Dombi Power Aggregation-Based Decision Framework for Smart City Initiative Prioritization under t-Arbicular Fuzzy Environment

1  Introduction

Multi-criteria decision-making (MCDM) refers to the systematic evaluation and selection of the most suitable alternative from a set of options, each assessed against multiple, and often conflicting, criteria, constraints, and objectives [1–4]. This process is integral to a wide range of domains—from routine personal choices to complex strategic decisions in business and public policy—where it facilitates rational analysis and prioritization. As real-world decision problems frequently involve ambiguous or incomplete information, effective MCDM strategies must account for such uncertainty. To address this, Zadeh [5] introduced the fuzzy set (FS) theory, which allows partial membership values rather than strict binary categorization. Subsequently, Atanassov [6] enhanced this framework by proposing intuitionistic FS (IFS), incorporating degrees of non-membership alongside membership. Later, Yager and Abbasov [7] advanced the theory further through the introduction of Pythagorean FS (PyFS), offering an even more nuanced mechanism for uncertainty representation.

While IFS and PyFS have proven to be effective tools within the broader FS theory, they share a significant shortcoming: their frameworks do not fully accommodate ambiguous or indeterminate opinions that cannot be classified strictly in terms of membership or non-membership. To bridge this conceptual gap, Cuong and Kreinovich [8] proposed the Picture FS (PFS), a more expressive model that permits the representation of human judgments across four distinct dimensions—positive, negative, neutral (abstention), and refusal. This allows decision-makers to express more nuanced evaluations in complex environments. However, like IFS, the PFS model imposes a constraint that the sum of the three primary degrees must not exceed one, limiting its capacity to represent broader uncertainty. To overcome this structural limitation, Mahmood et al. [9] introduced the spherical FS (SFS), a generalization of PFS that defines its membership components on the surface of a unit sphere, thus relaxing the additive constraint and offering a richer modeling framework. Since then, SFS theory has gained traction in decision sciences. Al-Shamiri et al. [10] extended the SFS framework to spherical fuzzy bipolar soft expert sets by introducing certain set-theoretic operations and properties for the proposed model, and addressed several important related problems. Sarwar et al. [11] introduced a quantum spherical fuzzy approach to the technique for order preference by similarity to ideal solution method, incorporating elements from quantum mechanics and fuzzy logic. Further advancements include Ali and Garg’s [12] formulation of a norm-based distance metric specific to SFS, and Akram et al.’s [13] exploration of outranking techniques under the same environment. Recently, Ali [14] proposed a symmetry-based ranking approach tailored to sustainable supplier evaluation using SFS. Building upon these developments, Ashraf et al. [15,16] introduced an extended version of the SFS model known as the disc SFS (DSFS), which incorporates an additional radius parameter. This enhancement allows the model to capture variable levels of fuzziness, providing a more flexible and realistic depiction of uncertainty in practical decision-making contexts. Unlike traditional SFS, where the degree values are constrained to lie on the unit sphere, the DSFS introduces a disc-like geometry that adjusts the scale of uncertainty representation. This additional parameter facilitates a more granular differentiation among alternatives and enhances the accuracy of aggregation operations—capabilities that conventional FS models often lack. However, DSFS imposes a rigid constraint that the sum of the squares of the three membership-related components must not exceed unity, i.e., N2+ꟽ2+M¯2≤1. This condition can be overly restrictive in practical scenarios. For instance, the triple (0.7,0.5,0.7) violates this condition, as 0.72+0.52+0.72=1.23>1, thus rendering such information invalid within the DSFS framework. To overcome this limitation, Ali and Popa [17] recently proposed the t-arbicular FS (t-AFS), which generalizes the spherical constraint by allowing the power in the inequality to be any natural number t≥1. In the previous example, while the triple fails the square-based condition, it satisfies the cubic constraint, i.e., 0.72+0.52+0.72=0.811<1, making it acceptable in the t-arbicular fuzzy (t-AF) context. This flexibility enables the model to accommodate a broader range of data points, which are otherwise excluded under DSFS. To illustrate the practical relevance of this advancement, consider a real-world decision-making scenario involving the selection of a healthcare solution for urban development. Decision-makers may evaluate not only the direct features of a proposed system—such as cost, efficiency, and technological readiness—but also broader contextual factors like long-term scalability, community adaptability, and integration with existing services. While standard models may effectively represent the core technical criteria, capturing these nuanced and often overlapping uncertainties requires a more flexible framework. The t-AF model, by extending the feasible boundary through a customizable power parameter, offers the capability to express such complex interrelations more effectively. It thus ensures that critical alternatives are not prematurely excluded due to rigid structural constraints, leading to more inclusive, accurate, and context-sensitive decision analysis. The existing literature has established the fundamental principles of t-AFS and examined their internal relationships, laying the groundwork for advancing their use in decision-making scenarios. Within this framework, two critical research inquiries arise: (i) What novel operators can be introduced, or how can limitations in current methodologies be resolved? and (ii) What approaches can effectively determine the most suitable option from among various alternatives?

