Fully Completed Spherical Fuzzy Approach-Based Z Numbers (PHI Model) for Enhanced Group Expert Consensus

1  Introduction

Recently, numerous scholars have focused on studying the representation of uncertain information, and one notable approach is the utilization of fuzzy set theory, which employs membership degrees (MD) to capture uncertainty. Many extended models have been developed based on classic fuzzy sets. Zadeh [1] introduced the idea of Fuzzy Sets (FSs) in 1965. The concept of FS is to use an MD (α with α∈[0,1]) to evaluate criteria. In several circumstances, the FSs cannot handle knowledge supplied to a person through truth and falsity grades. Therefore, Atanassov [2] developed the theory of Intuitionistic Fuzz Sets (IFSs) by adding the term of a non-membership degree (NMD) denoted by β such that β∈[0,1]. IFS is a comprehensive and robust strategy for dealing with complicated and unreliable data in decision-making settings. Numerous scholars indicated that IFS is a more comprehensive and robust strategy for dealing with complex and unreliable data in decision-making settings than FS. IFS theory has been used by many scholars in various fields [3,4]. However, the IFS cannot handle this if someone offers such values; the sum of MD and NMD exceeds the unit interval. Therefore, based on the weakness of IFS, Yager et al. [5] introduced the concept of Pythagorean fuzzy sets (PyFSs), which have a more flexible condition because they take the square of MD and NMD with α+β∈[0,1]. Due to the flexible conditions of objects, PyFSs can reduce information loss and are widely used by many scholars in various business fields [6,7]. However, if the square of MD and NMD exceeds 1, PyFs cannot handle this object. This is the reason why Yager [8] continuously developed the q-Rung Orthopair Fuzzy Sets (qROFs) with the restriction that the sum of the q-powers for the MD and NMD cannot be greater than the unit interval (αq+βq∈[0,1],q∈Z+). The q-ROFS has received much use and has attracted more interest from researchers because of its structure [9,10].

While q-ROF offers notable advantages, researchers may encounter challenges when assessing information. In numerous real-life scenarios, MD and NMD may fall short in accurately expressing information, often due to instances of abstention and refusal, similar to situations encountered in voting or collecting human opinions. Cuong et al. [11] proposed the Picture Fuzzy Sets (PFSs) to overcome these problems with four degrees, i.e., MD, NMD, an abstinence degree (AD), and refusal degree (RD), with the condition (α+β+γ∈[0,1]), where MD, AD, and NMD are denoted by α,β,and γ, respectively. PFS is a more robust method of handling complex and unreliable information in decision-making difficulties. Since its debut, PFS has drawn the fascination of numerous works [12,13]. Although PFSs can find more information loss than IFSs, PyFs, and q-ROFs, PFSs still have MD, AD, and NMD limitations, making it impossible for decision-makers to voice their opinions independently. Kutlu Gündoğdu et al. [14] recognized this problem and suggested an extension of PFS known as Spherical Fuzzy Sets (SFSs), such that the total of the squares of the MD, AD, and NMD is confined to [0,1] (or α2+β2+γ2∈[0,1]). Compared to PFS, DEs in SFS have more discretion when making decisions. SFSs is currently a helpful tool for evaluating information and has been used in several domains [15]. Since SFSs were introduced, they have attracted the attention of many researchers. Ashraf et al. [16] developed spherical fuzzy t’-norms and spherical fuzzy t’-conorms. It also presented a spherical fuzzy negator and several classes of spherical fuzzy t’-norms and t-conorms, which help develop the aggregation operator that aggregates the spherical fuzzy data. Ali et al. [17] proposed the complex spherical fuzzy Bonferroni mean (CSFBM) and complex spherical fuzzy weighted Bonferroni mean (CSFWBM) operators based on complex spherical fuzzy sets (CSFSs), integrating with the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) to examine the suggested procedures. Güner et al. [18] presented the generalized spherical fuzzy sets application of a notion about aggregation operators from SFSs by introducing Einstein sum, product, and scalar multiplication based on Einstein’s triangular norm and conorm. Ashraf et al. [19] developed Spherical fuzzy Dombi weighted averaging (SFDWA), Spherical fuzzy Dombi ordered weighted averaging (SFDOWA), Spherical fuzzy Dombi hybrid weighted averaging (SFDHWA), Spherical fuzzy Dombi weighted geometric (SFDWG), Spherical fuzzy Dombi ordered weighted geometric (SFDOWG), and Spherical fuzzy Dombi hybrid weighted geometric (SFDHWG) aggregation operators and discuss several properties of these aggregation operators. Furthermore, SFSs are usually combined with the MCDM method due to its advantages. Kutlu Gündoğdu et al. [14] extended traditional Weighted Aggregated Sum Product Assessment (WASPAS), VIseKriterijumska Optimizacija I KOmpromisno Resenje (VIKOR), Analytic Hierarchy Process (AHP), and TOPSIS methods to spherical fuzzy WASPAS (SF-WASPAS), spherical fuzzy VIKOR (SF-VIKOR), spherical fuzzy AHP (SF-AHP), and spherical fuzzy TOPSIS (SF-TOPSIS) methods to deal with the complexity of MCDM problems.

