MADM issue is a very common decision-making problem in real life (Arshad et al., 2022; Mefgouda and Idoudi, 2022; Rahman et al., 2023; Tang et al., 2022; Wang et al., 2024). The typical MADM task is the process of choosing the optimal among the limited alternatives according to certain evaluation criteria by using the decision-making approaches in the corresponding environment. The information of traditional decision problems is expressed by exact numbers. However, with the incompleteness of the data of practical problems and the uncertainty of the actual situation, it is often impossible to express the information by exact numbers. In order to better handle such MADM problems and better express the fuzziness of evaluation data, fuzzy set (FS) (Zadeh, 1965) is applied. In the past, a lot of scholars devoted themselves to the research of uncertain MADM issues. With the continuous deepening of research and the complicacy of the actual MADM tasks, how to effectively expand FS based on FS and solve the corresponding practical problems has always been a hot issue for academic scholars.

Although FS has shown strong advantages in expressing uncertain information, it is still unable to express some more complex cases in real life. In order to express evaluation information more effectively, some extended FSs dependent on FS were developed (Lima et al., 2021; Zhao et al., 2021). Several new types of FSs, such as the intuitionistic FS (IFS) (Atanassov, 1986), picture FS (PIFS) (De Miguel et al., 2017), hesitant FS (HFS) (Torra, 2010), picture HFS (PIHFS) (Hosseinpour and Martynenko, 2021), were developed, and applied in many fields (Aktas et al., 2022; Khatri et al., 2022; Mahmood et al., 2021; Senapati et al., 2023; Yazdi et al., 2022). In practical MADM problems, DMs often give more complex and fuzzy evaluation information when expressing decision information. For example, one class discussed the candidates for scholarships, 35 people agreed, 10 people disagreed, and the other 5 people chose to give up. It is obvious that FS can only express approval of such information, but cannot reflect disapproval and abstention. That is to say, FS cannot express the overall opinion of 50 students in this class. In order to better handle this situation, Atanassov (1986) proposed IFS that can reflect disapproval and waiver. It is obvious that the above situation can be well expressed in IFS. If the exact number in the membership degree (MD) and non-membership degree (NMD) is replaced by the interval number, the corresponding interval valued IFS (IVIFS) (Atanassov and Gargov, 1989) is developed. To deal with more complex practical situations, Ye (2015) proposed triangular IFSs and Liu and Yuan (2007) proposed trapezoidal IFSs. However, in some special cases, it is problematic to integrate decision attribute information with IFS and IVIFS. For example, when a class elects a monitor, generally speaking, the following four opinions will appear in the students’ voting opinions: approval, abstention, objection and refusal to vote (Garg, 2017). In response to this sobriety, Atanassov (1986) built the concept of PIFS, which exactly reflects this situation, so that PIFS can well express the opinions of DMs in the actual cases we put forward above. Many scholars have conducted detailed research on it (Garg, 2017; Son, 2016; Wei, 2016). The SFS (Kahraman and Gündoğdu, 2018) and SHFS (Khan et al., 2021) were proposed to tackle issues in some special and T-spherical fuzzy environments, and they have been utilized in various fields (Akram et al., 2023; Khan et al., 2021; Oezkan et al., 2022; Rajput and Kumar, 2024).

On the other hand, sometimes FS is difficult to accurately describe the membership of decision attribute information. Obviously, FS has some defects. Torra (2010) defined HFS, and MD can be expressed by several possible exact numbers, it can better reflect DMs’ hesitant attitudes. But we also found that HFS cannot reflect the hesitation degree of DM, so it is necessary to use an appropriate variable to describe the hesitation degree of DM. Integrating probabilistic information into HFS, PHFS was developed (Xu and Zhou, 2017). But we also found that HFS cannot reflect the degree of decision-maker hesitation, so it is necessary to use an appropriate variable to describe the hesitation degree of DM. Then if the exact number in the membership degree is replaced by the interval number, the interval-valued HFS (IVHFS) (Chen et al., 2013) was developed. On the basis of IFS and HFS and combining their respective advantages, Dual HFS (DHFS) (Zhu et al., 2012) was developed. Then by replacing the exact number in the MD and NMD by the interval number, the dual IVHFS (DIVHFS) (Farhadinia, 2014) was constructed, and by combining the probability information to DHFS, PDHFS was built (Hao et al., 2017), and it was utilized in many fields (Ning et al., 2022a, 2022b; 2022c). The SHFS (Ning et al., 2023) was proposed to tackle issues in some special fuzzy environments and utilized in various fields.

