Evaluation of mental disorder with prioritization of its type by utilizing the bipolar complex fuzzy decision-making approach based on Schweizer-Sklar prioritized aggregation operators

Literature Review

Roth (1955) interpreted the history of mental disorders. Harris & Barraclough (1998) investigated the mortality of mental disorders. Similarly, various other scholars have investigated mental disorders. For instance, Waern et al. (2002) studied mental disorders in elderly suicides, Verhaak et al. (2005) investigated chronic disease and mental disorder, Monahan & Steadman (1983) described mental disorder and crime, Morse (2011) investigated criminal law and mental disorder, Robbins, Monahan & Silver (2003) studied gender, violence, and mental disorder, Eaton, Muntaner & Sapag (1999) studied mental disorder and socioeconomic stratification. Silvana, Akbar & Audina (2018) classified the features of the mental disorder by utilizing fuzzy information. Liu et al. (2021) have diagnosed mental disorder in children by utilizing the fuzzy mathematical model. Chaudhuri, Chandrika & Kumari (2016) investigated mental health by employing neuro-fuzzy methods. van Draanen et al. (2022) investigated mental disorders and opioid overdose. The mental disorders due to COVID-19 and other epidemics were investigated by Leung et al. (2022). Denche-Zamorano et al. (2022) studied the enhancement of the risk of mental disorders in youth because of the lack of physical activities. Merikangas, Nakamura & Kessler (2022) investigated the epidemiology of mental disorders. Sumathi & Poorna (2017) predicted mental health problems through fuzzy clustering. The ECG-based mental depression evaluation through fuzzy computing was interpreted by Chiang (2015). Wang et al. (2017) investigated fuzzy maps on mental disorders. Kai, Vijayashree & Jayashree (2017) studied a diagnosis framework relying on fuzzy logic for mental disorders.

Zhang, Pandurangi & Peace (2007) studied major depressive and bipolar disorder. Han et al. (2018) investigated bipolar fuzzy cognition and bipolar disorder. Jun & Kavikumar (2011) studied finite state machines in bipolar fuzzy (BF) information. Bloch (2012) introduced a mathematical morphology under BFS. Wei et al. (2018), Jana, Pal & Wang (2019), and Riaz et al. (2022) introduced various aggregation operators (AOs) in the setting of BFS. Alghamdi, Alshehri & Akram (2018) studied a MADM (multi-attribute DM) approach for BFS. Akram & Akmal (2016) studied the application of BFS in the model of the graph. Samanta & Pal (2012) introduced the BF hypergraph. Bi et al. (2019) investigated arithmetic and Bi, Dai & Hu (2018) studied geometric AOs under CFS. The notion of BCFS is utilized in numerous fields such as pattern recognition (Ur Rehman & Mahmood, 2022) and medical diagnosis (Ur Rehman & Mahmood, 2022), and numerous DM approaches are in the structure of BCF information such as Mahmood et al. (2021) investigated the MADM procedure, Mahmood et al. (2022b) studied decision support systems, Mahmood & Ur Rehman (2022a) introduced MADM relying on Dombi AOs, and Rehman et al. (2022) devised the Analytical Hierarchy process. Furthermore, Mahmood, Rehman & Ali (2022) deduced Aczel-Alina, Mahmood & Ur Rehman (2022b) originated the Maclurin symmetric mean, Mahmood et al. (2022a) introduced the Bonferroni mean, and Mahmood, Rehman & Naeem (2023) introduced Heronian mean AOs based on the BCF information.

Motivation

The AOs depend on various sorts of operators and t-norms and t-conorms (TTs) like prioritized operators interpreted by Yager (2008), Schweizer-Sklar deduced by Deschrijver & Kerre (2002), the Bonferroni mean presented by Bonferroni (1950), the Maclaurin mean investigated by Maclaurin (1729), the Hamacher originated by Hamacher (1978) and the Dombi TTs deduced by Dombi (1982). In particular, the SS t-norm and t-conorm Deschrijver & Kerre (2002) contain a variable parameter that affects the related operations and makes them more flexible than the rest of the operators. Numerous authors utilized the SS t-norm and t-conorm to investigate AOs in various areas, as Liu & Wang (2018) introduced SS AOs for interval-valued intuitionistic FS (IVIFS), Biswas & Deb (2021) investigated SS AOs for Pythagorean FS, Tian et al. (2022) investigated for picture FS and Liu, Khan & Mahmood (2019) deduced for neutrosophic FS. Furthermore, the prioritized operators (PROs) can consider the prioritization relatedness over attributes or criteria and make the AOs more effective and useful. The prioritization among attributes can be structured by developing the weights linked to the attributes relying on the fulfillment of the superior priority criteria. Here, a concern arises of what would be happened if the decision-maker or expert has to consider both poles of the objects and extra fuzzy information in a single structure and need a variable parameter as well as the prioritization relatedness over attributes or criteria to make the aggregation procedure more effective and beneficial. None of the AOs and notions in the prevailing literature can cope with such situations and DM issues involving such sort of information and need variable parameters as well as the prioritization relatedness over attributes or criteria in the aggregation of the information. To remove this concern and fill this research gap in this script, we investigate:

