFNs [58], q-ROFNs [56], SFNs [57], IT2FNs [60,63], RNs [59,61] and SVNFNs [65], etc. However, the capabilities of these fuzzy numbers are not as good as TSULNs, and MARCOS has not been extended in the TSULSs environment. Thus, it is necessary to investigate the integration of MARCOS and TSULS in this paper. (2) Expert weights play an important role in MAGDM problems. From Table 1, many scholars adopted the ways of subjective assessment [56][57][58] and assumed weight [59][60][61][63][64][65] in the process of solving MAGDM problems, but these approaches are too subjective and arbitrary to fully reflect the objective importance of experts.
Although there are less distance-based approaches [62,66] in Table 1, these distance measures did not take into account the psychological behavior of decision-makers. Hence, it is necessary to propose a new distance measure to calculate expert weights in the TSUL environment, which can not only meet the axiomatic definition of distance measure, but also flexibly reflect the psychological behavior of decision-makers. (3) The attribute weight is critical and can influence the decision results in the process of MAGDM. In Table  1, scholars used the FUCOM [64][65][66], BWM [61][62][63], AHP [60], and PIPRECIA [59] methods to obtain the subjective attribute weights. These subjective weight techniques can make the decision results adhere to the subjectivity of the decision-makers. In addition, the CRITIC [56,57] approach was employed to obtain the attributes' objective weights using standard deviation and correlation coefficient. However, under the situation that the decision-makers are completely unknown to the weight information of attributes, an objective weight determination method considering experts' preferences has not been developed. Thus, the MDM is adopted to calculate the optimal attribute weight in a T-spherical uncertain linguistic context. (4) In existing MARCOS methods, the evaluation values under each attribute are aggregated to generate the comprehensive evaluation value of each alternative.
Unfortunately, the interrelationship between attributes is ignored, but the correlations between attributes do exist objectively in complex real-world decision problems, it can be more in line with the actual situation of decision-making. Therefore, we try to develop some new Heronian mean AOs with TSULNs based on the characteristics and advantages of the Heronian mean operator [12,47,48,68,69]. (5) In the existing MARCOS methods, the utility degrees (K i − and K i + ) of alternative solutions to ideal and antiideal solutions are calculated by division operation or division operation after defuzzification. However, the division operation may have two disadvantages: first, the value may be 0 when the anti-ideal solution is determined, which causes the division operation to fail. Second, the process of defuzzification may result in the loss of some evaluation information of various types of fuzzy numbers. Therefore, inspired by Euclidean distance in Rf. [34], we try to integrate a new distance measure in the MARCOS method to calculate the utility degrees. The new distance measure is employed to eliminate the above shortcomings, which not only includes the information of MD, AD, ND and refusal degree in TSULNs, but also can achieve the same effect as the division operation.
To sum up the above arguments and motivations, some contributions of this paper are presented as below: (1) A new notion of TSULSs is introduced to express the decision-maker's view and preferences in complex MAGDM problems, and the operation rules and generalized distance measures of TSULNs are defined. (2) Two classes of TSUL AOs considering the relationship (independent and correlative) between attributes are developed, their related properties are analyzed and these operators are reduced into some special cases. (3) Based on the TSUL generalized distance measure, the TSUL similarity is defined to calculate the decisionmakers' weight and the MDM is applied to obtain the weight of the attribute. (4) A novel TSUL-MARCOS method based on the TSUL-WHM operator and generalized distance measure is designed. We test the practicability and effectiveness of the proposed method by dealing with an investment decision for the community group-buying (CGB) platform.
The other segments are arranged as follows: some related basic concepts are briefly reviewed in "Preliminaries". "Some notions of T-spherical uncertain linguistic sets" defines the related notions of TSULSs, including the TSULSs, operation rules of TSULNs, and the TSUL generalized distance measure. "The TSUL-weighted AOs and TSULHM AOs" develops the TSULWA, TSULWG, TSULHM, and TSULWHM operators. " A novel MAGDM framework based on TSUL-MARCOS" builds the TSUL-MARCOS model for the MAGDM problems. A case of the investment decision for the CGB platform is employed to illustrate the practicability of the developed method in "A case study". Meanwhile, the sensitivity analysis and comparative study are performed. "Conclusions" summarizes the work of this article and introduces the plans.

