key: cord-0889246-vociaw7e
authors: Kirişci, Murat; Demir, İbrahim; Şimşek, Necip; Topaç, Nihat; Bardak, Musa
title: The novel VIKOR methods for generalized Pythagorean fuzzy soft sets and its application to children of early childhood in COVID-19 quarantine
date: 2021-09-15
journal: Neural Comput Appl
DOI: 10.1007/s00521-021-06427-3
sha: f231fe1fa253eda6b987b47b2fd5845f5b1f58fc
doc_id: 889246
cord_uid: vociaw7e

In this work, the new VIKOR methods are established using the generalized Pythagorean fuzzy soft sets (GPFSSs). For GPFSSs, the distance measures such as Hamming, Euclidean, and generalized are given. Further, the basic characteristics of these distance measures are examined. Fuzzy and soft sets are strong instruments for uncertainty. This strongness has been demonstrated by the GPFSS combining Pythagorean fuzzy sets and soft sets and applied to imprecise and ambiguous information. In this context, new remoteness index-based methods have been proposed, which are dissimilar from available VIKOR methods. The displaced and fixed ideals positive and negative Pythagorean fuzzy values (PFV) were defined. Thus, based on this definition, displaced positive ideal remoteness indices, negative ideal remoteness indices, and fixed positive ideal, negative ideal remoteness indices were discussed. Two different weights are used here: weights based on OF preference information and precise weights calculated with the expectation score function. The VIKOR method given here provides a different way from canonical VIKOR methods: rank candidate alternatives and determining a compromise solution based on different preference structures. The processes principles of the newly defined GPFSSs VIKOR methods are given by four algorithms. An example of these algorithms is given with the behavioral development and cognitive development of the children of Early Childhood children in the COVID-19 quarantine.

Uncertainty is a crucial concept for decision-making problems. It is not easy to make precise decisions in life since each information contains vagueness, uncertainty, imprecision. Fuzzy Set(FS) Theory, Zadeh's [49] pioneering work, proposed a membership function to solve problems such as vagueness, uncertainty, imprecision, and this function took value in the range of [0,1]. FS Theory had solved many problems in practice, but there was no membership function in real life, which only includes acceptances. Rejection is as important as acceptance in real life. Atanassov [3] clarified this problem and posed the Intuitionistic Fuzzy Set(IFS) Theory using the membership function as well as the non-membership function. In IFS, the sum of membership and non-membership grades is 1. This condition is also a limitation for solutions of vagueness, uncertainty, imprecision. Yager [46] has presented a solution to this situation and suggested Pythagorean Fuzzy Sets(PFS). PFS is more comprehensive than IFS because it uses the condition that the sum of the squares of membership and non-membership grades is equal to or less than 1. PFS is also a particular case of the Neutrosophic Set initiated by Smarandache [43] . There are many studies in the literature on FS, IFS, and PFS theories [1, 2, 13-20, 23-27, 29, 30, 38-41, 43-45, 47, 48, 51] . Despite all the possible solutions, these theories have limitations. How to set the membership function in each particular object and the deficiencies in considering the parametrization tool can be given as examples of these limitations. These limitations handicap decision-makers from making a correct decision during the analysis.

A new method, called Soft Set, was proposed by Molodtsov [33] , in which the preferences for each alternative were given in distinct parameters, and thus a solution was found for the limitations expressed. Immediately after the occurrence of SS theory, Fuzzy Soft Sets [29] and Intuitionistic Fuzzy Soft Set(IFSS) [30] were defined and their various properties were studied [31, 32] . Pythagorean Fuzzy Soft Set(PFSS) is defined by Peng et al. [38] . PFSS is a natural generalization of IFSS and is a parameterized family of PFSs. In [4, 5, 22, 34, 42] , the main features of PFSS were examined and applied to various areas such as medical diagnosis, selection of a team of workers for business, stock exchange investment problem. The benefit of these extended theories is that they are capable of simplifying the characterization of real-life cases with the help of their parameterized feature.

It is possible that people may hesitate during decisionmaking. In order to avoid human hesitations from adversely affecting the decision-making process, hesitation value is also taken into account in PFS, just like IFS. Thus, experts may have hesitations about membership grades. If the expert participating in the decision process is only one person, this expert's error or bias will affect the process negatively. However, hesitation is subjective and the expert's hesitation can be directed by her/his own perceptions. In this case, enriching the decision process, making the evaluation with alternative decisions more meaningful, combining the subjective evaluations of more than one expert instead of the subjective evaluation of a single expert will provide a healthier decision-making process. With this in mind, Agarwal et al. [1] defined Generalized Intuitionistic Fuzzy Soft Sets(GIFSS). Feng et al. [14] identified some problems and difficulties in the definition of GIFSS and operations related to GIFSS in the manuscript of Agarwal et al [1] . Kirisci [27] defined GPFSS, considering the fixes in [14] .

GPFSS ensures the frame for evaluating the reliability of the info in the PFSS so as to compensate for any distortion in the info given. The most important benefit of incorporation of the generalized parameter into the analysis is to decrease the likelihood of errors induced by the imprecise info by taking the chairperson's view on the same. For example, a patient may give wrong information to a physician about her/his symptoms. If the physician does not notice this wrong information, errors in diagnosis and treatment will occur. In this case, an experienced physician can measure the reliability of the information given by the patient with a generalization parameter. So, there is a requirement for a generalization parameter, demonstrating an expert's level of confidence in the reliability of the info, respectably making the approach quite close to real-world cases. This assists in extracting the singular bias from the input data and gets more credibility to the final decision. GPFSS has a generalization parameter to states the uncertainties.