• the prioritized AOs by employing SS t-norm and t-conorm in the setting of BCFS such as BCF Schweizer-Sklar prioritized weighted averaging (BCFSSPRWA), BCF Schweizer-Sklar prioritized ordered weighted averaging (BCFSSPROWA), BCF Schweizer-Sklar prioritized weighted geometric (BCFSSPRWG), and BCF Schweizer-Sklar prioritized ordered weighted geometric (BCFSSPRWG) operators;

• a technique of DM under the environment of BCFS by employing the investigated AOs.

• Further, we analyze mental disorders and rank the most prevalent types of mental disorders by employing the notion of BCF decision-making technique depending on the introduced operators.

The introduced work is the generalization of various prevailing ideas in the literature such as FS, BFS, and CFS. Thus, the introduced AOs can easily aggregate the information interpreted in the setting of FS, BFS, and CFS and the introduced DM technique can cope with DM issues under the environment of FS, BFS, and CFS.

‘Literature Review’ contains the basic review of the mental disorder, BCFS, prioritized operators, and SS t-norm and t-conorms. ‘Preliminaries’ contains the SS operational laws relying on BCF numbers (BCFNs) and BCFSSPRWA, BCFSSPROWA, BCFSSPRWG, and BCFSSPROWG operators based on these SS operational laws. ‘BCF Schweizer-Sklar Prioritized AOs’ contains the ranking of the kinds of mental disorder with the help of the approach of DM based on the introduced operators in the setting of BCF information. ‘Comparative Study’ contains the comparative study and ‘Conclusion’ contains the conclusion of this article.

Mental disorder

A broad variety of psychiatric conditions that influence your feelings, thoughts, and conduct patterns are described as mental diseases, occasionally identified as mental health disorders. Recession, social phobia, schizophrenic psychosis, eating disorders, and behavioral addictions are certain examples of mental disorders. In certain circumstances, many people fight for their psychological health. But when alarming signs persist and frequently causes alarm and weaken your ability to execute tasks, a mental health issue becomes a mental disorder. One may experience unhappiness as a consequence of a mental disorder, which can also affect everyday activities involving interpersonal relationships, jobs, and academics. For the majority of the time, a mix of drugs and behavioral therapy improves managing symptoms (psychotherapy).

Symptoms

Varying on the problems, the setting, and other elements, there can be a broad range of indications and manifestations of mental disorders. Emotional responses, beliefs, and behavior patterns can be impacted by the symptoms of mental disorders. Indicators and symptoms, for instance, include (1) issues with comprehension and interpersonal relationships, (2) intense wrath, fury, or aggression, (3) disruption of realism (fantasies), doubt, or mental confusion, (4) suicidality, (5) extreme concern, uneasiness, or regret (6), significant dietary alterations, (7) excessive exhaustion, extreme fatigue, or sleeping issues, (8) being unable to deal with stress or daily difficulties, (9) feeling low or sad, (10) up and downs of mood disturbances, (11) sexual desire modification, (12) alcoholism or drug use issue, (13) thinking that is jumbled up or has diminished focus, (14) abandoning people and interests.

Medical ailments like headaches, backaches, stomachaches, or other inexplicable sensations and discomfort can occasionally be mistaken for signs of mental illness.

Causes

Numerous social and hereditary variables are known to play a role in the development of mental diseases in general:

• Cognitive chemistry: The brain’s chemical messengers, known as neurotransmitters, send indicators to several regions of the body and mind. The functionality of neurotransmitters and nervous structures shifts when the brain pathways containing these substances are negotiated, which creates anxiety and other mental issue.

