Many management decision-making problems in the real world, such as logistics park location, supplier selection, medical service evaluation, fault diagnosis, etc., can be considered from the perspective of MADM (Garg et al., 2018; Akram et al., 2021; Waseem et al., 2019; Lu et al., 2021; Wei et al., 2022). As an important branch of management science and modern decision science, MADM theory and methods have been widely used in many practical decision-making problems (Yang and Pang, 2019; Xu and Zhang, 2019; Zavadskas et al., 2013; Ning et al., 2022). In the actual MAGDM process, due to the complexity and uncertainty of objective things, the limitations of human cognition and the ambiguity of thinking, it is difficult to use quantitative and accurate information to describe decision objects (Wang et al., 2022; Zhang H. et al., 2022; Liu et al., 2019; Li et al., 2021). In 1965, Zadeh (1965) was the first to define a novel fuzzy sets (FSs) to cope with information in the fuzzy new domain (Garg and Kumar, 2018; Garg, 2018b; Zhang and Xu, 2015; Su et al., 2022; Jiang et al., 2022; Lei et al., 2022). To extend the FSs, the intuitionistic fuzzy sets (IFSs) (Atanassov, 1989) were also defined. Subsequently, FSs and its related extension knowledges were predominantly exploited in decision analysis domains (Yu et al., 2017; Wan and Li, 2014; Zhang D. et al., 2022; Zhang et al., 2022a). Su et al. (2011) proposed the interactive method for dynamic IF-MAGDM. Arya and Yadav (2018) defined the intuitionistic fuzzy super-efficiency slack-based measure. Tian et al. (2017) studied the partial derivative and complete differential of binary IF-mathematical functions. Garg (2018a) proposed the improved cosine similarity measure for IFSs. Tan (2011) constructed the Choquet integral-based TOPSIS method for IF-MADM. Zhao et al. (2017) defined the Interactive intuitionistic fuzzy algorithms for multilevel programming problems. Li (2011) built the GOWA operator to MADM using IFSs. Buyukozkan et al. (2018) selected the transportation schemes with integrated intuitionistic fuzzy Choquet integral method. Joshi et al. (2018) defined the Jensen-alpha-Norm dissimilarity measure for IFSs. De and Sana (2018) defined the The ( p , q , r , l ) method for random demand with Bonferroni mean under IFSs. Li et al. (2018) defined the time-preference and VIKOR-based dynamic method for IF-MADM. Niroomand (2018) defined the multi-objective based direct solution method for linear programming along with intuitionistic fuzzy parameters. Zhao et al. (2021) perfected TODIM for IF-MAGDM on the strength of cumulative prospect theory. Yu S. et al. (2017) defined the derivatives and differentials for multiplicative IFSs. Yu (2012) defined the generalized prioritized geometric operators under IFSs. Xiao et al. (2020) built the intuitionistic fuzzy taxonomy method. Wu and Zhang (2011) built the IF-MADM based on weighted entropy. Verma and Sharma (2014) defined the measure of inaccuracy IF-MADM. Iakovidis and Papageorgiou (2011) defined the cognitive maps for medical decision making under IFSs.

Then the hesitant fuzzy element (HFE) proposed by Xia and Xu (2011) is to solve the problem of determining the element’s membership to a set on account of the uncertainty between different numbers and then prove the intuitionistic fuzzy set and hesitant fuzzy set. With the proposition of the HFE, the idea of correspondent operators to aggregate hesitant fuzzy information was obtained. Not long after this, Xu and Xia (2011) raised the idea of the score function, deviation function and the comparison rule, and set the basis on the calculation. Xu and Cai (2010) provided the aggregating operators to integrate the hesitant fuzzy information. Nevertheless, HFE can be regarded as a particular equivalent form whose occurring probabilities of the possible value are equal. The probabilistic hesitant fuzzy set and the corresponding score function, deviation function and its comparison law were proposed by Xu and Zhou (2017). Moreover, the probabilistic hesitant fuzzy weighted averaging geometric operators were introduced by Xu and Zhou (2017) to process PHFE information. Then the improved PHFS was introduced by Zhang et al. (2017) to give more space for hesitation, the integrations of the improved PHES can be calculated by the improved operators. Farhadinia and Xu (2021) gave the comparison techniques of PHFEs. Krishankumar et al. (2021a) built the COPRAS approach to PHFSs. Krishankumar et al. (2021b) proposed to extend a well-known VIKOR method to the PHFS context. Lin et al. (2021) put forward a novel probabilistic hesitant fuzzy MULTIMOORA method. Liu et al. (2021) defined the DEA cross-efficiency with probabilistic hesitant fuzzy preference relations. Yang and Xu (2021) defined the measure of probabilistic hesitant I-fuzzy sets and decision making for strategy choice. Song and Chen (2021) extended the COPRAS method to solve MADM problems under probabilistic hesitant fuzzy environment. Liu and Guan (2021) devised a new PHFE comparison method and then defined the comprehensive characteristic distance measure based on four characteristics.