Usually, the most present multiple criteria decision analysis (MCDA) methods goal to discover a unique solution that maximum achieves the general criteria as much as possible. The VIKOR method is skilled in specifying a set of compromise solutions(CS) in the existence of contradictory criteria according to the choices of the decision-makers. The VIKOR method developed by Opricovic and Tzeng [35, 36] focuses on ranking and selecting from a set of alternatives, and determines compromise solutions for a problem with conflicting criteria, which can help the decision makers to reach a final decision. This abbreviation comes from the Serbian name of the method: Višekriterijumska Optimizacija I Kompromisno Resenje(multicriteria optimization and compromise solution). Here, the CS is a feasible solution that is the closest to the ideal solution, and a compromise means an agreement established by mutual concessions [21, 35, 37] .

VIKOR focalizes on the specification of a set of CSs that are built by reciprocal compensations [6, 7, 28, 35, 37] to efficiently help MCDA and improve the quality of DM [50] . In recent years, VIKOR method studies have acquired popularity in varied MCDA areas because of their large ability to compromise rank performance by comparing the measure of closeness to the ideal solution [21] . Because of these properties, the VIKOR method has become popular and has been used in many areas [21] . The VIKOR method usually utilizes the positive-and negative-ideal solutions to serve as points of reference, and the specification of these ideal solutions is immediately produced from the evaluative ratings of all alternatives according to various criteria [35, 37] . Chen [12] defined the VIKOR method based on distance index for Pythagoras fuzzy clusters and applied it to various problems.

PF sets include more uncertainties and are generally able to accommodate higher degrees of uncertainty. To efficiently administrate the complex uncertainty in human cognitive and DM activities, it is necessary to expand the valid VIKOR methods that previously focalized on certain settings and fuzzy environments to the PF context. Although the VIKOR method is a very qualified approach in terms of DM processes, it has been used very little in medical DM, health services, and child development.

The aim of this study is to develop a new approach to solve MADM problems by integrating the VIKOR method with GPFSS. It is known that the importance of the VIKOR method is to determine the CS and to improve the standard of the DM process. The VIKOR method is balancing the majority's maximum group utility and the opponent's minimum individual regret. In the VIKOR method, the best and worst solutions serve as the point of reference and the distance between the best solution and evaluative ratings does not have an upper bound. The new idea of remoteness index is considerable because it ensures the upper bounds by dividing the distance between the best solution and evaluative ratings by the distance between the best solution and the worst solution. The novel ranking indexes based on the new idea of remoteness indexes are debated to ensure more efficient compromise rankings. Thus, a remoteness index-based VIKOR method for GPFSS is improved.

The negatives caused by early childhood children staying at home during quarantine times are measured with new algorithms. All areas of development are in interaction with each other, and the problem that may occur in one area has reflections on other areas. In this context, problems in the cognitive field may cause problems in other areas, as well as problems in other areas that may reflect on cognitive development. Therefore, it is necessary to know the development characteristics of each field well and to identify the problems that may arise in a timely and accurate manner. The pre-school period has a very important place in human life. In this period, the child's personality development and emotional, social, cognitive, language, and academic development are largely completed. The child completes her/his cognitive and behavioral development in a healthy way with the contribution of her family and school environment. That's why new algorithms have been applied to such an important subject.

The originality of this study can be expressed as follows: New distance measures such as Euclidean, Hamming, and Generalized for GPFFS are defined. The properties of these distance measures were examined, and an example was given. The displaced and fixed ideals are defined for PFVs. In addition, the displaced and fixed remoteness indexes are given for PFVs. Two types of weights called precise weights and PF weights are defined and discuss in their methods of generation. The precise importance weights are obtained by the expectation score function. The PF importance weights represented the importance and unimportance degrees of the criteria. The displaced and fixed ideals, the displaced and fixed remoteness indexes are identified as ranking indexes (four groups). Based on these definitions, four algorithms expressing VIKOR methods have been proposed. The problem of the cognitive and behavioral development of early childhood children in quarantine is examined with the suggested remotenessbased VIKOR method.

In this section, basic definitions, theorems, propositions and properties that will be used in the whole study will be given. Throughout the study, the initial universe, parameters sets will denote ! , R, respectively.

The theory of IFS was developed by Atanassov [3] and is a natural generalization of the theory of FS. IFS is defined by MS and NMS degrees and, for this reason, can indicate the fuzzy character of data in more elaboration comprehensively. The significant qualification of IFS is that it appoints to each element MS and NMS degrees (MS þ NMS 1). However, in some DM processes, the sum of the MS and NMS values obtained may be greater than 1. In this case, we can take the sum of the squares of these MS and NMS values obtained, which will be less than or equal to 1.

