Computational Assessment of Energy Supply Sustainability Using Picture Fuzzy Choquet Integral Decision Support System

1  Introduction

One of the most important energy carriers, electricity, is crucial for maintaining economic growth [1]. However, manufacturing electricity has become extremely reliant on fossil fuels, resulting in several serious issues, including air pollution, significant greenhouse gas emissions, imported energy, and inadequate energy security [2,3]. To improve the security of power, numerous nations implemented a variety of policies. For example, China established the renewable energy development plan with the aim of reducing harmful emissions, reducing reliance on energy imports, and improving air and water quality [4]. In order to reduce its reliance on fossil fuels for electricity generation, improve environmental performance, and potentially increase the security of the electricity supply, Brazil has implemented or planned a number of biofuel projects. These include the production of electricity from sugarcane bagasse [5], hybrid concentrated solar power (CSP)–biomass plants [6], and the production of rice husk electricity [7]. The goal of each study was to improve the reliability of the electrical supply. The basis for recommending practical policies and initiatives to improve a nation’s electrical supply sustainability and security (ESSS) is the examination of ESSS and its evolving trend [8]. Nonetheless, the quantity of research works that concentrate on creating the techniques for ESSS state analysis is constrained. The creation of the ESSS evaluation framework is, therefore, a necessary and crucial step. Multiple dimensions are typically taken into account when evaluating ESSS. Power system reliability analysis and power market analysis were linked by Kjølle and Gjerde [9] to create an integrated framework for the security of energy supply analysis. Numerous studies have also taken into account more than three variables for the analysis and evaluation of ESSS.

Especially in the field of energy planning, multi-criteria decision-making techniques have proven to be a very helpful instrument for resolving conflicts between disparate criteria. Nevertheless, social and environmental factors frequently conflict with technical and financial factors in the design of rural electrification plans. To choose the best option for remote rural locations’ electrical supply while taking into account social, economic, environmental, and technical factors, we suggest advanced multi-attribute decision-making (MADM) technique. The MADM problem has been applied to various complex issues. We assess different decision-making problems in daily life and important goods, and we can learn how to make the best decisions. We saw several research scholars working on the decision-making problems under different mathematical models [10–12].

To handle unpredictable situations of crisp theory, an innovative theory of fuzzy set (FS) was developed by Zadeh [13] with a certain membership degree (MD) of an object. The concept of intuitionistic fuzzy sets (IFSs) was introduced by Atanassov [14], who modified a version of fuzzy theory. The IFS contains two different aspects of an object, like membership degree (MD) and non-membership degree (NMD). The sum of MD and NMD lies in a close interval [0,1] with subject to the condition 0≤μ(x)+v(x)≤1. Yager and Basson [15] utilized the theory of an IFS to derive new operations and mathematical approaches. Yager [16] introduced the concept of the pythagorean fuzzy set (PyFS) by taking the sum of the squares of MD and NMD with the mathematical shape 0≤μ(x)2+v(x)2≤1. The enlargement of PyFS in the shape of complex numbers was introduced by Yager and Abbasov [17]. Some important results for PyFS improved by Peng and Yang [18]. Yager presented the concept of q-rung orthopair fuzzy sets (q-ROFSs), which is an effective method to explain the vagueness and ambiguous information in the MADM problem. The uniqueness of q-ROFS is to handle uncertainty and vagueness more effectively by incorporating multiple degrees to represent the MD and NMD of an element. Decision makers face some crucial challenges due to incomplete information and a lack of information provided by experts. To overcome such a scenario, Cường [19] presented the idea of a picture fuzzy set (PFS) with four possible terms, such as MD, abstinence degree (AD), NMD and refusal degree (RD). The mathematical expression of PFS is given by 0≤μ(x)+α(x)+v(x)≤1. A PFS is more effective and advanced framework of FS, IFS, PyFS and q-ROFSs. The PFS is a relatively recent extension of fuzzy set theory that aims to handle uncertainties with a spatial context, particularly in scenarios involving image processing, computer vision, and spatial data analysis. The application of picture fuzzy sets in pattern recognition and classification in scenarios where imprecision is inherent in the image data.