Aggregation operators (AOs) serve as essential tools in real-world decision-making by enabling the synthesis and prioritization of fuzzy information. Over time, researchers have proposed numerous AOs tailored to various fuzzy frameworks. For example, Jana et al. [18] formulated Dombi-based operators within the context of Pythagorean fuzzy sets. Senapati et al. [19] advanced the field by introducing Aczel-Alsina operators for interval-valued intuitionistic fuzzy settings, highlighting their practical relevance in decision analysis. Qiyas et al. [20] explored trigonometric sine-based operators for spherical fuzzy sets to address decision-making challenges, while in a separate study, Qiyas et al. [21] presented Hamacher-type operators for spherical uncertain linguistic environments, emphasizing their applicability in group decision scenarios to unify divergent expert opinions. Similarly, Abdullah et al. [22] conducted an in-depth assessment of decision support systems grounded in 2-tuple spherical fuzzy linguistic aggregation methods.

The Dombi AO, originally formulated by Dombi in 1982 [23], is characterized by its parameter-dependent adaptability, enabling it to function in either a conjunctive or disjunctive mode. This intrinsic flexibility has led to its extensive adaptation within a range of fuzzy set extensions. For instance, Chen and Ye [24] introduced weighted Dombi operators under single-valued neutrosophic sets, establishing a foundational approach for subsequent enhancements. Expanding on this, Liu et al. [25] incorporated the Dombi Bonferroni mean into IFS theory and applied it to MCDM. Similarly, Shi and Ye [26] explored the application of Dombi operators in neutrosophic cubic environments to address MCDM scenarios. More recent advancements include the work of Seikh and Mandal [27], who developed interval-valued Fermatean fuzzy Dombi weighted operators and extended them to interval-valued spherical fuzzy domains, with practical implementation in areas such as plastic waste management [28]. Furthermore, Jana et al. [18] adapted the Dombi operator to picture FSs, thereby broadening its operational landscape. In a recent development, Seikh and Mandal [29] introduced a new class of Dombi averaging and geometric AOs for IFS, applying them to MCDM problems. Despite extensive research on Dombi aggregation operations across various FS extensions, disc spherical fuzzy (DSF) operations have not yet been defined in terms of Dombi operations. Consequently, existing DSF AOs primarily rely on the algebraic product and sum of DSFs, making them less generalized. Moreover, conventional operators often overlook the mutual dependencies among the input values being aggregated, which significantly undermines their suitability for handling intricate decision-making problems. To overcome this shortcoming, and guided by the foundational principles of power aggregation [30–32], it becomes essential to construct a new class of aggregation operators under the DSF environment, wherein the weights are adaptively determined based on the characteristics of the input data. Incorporating Dombi-based operations into this scheme enhances the mutual reinforcement among the aggregated inputs, yielding more coherent and context-sensitive results. Additionally, these advancements call for the application of the weighted aggregated sum product assessment (WASPAS) method in conjunction with the newly proposed operators. A significant benefit of this integrated strategy lies in its capacity to reduce the impact of outliers and unfavorable inputs by leveraging power-based weights within a flexible, hybrid MCDM framework. This methodology combines the advantages of the weighted sum model (WSM) and the weighted product model (WPM), enabling a more stable and discriminative ranking of alternatives. WASPAS produces final scores by synthesizing the outcomes from both models, with its operational mode governed by a threshold parameter. Depending on the selected value, the method can emulate either WSM or WPM behavior, thus drawing on the respective merits of both approaches.