Drawing upon an extensive literature review, this study highlights a significant gap in the current decision-making methodologies, particularly in MCDM. The intrinsic challenge lies in the imperfect nature of available information, characterized by its ambiguity, unreliability, and partiality [18]. While conventional fuzzy numbers offer a mechanism for grappling with uncertainty, they often overlook the crucial aspect of certainty within the data [19]. Addressing this shortfall, Z-numbers emerge as a promising avenue, providing a more nuanced portrayal of incomplete information [20]. These Z-numbers, represented as Z (A, R), encapsulate two critical components: A, denoting the variable limitation, and R, signifying the reliability degree. This conceptualization presents a more profound and adaptable framework for formalizing the intricacies of decision-making processes [21]. As interpreted by Nguyen et al. [22], the Z number generalizes real, interval, random, and fuzzy numbers, thus offering a heightened level of effectiveness in modeling real-world systems. The versatility of Z-number theory extends to its ability to accurately encapsulate incomplete information in a manner akin to natural language expressions [20]. However, the prevailing Delphi method, whether traditional or spherical fuzzy, often falters in adequately factoring in the reliability of expert opinions, thereby jeopardizing the integrity of analysis outcomes. For instance, Mohandes et al. [23] constructed a Pentagonal Fuzzy Delphi Method (PFDM) to determine possible causes of associated accidents on building sites. Similarly, Nguyen’s work utilized the SF-Delphi technique to establish expert consensus on the criteria governing employee satisfaction in the logistics service sector. [24], as well as to validates critical criteria influencing Vietnamese customers’ apartment selection [25].

Nevertheless, both traditional Delphi and SF-Delphi models ignore the inherent uncertainty surrounding expert opinions, which can significantly sway analysis results and influence [22]. Therefore, the primary objective of this study is to lay the groundwork for the pioneering spherical fuzzy Z-number model. This model showcases exceptional adeptness in articulating ambiguous information, thus serving as a valuable tool for informed decision-making amidst uncertainty. Notably, it not only enhances the precision of decision-making data but also embodies qualities of fuzziness, flexibility, and applicability, as evidenced in Table 1. Furthermore, this research introduces the PHI model, founded on fully developed Z-SF laws, and presents it through two distinct approaches: the PHI-based SF approach and the PHI-based Z-numbers approach. Finally, to validate the efficacy of the new model, the study applies it to identify critical barriers to adopting logistics supply chain management (SCM) simulation software within Vietnamese universities.

(i) This research introduces a novel method that surpasses previous Delphi approaches by integrating the advantages of SFSs and Z-numbers. Unlike prior methods [22,25], this new approach not only adeptly handles the ambiguity and uncertainty of information using SFSs but also accounts for the information’s reliability level.

(ii) The proposed combined approach enhances established techniques, analytical precision, and decision-making abilities when investigating MCDM problems. Scholars and policymakers can utilize the two suggested approaches to the new method as a guide when applying them to various study areas, thus advancing the field’s analytical capabilities and decision-making efficacy.

2  Preliminaries and Basic Theory

This section introduces several fundamental definitions and operations that played a crucial role in shaping the suggested work.

Definition 1 [1]: Suppose F is the universal set, then the fuzzy set is defined as:

X¯={⟨f,μF¯(f)⟩|f∈F}

where μF¯(f) is a membership degree of f in X¯ and μF¯:N→[0,1].

Definition 2 [27]: A Z-number is an ordered pair of fuzzy numbers (A,R). The A component is the membership function, while R is the reliability of the A.

Definition 3 [22]: By applying the concept of fuzzy expected value, the Z-number can be converted to a fuzzy number:

EA(f)=∫FfμF¯(f)df(1)

First, the Z-number (R) is transformed into a numerical value as the following geometrical:

ξ=∫fμR(f)df∫μR(f)df(2)

The computed weight ξ is added to the first component (A):

X¯ξ={(f,μAξ(f)),μAξ(f)=ξ.μA(f),f∈ξF}(3)

Then, the obtained fuzzy number X¯′:

X¯′={(f,μX¯′(f)),μX¯′(f)=(fξ).μX¯′(f),f∈ξF}(4)

Definition 4 [14]: Let F be the universal set; the spherical fuzzy set is defined as follows:

X¯={⟨f,(σ(f),ς(f),τ(f))⟩|f∈F},

where σ(f)→[0,1],ς(f)→[0,1], and τ(f)→[0,1] be the degree of membership, non-membership, and hesitancy of f to X¯ for each f, respectively, with the condition that 0≤σ2(f)+ς2(f)+τ2(f)≤1.

Definition 5 [21]: Let F be the universal set, then Spherical fuzzy Z-numbers are defined as the subsequent form:

Xz={⟨f,(σXz,ςXz,τXz),(ϖXz,ρXz,ηXz)⟩|f∈N},

where σAz,ςAz,τAz are the membership, non-membership, and hesitancy degrees of A component, while ϖAz,ρAz,ηAz are membership the membership, non-membership, and hesitancy degrees of R component; σAz→[0,1],ςAz→[0,1], and τAz→[0,1]; and ϖXz→[0,1],ρXz→[0,1], and ηXz→[0,1]. And:

0≤σXz2+ςXz2+τXz2≤1,and

0≤ϖXz2+ρXz2+τXz2≤1

Definition 6 [21]: Suppose Xz=((σXz,ςXz,τXz),(ϖXz,ρXz,ηXz)) and Yz=((σYz,ςYz,τY),(ϖYz,ρYz,ηYz)) be two SFZNs and λ>0. Then, by the following relations:

Union

Xz∪Yz=((max(σXz,σYz),min(ςXz,ςYz),min(1−max(σXz,σYz)2+min(ςXz,ςYz)2),max(τXz,τY)),(max(ϖXz,ϖYz),min(ρXz,ρYz),min(1−max(ϖXz,ϖYz)2+min(ρXz,ρYz)2),max(ηXz,ηYz)))(5)

Intersection

Xz∩Yz=((min(σXz,σYz),max(ςXz,ςYz),max(1−min(σXz,σYz)2+max(ςXz,ςYz)2),min(τXz,τY)),(min(ϖXz,ϖYz),max(ρXz,ρYz),max(1−min(ϖXz,ϖYz)2+max(ρXz,ρYz)2),min(ηXz,ηYz)))(6)

Addition

Xz⊕Yz=(((σXz2+σYz2−σXz2.σYz2),ςXz.ςYz,(1−σYz2)τXz2+((1−σXz2)τYz2)−τXz2.τYz2),((ϖXz2+ϖYz2−ϖXz2.ϖYz2),ρXz.ρYz,(1−ϖYz2)ηXz2+((1−ϖXz2)ηYz2)−ηXz2.ηYz2))(7)

Multiplication

Xz⊗Yz=((σXz.σYz,ςXz2+ςYz2−ςXz2.ςYz2,(1−ςYz2)τXz2+(1−ςXz2)τYz2−τXz2.τYz2),(ϖXz.ϖYz,ρXz2+ρYz2−ρXz2.ρYz2,(1−ρYz2)ηXz2+(1−ρXz2)ηYz2−ηXz2.ηYz2))(8)

Multiplication by a scalar; λ>0

λ⋅Xz=((1−(1−σXz2)λ,ςXzλ,(1−σXz2)λ−(1−σXz2−τXz2)λ),(1−(1−ϖXz2)λ,ρXzλ,(1−ϖXz2)λ−(1−ϖXz2−ηXz2)λ))(9)

Power of Xz;λ>0

Xzλ=((σXzλ,1−(1−ςXz2)λ,(1−ςXz2)λ−(1−ςXz2−τXz2)λ),(ϖXzλ,1−(1−ρXz2)λ,(1−ρXz2)λ−(1−ρXz2−ηXz2)λ))(10)

Definition 7 [21]: Spherical fuzzy Z-number weighted arithmetic (SFZNWA) with respect to λ=(λ1,λ2,…,λn);λi∈[0,1];∑i=1nλi=1, SFZNWA is defined as:

SFZNWAλ(Xz1,…,Xzn)=∑i=1nλ1.Xzi=λ1.Xz1+λ2.Xz2+…+λn.Xzn(11)

=((1−∏i=1n(1−σXzi2)λi,∏i=1nςXziλi,∏i=1n(1−σXzi2)λi−∏i=1n(1−σXzi2−τXzi2)λi),(1−∏i=1n(1−ϖXzi2)λi,∏i=1nρXziλi,∏i=1n(1−ϖXzi2)λi−∏i=1n(1−ϖXzi2−ηXzi2)λi))

Definition 8 [21]: Spherical fuzzy Z-number weighted geometric (SFZNWG) with respect to λ=(λ1,λ2,…,λn);λi∈[0,1];∑i=1nλi=1, SFZNWG is defined as:

SFZNWGλ(Xz1,…,Xzn)=∏i=1nXziλi=Xz1λ1⋅Xz2λ2⋅…⋅Xziλi(12)

=((∏i=1nσXziλi,1−∏i=1n(1−ςXzi2)λi,∏i=1n(1−ςXzi2)λi−∏i=1n(1−ςXzi2−τXzi2)λi),(∏i=1nϖXziλi,1−∏i=1n(1−ρXzi2)λi,∏i=1n(1−ρXzi2)λi−∏i=1n(1−ρXzi2−ηXzi2)λi))

Definition 9 [14]: SFZNs are transformed into Spherical fuzzy numbers XF

XF=((σXz,ςXz,τXz)⊗(ϖXz,ρXz,ηXz))

=(σXz.ϖXz,ςXz2+ρXz2−ςXz2.ρXz2,(1−ρXz2)τXzi2+(1−ςXzi2)ηXzi2−τXzi2.ηXzi2)(13)

3  Proposed Model of PHI Model-Based SFZNs

3.1 PHI-Based SFSs Approach

The expert’s evaluations can be displayed as Z-numbers, indicating the boundary’s certainty and value for the required problem. With the combination of SFZNs, the author gathered the experts’ views of the concerned topic and the reliability level of their rate using the linguistic scales in Tables 2 and 3.

Step 1: Experts Ei=(1,2,…,n) declare their evaluations, giving SFZ numbers using Eq. (14):

Xzi=((σXzi,ςXzi,τXzi),(ϖXzi,ρXzi,ηXzi))(14)

where the (σXz,ςXz,τXz) component is the rank of the criteria using the linguistic terms listed in Table 2, while (ϖXz,ρXz,ηXz) component is the level of reliability using the linguistic terms listed in Table 3.