As an original idea, PFSs were created by Yager [45] . PFS is a very useful tool for uncertainty. PFS offers good results especially in solutions where IFS is insufficient. The differences between PFSs and IFSs can be mentioned as follows:

For, the initial universe set !, the function u A ðxÞ : ! ! ½0; 1 is called FS on !. The FS can be indicated by

Choose the MS function u B : ! ! ½0; 1 and the NMS function v B : ! ! ½0; 1. Let's assume that the condition 0 u B ðaÞ þ v B ðaÞ 1 for any a 2 ! is satisfied. Then, the set B ¼ fða; u B ðaÞ; v B ðaÞÞ : a 2 !g is said to be an IFS B on ! . In this case, the condition p B ¼ 1 À u B ðxÞ À v B ðxÞ holds [3] . Suppose that condition 0 ½u C ðxÞ 2 þ ½v C ðxÞ 2 1 satisfies for u C : ! ! ½0; 1 and v C : ! ! ½0; 1. Then, an PFS C in ! is defined by C ¼ fðx; u C ðxÞ; v C ðxÞÞ : x 2 !g. In this case, the condition p C ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ½u C ðxÞ 2 À ½v C ðxÞ 2 q holds [45, 46, 48] .

Let C be a PFS over R. In this definition, (F, C) can be represented as

The set of all PFS on ! will be denoted by Xð!Þ. Let ! ,R be an initial universe and a parameters sets, respectively. For D R, consider a set-valued mapping F : D ! xð!Þ, where the power set of ! is showed by xð!Þ. Therefore, a pair (F, D) is called a SS on !.

For E R, choose F : E ! xð!Þ, where the set of all PFSs over ! is indicated by xð!Þ. Then, a pair (F, E) is called a PFSS on xð!Þ [38] .

Choose the Pythagorean fuzzy numbers (

some fundamental operations are as follows [45, 48] :

Þ. The PFS defined by MS and NMS which satisfies the condition MS 2 þ NMS 2 1 is initiated by Yager [45] A new PFN formula consisting of the strength of commitment r p and the direction of commitment d p was also given by Yager [45] . The new PFN is denoted by P ¼ ðr p ; d p Þ, where r p 2 ½0; 1. The stronger commitment is defined by the larger the value of r p , in this case the lower the uncertainty of the commitment. r p and d p are characterized by u p ¼ r p cosðh p Þ, v p ¼ r p sinðh p Þ, i.e., these values are related to MS and NMS grades. Here h p is expressed in radian and calculate h p ¼ ð1 À d p Þðp=2Þ. If P ¼ ðr p ; d p Þ is defined in terms of polar coordinates, it is denoted as ðr p ; h p Þ. In this case, d p has a formula in the form of d p ¼ 1 À ð2:h p =pÞ. It can then be said that PFNs consists of u p ; v p ; p p ; r p ; d p and h p parameters. If u p ; v p 2 ½0; 1, then it is clear that u 2 p u p , v 2 p v p .

Consider the L ¼ fða; bÞ : a; b 2 ½0; 1; a 2 þ b 2 \1g with the partial order L . The complete lattice ðL; L Þ is defined by ða; bÞ L ði; jÞ , a i and b ! j, for all ða; bÞ; ði; jÞ 2 L.

The ða; bÞ 2 L given in this way is called Pythagorean fuzzy value (PFN)(or Pythagorean fuzzy number(PFN)) [16] . 

The values f C ðb 1 Þ ¼ ð0:77; 0:35Þ, f C ðb 3 Þ ¼ ð0:65; 0:42Þ and f C ðb 4 Þ ¼ ð0:88; 0:24Þ express the diagnostic view of a physician. The (F, C, f) is shown in Table 1 .

For all R ¼ ðu R ; v R Þ 2 L , the score function for PFNs ie described Therefore, if only the scoring function is used for comparison, it is not possible to make a comparison between these numbers. To overcome this problem, we can define a new function [1] , as follows:

The mapping AC : L ! ½0; 1 is called accuracy function, if By replacing the approval rates in Definition 2.5, the other relation ðES;uÞ is written as follows:

Definition 2.7 [27] For two PFNs R; T 2 L and the relation ðES;uÞ on L,

The relationship with each other of ðSF;ACÞ and ðES;uÞ can be given as follows:

Proposition 2.8 [27] Let R and T be PFNs in L. Then, R ðSC;ACÞ T , R ðES;uÞ T.

If we consider the debates of [14, 44] , then the following proposition is given:

Then, the following conditions are equivalent:

be two PFVs. Then, for a [ 0, we have the following operations:

: ð2:15Þ

Neural Computing and Applications Proposition 2.12 [27] For

ð2:16Þ

i. If R ðu;ESÞ T, then aR ðu;ESÞ aT, ii. If a 1 a 2 , then a 1 R ðu;ESÞ a 2 R.

The Hamming, Euclidean and generalized distance measures for GPFSSs are given in this section. Some properties of distance measures are discussed. The new distance measures offered in this section will be then applied to build useful concepts of remoteness indices, consisting of displaced positive-and negative-ideal remoteness indices as well as fixed positive-and negative-ideal remoteness indices.

Let ! ¼ fx 1 ; x 2 ; Á Á Á ; x m g, P ¼ fe 1 ; e 2 ; Á Á Á ; e n g be universe set and parameter set, respectively. Choose the two GPFSSs

; gÞ, the normalized Hamming distance, normalized Euclidean distance and generalized distance measures between X 1 and X 2 are defined, respectively, as follows:

þðvðf Þ e j ðx i Þ À vðgÞ e j ðx i ÞÞ k þ ðpðf Þ e j ðx i Þ À pðgÞ e j ðx i ÞÞ k þðrðf Þ e j ðx i Þ À rðgÞ e j ðx i ÞÞ k þ ðdðf Þ e j ðx i Þ À dðgÞ e j ðx i ÞÞ k 8 > > > > > > > > < > > > > > > > > :

ð3:1Þ

ð3:2Þ

where k ! 1.