Technique for order performance by similarity to ideal solution (TOPSIS) was originally developed by Hwang and Yoon (1981) for the sake of addressing a MADM problem. Lai et al. (1994) expanded TOPSIS to deal with a diverse objective decision making problem for Bow River Valley water quality management. Chen (2000) proposed the TOPSIS approach for group decision-making within the fuzzy environment. Wang and Elhag (2006) employed the fuzzy TOPSIS approach on the basis of alpha level sets for bridge risk assessment. Taleizadeh et al. (2009) designed a novel method which combined Pareto, TOPSIS and genetic algorithm to solve the multi-product multi-constraint inventory control systems with random fuzzy replenishments. Zhang et al. (2022b) defined the TOPSIS method for spherical fuzzy MAGDM based on cumulative prospect theory. Wei (2010) developed the TOPSIS method to cope with 2-tuple linguistic MAGDM with incomplete weight information. Nilashi et al. (2019) used two MADM techniques, Decision Making Trial and Evaluation Laboratory (DEMATEL) and Fuzzy TOPSIS, to reveal the interrelationships among the factors and to find the relative importance of these factors in the decision making model. In this paper, we extend the TOPSIS method to probabilistic hesitant fuzzy (PHF) environment based on the FUCOM method to deal with the flexible and complicated decision-making circumstance. The following is the innovation of this paper: (1) a novel PHF-TOPSIS method is proposed; (2) the extended TOPSIS method is developed in the probabilistic hesitant fuzzy environment; (3) the FUCOM method is used to obtain the attribute weight; (4) the normalization process of the original data has adapted the latest method to verify the precision.

The whole thread of the article is as follows: Section 2 gives a simple introduction of the PHF information, Section 3 structures the model of TOPSIS and Section 4 illustrates an example for performance evaluation of public charging service quality to prove the practicability of this new method. Section 5 gives a sensitivity analysis and comparison analysis with other existing models.

The first and significant step is the normalization process and we adapt the normalization approach proposed by Li et al. (2020) to break the limitation when processing multiplication of the sets which include different probabilities. Definition 2

Let v ( t i | g i ) = { t i ( g i ) }, v 1 ( t a | g a ) = { t 1 a ( g 1 a ) } and v 2 ( t b | g b ) = { t 2 b ( g 2 b ) } be three PHFEs, i = 1 , 2 , … , # t, a = 1 , 2 , … , # t 1 , b = 1 , 2 , … , # t 2 .

Step 1. Define the first element. If g 1 1 < g 2 1 , then t 1 1 ( g 1 1 ) = t 1 1 ( g 1 1 ) and t 2 1 ( g 2 1 ) = t 2 1 ( g 2 1 ), otherwise, t 1 1 ( g 1 1 ) = t 1 1 ( g 2 1 ) and t 2 1 ( g 2 1 ) = t 2 1 ( g 2 1 ).

Step 2. Determine the second element. If g 1 1 < g 2 1 and g 2 1 − g 1 1 ⩽ g 1 2 , then t 1 2 ( g 1 2 ) = t 1 2 ( g 2 1 − g 1 1 ) and t 2 2 ( g 2 2 ) = t 2 1 ( g 2 1 − g 1 1 ). If g 1 1 < g 2 1 and g 2 1 − g 1 1 > g 1 2 {g_{1}^{2}}$]]>, then t 1 2 ( g 1 2 ) = t 1 2 ( g 1 2 ) and t 2 2 ( g 2 2 ) = t 2 1 ( g 1 2 ). If g 1 1 ⩾ g 2 1 and g 1 1 − g 2 1 ⩽ g 2 2 , then t 1 2 ( g 1 2 ) = t 1 1 ( g 1 1 − g 2 1 ) and t 2 2 ( g 2 2 ) = t 2 2 ( g 1 1 − g 2 1 ). If g 1 1 ⩾ g 2 1 and g 1 1 − g 2 1 ⩾ g 2 2 , then t 1 2 ( g 1 2 ) = t 1 1 ( g 2 2 ) and t 2 2 ( g 2 2 ) = t 2 2 ( g 2 2 ).