Step 2: The SFZNWA Eq. (11) or SFZNWG operator Eq. (12) is used to obtain the significance level of each indicator, which is displayed in Eq. (15):

Uzagg=[((σXz11,ςXz11,τXz11),(ϖXz11,ρXz11,ηXz11))⋯((σXz1m,ςXz1m,τXz1m),(ϖXz1m,ρXz1m,ηXz1m))⋮⋮⋮((σXzn1,ςXzn1,τXzn1),(ϖXzn1,ρXzn1,ηXzn1))⋯((σXznm,ςXznm,τXznm),(ϖXznm,ρXznm,ηXznm))](15)

Step 3: Transform SFZNs into SFSs numbers using Eqs. (13) and (16).

USagg=[(μ11,ν11,π11)⋯(μ1m,ν1m,π1m)⋮⋱⋮(μn1,νn1,πn1)⋯(μnm,νnm,πnm)](16)

Step 4: Transforming SFSs numbers into crisp numbers by using Eq. (17):

Score(di)=(2μij−πij)2−(νij−πij)2(17)

Step 5: Calculate the threshold by Eq. (18) and validate the criteria.

D=∑i=1ndim(18)

If di<D, criterion Ci is removed, and if di>D, criterion Ci is valid.

3.2 PHI-Based Z-Number Approach

Step 1: This step is the same as Step 1 of the PHI-Based SFSs Approach.

Step 2: The Z-numbers are converted into SFSs numbers by Eqs. (2)–(4). The resulting fuzzy numbers take the following form:

X'¯zi=(μX'¯zi,vX'¯zi,πX'¯zi)(19)

Step 3: Eq. (16) presents the SFZ decision matrix.

Step 4: The utilization of either the SFZNWA Eq. (11) or the SFZNWG operator Eq. (12) is applied to determine the significance level of the criterion, as illustrated in Eq. (20):

USagg=[(μ11agg,ν11agg,π11agg)⋯(μ1magg,μ1magg,μ1magg)⋮⋱⋮(μn1agg,νn1agg,πn1agg)⋯(μnmagg,μnmagg,μnmagg)](20)

Step 5: Transforming Spherical fuzzy numbers into crisp numbers by using Eq. (17).

Step 6: Calculate the threshold by Eq. (18) and validate the criteria. If di<D, criterion Ci is removed, and if di>D, criterion Ci is valid.

4  Case Study

Adopting logistics Supply Chain Management (SCM) simulation software in Vietnamese universities has encountered various barriers, hindering its widespread implementation. Many universities in Vietnam may not fully comprehend how these software solutions can enhance the learning experience for students in logistics and supply chain management programs. Financial constraints pose a significant challenge, as investing in advanced simulation software requires a substantial upfront cost. Moreover, there is a shortage of skilled professionals who can effectively integrate and utilize SCM simulation software in educational settings. Overcoming these barriers will necessitate targeted efforts to raise awareness, secure funding, and provide adequate training to educators, thereby fostering a conducive environment for integrating logistics SCM simulation software in Vietnamese universities. Through a comprehensive literature review, Table 4 highlights various challenges organizations face when trying to adopt simulation software, such as EPR (Enterprise Resource Planning) or logistics Supply Chain Management.

4.1 Demographic of Experts

A questionnaire comprising 20 barriers associated with adopting the logistic SCM simulation platform was distributed to 13 experts within the educational domain. Responses were received from 10 experts, and Table 5 provides an overview of their demographics.

Based on the information provided in Table 5, it can be affirmed that the selected experts possess diverse experiences, educational backgrounds, and professional positions within the educational domain. This diversity contributes to a well-rounded and comprehensive evaluation of the barriers to adopting the logistic SCM simulation platform. The linguistic evaluations and weights assigned by each expert further underscore the suitability of this expert panel in offering valuable insights for this study.

4.1.1 Results of PHI-Based SFSs Approach

After collecting experts’ evaluations, the author transforms this information into SFZNs numbers based on Tables 2 and 3. The SFZNWA–Eqs. (11) and (15) are used to obtain the significance vector for each barrier. The SFZN significance vectors are transformed into SFSs numbers by Eq. (16), and SFSs numbers are continually transformed into crisp numbers using Eq. (17). Finally, the threshold is calculated by Eq. (18) to validate the barriers. The results are displayed in Table 6. Similarly, the SFZNWG results are shown in Table 7.

The Spearman correlation (Eq. (21)) is applied to investigate the relationship between two operators, SFZNWA and SFZNWG, rs= 0.727 (higher than 0.7). Therefore, the ranking results of SFZN-Delphi Using the SFs Approach by two operators have a high level of consistency. Furthermore, according to Tables 5 and 6, both SFZN-Delphi of SFZNWA and SFZNWG indicated that there are 15 barriers appropriate to the research scope, and five are eliminated. Specifically, the barriers C2, C4, C6, C18, and C20 were not suitable for adopting SCM simulation platforms for logistics in universities in the context of Vietnam.

rs=1−6∑(Rxi−Ryi)2n(n2−1)(21)

4.1.2 Results of PHI-Based Z-Numbers Approach

Unlike SFZN-Delphi, using the SFs approach, the reliability components were presented through Eq. (19) after collecting expert evaluation. Then, the author took the square of ξ to multiply with restriction components by Eq. (16). Then, the SFZNWA (Eq. (11)), SFZNWG operator (Eq. (12)), and Eq. (20) obtain the significance vector for each barrier. The results are shown in Tables 8 and 9. Like SFZN-Delphi Using SFs Approach, The Spherical fuzzy numbers are transformed into crisp numbers using Eq. (17). Finally, the threshold is calculated by Eq. (18) to validate the barriers.