The weighted normalized Hamming distance, weighted normalized Euclidean distance and weighted generalized distance measures between X 1 and X 2 are defined, respectively, as follows:

where k ! 1, 0 x i 1ði ¼ 1; 2; Á Á Á ; nÞ and P n i¼1 x i ¼ 1. Theorem 3.3 Let ðL; L Þ be a partially ordered set. For the GPFSSs X 1 ; X 2 ; X 3 belonging to L with the partial order L . A generalized distance measure of U G satisfies the following conditions:

The proof of this theorem is obvious.

ð3:4Þ

ð3:6Þ Neural Computing and Applications Note 1: If k ¼ 1 and k ¼ 2, U H ¼ U G and U E ¼ U G , respectively. Then U H satisfies the conditions of Theorem 3.3 with separability and triangle inequality. Therefore, the Hamming distance measure U H is a metric. Further, U E satisfies the conditions of Theorem 3.3 with separability. Fulfilling these conditions indicates that the U E is a semi-metric. The conditions separability and triangle inequality conditions, respectively, are as follows:

We will only prove for normalized Hamming distance. It can be proved similarly for normalized Euclidean distance and generalized distance measures. The distance between two GPFSSs X 1 ; X 2 can be written as follows:

From the complement of GPFSS, the MS degrees of X t 1 relate to the NMS degrees of X t 2 . Therefore, the distance between X 1 ; X 2 is the same as the distance between X t 1 ; X t 2 . h Example 3.5 Let ! ¼ fx 1 ; x 2 g, M ¼ N ¼ fe 1 ; e 2 ; e 3 g be universe set and parameter sets, respectively. Choose two GPFFSs X 1 ¼ ðF; M; f Þ, X 2 ¼ ðG; N; gÞ as follows: 

The purpose of this section is to develop new remoteness index-based GPFSS-VIKOR methods that utilize the concepts of displaced and fixed remoteness indices based on new distance measures to address MCDA problems within the GPFSS environment. The definition of positive and negative ideal points for a solution is the basic idea of the VIKOR method. It focalizes on ranking and selecting from a finite set of possible alternatives in the presence of contradictory and non-measurable criteria. It interprets a multicriteria ranking index based on the nearness to the ideal solution. By evaluating the alternatives according to each criterion, a compromise ranking can be obtained when the relative closeness measure is compared with the ideal alternative. Thus, the derived compromise solution is a feasible solution, which is the closest to the positive ideal solution and farthest from the negative ideal solution, and a compromise means an agreement established by mutual concessions made between the alternatives. The acquired CS can be admitted because it ensures a maximum group utility of the majority and a minimum of the individual regret of the opponent. The CSs can be the basis for negotiations, involving the decision maker's preference on criteria weights. In the VIKOR method, the performance ratings of the alternatives according to a set of criteria are quantified as crisp values [11] .

In this section, we define the displaced positive ideal PFV (DPI) and the displaced negative ideal PFV (DNI) which help to define the concept of displaced remoteness index. The fixed remoteness index is defined based on the fixed positive ideal PFS (FPI) and the fixed negative ideal PFV (FNI). The displaced and fixed group utility, individual regret and compromise indexes are the precise importance and Pythagorean fuzzy importance weights. The extra parameter parametric Pythagorean fuzzy set in GPFSS, which given by the head or director of decisionmaking committee, is used to define the precise importance and Pythagorean fuzzy importance weights. Each algorithm shows the complete procedure of the remotenessbased VIKOR method based on the displaced and fixed terminologies.

Let's give the general framework of the MCDA problem about PF information:

Choose the set of candidate alternatives D ¼ fd 1 ; d 2 ; Á Á Á ; d m g, m ðm ! 2Þ and the set of criteria K ¼ fk 1 ; k 2 ; Á Á Á ; k n g n ðn ! 2Þ as a discrete set of m and a finite set of n, respectively. Usually, the criteria set K is divided into two disjoint sets as K i and K ii . K i and K ii refer to a collection of benefit criteria and cost criteria, respectively, which means a larger value better performance and a larger value worse performance, respectively. The assessment rating of alternative d i 2 D with respect to criterion k j 2 K is stated as a PF value p ij ¼ ðu ij ; v ij Þ. u ij and v ij indicate the degrees that alternative d i satisfies and dissatisfies, respectively, criterion k j . The hesitation degree p ij that corresponds to each p ij is calculated as

. Describing this situation, the PF matrix is as follows:

:72 À :66j þ j0:68 À 0:76j þ j0:84 À 0:77j þ j0:75 À 0:83j þ j0:91 À 0:80j þ j0:83 À 0:74j þ j0:4232 À 0:512j þ j0:6772 À 0:56j þ j0:38 À 0:37j þ j0:1225 À 0:5j þ j0:315 À 0:47j þ j0:33 À 0:38j þ j0:906 À 0:86j þ j0:736 À 0:56j þ j0:93 À 0:93j þ j0:992 À 0:87j þ j0:95 À 0:88j þ j0:994 À 0:93j þ j0:585 À 0:56j þ j0:5331 À 0:73j þ j0:7216 À 0:62jj0:544 À 0:8j þ j0:8 À 0:724j þ j0:32 À 0:586j þ j0:48 À 0:46j þ j0:91 À 0:82j þ j0:69 À 0:67j þ j0:83 À 0:63j þ j0:38 À 0:56j þ j0:54 À 0:47j In the PF environment, the importance weight of criterion k j 2 K is stated as a PF value w j ¼ ðw j ; w j Þ, where w j and w j denote the degrees of importance and unimportance, respectively, of k j . Then the hesitation degree

The appropriate reference points need to be designated for the PF decision process. These points will help define displaced and fixed remoteness indices. In other words, locating the PIPF and NIPF values from two different perspectives of displaced and fixed ideals is the main problem. Here, PIPF and NIPF values will be used as points of reference because these points can effectively form anchored judgments in subjective decision-making processes.