1.1 Literature Review

Yager [20,21] explored the aggregation operators (AOs) made use of in view of humans under the consideration of picture fuzzy informing by different researchers from different states. Cuong et al. [22] explored the idea of picture triangular norm (t-corm) and picture triangular co-norms (t-conorms) and discussed the properties of PFS. Hussain et al. [23] discussed the significance of vendor management enterprises by exploring the characteristics of the Hamy mean model with advanced picture fuzzy theory. Wang et al. [24] suggested a possible plan for the Choquet integral and proposed operator. Hussain et al. [25] classified diverse construction materials, taking into account Hamy mean models and Aczel-Alsina operators in the decision-making process. Sahoo et al. [26] resolved energy energy-related real-life problem using multi-criteria and mathematical aggregation operators. Petchimuthu et al. [27] constructed decision-making models to investigate a suitable optimal option using unknown weights of criteria.

Ali et al. [28] described some methods by using the basic operation of the Aczel-Alsina aggressions operator and the system of PFSs. Mahmood et al. [29,30] explored the PFAAWG operator bases on the Azcel Alsina operation to find effective and reliable proposed aggression operators. Choquet integral is a useful technique for computing rank. Hussain et al. [31] discussed mathematical results of Aczel Alsina Heronian mean operators. Ahmmad et al. [32] initiated a list of Aczel-Alsina averaging operators to resolve a real-life application related to medical diagnosis with decision analysis process. Hussain et al. [33] elaborated on Dombi Bonferroni mean operators for choosing reliable recycling machines to reduce the impact of waste materials.

In order to construct the intuitionistic fuzzy aggregation operators previously stated, the criteria are expected to be independent. It should be noted that correlated criteria, as opposed to independent ones, are used in most MCDM problems. In these cases, the evaluations cannot be aggregated using weighted aggregation operators, and the criterion’s relevance levels are described as fuzzy measures as opposed to weights. Fuzzy measures can effectively capture the correlative relationships between criterion sets, including independent, redundant, and complementary ones. Since the total of all fuzzy measures of a criterion set may be greater than one, fuzzy measures are really extensions of weights because the fuzzy measure of the complete set is only one. Thus, the fuzzy measures have greater versatility. Demirel et al. [34] employed Choquet integral mathematical models to investigate a suitable location for a warehouse under different key criteria. Tan and Chen [35] generalized the concepts of Choquet integral operators to acquire an authentic results from decision analysis process under different criteria. Furthermore, a family of Sugeno-Weber aggregation operators also utilized to investigate unknown degree of weights to assess a suitable security techniques [36]. The intuitionistic fuzzy Choquet averaging and geometric operators based on Einstein were created by Yu [37]. Yu and Xu [38] presented the intuitionistic fuzzy interaction aggregation operators based on the Choquet integral and created a two-sided matching decision-making model based on aggregation operators.

Menger [39] introduced the theoretical concepts of triangular norms with prominent characteristics. Algebraic t-norm and t-conorm are flexible and efficient generalizations of triangular norms. A diverse generalization and extension of triangular norms developed by different scholars, such as Dombi aggregation operators [40], Frank t-norm and t-conorm [41], and Hamacher aggregation operators [42]. Numerous mathematicians employed these aggregation models to acquire smooth aggregated results during the decision analysis process. Aczél and Alsina [43] deduced a family of robust aggregation models by exploring the characteristics of triangular norms. Babu and Ahmed [44] characterized various properties to derive effective mathematical strategies for designing and developing to resolve different applications such as diagnostics, article intelligence, forecasting and pattern recognition. Farahbod and Eftekhari [45] compared various classifications of t-norms and t-conorms to evaluate the best aggregation model. After the evaluation and aggregation process, they concluded that Aczel-Alsina is the best one and Dombi aggregation is the second one. Hussain et al. [46] deliberated a list of Aczel-Alsina aggregation models to handle awkward and incomplete information of human opinion. Senapati et al. [47] proposed Aczel-Alsina aggregation models, taking into account an intuitionistic fuzzy context. Senapati et al. [48] derived innovative mathematical strategies by incorporating the theory of an interval values intuitionistic fuzzy theory. Mu et al. [49] investigated a family of efficient and effective models of Maclaurin symmetric operators with a decision support system. Hussain et al. [50] derived new methodologies of Aczel-Alsina operators to evaluate an electric motor car taking into account the complex spherical fuzzy circumstances. De et al. [51] developed a decision support system with a doubt-fuzzy environment.