In MADM problem, we can generally aggregate multiple evaluation information into comprehensive evaluation information by aggregated operator according to the obtained assessment information of decision attributes given by a DM. Therefore, aggregation operator acts a very important role in aggregation information method. For this hot issue, many scholars have carried out detailed research on aggregation problems under various fuzzy information. Such as, Xu and Yager (2006) built the operation of IF number (IFN) and proposed IF geometric aggregation operator. Subsequently, Xu (2007a) proposed an IF weighted average aggregation operator to aggregate IFNs. Next, several IVIF aggregation operators are constructed to deal with MCDM issues (Xu, 2007b; Xu and Chen, 2007). Regarding the evaluation information of picture fuzziness, Wei (2017) defined the operation of PIF number (PIFN) dependent on the research of Xu (2007a), and built a PIF weighted aggregated operator. Moreover, some valued PIF aggregated operators are proposed depended on various operations (Wang et al., 2017). In addition, a large number of HF aggregated operators and their generalized aggregated operators (Xia and Xu, 2011) were built, and some valued aggregated operators (Yu et al., 2016; Zhao and Xu, 2018) under DHF and dual IVHF environments are developed. And some aggregation operators were developed in SF and SHF environments (Ashraf and Abdullah, 2019; Bonab et al., 2023; Diao et al., 2022; Donyatalab et al., 2020; Rajput and Kumar, 2022; Yan et al., 2024).

In practical decision-making problems, evaluation indicators often have different priorities. For example, when parents buy toys for their children, when considering both price and safety, it is obvious that safety is more important than price. However, for MADM problems with different priorities of decision attributes, the aggregation operators mentioned above cannot handle such MADM problems. In order to better handle such MADM problems, the well-known priority average (PA) operator was built by professor Yager (2008). Inspired by PA operator, Yager (2008); Yu (2013); Yu et al. (2012) constructed IF priority fuzzy and IVIF priority fuzzy aggregation operators. In addition, a HF priority aggregation operator was built (Wei, 2012).