To be more precise, for the anchored judgments, the displaced PIPF and NIPF values, which consist of all the best and worst criterion values, respectively, will be defined. Firstly in light of anchored judgments with displaced ideals, the characterization of the PIPF and NIPF values with respect to each criterion can be identified according to all of the assessment ratings in the PF decision matrix p. This means that the PIPF and NIPF values can be often displaced because they are sensitive to changes in the available set of candidate alternatives. Definition 4.1 Take a PF decision matrix p ¼ ½p ij ðmÂnÞ . For the each criterion k j 2 K ðK ¼ K i [ K ii Þ, the displaced positive-ideal and negative-ideal PF values are defined as follows:

and also pðþÞ j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 À ðuðþÞ j Þ 2 À ðvðþÞ j Þ 2 q ð4:4Þ pðÀÞ j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 À ðuðÀÞ j Þ 2 À ðvðÀÞ j Þ 2 q ð4:5Þ

Consider anchored judgments with fixed ideals. It is convenient to designate (1, 0) and (0, 1) as PIPF and NIPF values, respectively, with respect to benefit criteria, because (1, 0) and (0, 1) are the top and bottom elements, respectively, of the lattice ðL; L Þ. Conversely, (0, 1) and (1, 0) can be considered the PIPF and NIPF values, respectively, with respect to cost criteria for simplicity. These ideal PF values are called fixed ideal PF values. Definition 4.2 Take a PF decision matrix p ¼ ½p ij ðmÂnÞ . For the each criterion k j 2 K ðK ¼ K i [ K ii Þ, the displaced PIPF and NIPF values are defined as follows:

and also pð b

In this subsection, it will be constructed the displaced and fixed RIs. Usually, the larger and the smaller the assessment rating p ij is, the greater the preference is for the criteria k j 2 K i and k j 2 K ii , respectively. Therefore the displaced PIPF value pðþÞ j and the fixed PIPF value pð b þÞ j can be deemed the most favorable values for k j on the basis of anchored judgments with displaced and fixed ideals, respectively. In this way, the characterization of the proposed displaced RI can be reasonably designated via the generalized distance measure U G ðp ij ; pðþÞ j Þ. The smaller U G ðp ij ; pðþÞ j Þ value is, the closer to pðþÞ j it is, and the greater the preference is for each assessment rating p ij . In a similar manner, the characterization of the proposed fixed RI can be defined using the generalized distance measure U G ðp ij ; pð b þÞ j Þ.

Moreover, the smaller U G ðp ij ; pð b þÞ j Þ value is, the closer pð b þÞ j it is, and the greater the preference is for each assessment rating p ij .

However, it is worthwhile to mention that the U G ðp ij ; pðþÞ j Þ values for all k j 2 K do not have a consistent upper bound in most situations. The main reason is that the displaced PIPF values are most favorable PF values with all of the currently considered assessment rating with respect to each criterion; thus, they are usually different among criteria. As a result, the maximal possible value of the generalized PF distances, i.e., the U G ðpðÀÞ j ; pðþÞ j Þ values, can be regarded as the upper bounds of the U G ðp ij ; pðþÞ j Þ values for all k j 2 K. Definition 4.3 Let p ij ; pðþÞ j ; pðÀÞ j denote an assessment rating, the displaced PIPF value, and the displaced NIPF value, respectively, in the PF decision matrix p ¼ ½p ij ðmÂnÞ . Moreover, without loss of generality, assume that pðÀÞ j 6 ¼ pðþÞ j for all k j 2 K. The displaced RI RIðp ij Þ of p ij based on the generalized distance measure U G is defined as follows: 

Proof (i) ): If RIðp ij Þ ¼ 0, then it implies that U G ðp ij ; pðþÞ j Þ ¼ 0. According to the separability condition of Theorem 3.3, it is obtained that p ij ¼ pðþÞ j . ( If p ij ¼ pðþÞ j , it is obvious that RIðp ij Þ ¼ 0, because of the reflexivity condition ðU 1 Þ of Theorem 3.3. (ii) ): If RIðp ij Þ ¼ 1, then it implies that U G ðp ij ; pðþÞ j Þ ¼ U G ðpðÀÞ j ; pðþÞ j Þ. It follows that p ij ¼ pðÀÞ j . ( If p ij ¼ pðÀÞ j , it is trivial to obtain that RIðp ij Þ ¼ 1 by employing (4.8).