Preliminaries
We briefly describe some basic concepts of ULS and T-SFS in this section.
For any linguistic set S, the following conditions should be satisfied [70]:

Definition 1 [8]. Supposes
[s α , s β ], s α , s β ∈S and α ≤ β, in which s α and β is the lower limit and the upper limit ofs, thens is called the uncertain linguistic variable (ULV). SupposeS is the set of all ULVs, namely uncertain linguistic set (ULS).
To eliminate some shortcomings of Xu [8] proposed operation rules, Liu and Zhang [12] proposed some new ULVs operation rules and defined them as follows: Definition 2 [12]. Supposes 1 [s α 1 , s β 1 ],s 2 [s α 2 , s β 2 ] are any two ULVs inS {s 0 , s 1 , s 2 , . . . , s k−1 }, then the new operational laws of ULVs as follows: Definition 3 [34]. Suppose X is a universe set, and then the form of T-SFS is described as below: in which τ (x), η (x), ϑ (x) are respectively the MD, AD and ND of element is named the refusal degree. For simplicity, the T-spherical fuzzy number (T-SFN) is represented as a triplet of τ , η and ϑ, and denoted as χ (τ , η, ϑ).
The HM σ , ρ is known as the Heronian mean operator.

Some notions of T-spherical uncertain linguistic sets
In this segment, we introduce a notion of TSULs based on the ULVs and T-SFSs, and define a series of relevant definitions, including operation rules and generalized distance measure of TSULNs.
). If q 1, then the TSULN reduces to the PULS; If q 2, then the TSULN reduces to the SULN.
We propose a new generalized distance to measure the difference between two TSULNs. This distance measure contains parameter, and the decision-makers can adjust parameter value according to the actual situation of decision-making to represent the decision-makers' psychological preference behavior. The definition of this distance is as below.
(ii) According to Definitions 7~10, we have Therefore, the proof of property (4) is complete.

The TSUL-weighted AOs and TSULHM AOs
We propose the TSULWA and TSULWG operators based on the operation rules of TSULNs, and then we devise the TSULHM and TSULWHM operators considering the interrelationship between input arguments.

The TSUL-weighted AOs
In this sub-section, we define two TSUL-weighted AOs, namely TSULWA and TSULWG operators. They do not concern the correlation between input arguments.
, with w j > 0 and n j 1 w j 1. The TSULWA and TSULWG operators are defined as.
Proof: it is easy to prove that the aggregated results of TSULWA and TSULWG operators are still TSULNs, and the proof process is omitted here. Next, we utilize the mathematical induction method to prove the Eqs. (18) and (19). We first prove the TSULWA operator, and the proof process is as follows: (1) when n 1, the Eq. (18) is clearly true. And when n 2, according to Definition 9, we can get. Then When n 2, the Eq. (18) holds.
(2) when n m, the Eq. (18) holds. That is, (3) when n m + 1, we can get Obviously, when n m + 1, the Eq. (18) also holds. From the above proof, we can know that Eq. (18) holds for any j. In the same way, the Eq. (19) holds for any j can be proved.
Note that Eq. (27) differs from the SLFWA operator [72] in the linguistic part. (3) If η j 0, the TSULWA and TSULWG operators are reduced into the q-rung orthopair uncertain linguistic weighted averaging and geometric operators, i.e., q-ROULWA and q-ROULWG.
Note that Eq. (36) differs from the ILWA operator [6] in the linguistic part.