Step 3. Determine the third element. If g 1 1 ⩾ g 2 1 , g 1 1 − g 2 1 ⩽ g 2 2 and g 2 1 ⩽ g 2 2 − g 1 1 + g 2 1 , then t 1 3 ( g 1 3 ) = t 1 2 ( g 1 2 ) and t 2 3 ( g 2 3 ) = t 2 2 ( g 1 2 ). If g 1 1 ⩾ g 2 1 , g 1 1 − g 2 1 ⩽ g 2 2 and g 2 1 > g 2 2 − g 1 1 + g 2 1 {g_{2}^{2}}-{g_{1}^{1}}+{g_{2}^{1}}$]]>, then t 1 3 ( g 1 3 ) = t 1 2 ( g 2 2 + g 2 1 − g 1 1 ) and t 2 3 ( g 2 3 ) = t 2 2 ( g 2 2 + g 2 1 − g 1 1 ). If g 1 1 ⩾ g 2 1 , g 1 1 − g 2 1 > g 2 2 {g_{2}^{2}}$]]> and g 1 2 ⩾ g 2 2 + g 2 3 , then t 1 3 ( g 1 3 ) = t 1 2 ( g 2 3 ) and t 2 3 ( g 2 3 ) = t 2 3 ( g 2 3 ). If g 1 1 ⩾ g 2 1 , g 1 1 − g 2 1 > g 2 2 {g_{2}^{2}}$]]> and g 1 2 < g 2 2 + g 2 3 , then t 1 3 ( g 1 3 ) = t 1 2 ( g 1 2 − g 2 2 ) and t 2 3 ( g 2 3 ) = t 2 3 ( g 1 2 − g 2 2 ). If g 1 1 < g 2 1 , g 2 1 − g 1 1 ⩽ g 1 2 and g 1 2 + g 1 1 ⩽ g 2 2 + g 2 1 , then t 1 3 ( g 1 3 ) = t 1 2 ( g 1 2 − g 2 1 + g 1 1 ) and t 2 3 ( g 2 3 ) = t 2 2 ( g 1 2 − g 2 1 + g 1 1 ). If g 1 1 < g 2 1 , g 2 1 − g 1 1 > g 1 2 {g_{1}^{2}}$]]> and g 2 1 + g 1 2 ⩽ g 1 3 + g 1 1 , then t 1 3 ( g 1 3 ) = t 1 3 ( g 2 1 − g 1 1 − g 1 2 ) and t 2 3 ( g 2 3 ) = t 2 1 ( g 2 1 − g 1 1 − g 1 2 ). If g 1 1 < g 2 1 , g 2 1 − g 1 1 > g 1 2 {g_{1}^{2}}$]]> and g 2 1 + g 1 2 > g 1 3 + g 1 1 {g_{1}^{3}}+{g_{1}^{1}}$]]>, then t 1 3 ( g 1 3 ) = t 1 3 ( g 1 3 ) and t 2 3 ( g 2 3 ) = t 2 1 ( g 1 3 ), where g 1 1 + g 1 2 + ⋯ + g 1 # t 1 = 1 and g 2 1 + g 2 2 + ⋯ + g 2 # t 2 = 1, # t 1 = # t 2 .

The MAGDM decision matrix is V c = [ v r x c ( g r x ) ] s × y , and the optional alternatives are defined as Q r = { Q 1 , Q 2 , … , Q s } and the attribute is shown as I x = { I 1 , I 2 , … , I y } and the decision makers (DMs) are defined as c = { c 1 , c 2 , … , c l }, while the weighting vector between the DMs is defined as u c = { 0.3 , 0.3 , 0.4 } and the weighting vector among the criterions is j x which is unknown, ∑ c = 1 l u c = 1, ∑ x = 1 y j x = 1 ( c = i , 2 , … , l ). V c = v 11 c … v 1 x c … v 1 y c ⋮ ⋱ ⋮ ⋱ ⋮ v r 1 c … v r x c … v r y c ⋮ ⋱ ⋮ ⋱ ⋮ v s 1 c … v s x c … v s y c s × y ; r = 1 , … , s , x = 1 , … , y , c = 1 , … , l .

With the above conventions, the operation of the PHF-TOPSIS is as follows: the whole operation flow chart is shown in Fig. 1.