Despite notable progress in fuzzy MCDM, several methodological and practical limitations persist, which serve as the motivation for this study:

i.   Lack of Dombi AOs in DSF environments: Although Dombi AOs are recognized for their parametric flexibility and generalization capabilities [23], they have not yet been incorporated into DSF frameworks. Existing DSF aggregation models primarily rely on conventional algebraic operations, limiting their adaptability in handling complex uncertainties.

ii.   Static weighting in DSF aggregation: Current DSF-based AOs utilize fixed weighting schemes, failing to reflect the contextual influence of input data. Inspired by power aggregation mechanisms [30–32], there exists a need to develop DSF-based AOs that dynamically adjust weights in response to varying input arguments, thereby better representing expert judgment and preferences.

iii.   Lack of hybrid aggregation strategies: Many DSF-based MCDM approaches are restricted to either WSM or WPM, which can limit their robustness in multifactorial decision environments. The hybrid WASPAS method, which balances additive and multiplicative reasoning, has not been adapted to the DSF context.

iv.   Underexplored application in smart city prioritization: To the best of the authors’ knowledge, no prior study has applied DSF-based MCDM frameworks—particularly those involving Dombi operators and hybrid power aggregation—to the domain of smart city initiative prioritization. Given the inherent complexity, uncertainty, and interdependence of smart city components, this application provides an ideal platform to validate the practical value and broader applicability of the proposed model.

Based on the above gaps, this research presents the following contributions:

i.   A novel class of DSF AOs based on Dombi functions, enhancing flexibility and generalization in fuzzy environments.

ii.   Integration of power aggregation into DSF AOs, allowing context-sensitive weight adjustment that better reflects the influence of input data.

iii.   Extension of the WASPAS methodology to the DSF framework using Dombi-based AOs, resulting in improved ranking stability and decision accuracy.

iv.   Implementation of the proposed model in a real-world case study on smart city initiative prioritization—an application area not previously addressed in the DSF-MCDM literature—to demonstrate its practical relevance and effectiveness.

The structure of this article is as follows: Section 2 reviews the foundational concepts necessary to comprehend the proposed framework. In Section 3, the formulation and properties of Dombi operations under the DSF environment are explored in detail. Section 4 introduces a set of aggregation operators based on these operations. Section 5 describes the procedural steps of the WASPAS technique, which is then employed in a practical case study in Section 6. Section 7 provides a comparative evaluation to assess the effectiveness of the proposed model. Concluding remarks are offered in Section 8.

2  Prerequisite Knowledge

In this article, the set ∪ is consistently regarded as a finite, non-empty collection, defined as the universal set of alternatives. Additionally, the symbol Γ represents the unit interval [0,1].

This section reviews the definitions of DTN, DTCN, and power operators, as well as t-SFS and t-AFS, along with their corresponding theoretical concepts.

Definition 1. [23] For β1,β2∈[0,1] with p≥1, the DTN D and DTCN Dc are defined as follows:

D(β1,β2)=11+((1−β1β1)p+(1−β2β2)p)1/p,(1)

Dc(β1,β2)=1−11+((β11−β1)p+(β21−β2)p)1/p.(2)

Definition 2. [30] Given a sequence of non-negative real numbers δ1,δ2,…,δm, the power average (PA) and the weighted power average (WPA) are defined as follows:

PA(δ1,δ2,…,δm)=∑l=1m(1+S(δl))∑j=1m(1+S(δj))δl,(3)

WPA(δ1,δ2,…,δm)=∑l=1m℧l(1+S(δl))∑j=1m℧j(1+S(δj))δl.(4)

Here, ℧l∈[0,1] denotes a component of the weight vector, and S(δj) is given by ∑i=1msupp(δj,δi), where supp(δj,δi) represents the support measure between δj and δi. This support measure satisfies the following conditions:

i) 0≤supp(δj,δi)≤1, ii) supp(δj,δi)=supp(δi,δj), iii) If |δj−δi|≤|δl−δl|, then supp(δj,δi)≥supp(δl,δl).

Moreover, the support measure is defined as supp(δj,δi)=1−d(δj,δi), where d(δj,δi) represents the distance between δj and δi.

Definition 3. [31] Consider a sequence of non-negative real numbers δ1,δ2,…,δm. The power geometric (PG) and weighted power geometric (WPG) operators are defined as follows:

PG(δ1,δ2,…,δm)=∏l=1mδl(1+S(δl))∑j=1m(1+S(δj)),(5)

WPG(δ1,δ2,…,δm)=∏l=1mδl℧l(1+S(δl))∑j=1m℧j(1+S(δj)).(6)

The variables are specified in the same manner as in Definition 2.