• Inherited traits: Individuals with mental disorders possibly have biological relations who are equally suffering. Your situation as well as certain traits may both improve your risk of mental disease.

• Social exposure before birth: The mental disease may occasionally be connected to prenatal contact with external chronic stress, inflammatory disorders, chemicals, alcoholism, or narcotics.

Prevention

Mental disorders cannot be entirely avoided. However, if you suffer from a mental disorder, managing anxiety, building tolerance, and improving confidence can all assist you in keeping your symptoms in check. The following actions should be taken:

• Ask for help when in need: If you hesitate unless signs become extreme, it may be more difficult to manage mental health issues. Maintenance period therapy may also actually prevent the return of a symptom.

• Take good care of yourself: A healthy lifestyle includes sleeping properly, consuming food well, and working out frequently. Maintain a consistent plan as much as you can. If you have worries about food and exercise, or when you have problems dropping asleep, ask your healthcare doctor.

• Routine check-ups: If you do not feel healthy, do not avoid checkups or appointments with your primary care physician. It is possible that you will have to be examined for brand-new health issues or that you are dealing with pharmaceutical side effects.

• Considering alarming signs: To determine what might cause you problems, talk to a physician or psychotherapist. Develop a plan thus you will recognize how to proceed if your signs reoccur. If your signs or feelings alter, consult with your physician or psychotherapist. To look out for warning indicators, think about enlisting the help of family or friends.

Bipolar complex fuzzy set

Definition 1: (Mahmood & Rehman, 2022) Consider the following structure:

(1)${\displaystyle \Xi =\left\{\left(\zeta ,\left({\mathfrak{Y}}_{\Xi }^P\left(\zeta \right),{\mathfrak{Y}}_{\Xi }^N\left(\zeta \right)\right)\right)|\zeta \in \Pi \right\}=\left\{\left(\zeta ,\left({\mathfrak{Y}}_{\Xi }^{RP}\left(\zeta \right)+\iota {\mathfrak{Y}}_{\Xi }^{IP}\left(\zeta \right),{\mathfrak{Y}}_{\Xi }^{RN}\left(\zeta \right)+\iota {\mathfrak{Y}}_{\Xi }^{IN}\left(\zeta \right)\right)\right)|\zeta \in \Pi \right\}}$ where Π is a set, ${\mathfrak{Y}}_{\Xi }^P\left(\zeta \right)$ is a positive truth degree, ${\mathfrak{Y}}_{\Xi }^N\left(\zeta \right)$ is a negative truth degree contained in a unit square of a complex plane, then Ξ is interpreted as BCFS. The set $\Xi =\left({\mathfrak{Y}}_{\Xi }^P,{\mathfrak{Y}}_{\Xi }^N\right)=\left({\mathfrak{Y}}_{\Xi }^{RP}+\iota {\mathfrak{Y}}_{\Xi }^{IP},{\mathfrak{Y}}_{\Xi }^{RN}+\iota {\mathfrak{Y}}_{\Xi }^{IN}\right)$ is interpreted as BCFN.

Definition 2: (Mahmood & Ur Rehman, 2022a) Consider ${\Xi }_1=\left({\mathfrak{Y}}_{{\Xi }_1}^P,{\mathfrak{Y}}_{{\Xi }_1}^N\right)$ $=\left({\mathfrak{Y}}_{{\Xi }_1}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_1}^{IP},{\mathfrak{Y}}_{{\Xi }_1}^{RN}+\iota {\mathfrak{Y}}_{{\Xi }_1}^{IN}\right)$ and ${\Xi }_2=\left({\mathfrak{Y}}_{{\Xi }_2}^P,{\mathfrak{Y}}_{{\Xi }_2}^N\right)=\left({\mathfrak{Y}}_{{\Xi }_2}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_2}^{IP},{\mathfrak{Y}}_{{\Xi }_2}^{RN}+\iota {\mathfrak{Y}}_{{\Xi }_2}^{IN}\right)$, as two BCFNs with α′ ≥ 0, then we have