1.2 Research Gap

Several mathematical approaches and terminologies were established to aggregate expert’s opinions or judgments. Most of them failed to integrate multiple conflicting criteria without any external weights. To investigate the weight of criteria or attributes, Choquet integral operators are authentic mathematical approaches under the system of picture fuzzy environment and Aczel-Alsina triangular norms. We examined that no one worked on this robust mathematical framework under the consideration of a picture fuzzy situation and Choquet integral operators.

1.3 Novelty and Contributions

The PFS has extended the capabilities of traditional FSs and IFSs by introducing a third dimension: neutrality. In many decision-making scenarios, it’s common to encounter not only positive (supportive) and negative (opposing) opinions but also neutral stances where no clear judgment is made. Traditional fuzzy sets and even intuitionistic fuzzy sets, which include membership and non-membership degrees, fall short in accurately capturing this neutral perspective. For instance, voter express their opinion about any candidate in a specific way, vote to favor (MD), against (NMD), neutral (AD) and refuse to vote (RD). The FSs and IFSs unable to handle such a scenario, which has more than three components of human opinion. However, the PFS is a more abundant and flexible fuzzy framework that contains four aspects of human opinion. Furthermore, the PFSs address the limitation by incorporating a degree of neutrality alongside the traditional membership and non-membership degrees. This three-dimensional approach allows for a more nuanced representation of complex scenarios where neutrality plays a significant role, enabling better modelling and decision-making in situations with uncertain, hesitant, or indeterminate responses.

The Choquet integral operators are used to overcome the limitations of traditional aggregation methods, particularly in contexts where criteria are interdependent. Traditional aggregation operators, like weighted averages, typically assume that criteria are independent, which often oversimplifies the decision-making process. In reality, criteria frequently interact with one another, with some being complementary (synergistic) or substitutive (antagonistic). The Choquet integral was designed to handle such interactions effectively using a fuzzy measure that captures the relationship between criteria. This allows for a more accurate and nuanced aggregation of information, particularly in multi-criteria decision-making situations where the interplay between factors significantly influences the outcome. By accounting for these interactions, the Choquet integral provides a more sophisticated tool for decision-makers, leading to better-informed and more reliable decisions. The primary features of this presentation are given by:

(a)   To explore the picture fuzzy theory for handling awkward and unpredictable situations of human opinion. The PFS uniquely captures extensive information, enabling smoother approximations during decision analysis.

(b)   Choquet integral operators are more efficient and effective, used to aggregate approximated information with fuzzy measures of criteria and the decision analysis process.

(c)   We derive a list of new aggregation operators by incorporating the picture fuzzy theory and Choquet integral operators, namely PFCIAAA and PFCIAAG operators with prominent properties and special cases.

(d)   To resolve complicated, genuine real-life dilemmas, an algorithm of the MADM problem is utilized by incorporating derived mathematical methodologies.

(e)   To validate the robustness and compatibility of the diagnosed mathematical methodologies, we gave a numerical example to evaluate a suitable electric transformer for electricity supply.

1.4 Layout of the Manuscript

In order to maintain research work, divide the remaining parts as follows: Section 2 briefly discusses fundamental concepts and basic preliminaries of triangular norms, Aczel-Alsina operations, and PFS with dominant rules. Section 3 illustrates a family of averaging Choquet integral operators. Moreover, we also derive mathematical methodologies of geometric Choquet integral operators in the light of picture fuzzy information. The robust technique of the MCDM problem is applied to resolve genuine real-life applications with derived mathematical methodologies in Section 5. An experimental case study is also adopted to reveal the feasibility of the invented approaches. Section 6 conducted an extensive comparative study to mitigate the flexibility of pioneered approaches with previously developed approaches. Section 7 enclosed some remarkable remarks related to the developed research work. Additionally, we also portrayed the main aspects of this article in Fig. 1.

Figure 1: The primary features of the article

2  Preliminaries

This section presents basic and necessary notions of triangular norms, Aczel-Alsina operations, and PFSs with dominant rules.

Definition 1 [52]: A mapping T:[0,1]2→[0,1] is known as t-norm if underlying the following axioms:

(a)   T(ϰ,ℴ)=T(ℴ,ϰ)

(b)   T(ϰ,ℴ)≤T(ϰ,r) if ℴ≤𝕔

(c)   T(ϰ,T(ℴ,𝕔))=T(T(ϰ,ℴ),𝕔)

(d)   T(ϰ,1)=ϰ

For all ϰ,ℴ,𝕔∈[0,1].