Definition 4. [9] A t-SFS S on ℑ is defined as follows:

S={(N(u),ꟽ(u),M¯(u))|u∈ℑ},(7)

where N(u),ꟽ(u),M¯(u)∈Γ refer to degrees of association, neutrality and non-association, respectively, such that the condition (N(u))t+(ꟽ(u))t+(M¯(u))t≤1 fulfils for all u∈ℑ. Further the degree of indeterminacy is expressed by πs(u)=1−((N(u))t+(ꟽ(u))t+(M¯(u))t)t. The triplet (N,ꟽ,M¯) is referred to as t-spherical fuzzy number (t-SFN).

Definition 5. [9,33] Let S1=(N1,ꟽ1,M¯1) and S2=(N2,ꟽ2,M¯2) represent two t-SFNs with ⊤>0. Then

1.   S1⊕S2=(N1t+N2t−N1tN2tt,ꟽ1ꟽ2,M¯1M¯2);

2.   S1⊗S2=(N1N2,ꟽ1t+ꟽ2t−ꟽ1tꟽ2tt,M¯1t+M¯2t−M¯1tM¯2tt);

3.   ⊤S1=(1−(1−N1t)⊤t,ꟽ1⊤,M¯1⊤);

4.   S1⊤=(N1⊤,1−(1−ꟽ1t)⊤t,1−(1−M¯1t)⊤t).

Definition 6. [34] Consider S1=(N1,ꟽ1,M¯1) as a t-SFN. The corresponding score function is defined as:

S(S1)=1+N1t−ꟽ1t−M¯1t2.(8)

The greater the score function value, the higher the ranking of the corresponding t-SFN.

Definition 7. [17] A t-AFS on ℑ is expressed mathematically as:

ϝ={(u,N(u),ꟽ(u),M¯(u);r(u))|u∈ℑ}={Dr(N(u),ꟽ(u),M¯(u))|u∈ℑ},(9)

where N,ꟽ,M¯∈Γ, r∈[0,3t] such that Nt(u)+ꟽt(u)+M¯t(u)≤1, and Dr(N(u),ꟽ(u),M¯(u)) ={(p,q,s)|p,q,s∈Γ & (N(u)−p)t+(ꟽ(u)−q)t+(M¯(u)−s)tt≤r(u)}⋂R; R={(p,q,s)∈Γ & pt+qt+st≤1}.

The expression π(u)=(1−(Nt(u)+ꟽt(u)+M¯t(u)))1/t represents the degree of refusal of an element u∈ℑ. Further the elements of the t-AFS are termed as the t-AF numbers (t-AFNs), denoted by ϝ=(N,ꟽ,M¯;r).

Definition 8. [17] Let ϝ1=(N1,ꟽ1,M¯1) and ϝ2=(N2,ꟽ2,M¯2) be two t-AFNs, with ⊤>0 and ‡∈{min,max}. Then

1.   ϝ1⊕ϝ2=(N1t+N2t−N1tN2tt,ꟽ1ꟽ2,M¯1M¯2;‡{r1,r2});

2.   ϝ1⊗ϝ2=(N1N2,ꟽ1t+ꟽ2t−ꟽ1tꟽ2tt,M¯1t+M¯2t−M¯1tM¯2tt;‡{r1,r2});

3.   ⊤ϝ1=(1−(1−N1t)⊤t,ꟽ1⊤,M¯1⊤;r1);

4.   ϝ1⊤=(N1⊤,1−(1−ꟽ1t)⊤t,1−(1−M¯1t)⊤t;r1);

5.   ϝ1c=(M¯1,ꟽ1,N1;r1).

Definition 9. [17] Let ϝ1=(N1,ꟽ1,M¯1;r1) be a t-AFN. Then the score function is given by

S(ϝ1)=(N1t−M¯1t−ln⁡(1+π1t)+r1+233t),(10)

where S(ϝ1)∈[0,1]. The greater value of S(ϝ1) signifies a greater t-AFN ϝ1.

3  Dombi Operational Laws

In this section, we establish the Dombi operational laws tailored for t-AFNs, laying the groundwork for subsequent aggregation mechanisms.