The Spearman correlation in Eq. (21) is applied to investigate the relationship between two operators, SFZNWA and SFZNWG, rs= 0.883 (higher than 0.7). Therefore, the ranking results of SFZN-Delphi Using the Z-Numbers Approach by two operators have a high level of consistency. However, based on Table 9, the results of the SFZN-Delphi-Based Z-Numbers Approach by the two operators are slightly different. There are seven barriers eliminated by the SFZNWA operator (C2, C3, C4, C5, C6, C18 and C20) and eight barriers eliminated by the SFZNWG operator (C2, C3, C4, C5, C6, C13, C18 and C20).

4.2 Comparative Analysis

To investigate the appropriateness of the newly proposed methods, the author used three spherical Z-number approaches proposed by Alkan et al. [28] to assess the relationships between various Delphi techniques.

4.2.1 The First Is Named SFZN-Delphi, and It Has Defuzzified Restriction and Reliability Functions

First, after collecting expert evaluations, we defuzzify the reliability components in each pairwise comparison matrix using Eq. (22) and normalize the obtained values using Eq. (23). Then, we multiply the SF restriction values in each pairwise comparison matrix by the square root of the normalized reliability values using Eq. (24). Third, Eqs. (11) or (12) is used to aggregate the values in pairwise comparison matrices obtained in the previous step. Finally, the score function is found using Eq. (22) and compared with the threshold by Eq. (18) to validate the criteria.

S(w~js)=|100⋅[(3μA~s−πA~s2)2−(ϑA~s2−πA~s)2]|(22)

w~js=S(w~js)∑j=1nS(w~js)(23)

kA~s=(1−(1−μA~s2)k,ϑA~sk,(1−μA~s2)k−(1−μA~s2−πA~s2)k)(24)

4.2.2 The Second Was Named SFZN-Delphi with Aggregated and Defuzzified Reliability Function

First, the components in each pairwise comparison matrix are aggregated and defuzzified using Eqs. (11) or (12) and (22), respectively, and normalize the obtained values using Eq. (23). Secondly, we multiply the aggregated restriction vector by the square root of the normalized reliability values using Eq. (24). Finally, the score function is found using Eq. (22) and compared with the threshold by Eq. (18) to validate the criteria.

4.2.3 The Third Is Named Fully Completed PHI-Based SFZNs Models

Firstly, we compute the square root of each SF number in the reliability matrix. Then, we multiply the SF restriction values in pairwise comparison matrices by the values obtained before using Eq. (25). Similarly to the previous methods, the score function is found using Eq. (22) and compared with the threshold using Eq. (18) to validate the criteria.

A~s⊗B~s=(μA~s,μB~s,ϑA~s2+ϑB~s2−ϑA~s2ϑB~s2,(1−ϑB~s2)πA~s2+(1−ϑA~s2)πB~s2−πA~s2πB~s2)(25)

The results of the three approaches for using SWAM and SWGM operators are presented in Tables 10 and 11, respectively. The results show that, from the perspective of the SWAM operator, the first and second approaches have 14 appropriate and six inappropriate barriers (C2, C3, C4, C6, C18 and C20). The third approach has 15 appropriate and five unacceptable barriers (C2, C4, C6, C18, and C20). The results of the third approach are consistent with those of the SFZN-Delphi approach. In SWGM, the three approaches have 15 appropriate and five inappropriate barriers (C2, C4, C6, C18, and C20).

Figs. 1 and 2 compare the results obtained using these three methods. The comparison reveals that the value of the crisp numbers of these barriers is similar across the approaches. However, there are some slight differences in the ranking of these barriers. Specifically, the ranking of the SWAM operator is more stable than that of the SWGM operator.

Figure 1: Comparative analysis based on rank

Figure 2: Comparative analysis based on crisp numbers

To comprehensively understand comparative analysis, we used the Spearman correlation to compare the ranking of five approaches. Table 12 presents the correlation of the PHI-Based SFSs Approach with the other approaches, and Table 13 demonstrates the correlation of SFZN-Delphi Using the Z-numbers Approach with the different approaches. The results show that all correlation values above 0.7 indicate a strong relationship among the five approaches. The smallest correlation value of the PHI-Based SFSs Approach with the other approaches is 0.867 and 0.819 with the PHI-Based Z-numbers Approach with the different approaches.

4.3 Discussions

This study combined the advantages of the Spherical Fuzzy Sets and Z-Numbers to develop the new PHI model with two approaches. Based on the results in Section 4.2, the SFS approach indicated 15 appropriate and five rejected barriers in both SWAM and SWGM ways. However, the Z-numbers approach revealed 13 appropriate barriers, seven left barriers in SWAM, and only 12 appropriate barriers and eight rejected barriers to adopting logistics SCM simulation software in Vietnamese universities. The PHI-based Z-numbers approach utilized Kang et al. [29] methodology, which involves converting a Z-number into a fuzzy number. This method is advantageous due to its easy computational and analytic complexity, which permits a broad scope of application. Unfortunately, when Z-numbers are converted to imprecise numbers, a substantial amount of original information is lost, which diminishes the utility of Z-number-based details in its initial situation [30,31]. Therefore, the SFZN-Delphi using the Z-numbers app PHI-based Z-numbers approach roach eliminates more factors than the other method.