Through years of practice and exploration, although FS can better express the evaluation information in MADM tasks, it still cannot handle some more complex MADM problems. The ultimate aim of this study is to act a novel type of FS, which can more accurately express the decision-making information. The SFS (Gündogdu and Kahraman, 2019) has good performance ability in expressing decision information, but faces some special MADM problems, there are still problems that cannot be solved. Therefore, combining the respective advantages of the two FSs (Mahmood et al., 2019; Torra, 2010), the concept of SHFS (Ning et al., 2023) was proposed as a generalized form of FS (Zadeh, 1965), IFS (Atanassov, 1986), HFS (Torra, 2010) and SFS (Gündogdu and Kahraman, 2019). Although both SFS and SHFS have played a very important role and achieved good results in the actual decision-making problem. For example, SFN ( 0.35 , 0.5 , 0.7 ), DMs or experts are skeptical that they can give such accurate values when making decisions. For the hesitation attitude of DMs or experts, it is obvious that SFS cannot cope with such a psychological situation. For such a situation, SHFS can cope well, such as SHFE ⟨ { 0.36 , 0.35 } , { 0.5 , 0.51 } , { 0.6 , 0.7 } ⟩. However, there is a very realistic situation that although SHFE has solved the hesitation of DMs or experts, the DMs or experts are uncertain about the possibility of taking the exact value of the three membership degrees, that is, the probability of several values in SHFE ⟨ { 0.36 , 0.35 } , { 0.5 , 0.51 } , { 0.6 , 0.7 } ⟩. Therefore, it is necessary to propose a new type of fuzzy set that includes both the advantages of SHFS and reflects probability information. This paper proposes a PSHFS that integrates the respective advantages of PHFS and SHFS, which is in line with such advantages, including both the advantages of SHFS and reflecting probability information. It is clear that the PSHFS proposed in this study can well solve this complex problem, this can be well reflected, such as PSHFE { { 0.36 | 0.4 , 0.35 | 0.6 } , { 0.5 | 0.8 , 0.51 | 0.2 } , { 0.6 | 0.3 , 0.7 | 0.7 } }, which can more precisely indicate the indefinite decision-making information in MADM tasks, and better express the uncertainty, fuzziness and complexity in MADM problems. Furthermore, the possible positive-membership hesitant degree, neutral-membership hesitant degree, negative-membership hesitant degree and refusal membership hesitant degree can be represented by multiple values in [ 0 , 1 ] according to the needs of DM. Due to the ripe applications of PA operator in various fuzzy settings, this paper effectively fuses PA operator with the newly proposed PSHFS. Finally, this paper develops PSHFS based on SHFS and PHFS. At the same time, inspired by the operation rules of the above two FSs, the operation rules of PSHFE are developed. Then, based on these operation rules, the GPSHFWA, GPSHFWG, GPSHFPWA and GPSHFPWG operators are proposed, and invest their excellent properties and some special operators. Finally, two new MADM techniques are built dependent on the two built types of aggregated operators to choose the best alternative in practical MADM issues. The effectiveness of the two MADM techniques built is tested through an application example of GECS.

In addition to the above important content, the rest sections are shown as: Section 2 reviews the SHFS, PHFS and PA operator. Section 3 puts forward the PSHFS, and develops the operations of PSHFE, scoring function, accurate function, and comparison method. Section 4 proposes the GPSHFWA, GPSHFWG, GPSHFPWA and GPSHFPWG operators. Two MADM techniques are developed based on the newly built aggregation operator in Section 5. The Section 6 implements the application of two algorithms to the GECS, the comparative analysis and sensitivity analysis are implemented to verify the effectiveness and flexibility of the two built MADM techniques. Finally, Section 7 summarizes all research work and prospects the future work.

Before proposing PSHFS, we first review the PHFS (Xu and Zhou, 2017) and SHFS (Naeem et al., 2021) that can be used later in this paper. The famous PHFS (Xu and Zhou, 2017) was proposed by integrating probability information into HFS, which not only reflects the hesitation of DMs in multiple choices, but also gives consideration to the degree of hesitation. The SHFS (Naeem et al., 2021) combines the common advantages of SFS (Gündogdu and Kahraman, 2019) and HFS (Torra, 2010), and has been widely used in many fields.

For the aggregation of decision information, (Xu and Zhou, 2017) proposes the operation rules of the PHFEs.

Generally, a = { M , N , K } is called an SHFN. In order to facilitate the calculation between SHFNs, inspired by literature (Torra, 2010), Ren et al. (2017) gave the basic operation rules of SHFNs. Definition 4

For three SHFNs α = ⟨ M , N , K ⟩, α 1 = ⟨ M 1 , N 1 , K 1 ⟩ and α 2 = ⟨ M 2 , N 2 , K 2 ⟩, and α C represent the complementary set of α, the followings are valid under the condition λ > 0. α 1 ⊕ α 2 = ⋃ μ 1 ∈ M 1 , v 1 ∈ M 1 , o 1 ∈ M 1 μ 2 ∈ M 2 , v 2 ∈ M 2 , o 2 ∈ M 2 { ⟨ μ 1 2 + μ 2 2 − μ 1 2 μ 2 2 , v 1 v 2 , o 1 o 2 ⟩ } ; α 1 ⊗ α 2 = ⋃ μ 1 ∈ M 1 , v 1 ∈ M 1 , o 1 ∈ M 1 μ 2 ∈ M 2 , v 2 ∈ M 2 , o 2 ∈ M 2 { ⟨ μ 1 μ 2 , v 1 v 2 , o 1 2 + o 2 2 − o 1 2 o 2 2 ⟩ } ; λ · α = ⟨ 1 − ( 1 − μ 2 ) λ , v λ , o λ ⟩ ; α λ = ⟨ μ λ , v λ , 1 − ( 1 − π 2 ) λ ⟩ ; α C = ⟨ K , N , M ⟩ .