Example 1: The following expressions are examples of Definition 1:

(a)   TM(κ,ℴ)=min(κ,ℴ)

(b)   Tp(ϰ,ℴ)=κ.ℴ

(c)   TL(ϰ,ℴ)=max(ϰ+ℴ−1,0)

(d)   TD(ϰ,ℴ)={ϰ,ifℴ=1ℴ,ifϰ=10,otherwise

For any ϰ,ℴ∈[0,1].

Definition 2 [52]: A mapping S:[0,1]2→[0,1] is known as t-norm if underlying the following axioms:

(a)   S(ϰ,ℴ)=S(ℴ,ϰ)

(b)   S(ϰ,ℴ)≤S(ϰ,𝕔) if ℴ≤𝕔

(c)   S(ϰ,S(ℴ,𝕔))=S(S(ϰ,ℴ),𝕔)

(d)   S(ϰ,0)=ϰ

For any ϰ,ℴ,𝕔∈[0,1].

Example 2: The following expressions are examples of Definition 2:

(a)   SM(ϰ,ℴ)=max(ϰ,ℴ)

(b)   Sp(ϰ,ℴ)=ϰ+ℴ−ϰ.ℴ

(c)   SL(ϰ,ℴ)=min(ϰ+ℴ,1)

(d)   SD(ϰ,ℴ)={ϰ,if ℴ=0ℴ,if ϰ=01,otherwise

For any ϰ,ℴ∈[0,1]

Definition 3 [43]: The theory of the Aczel-Alsina t-nom (SAℵ) and t-conorm SAℵ(ϰ,ℴ) is given by:

TAℵ(ϰ,ℴ)={TD(ϰ,ℴ)  if ℵ=0min(ϰ,ℴ)  if ℵ=∞e−((−ℒnϰ)ℵ)+(−ℒnℴ)ℵ)1ℵotherwise}

and

SAℵ(ϰ,ℴ)={SD(ϰ,ℴ)  if ℵ=0max(ϰ,ℴ)  if ℵ=∞1−e−((−ℒn(1−ϰ))ℵ+(−ℒn(1−ℴ))ℵ)1ℵ otherwise}

for each ℵ∈[0,∞].

Definition 4 [14]: An IFS K  on X is defined as K ={⟨ℸ,μ(ℸ),v(ℸ)⟩|ℸ∈X}. Where μ(ℸ):X→[0,1] and v(ℸ):X→[0,1] represent MD and NMD with subject to the condition:

0≤μ(ℸ)+v(ℸ)≤1

Additionally, a hesitancy degree of ℸ in K  is denoted by π(ℸ)=1−(μ(ℸ)+v(ℸ)) and a pair (μ(ℸ),v(ℸ)) is known as an intuitionistic fuzzy value (IFV).

Definition 5 [53]: A PFS K  on X is defined as K ={⟨ℸ,μ(ℸ),α(ℸ),v(ℸ)⟩|ℸ∈X}. Where μ(ℸ):X→[0,1],α(ℸ):X→[0,1] and v(ℸ):X→[0,1] indicate MD, AD and NMD, respectively, subject to the condition:

0≤μ(ℸ)+α(ℸ)+v(ℸ)≤1

Additionally, the RD of ℸ in K  is denoted by π(ℸ)=1−(μ(ℸ)+α(ℸ)+v(ℸ)) and a triplet g=(μ(ℸ),α(ℸ),v(ℸ)) is known as a picture fuzzy value (PFV).

Definition 6 [33]: Consider any three PFVs g=(μ,α,ν),g1=(μ1,α1,ν1) and g2=(μ2,α2,ν2). Then some prominent rules are given bellow:

(a)   g1⊆g2, if μ1≤μ2,α1≤α2 and ν1≤ν2

(b)   g1=g2 iff g1⊆g2 and g2⊆g1

(c)   g1∪g2=(max{μ1,μ2},min{α1,α2},min{ν1,ν2})

(d)   g1∩g2=(min{μ1,μ2},min{α1,α2},max{ν1,ν2})

(e)   g¯=(μ,α,ν)

(f)   g1⨁g2=(μ1+μ2−μ1μ2,α1α2,ν1ν2)

(g)   g1⊗g2=(μ1μ2,α1α2,ν1+ν2−ν1ν2)