(iii) Applying (4.2) and (4.3), it is obvious that uðÀÞ j u ij uðþÞ j and vðÀÞ j ! v ij ! vðþÞ j for each k j 2 K i . Moreover uðþÞ j u ij uðÀÞ j and vðþÞ j ! v ij ! vðÀÞ j for each k j 2 K ii . Thus, one can easily obtain pðÀÞ j L p ij L pðþÞ j and pðþÞ j L p ij L pðÀÞ j for k j 2 K i and k j 2 K ii , respectively. According to ðU 5 Þ of Theorem 3.3, it is known that U G ðp ij ; pðþÞ j Þ ðU G ðpðÀÞ j ; pðþÞ j Þ and U G ðpðþÞ j ; p ij Þ ðU G ðpðþÞ j ; pðÀÞ j Þ for k i 2 K i and k i 2 K ii , respectively. Next, combining the condition U G ðp ij ; pðþÞ j Þ ðU G ðpðÀÞ j ; pðþÞ j Þ and the boundedness property of Theorem 3.3, one can obtain 0 RIðp ij Þ 1. (iv) From the proof of (iii.), for each k i 2 K i , it is easily seen that p i 1 j L p i 2 j L pðþÞ j . According to the condition (v.) of Theorem 3.3, the condition p i 1 j L p i 2 j L pðþÞ j implies U G ðp i 2 j ; pðþÞ j Þ ðU G ðp i 1 j ; pðþÞ j Þ. Note that in the denominator of (4.8), one has U G ðpðÀÞ j ; pðþÞ j Þ [ 0 because of the assumption of pðÀÞ j 6 ¼ pðþÞ j for k i 2 K i . Thus, it is easily proven that RIðp i 2 j Þ RIðp i 1 j Þ for k i 2 K i .

The proof of condition (v) can be proven as similar to (iv). h Example 4.5 Take the set of candidate alternatives D ¼ fd 1 ; d 2 ; d 3 g, and the criteria set K ¼ fk 1 ; k 2 g, where K i ¼ fk 1 g and K ii ¼ fk 2 g. The PF decision matrix can be given as follows: þÞ j ; pð b ÀÞ j denote an assessment rating, the fixed PIPF value, and the fixed NIPF value, respectively, in the PF decision matrix p ¼ ½p ij ðmÂnÞ . The fixed RI c RI ðp ij Þ of p ij based on the generalized distance measure U G is defined as follows:

Theorem 4.7 Choose the three assessment ratings p ij ; p i 1 j ; p i 2 j . The fixed remoteness index satisfies the following properties: As seen in this example, the oppositions between the remoteness indices become less discernible when anchoring subjective judgments with fixed ideal PF values. This situation is not so noticeable in Example 4.5.

When decision-making in life, not all of the characteristics that affect these decisions are evaluated equally. Therefore, decision-making is determining alternatives according to the preferences and values of the decision-maker and choosing among them. Each situation has a weight for this choice. In DM operates, weights are assigned to the features. Two types of weights will be used in this study: 

The PF Importance Weights: The PF importance weights of criteria e j 2 R are displaced as the PFV x j ¼ ðx j ; x j Þ, where x j and x j represent the importance and unimportance degrees of the criteria e j 2 R, respectively. Throughout the DM process, the expert group examines the steps of the process and records their evaluations as a parametric Pythagorean fuzzy set (PPFS) at the end of the process. These PPFS built by experts will be used as PF importance weights in the DM process.

New multiple criteria ranking indices for RIðp ij Þ and c RI ðp ij Þ measures of remoteness to the displaced and fixed ideal PF values, respectively, will be offered. In this way, the remoteness index-based new VIKOR method will be obtained. These multiple criteria ranking indices are: remoteness-based group utility indices, individual regret indices, and compromise indices. They can provide a mechanism to trade off a maximum group utility for the majority and a minimum individual regret for the opponent and can be used as an aggregating function for a compromise ranking among alternatives. The smaller the value obtained in the displaced distance index RIðp ij Þ, the more priority the assessment rating of the p ij will be. That is, the criterion-wise priority relationship among alternatives can be ordinarily identified according to ascending order of each RIðp ij Þ value. This is because displaced ideal PF values are to facilitate anchored judgments. In this study, a displaced remoteness-based compromise index by combining the suggested new definitions of a group utility index and an individual regret index is offered. The reason for this is to synthesize criteria-based priority relationships to obtain a compromise ranking on all criteria.

The displaced remoteness-based group utility index, the displaced remoteness-based individual regret index and the displaced remoteness-based compromise index are defined as follows: 

where p ¼ ½p ij mÂn and x j are PF decision matrix and a set of precise importance weights, respectively, and d 2 ½0; 1.

Further, the conditions 0 x j 1 and P n j¼1 x j ¼ 1 are the normalization conditions of precise weights. h Next, consider an MCDA problem involving the PF decision matrix and PF importance weights. In a similar manner, this paper defines the multiple criteria ranking indices Sðd i Þ; Rðd i Þ; Qðd i Þ under the condition of PF preference information. Analogous to Definition 4.10 , the product of RIðp ij Þ multiplied by x j must be calculated before determining these multiple criteria ranking indices. However, a difficulty in comparing the multiple criteria ranking indices among alternatives is encountered because RIðp ij Þ:x j s a PF value when the PF importance weight x j is taken into account. Here, a score function approach to acquire a comparable value of RIðp ij Þ:x j and then to identify the multiple criteria ranking indices is offered. It is worth noting that the score function approach leads to different ranges among Sðd i Þ; Rðd i Þ; Qðd i Þ because the score function defined as in (2.4) is between À1 and 1.

The displaced remoteness-based group utility index, the displaced remoteness-based individual regret index and the displaced remoteness-based compromise index are defined as follows:

; ðk j 2 KÞ;

ð4:14Þ

; ðk j 2 KÞ;

ð4:15Þ

where p ¼ ½p ij mÂn and x j are PF decision matrix and a set of PF importance weights, respectively, and d 2 ½0; 1.