Definition 10. Consider two t-AFNs, ϝ1=(N1,ꟽ1,M¯1;r1) and ϝ2=(N2,ꟽ2,M¯2;r2), where ⊤>0 and ‡ belongs to the set {min,max}. Then,

1.   ϝ1⊕ϝ2=(1−11+((N1t1−N1t)p+(N2t1−N2t)p)1pt,11+((1−ꟽ1tꟽ1t)p+(1−ꟽ2tꟽ2t)p)1pt,11+((1−M¯1tM¯1t)p+(1−M¯2tM¯2t)p)1pt;‡{r1,r2});

2.   ϝ1⊗ϝ2=(11+((1−N1tN1t)p+(1−N2tN2t)p)1pt,1−11+((ꟽ1t1−ꟽ1t)p+(ꟽ2t1−ꟽ2t)p)1pt1−11+((M¯1t1−M¯1t)p+(M¯2t1−M¯2t)p)1pt;‡{r1,r2});

3.   ⊤ϝ1=(1−11+(⊤(N1t1−N1t)p)1pt,11+(⊤(1−ꟽ1tꟽ1t)p)1pt,11+(⊤(1−M¯1tM¯1t)p)1pt;r1);

4.   ϝ1⊤=(11+(⊤(1−N1tN1t)p)1pt,1−11+(⊤(ꟽ1t1−ꟽ1t)p)1pt,1−11+(⊤(M¯1t1−M¯1t)p)1pt;r1).

4  t-AF Dombi Power Operators

Building on the t-AFN operations outlined in Section 3, we introduce a set of Dombi power operators and examine their resulting properties.

4.1 t-AF Dombi Power Average Operator

In the present part, we establish the theoretical foundation of the t-AFDPA operator and explore its characteristics.

Definition 11. For a given set of t-AFNs ϝl=(Nl,ꟽl,M¯l;rl), where l=1,2,…,m, the t-AFDPA operator is defined as

t−AFDPA(ϝ1,ϝ2,…,ϝm)=⊕l=1m((1+S(ϝl))∑1≤l≤m(1+S(ϝl)))ϝl=⊕l=1m€lϝl.(11)

Here, €l represents the power weight of ϝl, where

S(ϝl)=∑1≤l≤m,l≠lsupp(ϝl,ϝl).

Theorem 1. Given a set of t-AFNs ϝl=(Nl,ꟽl,M¯l;rl) for l=1,2,…,m, we obtain the following result based on Definition 10:

t−AFDPA(ϝ1,ϝ2,…,ϝm)=(1−11+(∑1≤l≤m€l(Nlt1−Nlt)p)1/pt,11+(∑1≤l≤m€l(1−ꟽltꟽlt)p)1/pt,11+(∑1≤l≤m€l(1−M¯ltM¯lt)p)1/pt;‡{r1,r2,…,rm}).(12)

Proof. This can be proven using mathematical induction as follows:

For m=2,

t−AFDPA(ϝ1,ϝ2)=€1ϝ1⊕€2ϝ2=(1−11+(€1(N1t1−N1t)p)1pt,11+(€1(1−ꟽ1tꟽ1t)p)1pt,11+(€1(1−M¯1tM¯1t)p)1pt;r1)⊕(1−11+(€2(N1t1−N1t)p)1pt,11+(€2(1−ꟽ1tꟽ1t)p)1pt,11+(€2(1−M¯1tM¯1t)p)1pt;r2)=(1−11+(∑1≤l≤2€l(Nlt1−Nlt)p)1/pt,11+(∑1≤l≤2€l(1−ꟽltꟽlt)p)1/pt,11+(∑1≤l≤2€l(1−M¯ltM¯lt)p)1/pt;‡{r1,r2}).

Assume that Eq. (12) holds for m=x,

t−AFDPA(ϝ1,ϝ2,…,ϝx)=(1−11+(∑1≤l≤x€l(Nlt1−Nlt)p)1/pt,11+(∑1≤l≤x€l(1−ꟽltꟽlt)p)1/pt,11+(∑1≤l≤x€l(1−M¯ltM¯lt)p)1/pt;‡{r1,r2,…,rx}).