The TSULHM AOs
In this sub-section, we define the TSULHM operator and its weighted form. They have the ability to capture the interrelationship between input arguments.
Proof: On the basis of Definition 9, we can obtain.
According to Theorem 1 and Theorem 5, the following properties of the TSULHM operator can be easily proved: (1) (Idempotency). If δ j δ for all j, then (2) (Monotonicity). If δ j * (j 1, 2,…, n) is also a set of TSULNs, and δ j ≤δ j * , then Next, some particular cases of the TSULHM operator are discussed about the parameters σ and ρ.
Similar to Theorem 4, Theorem 5 is true and can be proved.
It is worth noting that we also can prove that the TSUL-WHM operator has the Boundedness and Monotonicity, but it has not the property of Idempotency.
For attribute a j ∈ A, we can construct the similarity matrix S (j) by using the TSUL similarity measure sim Based on the similarity matrix S (j) , we can use Eq. (52) to calculate the overall similarity degree of expert e k for attribute a j ∈ A. (52) Lastly, the weight of expert e k on the attribute a j ∈ A can be obtained as below:

Calculate attribute weights based on MDM
When attribute weight information is completely unknown, the uncertainty of attribute weight can affect the final ranking result of alternatives. Generally, if there is a small difference between the attribute values r ij (i 1,2,…,m; j 1,2,…,n) in attribute a j of all alternatives, it means that the attribute a j is of low importance in the ranking of all alternatives, and we can assign a small weight to this attribute. Otherwise, a larger weight is assigned. Therefore, we can adopt the MDM [74,75] to obtain the attribute weight vector.
For the attribute a j ∈ A, the deviation of alternative h i to all other alternatives h l (l 1,2,…,m, l i) can be expressed as: D j (w) represents the deviation value from all alternatives to other alternatives for attribute a j ∈ A, so it can be expressed as: where D ϕ (r i j , r l j ) represents the TSUL generalized distance measure between r ij and r lj , which can be obtained from Eq. (12). Therefore, we can build a linear mathematical model that maximizes all deviations for all attributes to assign weight vector W , as follows: To solve Eq. (56), we apply the Lagrange function with the Lagrange multiplier λ. Then we have n j 1 w 2 j − 1 0 Thus, we can obtain a formula for the optimal weight of attributes as follows: By normalizing w j *, we get the attribute weight w j (j 1,2,…,n).

Rank the alternatives by extended MARCOS
The existing MARCOS methods cannot handle TSULNs. In this sub-section, we extend the MARCOS method to the TSUL environment. The specific steps are as below: Step 1: the TSUL evaluation matrix D k [d ij k ] m×n ,(i 1, 2 ,…,m; j 1, 2 ,…,n; k 1, 2,…,p) is constructed and normalized to the normalized decision matrix R k [r ij k ] m×n ,(i 1, 2 ,…,m; j 1, 2 ,…,n; k 1, 2 ,…,p).
where (d k i j ) c is the complement set of TSULN d k i j , 1, and 2 indicate benefit and cost attributes, respectively.
Step 3: the TSULWA (Eq. (18)) or TSULWG (Eq. (19)) operator is utilized to fuse the evaluation information given by different experts under the attribute a j of alternative h i , then the individual TSUL decision matrix R k is aggregated into group TSUL decision matrix G [g ij ]m × n, g i j [s θ i j , s ρ i j ], τ i j , η i j , ϑ i j (i 1,2 ,…,m; j 1, 2 ,…,n). Further, we can obtain the extended group TSUL decision matrixG by Eq. (60). (60) where the anti-ideal solution h AAI represents the worst alternative, which can be obtained by the ideal solution h AI means the best alternative, which can be obtained by Step 4: we apply the attribute weight determination method in sub-Sect. 5.2 to get the optimal attribute weight w j (j 1,2,…,n).
Step 5: all attribute evaluation values g ij corresponding to alternative h i are aggregated by the TSULWHM operator (Eq. (38)), then the comprehensive evaluation values of each alternative x i (i 1, 2 ,…,m). Similarly, we can get the comprehensive evaluation values x AAI of h AAI and x AI of h AI .
Step 6: calculate the utility function f (K i ) of the alternative h i (i 1, 2,..m).
The utility degrees K i − and K i + of each alternative are obtained from the TSUL generalized distance measure between the comprehensive evaluation value x i and x AAI , x AI , respectively, that is, K + D ϕ (x i , x AI ) and K − D ϕ (x i , x A AI ). The calculation formulas of ideal utility function f (K i + ) and anti-ideal utility function f (K i − ) are shown below: Step 7: we rank the alternatives according to the utility function value. The alternative with the maximum utility function value is the optimal option.