1. ${\Xi }_1\oplus {\Xi }_2=\left(\begin{array}{c}\hfill {\mathfrak{Y}}_{{\Xi }_1}^{RP}+{\mathfrak{Y}}_{{\Xi }_2}^{RP}-{\mathfrak{Y}}_{{\Xi }_1}^{RP}{\mathfrak{Y}}_{{\Xi }_2}^{RP}\hfill \\ \hfill +\iota \left({\mathfrak{Y}}_{{\Xi }_1}^{IP}+{\mathfrak{Y}}_{{\Xi }_2}^{IP}-{\mathfrak{Y}}_{{\Xi }_1}^{IP}{\mathfrak{Y}}_{{\Xi }_2}^{IP}\right),\hfill \\ \hfill -\left({\mathfrak{Y}}_{{\Xi }_1}^{RN}{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right)+\iota \left(-\left({\mathfrak{Y}}_{{\Xi }_1}^{IN}{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right)\right)\hfill \end{array}\right)$

2. ${\Xi }_1\otimes {\Xi }_2=\left(\begin{array}{c}\hfill {\mathfrak{Y}}_{{\Xi }_1}^{RP}{\mathfrak{Y}}_{{\Xi }_2}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_1}^{IP}{\mathfrak{Y}}_{{\Xi }_2}^{IP},\hfill \\ \hfill {\mathfrak{Y}}_{{\Xi }_1}^{RN}+{\mathfrak{Y}}_{{\Xi }_2}^{RN}+{\mathfrak{Y}}_{{\Xi }_1}^{RN}{\mathfrak{Y}}_{{\Xi }_2}^{RN}\hfill \\ \hfill \iota \left({\mathfrak{Y}}_{{\Xi }_1}^{IN}+{\mathfrak{Y}}_{{\Xi }_2}^{IN}+{\mathfrak{Y}}_{{\Xi }_1}^{IN}{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right)\hfill \end{array}\right)$

3. ${\Xi }_1=\left(\begin{array}{c}\hfill 1-{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{{\alpha }^{\prime }}+\iota \left(1-{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{{\alpha }^{\prime }}\right),\hfill \\ \hfill -{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{{\alpha }^{\prime }}+\iota \left(-{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{{\alpha }^{\prime }}\right)\hfill \end{array}\right)$

4. ${\Xi }_1^{{\alpha }^{\prime }}=\left(\begin{array}{c}\hfill {\left({\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{{\alpha }^{\prime }}+\iota {\left({\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{{\alpha }^{\prime }},\hfill \\ \hfill -1+{\left(1+{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right)}^{{\alpha }^{\prime }}+\iota \left(-1+{\left(1+{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right)}^{{\alpha }^{\prime }}\right)\hfill \end{array}\right)$.

Definition 3: (Mahmood & Ur Rehman, 2022a) Consider a BCFN $\Xi =\left({\mathfrak{Y}}_{\Xi }^P,{\mathfrak{Y}}_{\Xi }^N\right)=\left({\mathfrak{Y}}_{\Xi }^{RP}+\iota {\mathfrak{Y}}_{\Xi }^{IP},{\mathfrak{Y}}_{\Xi }^{RN}+\iota {\mathfrak{Y}}_{\Xi }^{IN}\right)$, then the score value is introduced as (2)${\displaystyle {\mathfrak{S}}_{\mathfrak{B}}\left(\Xi \right)=\frac{1}{4}\left(2+{\mathfrak{Y}}_{\Xi }^{RP}+{\mathfrak{Y}}_{\Xi }^{IP}+{\mathfrak{Y}}_{\Xi }^{RN}+{\mathfrak{Y}}_{\Xi }^{IN}\right),{\mathfrak{S}}_{\mathfrak{B}}\left(\Xi \right)\in \left[0,1\right]}$

Definition 4: (Mahmood & Ur Rehman, 2022a) Consider a BCFN $\Xi =\left({\mathfrak{Y}}_{\Xi }^P,{\mathfrak{Y}}_{\Xi }^N\right)=\left({\mathfrak{Y}}_{\Xi }^{RP}+\iota {\mathfrak{Y}}_{\Xi }^{IP},{\mathfrak{Y}}_{\Xi }^{RN}+\iota {\mathfrak{Y}}_{\Xi }^{IN}\right)$, then the accuracy value is introduced as (3)${\displaystyle {\mathfrak{H}}_{\mathfrak{B}}\left(\Xi \right)=\frac{{\mathfrak{Y}}_{\Xi }^{RP}+{\mathfrak{Y}}_{\Xi }^{IP}-{\mathfrak{Y}}_{\Xi }^{RN}-{\mathfrak{Y}}_{\Xi }^{IN}}{4},{\mathfrak{H}}_{\mathfrak{B}}\left(\Xi \right)\in \left[0,1\right]}$