According to the PHFS and SHFS, we define a PSHFS composed of positive, neutral, negative and refusal membership hesitant functions belonging to [ 0 , 1 ]. The uncertainty and hesitation of DMs can be expressed better. Definition 6.

A PSHFS M on a non-empty and finite set X is defined by (1) M = { ⟨ x , w ( x ) | p ( x ) , e ( x ) | q ( x ) , r ( x ) | l ( x ) ⟩ | x ∈ X } . The components w ( x ) | p ( x ), e ( x ) | q ( x ) and r ( x ) | l ( x ) are three sets of some possible elements, where w ( x ), e ( x ) and r ( x ) represent the possible positive-membership hesitant degree, neutral-membership hesitant degree, and negative-membership hesitant degree to the set X of x, respectively. p ( x ), q ( x ) and l ( x ) are the probabilities of w ( x ), e ( x ) and r ( x ). There is: 0 ⩽ μ , ν, o ⩽ 1, 0 ⩽ ( μ + ) 2 + ( ν + ) 2 + ( o + ) 2 ⩽ 1; p i ∈ [ 0 , 1 ], q j ∈ [ 0 , 1 ], r k ∈ [ 0 , 1 ]; ∑ i = 1 # w p i ⩽ 1, ∑ j = 1 # e q j ⩽ 1, ∑ k = 1 # r l k ⩽ 1, where μ ∈ w ( x ), ν ∈ e ( x ), o ∈ r ( x ). μ + ∈ w + ( x ) = ⋃ μ ∈ w ( x ) max μ, ν + ∈ e + ( x ) = ⋃ ν ∈ e ( x ) max ν, o + ∈ r + ( x ) = ⋃ o ∈ r ( x ) max o, p i ∈ p, q j ∈ q, l k ∈ l. The symbols # w, # e and # r are total numbers of elements in w ( x ) | p ( x ), e ( x ) | q ( x ) and r ( x ) | l ( x ).

We call m = ⟨ w ( x ) | p ( x ) , e ( x ) | q ( x ) , r ( x ) | l ( x ) ⟩ a probabilistic spherical hesitant fuzzy element (PSHFE) denoted by m = ⟨ w | p , e | q , r | l ⟩, and π = 1 − ∑ i = 1 # w μ i 2 p i − ∑ j = 1 # e v j 2 q j − ∑ k = 1 # r o k 2 l k is a refusal-membership hesitant degree of m. Next, an example of supplier selection is used to explain PSHFS: there are the following four alternatives X a ( a = 1 , 2 , 3 , 4 ) with four criteria: supply capacity ( C 1 ), logistics speed ( C 2 ), product quality ( C 3 ), product price ( C 4 ); all evaluation values are listed in Tables 1, 2, 3 and 4. Each table is named as probabilistic spherical hesitant fuzzy decision matrix (PSHFDM).

Table 1

A PSHFDM M 1 with respect to X 1 .

Table 2

PSHFDM M 2 with respect to X 2 .

Table 3

PSHFDM M 3 with respect to X 3 .

Table 4

PSHFDM M 4 with respect to X 4 .

In the process of using PSHFE for actual MADM problems, we need to rank PSHFEs. Therefore, we will propose indicators for comparing two PSHFEs, namely score function and accurate function of PSHFE, and a theorem for determining the size relationship of two PSHFEs is given.

The GPSHFWA, GPSHFWG, GPSHFPWA and GPSHFPWG operators are built by combining with the definition and operations of PSHFE. Then, some valued properties of them are studied, and some special operators of the above built generalized operators under PSHF setting are presented.