Step 1. Normalize the original decision matrices through Eq. (8). (8) v ¯ r x ( g ¯ r x ) = { t e r ( g e r ) } , if the attribute is positive attribute , v ¯ r x ( g ¯ r x ) = { ( 1 − t r x ) ( g r x ) } , if the attribute is negative attribute . Then using the introduction in Definition 2 to get normalized matrices.

Step 2. Acquire the criterion weights p ˆ x using FUCOM method.

The full consistency (FUCOM) method proposed by Pamučar et al. (2018) is the latest model for weighting the coefficients of attributes. Compared to other methods, such as the best worst method (BWM) and the analytic hierarchy process (AHP) method, the FUCOM method can give more appropriate results when considering the relation between the criterions and the number of comparisons (only y − 1). The application of this new model for criterion weight has been used in numerous fields, such as assessment of alternative fuel vehicles for sustainable road transportation (Pamucar et al., 2021), safety evaluation of road sections (Simić et al., 2020), sustained academic quality assurance and ABET accreditation (Ahmad and Qahmash, 2020) and so on. Bozanic et al. (2020) built the MADM with Z-numbers based on FUCOM and MABAC. Durmić et al. (2020) combined FUCOM-Rough SAW model. Bozanić et al. (2021) built the FUCOM-Fuzzy RAFSI model for selecting the group of construction machines for enabling mobility with D-numbers. Pamucar and Ecer (2020) prioritized the weights of the evaluation criteria under fuzziness through the fuzzy full consistency method-FUCOM-F. Stević and Brković (2020) built novel integrated FUCOM-MARCOS model for evaluation of human resources in a transport company. Simić et al. (2020) built the CRITIC-fuzzy FUCOM-DEA-fuzzy MARCOS model for safety evaluation of road sections based on geometric parameters of road. Fazlollahtabar et al. (2019) defined the FUCOM method in group decision-making for selection of forklift in a warehouse. Durmić (2019) evaluated the criteria for sustainable supplier selection using FUCOM method. Baig et al. (2022) used FUCOM and FQFD for prioritizing the vulnerabilities and identifying those capabilities that can ensure protection against these vulnerabilities.

The specific process to get the weight is as follows:

(i) Rank the criterions and get the set from I x c = { I 1 c , I 2 c , … , I y c } which is according to the relative importance of the criteria. Thus, the parameters rank is obtained by the values of the weight coefficient: I y ( 1 ) c > I y ( 2 ) c > ⋯ > I y ( k ) c , {I_{y(2)}^{c}}>\cdots >{I_{y(k)}^{c}},\]]]> where k denotes the order of the criterions.

(ii) The comparison priorities between the adjacent attributes ψ x − 1 x c , x = 1 , 2 , … , y, which denotes the value of the N y ( x − 1 ) c relative to N y ( x ) c , then we get the set of the criterion comparative preference: ψ c = { ψ 1 2 c , ψ 2 3 c , … , ψ x − 1 x c } .

(iii) Get the final weight coefficients of the attributes p x c = { p 1 c , p 2 c , … , p y c }, which should meet the conditions showing as follows:

Thus, the inequality constraints for this model are shown in Eq. (11): (11) min χ c , s.t. | ψ x − 1 x c − p x − 1 c p x c | ⩽ χ c , | p x − 2 c p x c − ψ x − 2 x − 1 c ⊗ ψ x − 1 x c | ⩽ χ c , ∑ x = 1 y p x c = 1 , p x c ⩾ 0 .

With the help of the MATLAB software, we get the final result of the weighting vector of the evaluation criterions for each DM. Then the integrated weight p ˆ x is finally obtained by geometric means.

Step 3. Integrate the decision matrices by different DMS into one matrix v ˆ r x ( g ˆ r x ) = { t ˆ r x ( g ˆ r x ) ] s × y , using Eq. (12): (12) PHFWA ( v ¯ r x l , v ¯ r x 2 , … , v ¯ r x l ) = ⨁ c = 1 l ( v ¯ r x c u c ) = ⋃ t ¯ r s 1 ∈ v ¯ r s 1 , t ¯ r s 2 ∈ v ¯ r s 2 , … , t ¯ r s l ∈ v ¯ r s l { 1 − ∏ c = 1 l ( 1 − t ¯ r s c ) u c ( g ¯ ) } .

Step 4. Figure out the score function of the integrated decision matrix by using Eq. (13) (13) s ( v ˆ r s ( g ˆ r s ) ) = ∑ i = 1 # t t ˆ r s i g ˆ r s i .