(h)   τg=(1−(1−μ1)τ,αgτ,νgτ)

(i)   gτ=(μgτ,αgτ,1−(1−ν)τ)

Definition 7 [54]: Let g=(μ,α,ν), g1=(μ1,α1,ν1) and g2=(μ2,α2,ν2) be three PFVs with τ,τ1,τ2>0. Then:

(a)   g1⨁g2=g2⨁g1

(b)   g1⊗g2=g2⊗g1

(c)   τ(g1⨁g2)=τg1⨁τg2

(d)   (g1⊗g2)τ=g1τ⨁g2τ

(e)   τ1g⨁τ2g)=(τ1+τ2)g

(f)   gτ1⊗gτ2=g(τ1+τ2)

(g)   (gτ1)τ2=gτ1τ2

Definition 8 [33]: Let g=(μ,α,v) be a PFV, the score function S(g) and accuracy function H(g) are defined as follows:

Æ(g)=(13)(2+μ−α−v),S(g)∈[0,1](1)

and

H(g)=μ+α+v,H(g)∈[0,1](2)

Consider two PFVs g1 and g2. We make a comparison by using the following steps:

(a)   If S(g1)>S(g2), then g1 is superior to g2.

(b)   If S(g1)=S(g2). Then:

H(g1)=H(g2) Implies that g1 is equivalent to g2.

H(g1)>H(g2) Implies that g1 is superior to g2.

Example 1: Let g1=(0.12,0.32,0.27) and g2=(0.21,0.42,0.19) are two PFVs. Then, the score and accuracy functions are illustrated as follows:

Æ(g1)=(13)(2+0.12−0.32−0.27),S(g1)=0.51∈[0,1]

Æ(g2)=(13)(2+0.21−0.42−0.19),S(g2)=0.53∈[0,1]

H(g)=0.12+0.32+0.27,H(g)=0.71∈[0,1]

H(g)=0.21+0.42+0.19,H(g)=0.82∈[0,1]

Definition 9 [35]: A mapping χ:P(X)→[0,1] is known a fuzzy measure based on the following properties:

(a)   χ(ϕ)=0 and χ(X)=1.

(b)   If P⊆Q then χ(P)≤χ(Q). For any P,Q∈P(X).

We see above investigation process for finding fuzzy measures is quite complex. Therefore, Sugeno proposed a λ-fuzzy measure, which can be defined as follows:

χ(P∪Q)=χ(P)+χ(Q)+λχ(P)⋅χ(Q)

where χ(P∪Q)=ϕ with parameter λ∈[−1,∞) denote correlation among attributes. If λ=0, if χ(P∪Q)=χ(P)+χ(Q), P∪Q=ϕ.

Definition 10 [35]: Let g and χ be the functions of non-negative real numbers and fuzzy measure on X. The Choquet integral operator of g is expressed as follows:

Cμ(g)=∑i=1ng(i)[χ(P(i)−χ(P(i+1))](3)

here, the permutation on X is indicated by g(i)≤g(i+1),i=1,2,…,n−1 and P(i)={g(i),…,g(n)} with P(n+1)=ϕ.

Definition 11 [54]: Let g=(μ,α,v), g1=(μ1,α1,v1) and g2=(μ2,α2,v2) be the three PFVs with, ℵ≥1 and τ>0. A list of fundamental operations of Aczel-Alsina t-norm and t-conorm are expressed as follows:

(a)   g1⨁g2=(1−e−((−ℒn(1−μ1))ℵ+(−ℒn(1−μ2))ℵ)1ℵ,e−((−ℒn(α1))ℵ+(−ℒn(α2))ℵ)1ℵ,e−((−ℒn(v1))ℵ+(−ℒn(v2))ℵ)1ℵ)

(b)   g1⨂g2=(e−((−ℒn(μ1))ℵ+(−ℒn(μ2))ℵ)1ℵ,e−((−ℒn(α1))ℵ+(−ℒn(α2))ℵ)1ℵ,1−e−((−ℒn(1−v1))ℵ+(−ℒn(1−v2))ℵ)1ℵ)

(c)   τg=(1−e−(τ(−ℒn(1−μ))ℵ)1ℵ,e−(τ(−ℒn(α))ℵ)1ℵ,e−(τ(−ℒn(v))ℵ)1ℵ)

(d)   gτ=(e−(τ(−ℒn(1−μ))ℵ)1ℵ,e−(τ(−ℒn(1−α))ℵ)1ℵ,1−e−(τ(−ℒn(1−v))ℵ)1ℵ)

Remark: The above-discussed operations failed, if g1=(1,0,0) and g2=(0,0,1) are two PFVs. Finally, we can say the above-discussed operations of Aczel Alsian t-norms and t-conorms cannot handle extreme MD, AD, and NMD values in a triplet.