PRA operator

Definition 5: (Yager, 2008) Consider $\mathfrak{C}=\left\{{\mathfrak{C}}_1,{\mathfrak{C}}_2,\dots ,{\mathfrak{C}}_{\check{\mathrm{k}}}\right\}$ as a group of attributes and take a prioritization between these attributes interpreted by a linear order that is ${\mathfrak{C}}_1>{\mathfrak{C}}_2>\dots >{\mathfrak{C}}_{\check{\mathrm{k}}}$. This means that if ̦ e $<\breve{\mathrm{g}}$, then ℭ_̦e is prior than ${\mathfrak{C}}_{\breve{\mathrm{g}}}$. ${\mathfrak{C}}_{̦e}\left(\Xi \right)$ is the assessment argument describing the alternative’s performance by keeping in view the attribute ℭ_{𝔄𝔱−̦e} and holding ${\mathfrak{C}}_{\mathfrak{A}\mathfrak{t}-̦e}\in \left[0,1\right]$. If (4)${\displaystyle PRA\left({\mathfrak{C}}_1\left(\Xi \right),{\mathfrak{C}}_2\left(\Xi \right),\dots ,{\mathfrak{C}}_{\mathrm{J}}\left(\Xi \right)\right)=\begin{array}{c}\hfill \check{\mathrm{k}}\hfill \\ \hfill \oplus \hfill \\ \hfill ̦e=1\hfill \end{array}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\mathfrak{C}}_{̦e}\left(\Xi \right)}$

Schweizer-Sklar operations

Definition 6: (Deschrijver & Kerre, 2002) The underneath equations interpret the SS t-norm and t-conorm, respectively, where $\mathfrak{m},\mathfrak{n}\in \left[0,1\right]$ and ℷ < 0. (5)${\displaystyle {\mathfrak{V}}_{\mathbb{SS}}\left(\mathfrak{m},\mathfrak{n}\right)={\left({\mathfrak{m}}^{\mathrm{\gimel }}+{\mathfrak{n}}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}}$ (6)${\displaystyle {\mathfrak{V}}_{\mathbb{SS}}^{\ast }\left(\mathfrak{m},\mathfrak{n}\right)=1-{\left({\left(1-\mathfrak{m}\right)}^{\mathrm{\gimel }}+{\left(1-\mathfrak{n}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}}$

BCF Schweizer-Sklar prioritized AOs

This part of the article contains the SS operational laws relying on the SS t-norm and t-conorm under the environment of BCF information. Furthermore, we invent BCFSSPRWA, BCFSSPROWA, BCFSSPRWG, and BCFSSPROWG operators based on these SS operational laws.

Definition 7: Consider ${\Xi }_1=\left({\mathfrak{Y}}_{{\Xi }_1}^P,{\mathfrak{Y}}_{{\Xi }_1}^N\right)=\left({\mathfrak{Y}}_{{\Xi }_1}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_1}^{IP},{\mathfrak{Y}}_{{\Xi }_1}^{RN}+\iota {\mathfrak{Y}}_{{\Xi }_1}^{IN}\right)$ and ${\Xi }_2=\left({\mathfrak{Y}}_{{\Xi }_2}^P,{\mathfrak{Y}}_{{\Xi }_2}^N\right)=\left({\mathfrak{Y}}_{{\Xi }_2}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_2}^{IP},{\mathfrak{Y}}_{{\Xi }_2}^{RN}+\iota {\mathfrak{Y}}_{{\Xi }_2}^{IN}\right)$, are two BCFNs with ℷ < 0 and α′ > 0 then we investigate the BCF SS operational laws as

1. ${\Xi }_1{\oplus }_{\mathbb{SS}}{\Xi }_2=\left(\begin{array}{c}\hfill 1-{\left({\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}+{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{RP}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left({\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}+{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{IP}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left({\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}+{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left(-{\left({\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}+{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)$

2. ${\Xi }_1{\otimes }_{\mathbb{SS}}{\Xi }_2=\left(\begin{array}{c}\hfill {\left({\left({\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}+{\left({\mathfrak{Y}}_{{\Xi }_2}^{RP}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left({\left({\left({\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}+{\left({\mathfrak{Y}}_{{\Xi }_2}^{IP}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -1+{\left({\left(1+{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right)}^{\mathrm{\gimel }}+{\left(1+{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-1+{\left({\left(1+{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right)}^{\mathrm{\gimel }}+{\left(1+{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right)}^{\mathrm{\gimel }}-1\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)$