Now, to prove it for m=l+1,

t−AFDPA(ϝ1,ϝ2,…,ϝx,ϝx+1)=⊕l=1x€lϝl⊕€x+1ϝx+1=(1−11+(∑1≤l≤x€l(Nlt1−Nlt)p)1/pt,11+(∑1≤l≤x€l(1−ꟽltꟽlt)p)1/pt,11+(∑1≤l≤x€l(1−M¯ltM¯lt)p)1/pt;‡{r1,r2,…,rx})⊕(1−11+(€1(Nx+1t1−Nx+1t)p)1pt,11+(€x+1(1−ꟽx+1tꟽx+1t)p)1pt,11+(€x+1(1−M¯x+1tM¯x+1t)p)1pt;rx+1)=(1−11+(∑1≤l≤x+1€l(Nlt1−Nlt)p)1/pt,11+(∑1≤l≤x+1€l(1−ꟽltꟽlt)p)1/pt,11+(∑1≤l≤x+1€l(1−M¯ltM¯lt)p)1/pt;‡{r1,r2,…,rx+1}).

Therefore, Eq. (12) holds for all x∈N. ▪

Theorem 2. For a given set of t-AFNs ϝl=(Nl,ꟽl,M¯l;rl), where l=1,2,…,m, if ϝl=ϝ for all l, then

t−AFDPA(ϝ1,ϝ2,…,ϝm)=ϝ.(13)

Proof. Since ϝl=ϝ for all l, it follows from Eq. (12) that

t−AFDPA(ϝ1,ϝ2,…,ϝm)=(1−11+(∑1≤l≤m€l(Nlt1−Nlt)p)1/pt,11+(∑1≤l≤m€l(1−ꟽltꟽlt)p)1/pt,11+(∑1≤l≤m€l(1−M¯ltM¯lt)p)1/pt;‡{r1,r2,…,rm})=(1−11+(∑1≤l≤m€l(Nt1−Nt)p)1/pt,11+(∑1≤l≤m€l(1−ꟽtꟽt)p)1/pt,11+(∑1≤l≤m€l(1−M¯tM¯t)p)1/pt;‡{r,r,…,r})=(1−11+((Nt1−Nt)p)1/pt,11+((1−ꟽtꟽt)p)1/pt,11+((1−M¯tM¯t)p)1/pt;‡{r,r,…,r})=(N,ꟽ,M¯;r).◼

Theorem 3. For any two sets of t-AFNs, ϝl=(Nl,ꟽl,M¯l;rl) and ϝ¨l=(N¨l,ꟽ¨l,M¯¨l;r¨l), where l=1,2,…,m, if Nl≤N¨l, M¯l≥M¯¨l, and rl≤r¨l for all l, then

t−AFDPA(ϝ1,ϝ2,…,ϝm)≤t−AFDPA(ϝ¨1,ϝ¨2,…,ϝ¨m).(14)

Proof. As Nl≤N¨l, it follows that

11+(∑1≤l≤m€l(Nlt1−Nlt)p)1/p≥11+(∑1≤l≤m€l(N¨lt1−N¨lt)p)1/p1−11+(∑1≤l≤m€l(Nlt1−Nlt)p)1/pt≤1−11+(∑1≤l≤m€l(N¨lt1−N¨lt)p)1/pt.

Furthermore, given that ꟽl≥N¨l, M¯l≥M¯¨l, we can express it as

11+(∑1≤l≤m€l(1−ꟽltꟽlt)p)1/pt≥11+(∑1≤l≤m€l(1−ꟽ¨ltꟽ¨lt)p)1/pt,

11+(∑1≤l≤m€l(1−M¯ltM¯lt)p)1/pt≥11+(∑1≤l≤m€l(1−M¯¨ltM¯¨lt)p)1/pt.

Additionally, since rl≤r¨l, it follows that

‡{r1,r2,…,rm}≤‡{r¨1,r¨2,…,r¨m}.

Hence, we obtain

t−AFDPA(ϝ1,ϝ2,…,ϝm)≤t−AFDPA(ϝ¨1,ϝ¨2,…,ϝ¨m).◼

Theorem 4. For any sequence of t-AFNs given by ϝl=(Nl,ꟽl,M¯l;rl) for l=1,2,…,m, if ϝ−=(min1≤l≤mNl,max1≤l≤mꟽl,max1≤l≤mM¯l;min1≤l≤mrl) and ϝ+=(max1≤l≤mNl,min1≤l≤mꟽl,min1≤l≤mM¯l;max1≤l≤mrl), then

ϝ−≤t−AFDPA(ϝ1,ϝ2,…,ϝm)≤ϝ+.(15)