A case study
The CGB is a kind of shopping and consumption behavior with a low discount that a certain number of consumer groups in real living communities purchase goods online and pick up goods offline. Compared with online group buying, the CGB requires the establishment of a service center in the community or other specific places, where consumers can pay the money and obtain after-sales protection in case of problems with commodities. Meanwhile, group buying organizations can collect consumers' purchasing demands through the service center and contact appropriate merchants to provide commodities and services. At present, there are hundreds of large-scale CGB platforms in China, which are similar in the use of methods and play their due role in the speed of expansion, cross-regional management ability, supply chain ability, commodity system, technical ability, brand power, and other aspects. According to industry experts, the scale of CGB in China will reach 500 billion CNY in 2025.
Currently, there are many CGB platforms in the groupbuying market in China, such as Xingsheng, Meituan, Duoduo, Shihui, etc. Due to the fierce competition in the group-buying market, the investment and financing funds obtained by the CGB platform from the capital market increased from 2.402 billion CNY in 2016 to 20.073 billion CNY in 2020 to seize the market share. However, for investors, how to select a potential CGB platform as the investment object has become a challenging decision-making problem. LXC is a venture capital and private equity fund management company owned by Legend Holdings of China. At present, LXC has a fund to invest in the CGB platform. According to the market survey and preliminary screening, there are five CGB platforms as potential investment objects, for which the best investment projects need to be determined. A brief introduction of alternative CGB platforms is shown in Table 2.
To screen out the best CGB platform project, five options are evaluated from six attributes. There are six attributes used to evaluate the alternatives, including platform operation and maintenance ability (a 1 ), expected revenue (a 2 ), market competitiveness (a 3 ), risk resistance ability (a 4 ), supply chain management ability (a 5 ), and product and service innovation ability (a 6 ). Their weight information is completely unknown. To assess the five CGB platforms with six attributes, LXC company invited three senior investment

Changsha
As a well-known brand of community e-commerce, Xingsheng provides vegetables and fruits, meat, poultry, and aquatic products, rice, flour, grain and oil, daily necessities, and other selected commodities. Xingsheng relies on community convenience stores, through the "pre-sale + self-pickup" model to provide services for residents 2 Meituan (h 2 )

Beijing
The social group-buying business of Meituan adopts the mode of "pre-order + self-pickup" to select cost-effective vegetables, fruits, meat, poultry and eggs, drinks and snacks, household kitchen and toilet, instant frozen food, grain, oil and seasoning, and other products for residents in the community 3 Duoduo (h 3 )

Shanghai
Duoduo is the community group buying launched by PDD in August 2020. It adopts the semi-pre-purchase mode of "online order + offline self-pickup" and provides food buying service through PDD APP or Wechat grogram 4 Nicetuan (h 4 ) Beijing Nicetuan provides urban community families with fresh and delicious ingredients and daily necessities through the Wechat program, to bring convenient and fresh purchasing services for social families 5 Jingxi (h 5 )

Beijing
Jingxi is a community O2O group-buying service platform. Relying on the Jingdong supply chain system ensures that consumers can place orders on the same day and pick them up the next day. It also guarantees the daily supply of rice, flour, oil, meat, poultry, eggs and milk, fresh vegetables and fruits, epidemic prevention and elimination, leisure snacks, and other daily necessities for community residents experts E {e 1 ,e 2 ,e 3 }, and they are required to use the linguistic term set S {s 0 Extremely bad, s 1 Very bad, s 2 Bad, s 3 Medium bad, s 4 Medium, s 5 Medium good, s 6 Good, s 7 Very good, s 8 Extremely good}to assess the five alternatives, respectively. The evaluation information of the experts on the five alternatives can be obtained, which is presented in Table 3.