3. ${\alpha }^{\prime }{\Xi }_1=\left(\begin{array}{c}\hfill 1-{\left({\alpha }^{\prime }{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left(1-{\left({\alpha }^{\prime }{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left({\alpha }^{\prime }{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left(-{\left({\alpha }^{\prime }{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)$

4. ${\left({\Xi }_1\right)}^{{\alpha }^{\prime }}=\left(\begin{array}{c}\hfill {\left({\alpha }^{\prime }{\left({\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota {\left({\alpha }^{\prime }{\left({\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}},\hfill \\ \hfill -1+{\left({\alpha }^{\prime }{\left(1+{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right)}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left(-1+{\left({\alpha }^{\prime }{\left(1+{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right)}^{\mathrm{\gimel }}-\left({\alpha }^{\prime }-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)$

Definition 8: Consider ${\Xi }_{̦e}=\left({\mathfrak{Y}}_{{\Xi }_{̦e}}^P,{\mathfrak{Y}}_{{\Xi }_{̦e}}^N\right)=\left({\mathfrak{Y}}_{{\Xi }_{̦e}}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_{̦e}}^{IP},{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RN}+\iota {\mathfrak{Y}}_{{\Xi }_{̦e}}^{IN}\right),̦e=1,2,..,\check{\mathrm{k}}$ as a group of BCFNs, then (7)${\displaystyle BCFSSPRWA\left({\Xi }_1,{\Xi }_2,\dots ,{\Xi }_{\check{\mathrm{k}}}\right)=\begin{array}{c}\hfill \check{\mathrm{k}}\hfill \\ \hfill \oplus \hfill \\ \hfill ̦e=1\hfill \end{array}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\Xi }_{̦e}}$

is introduced as BCFSSPRWA operator, where Ť₁ = 1, ${\mathrm{Ť}}_{̦e}={\prod }_{̦e=1}^{\check{\mathrm{k}}-1}{\mathfrak{S}}_{\mathfrak{B}}\left({\Xi }_{̦e}\right),̦e=1,2,\dots ,\check{\mathrm{k}}$ and ${\mathfrak{S}}_{\mathfrak{B}}\left({\Xi }_{̦e}\right)$ would deduce the score value of BCFN Ξ_̦e.

Theorem 1: Consider ${\Xi }_{̦e}=\left({\mathfrak{Y}}_{{\Xi }_{̦e}}^P,{\mathfrak{Y}}_{{\Xi }_{̦e}}^N\right)=\left({\mathfrak{Y}}_{{\Xi }_{̦e}}^{RP}+\iota {\mathfrak{Y}}_{{\Xi }_{̦e}}^{IP},{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RN}+\iota {\mathfrak{Y}}_{{\Xi }_{̦e}}^{IN}\right),̦e=1,2,\dots ,\check{\mathrm{k}}$ as a group of BCFNs, then by utilizing BCFSSPRWA, we achieve the aggregated result and (8)${\displaystyle BCFSSPRWA\left({\Xi }_1,{\Xi }_2,\dots ,{\Xi }_{\check{\mathrm{k}}}\right)=\left(\begin{array}{c}\hfill 1-{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RP}\right)}^{\mathrm{\gimel }}\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left(1-{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_{̦e}}^{IP}\right)}^{\mathrm{\gimel }}\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RN}\right|}^{\mathrm{\gimel }}\right)}^{\frac{1}{\mathrm{\gimel }}}+\iota \left(-{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_{̦e}}^{IN}\right|}^{\mathrm{\gimel }}\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)}$

Proof: Equation (8) can be rewritten as underneath (9)${\displaystyle BCFSSPWA\left({\Xi }_1,{\Xi }_2,\dots ,{\Xi }_{\check{\mathrm{k}}}\right)=\left(\begin{array}{c}\hfill 1-{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RP}\right)}^{\mathrm{\gimel }}-\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}+1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_{̦e}}^{IP}\right)}^{\mathrm{\gimel }}-\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}+1\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RN}\right|}^{\mathrm{\gimel }}-\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}+1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-{\left(\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_{̦e}}^{IN}\right|}^{\mathrm{\gimel }}\right)}^{\frac{1}{\mathrm{\gimel }}}-\sum _{̦e=1}^{\check{\mathrm{k}}}\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^{\check{\mathrm{k}}}{\mathrm{Ť}}_{̦e}}+1\right)\hfill \end{array}\right)}$