The ranking indices Sðd i Þ; Rðd i Þ; Qðd i Þ satisfy the following properties:

i. Àn Sðd i Þ n, À1 Rðd i Þ 1 and 0 Qðd i Þ 1; ii. If p i 2 j L p i 1 j and p i 1 j L p i 2 j for all k i 2 K i and k i 2 K ii , then Sðd i 1 Þ Sðd i 2 Þ, Rðd i 1 Þ Rðd i 2 Þ and Qðd i 1 Þ Qðd i 2 Þ, respectively.

Proof According to the (2.9), the inequality À1 ESðRIðp ij Þ:x j Þ 1 ð8k j 2 KÞ holds. Then, À1 max n j¼1 ESðRIðp ij Þ:x j Þ 1 is valid. From here, À1 Rðd i Þ 1 is obtained. At the same time, Àn P n j¼1 ESðRIðp ij Þ:x j Þ n ) Àn Rðd i Þ n. Consider (4.16). Since Qðd i Þ is merely a linear combination of the normalized values Sðd i Þ, and Rðd i Þ, then, it is obvious that it will be 0 Qðd i Þ 1.

The proof of the condition (ii.) can be easily get from part (ii.) of Theorem 4.11 h

For fixed ideal PF values, new definitions are given as follows:

Definition 4.14 The fixed remoteness-based group utility index, the fixed remoteness-based individual regret index and the fixed remoteness-based compromise index are defined as follows:

where p ¼ ½p ij mÂn and x j are PF decision matrix and a set of precise importance weights, respectively, and d 2 ½0; 1. Further, the conditions 0 x j 1 and P n j¼1 x j ¼ 1 are the normalization conditions of precise weights.

The fixed remoteness-based group utility index, the fixed remoteness-based individual regret index and the fixed remoteness-based compromise index are defined as follows:

ð4:20Þ

; ðk j 2 KÞ;

ð4:21Þ

where p ¼ ½p ij mÂn and x j are PF decision matrix and a set of PF importance weights, respectively, and d 2 ½0; 1.

The ranking indices f SFðd i Þ; f RFðd i Þ; g QFðd i Þ satisfy the following properties:

i. 0 f SFðd i Þ 1, 0 f RFðd i Þ 1 and 0 g QFðd i Þ 1;

ii. If p i 2 j L p i 1 j and p i 1 j L p i 2 j for all k i 2 K i and

and g QFðd i 1 Þ g QFðd i 2 Þ, respectively.

The ranking indices SFðd i Þ; RFðd i Þ; QFðd i Þ satisfy the following properties:

i. Àn SFðd i Þ n, À1 RFðd i Þ 1 and 0 QFðd i Þ 1; ii. If p i 2 j L p i 1 j and p i 1 j L p i 2 j for all k i 2 K i and k i 2 K ii , then SFðd i 1 Þ SFðd i 2 Þ, RFðd i 1 Þ RFðd i 2 Þ and QFðd i 1 Þ QFðd i 2 Þ, respectively.

These theorems can be proved as Theorems 4.11 and 4.13.

The decision process coefficient is shown by d. One can modify decision making strategy by changing the value of d. The value of parameter d indicates the importance of maximum group utility, while 1 À d indicates the importance of individual regrets. In the classical VIKOR method, the higher the value of d (when d [ 0:5), the compromise ranking procedure is categorized as the procedure with voting by majority. The compromise ranking procedure is categorized as the procedure with veto when d:5. The consensus is achieved in the compromise ranking procedure at d ¼ 0:5. (ii). The PF importance weights are given as x 

In this section, firstly, using newly defined distance measurements, weights, displaced and fixed ideals, and displaced and fixed remoteness indices novel algorithms will be given. Later, these algorithms will be applied to the cognitive and behavioral development of early childhood children staying at home in the COVID-19 quarantine.

For the four different scenarios, algorithms are given as follows:

In natural disasters and especially in epidemic diseases, some measures are taken to protect people from the negative effects of the situation. One of the measures that can be taken is quarantine. The quarantine is used to indicate restrictions on the activities of people or animals exposed to infectious diseases during the infectious period. Children, who are members of society and cannot be isolated from society, should be informed correctly and sufficiently to prevent them from being affected by both the biological effect and the psychological effect of the epidemic. It is extremely important to inform and raise awareness of children beforehand in order to prevent a pandemic because children may face troubles due to the long duration of natural disasters and measures such as quarantine restricting people. Children may face personal losses, collective deaths, and discomfort caused by the diseases caught in natural disasters and outbreaks. These situations can cause adversities such as stress, anxiety, depression, and behavioral disorders in children.

Children learn a lot of the information they learn through environmental stimuli. The interaction of the child with his/her environment, social relationships, other people, especially adults, plays a very important role in cognitive development. The stimuli that it is exposed to in the pandemic process direct the perception of children to the pandemic. In this case, it is clear that children will pay more attention to the pandemic, quarantine, and related stimuli. In the process, the vast majority of stimuli around children, including parents and digital media, lead their perception of COVID-19. If this perception cannot be controlled properly, a false cognition and belief in children will be inevitable.

In this study, the survey model was used. The survey was prepared to be answered on the internet. Survey questions were asked to children aged 5-6 and their families. The linguistic terms and their numeric labels are: For Questions to be asked to the child: Yes(1), maybe/some (2), no (3) . For Questions to be asked to parents: too much(1), much (2), some (3), too little(4), none (5) .