Step 5. Determine the v ˆ x ∗ and v ˆ x − indexes by the following equation by using Eq. (14): (14) v ˆ x ∗ ( g ˆ x ∗ ) = max r s ( v ˆ r x ) , v ˆ x − ( g ˆ x − ) = min r s ( v ˆ r x ) .

Step 6. Calculate the positive and negative distance by using Eqs. (15)–(16) (15) d r ∗ ( v ˆ r x ( g ˆ r x ) , v ˆ x ∗ ( g ˆ x ∗ ) ) = ∑ x = 1 y p ˆ x 1 # t ∑ a = b = 1 # t | t ˆ x a ∗ g ˆ x a ∗ − t ˆ r x b g ˆ r x b | t ˆ x a ∗ g ˆ x a ∗ + t ˆ r x b g ˆ r x b , (16) d r − ( v ˆ r x ( g ˆ r x ) , v ˆ x − ( g ˆ x − ) ) = ∑ x = 1 y p ˆ x 1 # t ∑ a = b = 1 # t | t ˆ x a − g ˆ x a − − t ˆ r x b g ˆ r x b | t ˆ x a − g ˆ x a − + t ˆ r x b g ˆ r x b .

Step 7. Compute out the relative closeness to the ideal solution.

The relative closeness of alternative Q r with the probabilistic hesitant fuzzy positive ideal solution Q ∗ is described by using Eq. (17): (17) H r = d r − ( v ˆ r x ( g ˆ r x ) , v ˆ x − ( g ˆ x − ) ) d r ∗ ( v ˆ r x ( g ˆ r x ) , v ˆ x ∗ ( g ˆ x ∗ ) ) + d r − ( v ˆ r x ( g ˆ r x ) , v ˆ x − ( g ˆ x − ) ) .

Step 8. Get the final ranking.

The final rank is determined by the value of relative closeness, and the sort rule is: the bigger the relative closeness is, the more appropriate the scheme is.

Public service is provided by the public sector to the public at a professional level according to its own social responsibility. In my country, public service is provided to the external public by government departments and institutions with public management functions according to the law. If there are public services, the problem of public service charges will inevitably arise. The public service charges mentioned here do not include taxes, but include administrative management fees, which mainly refer to the specific fees charged by the departments and institutions that provide public services to the public in need of services, so that they can enjoy the right to and benefits of public services. The logic of public service charging is roughly as follows: first, to provide public services, costs will inevitably occur, and fees can compensate for the cost input of public services; second, public services are public goods, and public goods are noncompetitive and non-exclusive, and their external effect is obvious, and it is easy to generate “free-rider” behaviour. Charging fees can curb waste in public service consumption and improve the quality and efficiency of public services. Social investment can increase funding sources for public services. However, since currently in our country public services are mainly provided by government departments, and the government is also responsible for the examination and approval of charging items and charging standards for public services, as well as supervising and inspecting charging behaviours, public services are basically monopoly industries. With low efficiency and poor quality, people feel that public service charges and public service utility are asymmetrical, so there are many criticisms about public service charges. The performance evaluation of public charging service quality is frequently viewed as the multiple attribute group decision-making (MAGDM) issue. In this paper, an extended TOPSIS model is established to provide new means to solve the performance evaluation of public charging service quality. The TOPSIS method integrated with FUCOM method in probabilistic hesitant fuzzy circumstance is applied to rank the optional alternatives and a numerical example for performance evaluation of public charging service quality is used to test the newly proposed method’s practicability in comparison with other methods. Therefore, to illustrate the method presented in this paper, we will give a numeric-based example for performance evaluation of public charging service quality using the PHFSs in this part. Five applicable new public charging service sections A i ( i = 1 , 2 , 3 , 4 , 5 ) are considered. To evaluate the five applicable new public charging service sections by three experts d λ ( λ = 1 , 2 , 3 ), four attributes are given: ①G₁ is the infrastructure; ②G₂ is the urban-rural integration; ③G₃ is the economic development; ④G₄ is the resources and environment. For the performance evaluation of public charging service quality, there are four attributes to be chosen, in which three DMs select the suppliers whose expert weighting is u c = { 0.3 , 0.3 , 0.4 }. The following is the process of numerical example applying this model. The evaluation result is listed in Tables 1–3.

Then, the PHF-TOPSIS method is used to deal with the performance evaluation of public charging service quality with PHFNs.

Step 1. Obtain the normalized matrices (see Tables 4–6).