In contrast, the PHI-based SFSs approach completely applies the properties of Spherical Fuzzy Sets. Therefore, this method is more complicated than the PHI-based Z-numbers approach due to their equations and operators. In this approach, the researcher utilizes more time and assets to address the MCDM challenges. However, as previously declared regarding the advantages of SFS in Section 1, this method will provide an expanded outcome compared to the app PHI-based Z-numbers approach with a typical spherical observation angle. The data will be thoroughly analyzed to maximize the quality of the information obtained. Depending on the resources and research purposes, researchers can apply appropriate approaches.

Compared with Alkan’s methods, the PHI-based SFSs approach has results similar to all three. The similarities between all four methods can be explained by their reliance on the SFS formula. At the same time, their results provide additional evidence to eliminate 5 barriers unsuitable for adopting the logistics SCM simulation software in universities in Vietnam. A set of 15 barriers is confirmed to be the most critical barrier to preventing the process of adopting the logistics SCM simulation software in universities in Vietnam. Each barrier represents a significant obstacle that must be carefully addressed to facilitate successful adoption and integration. High upfront costs (C1) emerge as a prominent barrier, highlighting the financial strain associated with implementing such software solutions. Integration challenges (C3) further complicate matters, indicating the difficulty in seamlessly integrating the software with existing systems and processes. Complexity (C5) underscores the ins and outs involved in navigating and utilizing the software effectively, while data security concerns (C7) raise valid apprehensions regarding protecting sensitive information. Theoretical results (C8) signify a gap between theoretical understanding and practical application, suggesting enhanced training and support mechanisms are needed. Long sales cycles (C9) and implementation time (C10) reflect the prolonged and resource-intensive nature of the adoption process, posing logistical challenges for universities. Computational power (C11) emerges as a critical consideration, highlighting the necessity of robust infrastructure to support the software’s functionalities. The lack of industry-specific templates (C12) further exacerbates implementation challenges, necessitating customized solutions tailored to the unique needs of the education sector. The inability to adapt to business changes (C13) underscores the importance of software flexibility and scalability, while opportunity cost (C14) underscores the trade-offs associated with investment in the software. Uncertain ROI (C15) raises doubts regarding the software’s potential benefits, emphasizing the need for clear and quantifiable outcomes. The lack of trust (C16) and resistance to change (C17) represent psychological barriers that must be addressed through effective communication and stakeholder engagement. Customization complexity (C19) and automation concerns (C20) highlight technical challenges that must be navigated to ensure smooth adoption and utilization.

5  Conclusions, Implications, Limitations, and Future Research

5.1 Conclusions

The intricacies of information inherently challenge the decision-making process, encompassing complexity, vagueness, and uncertainty. This study represents a pioneering effort in considering the PHI method from dual perspectives, incorporating its restriction and reliability components. Through the presentation of two distinct approaches—the PHI-based SFSs approach and the PHI-based Z-numbers approach—the author adeptly delineates each methodology’s primary advantages and limitations. Furthermore, a comprehensive comparative analysis is conducted against three alternative approaches proposed by Alkan et al. [28], underscoring their appropriateness within the context of the study. The outcomes of this comparison reveal noteworthy insights. The SFZN-Delphi, using the Z-numbers approach, encounters 13 barriers when applying SWAM operators and 12 obstacles with SWGM operators, constituting 20 identified barriers in total. Conversely, the PHI-based SFSs approach, integrating both SWAM and SWGM operators, identifies 15 barriers, aligning closely with the proposed research framework. Notably, these findings parallel those observed in SFZN-Delphi with Defuzzified Restriction and Reliability Functions (SWGM operator), SFZN-Delphi with Aggregated and Defuzzified Reliability Function (SWGM operator), and Fully Completed PHI-based SFZNs models across both operator scenarios. This nuanced analysis contributes significantly to understanding decision-making methodologies, offering valuable insights for future research and practical applications.

5.2 Practical Implications

The study’s implications are far-reaching, offering valuable insights into enhancing decision-making processes by introducing a new method based on SFSs and Z-numbers. By comparing these approaches with existing methods, the study suggests they can improve decision quality, providing decision-makers with additional tools to navigate complexity, vagueness, and uncertainty. Beyond conventional domains, the interdisciplinary applications of these methodologies span a multitude of fields, including economics, engineering, healthcare, and environmental management. Organizations stand to benefit immensely from leveraging these findings to optimize their strategic decision-making processes, thereby gaining a competitive edge in volatile business environments. Moreover, policymakers can harness the power of advanced decision methodologies to inform policy frameworks, effectively addressing complex societal challenges and enhancing public governance structures. Educational endeavors are crucial in disseminating knowledge and fostering proficiency in applying SFSs and Z-numbers, ensuring their practical utilization in real-world decision-making scenarios.