Through mathematical induction, we would exhibit Eq. (9) is holds $\forall \check{\mathrm{k}}$. Let $\check{\mathrm{k}}=2$. Then ${\displaystyle \frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\Xi }_1=\left(\begin{array}{c}\hfill 1-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)}$

and ${\displaystyle \frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\Xi }_2=\left(\begin{array}{c}\hfill 1-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{RP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{IP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)}$

then, ${\displaystyle \frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\Xi }_1{\oplus }_{\mathbb{SS}}\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\Xi }_2=\left(\begin{array}{c}\hfill 1-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_1-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right){\oplus }_{\mathbb{SS}}\left(\begin{array}{c}\hfill 1-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{RP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{IP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill 1-{\left(\begin{array}{c}\hfill {\left(1-\left(1-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\right)}^{\mathrm{\gimel }}\hfill \\ \hfill +{\left(1-\left(1-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{RP}\right)}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\right)}^{\mathrm{\gimel }}-1\hfill \end{array}\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\begin{array}{c}\hfill {\left(1-\left(1-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\right)}^{\mathrm{\gimel }}\hfill \\ \hfill +{\left(1-\left(1-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{IP}\right)}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\right)}^{\mathrm{\gimel }}-1\hfill \end{array}\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \end{array}\hfill \\ \hfill -{\left(\begin{array}{c}\hfill {\left|-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right|}^{\mathrm{\gimel }}\hfill \\ \hfill +{\left|-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right|}^{\mathrm{\gimel }}-1\hfill \end{array}\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \end{array}\hfill \\ \hfill +\iota \left(-{\left(\begin{array}{c}\hfill {\left|-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right|}^{\mathrm{\gimel }}\hfill \\ \hfill +{\left|-{\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right)}^{\frac{1}{\mathrm{\gimel }}}\right|}^{\mathrm{\gimel }}-1\hfill \end{array}\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)}$

As $\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right),\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$, $\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$, and $\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$ would be always non-negative so $\left|\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right|=\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$, $\left|\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right|=\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$, $\left|\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right|=\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$, $\left|\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\right|=\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_2}{{\sum }_{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)$. ${\displaystyle =\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill 1-{\left(\begin{array}{c}\hfill \frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{RP}\right)}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\hfill \\ \hfill +\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{RP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)-1\hfill \end{array}\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\begin{array}{c}\hfill \frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_1}^{IP}\right)}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)\hfill \\ \hfill +\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_2}^{IP}\right)}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)-1\hfill \end{array}\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \end{array}\hfill \\ \hfill -{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{RN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)+\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{RN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)-1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \end{array}\hfill \\ \hfill +\iota \left(-{\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_1}^{IN}\right|}^{\mathrm{\gimel }}-\left(\frac{{\mathrm{Ť}}_1}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}-1\right)+\frac{{\mathrm{Ť}}_2}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_2}^{IN}\right|}^{\mathrm{\gimel }}-\left({\mathfrak{K}}_2-1\right)-1\right)}^{\frac{1}{\mathrm{\gimel }}}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1-{\left(\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RP}\right)}^{\mathrm{\gimel }}-\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}+1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(1-{\left(\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left(1-{\mathfrak{Y}}_{{\Xi }_{̦e}}^{IP}\right)}^{\mathrm{\gimel }}-\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}+1\right)}^{\frac{1}{\mathrm{\gimel }}}\right),\hfill \\ \hfill -{\left(\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}{\left|{\mathfrak{Y}}_{{\Xi }_{̦e}}^{RN}\right|}^{\mathrm{\gimel }}-\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}+1\right)}^{\frac{1}{\mathrm{\gimel }}}\hfill \\ \hfill +\iota \left(-{\left(\sum _{̦e=1}^2{\mathfrak{K}}_{̦e}{\left|{\mathfrak{Y}}_{{\Xi }_{̦e}}^{IN}\right|}^{\mathrm{\gimel }}\right)}^{\frac{1}{\mathrm{\gimel }}}-\sum _{̦e=1}^2\frac{{\mathrm{Ť}}_{̦e}}{\sum _{̦e=1}^2{\mathrm{Ť}}_{̦e}}+1\right)\hfill \end{array}\right)}$