Decision process
Step 1: since all attributes are benefit types, the TSULDM D k does not need to be normalized in Table 1, that is, D k R k .
Step 2: first, the evaluation matrix D k provided by e k (k 1,2,…,p) is converted into an evaluation matrix for each attribute F (j) ( j 1,2,…,n). E.g. the evaluation matrix F (1) for attribute a 1 .
Then, Eq. (49) is used to calculate the evaluation mean value of alternative h i regarding attribute a j . E.g.ξ i . We employ the TSUL Hamming distance measure (ϕ 1)(Eq. (13)). Thus, the similarity matrix S (j) is constructed. E.g. the similarity matrix S (1) for attribute a 1 .
Lastly, Eq. (53) is used to calculate the weight of expert e k on attribute a j .  18)) operator is utilized to aggregate the evaluation information of alternative h i under attribute a j given by experts, then we can obtain the group TSUL decision matrix G. Further, we the G is expanded to the extended group TSUL decision matrixG according to Eq. (60). See Table 4.
Step 5: we use the TSULWHM operator (Eq. (38)) to obtain the comprehensive evaluation value x i (i 1,2,…,5), x AAI and x AI . Step 6: the Eqs. (61)(62) are used to calculate the utility function value of each alternative.
Step 7: according to the utility function value of each alternative, we rank the alternative: h 1 > h 2 > h 4 > h 3 > h 5 . Therefore, h 1 (Xingsheng) is the optimal option and h 2 (Meituan) is the second best solution.

Sensitivity investigation
The parameters q, ϕ, σ and ρ play an major role in the influence of alternative ranking results. We first investigate the effect of different values in q ∈ [3, 10] on alternative ranking results (ϕ 1, σ ρ 1). The alternative ranking results are shown in Table 5.
As the parameter q increases, each alternative's utility function decreases gradually in Table 3. When q takes different values, the ranking of alternatives also changes, see Fig. 1. Specifically, when q 3, the ranking of alternatives is h 1 > h 2 > h 4 > h 3 > h 5 , the best alternative is h 1 , and the second-best alternative is h 2 . When q 4, the ranking of alternatives is h 2 > h 4 > h 3 > h 1 > h 5 , and the optimal and second alternatives are h 2 and h 4 , respectively. When q ∈ [5,10], the ranking of alternatives changes from h 4 > h 3 > h 2 > h 5 > h 1 to h 4 > h 5 > h 3 > h 2 > h 1 , and tends to be stable. As we know, the value of parameter q not only reflects the ability of TSULSs to  model uncertain information but also reflects the evaluation preference of decision-makers in the actual decision-making environment. Therefore, the appropriate parameter q value needs to be chosen by the decision-makers. Generally, the parameter can be the smallest integer satisfying the condi- depending on the evaluation value of the attribute. Therefore, h 1 and h 2 are the best and second-best alternatives in this paper. Next, the effect of the parameter ϕ on the ranking results of each alternative is analyzed. When parameters q 3, σ ρ 1, we take various values for the parameterϕ, and the results of each alternative are shown in Table 6.
From Table 6, when the parameter ϕ in the TSUL generalized distance measure is set to different values, the alternative ranking changes from h 1 > h 2 > h 4 > h 3 > h 5 to h 1 > h 2 >   Fig. 2 The influence of parameter ϕ on the alternatives ranking changes slightly. On the whole, the effect of the parameter ϕ on the alternative ranking is not obvious, and the ranking of alternatives is stable in Fig. 2. We take various values for parameters σ and ρ, then analyze the effect of σ and ρ on alternative ranking in Step 5 of our method. The utility function values of each alternative are shown in Table 7(q 3, ϕ 1).
The ranking of all alternatives varies with the value of parameters σ and ρ, but the optimal option is h 2 in Table 7. The parameters σ and ρ represent the degree of correlation between attributes, which is also the main reason for the variation of alternative ranking. When σ 0,ρ 1 and σ 1,ρ 0, there is no correlation between attributes. Except for the best alternative h 2 , the order of other alternatives is completely different, namely h 2 > h 4 > h 3 > h 1 > h 5 and h 2 > h 3 > h 5 > h 4 > h 1 . As the values of σ and ρ increase, the correlation level between attributes increases. Therefore, once various values of parameters σ and ρ are chosen, the relationship structure between attributes simulated by the TSULWHM operator changes. In general, the decision-makers usually take σ ρ 1, which can reduce the complexity of calculation and capture the interrelationship between attributes.

