# A Study of Bipolar Fuzzy Soft Sets and Its Application in Decision-Making Problems

Authors: Saima Mustafa; Neelofar Safdar; Murrium Bibi; A.F. Sayed; Muhammad Ghaffar Khan; Zabidin Salleh
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We live in a society where we have to deal with so many social issues daily, and making a right choice is everyone’s main concern. This study is based on the selection of right university while getting admission which is the core issue for students nowadays. We are concerned with bipolar fuzzy multicriteria decision-making methods. The main purpose is to provide guidance to the students for determining the best university and evaluating the factors that are affecting while getting admission. We thus combined bipolar fuzzy and soft expert sets to give multicriteria decision-making approach that overcomes the issues that arise while taking decisions. This study involves the development of structural hierarchical models of parameters and the implementation of soft expert sets to make the decision-making problem much more precise by introducing a new algorithm. Thus, this model is helpful for multicriteria decision making and can be used for university selection and thus suitable for education sector as well.


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## Body

## 1. Introduction

Education is considered to be the core need and right of every individual and in every discipline, namely, in engineering and agriculture. The dream of a healthy, peaceful, and prosperous world can be made true on the basis of education. This strengthens an individual personality building and keeps updated with new knowledge, and experiences also help to understand the modern world and advancements. It is therefore considered as a tool for eradicating poverty [1]. A theory about human capital says that education is the source of bringing positive change to human life. It builds his personality by improving his intellectual capabilities and provides him with a path to upgrade his hidden talents and also provides a way of increasing his earning [2]. The students need better education and knowledge when they are getting admission for higher education. It is seen that mostly students select the university (such as engineering or medical) according to the opinion and experiences of their fathers and senior students but it may go wrong as with the passage of time, universities change their policies and there is a need to see the certain that a student look for while getting admission. There is a need of a decision-making tool to help student for the correct selection. Decision making of multiple parameters is dynamic and difficult. The difficulty of the decision-making process with multicriteria is due to multiple parameters. Selecting the best university (such as engineering or medical) is the multicriteria decision-making method which has many problems.The analytical hierarchy process (AHP) is a recent addition to various approaches used to determine the relative importance of a set of activities or criteria. The novel aspect and major distinction of this approach are that it structures any complex, multiperson, multicriteria, and multiperiod problem hierarchically.L. A. Zadeh [3] gave the idea of fuzzy set theory to deal with the vagueness and uncertainties occurring while decision making. It is human nature to have dual thinking. Zhang in [4] gave the idea of bipolarity to overcome the double-sided thinking nature of human while decision making. It provides more flexibility while modeling human thinking as compared with fuzzy.The idea of soft sets described by Molodtsov [5] is an extension of fuzzy sets which is used to describe uncertainties. Alkhazaleh and Salleh gave the idea of soft expert sets where they gave the idea of including the opinion of experts for decision making. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a famous approach to deal with the multicriteria decision-making method that was first proposed by Hwang and Yoon [6]. It is a multicriteria decision-making method to find a best choice among set of alternatives.The idea of the classical TOPSIS method is discussed by Chen Hwang and Lofti in which the rating and weights of the criteria are given by crisp values [7]. As in real life situations, we deal with vagueness and ambiguity. So, the drawback of the classical TOPSIS is that it does not deal with vague situations for decision making. Therefore, Chen [8] extended the classical TOPSIS to fuzzy TOPSIS to cover the vagueness and uncertainty occurring in real-life situations. To cover the uncertainty and give more precise decision, Dey et al. [9] extended TOPSIS for solving decision-making problems under bipolar neutrosophic environment. Apart from this, some tremendous progress has been extensively done by different researchers in different disciplines (see [10–13]).Recently, another new technique was introduced by Xiao [14] for improving the performance of fuzzy complex event processing-based decision-making systems. He addressed the issues of intrinsic uncertainty in dynamic input events and overcame the difficulties of operator distribution problem. He proposed the CAFtr system for operator scheduling on fuzzy complex event processing systems based on the technique for order preferences by similarity to an ideal solution. The efficiency of the proposed technique has been checked through an application on the stream base system.Lots of work have been done on decision-making processes and are readily available (see [15, 16]). Decision maker always gets confused on what technique to be applied to carry out the decision process. Choosing a wrong technique can lead to imprecise and ambiguous results. Therefore, it is very important to choose a right technique at right time. Though all the techniques that are presented so far are correct and accepted across the world, it is always hard to choose a best technique. The best method implies to be the method giving the fastest results or sometimes researchers look for cheapest methods to be the best method. A technique with lots of complicated calculations could be time consuming and needs expertise to solve one.The gaps in the previous methods were because of their inability to cover human judgmental thinking that is on both positive and negative sides. This drawback has been overcome by the bipolar fuzzy concept see [17, 18] which further innovates the new concept. On the other hand, by including soft sets, the order of preferences see, [19, 20] is created; thus, a more precise result can be obtained. Thus, the optimal decision is obtained by using bipolar fuzzy TOPSIS approach.Briefly speaking, the objective of this research is based on overcoming the hurdles and problems by using bipolar fuzzy which covers human dual behavior combined with soft sets which give order of preferences thus giving a bipolar fuzzy soft set decision approach.The significance of this study is that it provides a correct decision-making method when there is ambiguity and dual human responses. It gives us basis of dealing with human nature by implementing bipolar fuzzy numbers. It is the best way to deal with two-sided human behavior as bipolar has two sides like agree and disagree. Bipolar fuzzy set is defined in the interval (−1, +1), where −1 indicates the negative opinion whereas +1 indicates the positive opinion, so it provides a wide space of dealing with uncertainties [21–23].In this research to cover the vagueness and imprecision, a trapezoidal bipolar fuzzy number is assigned to each linguistic value. The role of bipolar is very important as bipolarity provides precision and flexibility to the system when compared with crisp and fuzzy models. The human decision making is vague and is double sided so it allows us to make decision making in bipolar fuzzy environment, and using soft sets gave us the order of preference thus making the data precise.