Theorem 1 [54]: Let g=(μ,α,v),g1=(μ1,α1,v1) and g2=(μ2,α2,v2) be the three PFVs with ℵ≥1 and τ>0. Then, a few fundamental OLs are defined as follows:

(a)   g1⊕g2=g2⊕g1

(b)   g1⨂g2=g2⨂g1

(c)   τ(g1⨁g2)=τg1⨁τg2,τ>0

(d)   (τ1+τ2)g=τ1g+τ2g,τ1,τ2>0

(e)   (g1⨂g2)τ=g1τ⨂g2τ,τ>0

(f)   gτ1⨂gτ2=g(τ1+τ2),τ1,τ2>0

3  Picture Fuzzy Choquet Integral Aczel-Alsina Averaging Operators

This section developed a family of robust mathematical approaches of Choquet integral operators and Aczel-Alsina operations such as PFCIAAA and PFCIAAOA operators.

Definition 12: For any PFVs gs=(μs,αs,vs),s=1,2,3,…,n and [μ(P(s)−μ(P(s+1))] be the fuzzy measure of gs. Then, the PFCIAAA operator is characterized as follows:

PFCIAAA(g1,g2,…,gn)=⊕s=1n([χ(P(s)−χ(P(s+1))]gs),PFCIAAA(g1,g2,…,gn)=[χ(P(s)−χ(P(s+1))]g1⊕[χ(P(s)−χ(P(s+1))]g2⊕…⊕[χ(P(s)−χ(P(s+1))]gn(4)

Theorem 2: For any PFVs gs=(μs,αs,vs),s=1,2,3,…,n and [χ(P(s)−χ(P(s+1))] be the fuzzy measure of gs. Then, the aggregated outcome of the PFCIAAA operator is still a PFV, so we can express as follows:

PFCIAAA=∑s=1n[χ(P(s)−χ(P(s+1))]gs=(1−e−(∑s=1n[χ(P(s)−χ(P(s+1))](−ℒn(1−μs))ℵ)1ℵ,e−(∑s=1n[χ(P(s)−χ(P(s+1))](−ℒn(αs))ℵ)1ℵ,e−(∑s=1n[χ(P(s)−χ(P(s+1))](−ℒn(vs))ℵ)1ℵ)(5)

Proof: To prove the aforementioned expression, follow the induction method by taking the value of p=2, and we have:

[χ(P1−χ(P2)]g1=(1−e−(∑s=1n[χ(P1−χ(P2)](−ℒn(1−μ1))ℵ)1ℵ,e−(∑s=1n[χ(P1−χ(P2)](−ℒn(α1))ℵ)1ℵ,e−(∑s=1n[χ(P1−χ(P2)](−ℒn(v1))ℵ)1ℵ)

[χ(P2−χ(P3)]g2=(1−e−(∑s=1n[χ(P2−χ(P3)](−ℒn(1−μ2))ℵ)1ℵ,e−(∑s=1n[χ(P2−χ(P3)](−ℒn(α2))ℵ)1ℵ,e−(∑s=1n[χ(P2−χ(P3)](−ℒn(v2))ℵ)1ℵ)

By using the above Definition 12, we have:

PFCIAAA(g1,g2)=((1−e−(∑s=1n[χ(P1−χ(P2)](−ℒn(1−μ1))ℵ)1ℵ,e−(∑s=1n[χ(P1−χ(P2)](−ℒn(α1))ℵ)1ℵ,e−(∑s=1n[χ(P1−χ(P2)](−ℒn(v1))ℵ)1ℵ),(1−e−(∑s=1n[χ(P2−χ(P3)](−ℒn(1−μ2))ℵ)1ℵ,e−(∑s=1n[χ(P2−χ(P3)](−ℒn(α2))ℵ)1ℵ,e−(∑s=1n[χ(P2−χ(P3)](−ℒn(v2))ℵ)1ℵ))