5.3 Limitations and Future Research

While this study offers significant insights, it is essential to acknowledge its limitations, which should be addressed in future research endeavors. Firstly, the study’s exclusive focus on comparing two new PHI methods based on SFSs and Z-numbers may limit its broader applicability by potentially overlooking other relevant decision-making methodologies. Moreover, methodological biases in selecting and implementing comparison methods could impact the efficacy of the proposed approaches. Additionally, the quality and quantity of available data for analysis may influence the validity of the conclusions, emphasizing the necessity for robust data collection and analysis techniques in future studies. Furthermore, the generalizability of the findings might be constrained by the specific context or domain in which they were tested. To mitigate these limitations, future research avenues could explore the integration of SFSs and Z-numbers with other MCDM methods such as DEMATEL, VIKOR, and TOPSIS. Furthermore, applying the proposed approaches to diverse decision-making problems across various industries and domains could enhance their applicability and effectiveness. Refinement of the methodologies to address inherent limitations and biases is also crucial for improving their efficacy. Longitudinal studies could be conducted to assess the long-term impacts of implementing these methodologies in real-world scenarios. Lastly, involving stakeholders in the development and validation process of these methodologies could provide valuable insights and ensure their practical relevance and acceptance.

Acknowledgement: The author appreciates Thu-Hien Tran from the National Taipei University of Technology, Taiwan, for her hard work and support.

Funding Statement: The author received no specific funding for this study.

Availability of Data and Materials: The data that support the findings of this study are available from the corresponding author, Phi-Hung Nguyen, upon reasonable request.

Conflicts of Interest: The author declares that there are no conflicts of interest related to this study.

References

1. L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, 1965. doi: 10.1016/S0019-9958(65)90241-X. [Google Scholar] [CrossRef]

2. K. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 20, no. 1, pp. 87–96, 1986. doi: 10.1016/S0165-0114(86)80034-3. [Google Scholar] [CrossRef]

3. G. Büyüközkan and F. Göçer, “Application of a new combined intuitionistic fuzzy MCDM approach based on axiomatic design methodology for the supplier selection problem,” Appl. Soft. Comput., vol. 52, no. 3, pp. 1222–1238, 2017. doi: 10.1016/j.asoc.2016.08.051. [Google Scholar] [CrossRef]

4. M. G. A. Malik, Z. Bashir, T. Rashid, and J. Ali, “Probabilistic hesitant intuitionistic linguistic term sets in multi-attribute group decision making,” Symmetry, vol. 10, no. 9, pp. 392, 2018. doi: 10.3390/sym10090392. [Google Scholar] [CrossRef]

5. R. R. Yager and A. M. Abbasov, “Pythagorean membership grades, complex numbers, and decision making,” Int. J. Intell. Syst., vol. 28, no. 5, pp. 436–452, 2013. doi: 10.1002/int.21584. [Google Scholar] [CrossRef]

6. K. Ullah, T. Mahmood, Z. Ali, and N. Jan, “On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition,” Complex Intell. Syst., vol. 6, no. 1, pp. 15–27, 2020. doi: 10.1007/s40747-019-0103-6. [Google Scholar] [CrossRef]

7. J. Mahanta and S. Panda, “Distance measure for Pythagorean fuzzy sets with varied applications,” Neur. Comput. Appl., vol. 33, no. 24, pp. 17161–17171, 2021. doi: 10.1007/s00521-021-06308-9. [Google Scholar] [CrossRef]

8. R. R. Yager, “Generalized orthopair fuzzy sets,” IEEE Trans. Fuzzy Syst., vol. 25, no. 5, pp. 1222–1230, 2016. doi: 10.1109/TFUZZ.2016.2604005. [Google Scholar] [CrossRef]

9. H. Garg, Z. Ali, and T. Mahmood, “Generalized dice similarity measures for complex q-rung orthopair fuzzy sets and its application,” Complex Intell. Syst., vol. 7, no. 2, pp. 667–686, 2021. doi: 10.1007/s40747-020-00203-x. [Google Scholar] [CrossRef]

10. G. Tang, F. Chiclana, and P. Liu, “A decision-theoretic rough set model with q-rung orthopair fuzzy information and its application in stock investment evaluation,” Appl. Soft. Comput., vol. 91, no. 1, pp. 106212, 2020. doi: 10.1016/j.asoc.2020.106212. [Google Scholar] [CrossRef]

11. B. C. Cuong and V. Kreinovich, “Picture fuzzy sets,” J. Comput. Sci. Cybern., vol. 30, no. 4, pp. 409–420, 2014. [Google Scholar]

12. F. Göçer, “A novel interval value extension of picture fuzzy sets into group decision making: An approach to support supply chain sustainability in catastrophic disruptions,” IEEE Access, vol. 9, pp. 117080–117096, 2021. doi: 10.1109/ACCESS.2021.3105734. [Google Scholar] [CrossRef]

13. V. Simić, R. Soušek, and S. Jovčić, “Picture fuzzy MCDM approach for risk assessment of railway infrastructure,” Mathematics, vol. 8, no. 12, pp. 2259, 2020. doi: 10.3390/math8122259. [Google Scholar] [CrossRef]

14. F. Kutlu Gündoğdu, and C. Kahraman, “A novel spherical fuzzy analytic hierarchy process and its renewable energy application,” Soft. Comput., vol. 24, no. 6, pp. 4607–4621, 2020. doi: 10.1007/s00500-019-04222-w. [Google Scholar] [CrossRef]