Comparative study
We applied some existing MADM approaches, including q-rung orthopair uncertain linguistic weighted geometric Heronian mean (q-ROULWGHM) [76], picture uncertain linguistic weighted averaging and geometric (PULWA and PULWG) [30], picture uncertain linguistic weighted Bonferroni mean and geometric form (PULWBM and PULWGBM) [31] operators and linguistic T-spherical fuzzy ARAS (Lt-SF-ARAS) method [77] to the case in this paper, and the results of alternatives are shown in Table 8.
We apply the existing AOs (q-ROULWGHM, PULWA, PULWG, PULWBM and PULWGBM) and ranking technique (Lt-SF-ARAS) to this case. This results in Table 8 show that the existing methods are not applicable to this case. Due to the lack of AD in q-ROULS, the ability of Lt-SFS to represent uncertain information is not as good as TSULS, and the PULS cannot process the TSULNs in Table 3 and these TSULNs require the parameter q to take the minimum integer as 3. Therefore, these existing methods cannot be used to solve the decision-making problem in above case. However, the proposed method based on the TSULWA (or TSULWG), TSULWHM operators, and TSUL-MARCOS can process the evaluation information in Table 3. It can be seen that our method has a wider scope of application than the existing methods. For this reason, we further use an example from Ref. [31] to demonstrate that the proposed method is more generalized.
Example 4 [31] . Assume that there are five service outsourcing providers (P 1 , P 2 ,…, P 5 ) to select for the communication industry. The panel selected four criteria (Business reputation C 1 , technical capability C 2 , Management capability C 3 , Quality of Service C 4 ) to evaluate the five possible service outsourcing providers (P 1 , P 2 ,…, P 5 ). The PULNs are applied to represent the evaluation values of the alternative concerning criteria, and the weight vector of the attribute is W (0.2,0.1,0.3,0.4) T . Thus, the decision matrix is constructed as shown in Table 9.   Table 9 The decision matrix with PULNs (k 7) We adopt Steps 5~7 in sub-Sect. 5.3 to solve the selection of service outsourcing providers, and compared the results of the q-ROULWGHM, PULWA, PULWG, PULWBM, PUL-WGBM operators and Lt-SF-ARAS method, as listed in Table 10.
From Table 10, the q-ROULWGHM operator and Lt-SF-ARAS method are not applicable to Example 2, because q-ROULS and Lt-SFS cannot process the data in Table 9, although the parameter q in these TSULNs is 1. The best alternative obtained by our method is P 4 and the sub-best alternative is P 3 , while P 3 is obtained by the PULWA, PULWG, PULWBM, and PULWGBM operators as the optimal option, the second-best alternative is P 1 and P 4 respectively. This is completely inconsistent with the results of this paper. The specific reasons are as follows: (1) The operation rules of the linguistic part in PULWA, PULWG, PULWBM, and PULWGBM operators are based on Rf. [8], while the operation of the linguistic part of TSULNs in this paper is based on Definition 2 and can ensure that the calculation results are still within the range of linguistic level. (2) The PULWA and PULWG operators ignore the correlation between attributes, while the PULWBM and PULWGBM operators are similar to the TSULWHM operator, which can not only concern the interrelationship between attributes but also reflect the preferences of decision-makers by adjusting parameters. However, the PULWA, PULWG, PULWBM, and PULWGBM operators all use score functions that do not contain the degree of refusal to defuzzify, but we apply the TSUL generalized distance measure to compute the utility of alternative in the improved MARCOS. (3) The existing approaches aggregate the evaluation information and obtain the comprehensive evaluation value of the alternative. This information fusion process is relatively direct and rigid, while the deviation between the alternative and the ideal and anti-ideal solution is considered in the TSUL-MARCOS method, which reflects the characteristics of compromise in the calculation of the alternative's utility function.
Therefore, in addition to being more generalized than the existing methods, the methodology of this paper is more rationality.
Next, we further compare with the traditional MARCOS method. For the multi-criteria decision-making problems, we can have the general ideal of traditional MARCORS method to solve the TSUL decision-making problems based on the existing works [56-58, 63-65, 67]: (1) Use the Eq. Then, we can get the results of the traditional MARCOS method and the improved MARCOS method, as shown in Table 11.
From Table 11, the ranking result of traditional MAR-COS method is h 1 h 2 > h 3 , while the ranking result of improved MARCOS method is h 1 > h 2 > h 3 . The former cannot distinguish between the optimal alternative h 1 and h 2 , but the latter can clearly obtain the optimal alternative h 1 . The detailed reasons are as follows: first, we used the score function (Eq. (6)) in the traditional MARCOS method to de fuzzify the evaluation matrix in the early stage. In this process, it is impossible to distinguish the TSUL evaluation values of h 1 and h 2 under each attribute, such as the score function values of d 11 and d 21 are s 0.750 . In the later stage of the improved MARCOS method, the proposed TSUL Hamming distance measure was applied for de fuzzification, and this measure contains refusal degree information of TSULN. In contrast, the traditional MARCOS method loses part of the decision information in the process of de fuzzification. Second, the standardized evaluation values of alternative were summed simply in the traditional MARCOS method to obtain the comprehensive evaluation vale x i of alternative, while the TSULWHM operator was applied to aggregate the TSUL evaluation values of alternative in the improved MARCOS method, which takes into account the correlation between attributes. In contrast, the former method ignores the relationship between attributes, which makes it impossible to mine the potential information in decision information. To sum up, our improved MARCOS method can rank the alternative more effectively in the TSUL environment. Therefore, the improved method in this paper has more superiority compared with the traditional MARCOS method.