=(1−e−((∑s=1n[χ(P1−χ(P2)])(−ℒn(1−μ1))ℵ)⨁((∑s=1n[χ(P2−χ(P3)])(−ℒn(1−μ2))ℵ)1ℵ,e−((∑s=1n[χ(P1−χ(P2)])(−ℒn(α1))ℵ)⨁((∑s=1n[χ(P2−χ(P3)])(−ℒn(α2))ℵ)1ℵ,e−((∑s=1n[χ(P1−χ(P2)])(−ℒn(v1))ℵ)⨁((∑s=1n[χ(P2−χ(P3)])(−ℒn(v2))ℵ)1ℵ)

=(1−e−(∑s=12[χ(P(s)−χ(P(s+1))](−ℒn(1−μ1))ℵ)1ℵ,e−(∑s=12[χ(P(s)−χ(P(s+1))](−ℒn(α1))ℵ)1ℵ,e−(∑s=12[χ(P(s)−χ(P(s+1))](−ℒn(v1))ℵ)1ℵ)

Hence, this is true for s=2:

i. Now, suppose that this is true for s=k. Then, we have:

PFCIAAA(g1,g2,…,gn)=∑s=1k[χ(P(s)−χ(P(s+1))]gk=(1−e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(1−μk))ℵ)1ℵ,e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(αk))ℵ)1ℵ,e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(vk))ℵ)1ℵ)

Now, for s=k+1. We get:

PFCIAAA(g1,g2,…,gk,gk+1)=⊕s=1k([χ(P(s)−χ(P(s+1))]gk⊕[χ(P(s)−χ(P(s+1))]gk+1)

=((1−e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(1−μk))ℵ)1ℵ,e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(αk))ℵ)1ℵ,e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(vk))ℵ)1ℵ)⊕(1−e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(1−μk+1))ℵ)1ℵ,e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(αk+1))ℵ)1ℵ,e−(∑s=1k[χ(P(s)−χ(P(s+1))](−ℒn(vk+1))ℵ)1ℵ))

=(1−e−(∑s=1k+1[χ(P(s)−χ(P(s+1))](−ℒn(1−μk+1))ℵ)1ℵ,e−(∑s=1k+1[χ(P(s)−χ(P(s+1))](−ℒn(αk+1))ℵ)1ℵ,e−(∑s=1k+1[χ(P(s)−χ(P(s+1))](−ℒn(vk+1))ℵ)1ℵ)

□

Theorem 3: For any gs=(μs,αs,vs),s=1,2,3,…,n are equal, that is, gs=g for all g. Then, we have:

PFCIAAA(g1,g2,g3,…gn)=g

Proof: Since gs=(μs,αs,vs),s=1,2,3,…,n. Then:

PFCIAAA(g1,g2,g3,…gn)=∑s=1n([χ(P(s)−χ(P(s+1))]gs)

=(1−e−(∑s=1n[χ(P(s)−χ(P(s+1))](−ℒn(1−μs))ℵ)1ℵ,e−(∑s=1n[χ(P(s)−χ(P(s+1))](−ℒn(αs))ℵ)1ℵ,e−(∑s=1n[χ(P(s)−χ(P(s+1))](−ℒn(vs))ℵ)1ℵ)

=(1−e−(−ℒn(1−(μs))ℵ)1ℵ,e−((−ℒn(αs))ℵ)1ℵ,e−((−ℒn(vs))ℵ)1ℵ)=g

□

Theorem 4: Let gs=(μs,αs,vs),s=1,2,3,…,n be the collection of PFVs, with PA [χ(P(s)−χ(P(s+1))] operator for each gs. Let g−=min(g1,g2,g3,…,gn) and g+=max(g1,g2,g3,…,gn). So,

g−≤PFCIAAA(g1,g2,g3,…,gn)≤g+

Proof: Let gs=(μs,αs,vs),s=1,2,3,…,n be the collection of PFVs, with PA χ(P(s)−χ(P(s+1)) operator for each gs. Let g−=min(g1,g2,g3,…,gn)=(μ−,α−,v−) and g+=max(g1,g2,g3,…gn)=(μ+,α+,v+). We have, μ−=mins{μs},α−=maxs{αs} and v−=maxs{vs} and μ+=maxs{αs},α+=mins{αs},v+=mins{αs}.