15. F. Kutlu Gundogdu and C. Kahraman, “Extension of WASPAS with spherical fuzzy sets,” Informatica, vol. 30, no. 2, pp. 269–292, 2019. doi: 10.15388/Informatica.2019.206. [Google Scholar] [CrossRef]

16. S. Ashraf, S. Abdullah, M. Aslam, M. Qiyas, and M. A. Kutbi, “Spherical fuzzy sets and its representation of spherical fuzzy t-norms and t-conorms,” J. Intell. Fuzzy Syst., vol. 36, no. 6, pp. 6089–6102, 2019. doi: 10.3233/JIFS-181941. [Google Scholar] [CrossRef]

17. Z. Ali, T. Mahmood, and M. S. Yang, “TOPSIS method based on complex spherical fuzzy sets with Bonferroni mean operators,” Mathematics, vol. 8, no. 10, pp. 1739, 2020. doi: 10.3390/math8101739. [Google Scholar] [CrossRef]

18. E. Güner and H. Aygün, “Generalized spherical fuzzy Einstein aggregation operators: Application to multi-criteria group decision-making problems,” in Conf. Proc. Sci. Technol, Murat TOSUN, 2020, pp. 227–235. [Google Scholar]

19. S. Ashraf, S. Abdullah, and T. Mahmood, “Spherical fuzzy Dombi aggregation operators and their application in group decision making problems,” J. Ambient Intell. Humaniz. Comput., vol. 11, no. 7, pp. 2731–2749, 2020. doi: 10.1007/s12652-019-01333-y. [Google Scholar] [CrossRef]

20. H. G. Peng, X. K. Wang, and J. Q. Wang, “New MULTIMOORA and pairwise evaluation-based MCDM methods for hotel selection based on the projection measure of Z-numbers,” Int. J. Fuzzy Syst., vol. 24, no. 1, pp. 371–390, Feb. 2022. doi: 10.1007/s40815-021-01141-7. [Google Scholar] [CrossRef]

21. S. Ashraf, M. Sohail, A. Fatima, and S. M. Eldin, “Evaluation of economic development policies using a spherical fuzzy extended TODIM model with Ž-numbers,” PLoS One, vol. 18, no. 6, pp. e0284862, 2023. doi: 10.1371/journal.pone.0284862. [Google Scholar] [CrossRef]

22. P. H. Nguyen et al., “Z-number based fuzzy MCDM models for analyzing non-traditional security threats to finance supply chains: A case study from Vietnam,” Heliyon, vol. 10, no. 11, pp. e31615, Jun. 2024. doi: 10.1016/j.heliyon.2024.e31615. [Google Scholar] [CrossRef]

23. S. R. Mohandes et al., “Causal analysis of accidents on construction sites: A hybrid fuzzy Delphi and DEMATEL approach,” Saf. Sci., vol. 151, no. 10, pp. 105730, 2022. doi: 10.1016/j.ssci.2022.105730. [Google Scholar] [CrossRef]

24. P. H. Nguyen, “A fully completed spherical fuzzy data-driven model for analyzing employee satisfaction in logistics service industry,” Mathematics, vol. 11, no. 10, pp. 2235, May 2023. doi: 10.3390/math11102235. [Google Scholar] [CrossRef]

25. P. H. Nguyen, T. H. Tran, L. A. Thi Nguyen, H. A. Pham, and M. A. Thi Pham, “Streamlining apartment provider evaluation: A spherical fuzzy multi-criteria decision-making model,” Heliyon, vol. 9, no. 12, pp. e22353, Dec. 2023. doi: 10.1016/j.heliyon.2023.e22353. [Google Scholar] [CrossRef]

26. R. R. Yager, “Pythagorean fuzzy subsets,” in 2013 Joint IFSA World Cong. NAFIPS Annu. Meet. (IFSA/NAFIPS), 2013, pp. 57–61. [Google Scholar]

27. L. A. Zadeh, “A note on Z-numbers,” Inf. Sci., vol. 181, no. 14, pp. 2923–2932, 2011. doi: 10.1016/j.ins.2011.02.022. [Google Scholar] [CrossRef]

28. N. Alkan and C. Kahraman, “Fuzzy analytic hierarchy process using spherical Z-numbers: Supplier selection application,” in Int. Conf. Intell. Fuzzy Syst., Izmir, Turkey, Springer, 2022, pp. 702–713. [Google Scholar]

29. B. Kang, D. Wei, Y. Li, and Y. Deng, “A method of converting Z-number to classical fuzzy number,” J. Inf. Comput. Sci., vol. 9, no. 3, pp. 703–709, 2012. [Google Scholar]

30. K. W. Shen and J. Q. Wang, “Z-VIKOR method based on a new comprehensive weighted distance measure of Z-number and its application,” IEEE Trans. Fuzzy Syst., vol. 26, no. 6, pp. 3232–3245, Dec. 2018. doi: 10.1109/TFUZZ.2018.2816581. [Google Scholar] [CrossRef]

31. M. Abdullahi, T. Ahmad, and V. Ramachandran, “A review on some arithmetic concepts of Z-number and its application to real-world problems,” Int. J. Inf. Technol. Decis. Mak., vol. 19, no. 4, pp. 1091–1122, 2020. doi: 10.1142/S0219622020300025. [Google Scholar] [CrossRef]