Conclusions
In this paper, we define some new concepts of TSULSs inspired by q-rung orthopair uncertain linguistic sets, including operation rules and generalized distance measures of TSULNs. The TSULSs can express more freely and deal with uncertainty in data more effectively. We not only propose the TSULWA and TSULWG operators but also develop the TSULHM and TSULWHM operators which can capture the interrelationship of attributes. Furthermore, we construct the improved MARCOS-based MAGDM framework with TSUL information. In it, based on the generalized distance measure of TSULNs, the TSUL similarity is defined to calculate attribute weights and the MDM is applied to determine the attribute weights, respectively. At the same time, we integrate the TSULWHM operator and generalized distance measure into the MARCOS method, so that this method can capture the interrelationship of attributes and calculate the alternative's utility degree more accurately. Finally, an illustrative example of a CGB platform investment decision is presented using the proposed method, and sensitivity analysis and comparative study are performed to test the validity of the proposed method.
Although the proposed method can effectively solve the MAGDM problems with TSULNs, there are still some limitations: (1) The HM is included in our method, which can consider the correlation between two attributes, but the HM cannot capture the interrelationship among multiple attributes. (2) Furthermore, there is an objective priority relationship and degree between attributes in the real decision-making problems, but it is not concerned in this case. (3) In addition, the attribute subjective weight is not taken into account in the proposed method, which ignores the intuitive judgment of experts on the importance of attributes. To this end, a series of new AOs will be further developed in the TSULSs environment in the future. For example, Muirhead mean [40], Hamy mean [78], Maclaurin Symmetric mean [24]. These operators can capture the interrelationship among multiple attributes. Further, we will try to integrate Softmax function [79] into above operators, so that they can focus on the priority and degree between attributes. The attribute combined weight will be adopted in the future, in which the attribute subjective weight will be determined by SWARA (Step-wise Weight Assessment Ratio Analysis) [80] or BWM (Best-Worst Method) [81]. We will try to integrate the TSULSs with other decisionmaking techniques, such as WASPAS (Weighted Aggregated Sum Product ASsessment) [82], MABAC (Multi-Attribute Border Approximation area Comparison) [83], CoCoSo (Combined Compromise Solution) [84], and so on. In addition, we will expand the improved MARCOS method in various decision-making environments, such as probabilistic T-spherical hesitant fuzzy set [85], probabilistic linguistic term set [86,87], probabilistic uncertain linguistic term set [88]. Apart from these, we can also test our methodology with real-world decision scenarios, such as firm business decisions, capital item selection, and consumer purchasing decisions.