The survey included the following questions: Questions to be asked to the child:

• The target audience is children aged 5-6 in early childhood. Children of this age group are in the process of gaining skills in expressing their feelings during this period. In addition, the emotional responses of these children can be noticed by a careful observer or even an expert. This section is to investigate the effects of quarantine status due to the COVID-19 pandemic on the cognitions and behaviors of children who stay at home. For this, questions were asked to the children and their parents. Opinions of each child and each parent about the questions asked were got. The effect of quarantine on their own cognition in line with the answers given by the children and the effect of the behavior of their children in line with the observations of the parents have been revealed. Let C ¼ fc 1 ; c 2 ; c 3 g and D ¼ fd 1 ; d 2 ; d 3 g represent the sets of the questions related to children's cognition and the questions related to children's behavioral, respectively. Let the sets M ¼ fm 1 ; m 2 ; m 3 g ¼ fyes; maybe; nog and N ¼ fn 1 ; n 2 ; n 3 g ¼ fmuch; some; noneg represent the answer given by the children to the questions asked and answer given by the parents to the questions asked, respectively.

An expert group was formed to ask questions to children and evaluate their answers. The experts assessed the answers according to criteria. The experts gave their judgments based on their knowledge and previous statistical measures. The committee preferences are stored in the form of GPFSS. These results are shown in Tables 2 and 3 . f and g in these tables represent expert opinions. Algorithm-based solutions will be made for both cognitive and behavioral developments.

As demonstrated in Theorem 3.3 and Note 1, the Hamming distance X H is a metric because of its reflexivity, symmetry, separability and triangle inequality. Thus, when computing the generalized distance measure X G , this paper sets the distance parameter k ¼ 1 for the application. Moreover, assume that the decision-maker would like to achieve the goal of ''consensus,'' and thus, set d ¼ 0; 5.

A simple MCDA problem concerning three alternatives and three benefit criteria for evaluating the alternatives, where K ¼ K i ¼ fm 1 ; m 2 ; m 3 g and K ii ¼ ;. The displaced RIs are calculated as: Table 4 : Now, e Q ð c 2 Þ À e Q ð c 1 Þ ¼ 0:49 À 0:222 ¼ 0:268\ 1 3À1 ¼ 0:5. The first condition in S 8 of Algorithm 1 is not satisfied. Then, the ultimate compromise solution is proposed. Therefore the order is fc 1 ; c 2 g

Generalized intuitionistic fuzzy sets with applications in decision making

On some new operations in soft set theory

Intuitionistic fuzzy sets

Entropy and distance measures of Pythagorean fuzzy soft sets and their applications

A novel entropy measure of Pythagorean fuzzy soft sets

A comparative survey of the condition of tourism infrastructure in Iranian provinces using VIKOR and TOPSIS

Green supplier selection using fuzzy group decision making methods: a case study from the agri-food industry

Une méthode pour guider le choix en presence de points de vue multiples

A preference ranking organisation method (the PROMETHEE method for multiple criteria decisionmaking)

Multiple Criteria Decision Analysis

A comparative analysis of VIKOR method and its variants

Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis

Intuitionistic fuzzy parameterized soft set theory and its decision making

Another view on generalized intuitionistic fuzzy soft sets and related multi-attribute decision making methods

An overview of ELECTRE methods and their recent extensions

A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making

A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making process

Some series of intuitionsitic averaging aggregation operators

A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems

Distance and similarity measures for dual hesitant fuzzy soft sets and their applications in multicriteria decision-making problem

A state of the art literature review of VIKOR and its fuzzy extensions on applications

On Pythagorean fuzzy matrices, operations and their applications in decision making and medical diagnosis

Comparison the medical decision-making with Intuitionistic fuzzy parameterized fuzzy soft set and Riesz Summability

Neural Computing and Applications

An ANFIS perspective for the diagnosis of type II diabetes

Medical decision making with respect to the fuzzy soft sets

A case study for medical decision making with the fuzzy soft sets

XÀ sodt sets and medical decision-making application

A VIKOR-based method for hesitant fuzzy multi-criteria decision making

Fuzzy soft sets

Intuitionistic fuzzy soft sets

Soft set theory

Similarity measure of soft sets

Soft set theory-first results

Pythagorean fuzzy soft MCGDM methods based on TOPSIS, VIKOR and aggregation operators

Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS

Multicriteria planning of postearthquake sustainable reconstruction

Extended VIKOR method in comparison with outranking methods

Pythagorean fuzzy soft set and its application

Some results for Pythagorean fuzzy sets

Multiple attribute group decision making methods based on Pythagorean fuzzy linguistic set

Pythagorean fuzzy set: state of the art and future directions

Hypergraphs based on Pythagorean fuzzy soft model

Neutrosophic set, a generalisation of the intuitionistic fuzzy sets

Intuitionistic fuzzy aggregation operator

Pythagorean fuzzy subsets

Pythagorean membership grade in multicriteria decision making

Multicriteria decision making with ordinal linguistic intuitionistic fuzzy sets for mobile apps

Pythagorean membership grades, complex numbers and decision making

Fuzzy sets

VIKOR method with enhanced accuracy for multiple criteria decision making in healthcare management

Extension of TOPSIS to multi-criteria decision making with Pythagorean fuzzy sets

Acknowledgements We would like to thank the anonymous reviewers for their comments that helped us improve this manuscript.Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest The authors declare that they have no conflict of interest.

Stability analysis for the GPFSS (F, M, f)