## 2. Preliminaries

The role of bipolarity is very essential in many research fields. It provides more flexibility as compared with fuzzy and Boolean logic. Zhang [4] defined the concept of bipolarity as a wide variety of human decision making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side, for instance, cooperation and competition, common interests and conflict interests, friendship and hostility, likelihood and unlikelihood, and effect and side effect.The concept of soft expert sets is defined by Salleh (2011) as follows.Definition 1.
A pairF,A is called a soft expert set over U, where F is a mapping given by F:A⟶PU where PU denotes the power set of U.


## 3. Proposed Bipolar Fuzzy Multicriteria Decision-Making Technique and Soft Sets Algorithm

### 3.1. Bipolar Fuzzy Analytical Hierarchical Process

Analytical hierarchy processAHP is used to estimate criterion weights through pairwise comparison tables. Through this comparison, the decision makers are allowed to assign weight to the parameters, and thus alternatives are compared without any difficulty. We convert crisp value into trapezoidal bipolar fuzzy numbers. By using these numbers, we construct a pairwise decision matrix defined as follows:(1)B=M,N=m1,m2,m3,m4,n1,n2,n3,n4,which are the trapezoidal bipolar fuzzy numbers. Here, M and N represent the positive and negative responses of the bipolar fuzzy number.Then, aggregated bipolar fuzzy weights are assigned with respect to the positive and negative responses of each criterion. These are represented as follows:(2)μi=w1j,w2j,w3j,w4j,w1j′,w2j′,w3j′,w4j′,where j=1,2,…,m and μi is calculated by taking the average of the weights with respect to positive and negative responses.The defuzzification process to obtain crisp values of weights using the ranking function is as follows:(3)εij=w1j+w2j+w3j+w4j4+−w1j−w2j−w3j−w4j2−w1j′+w2j′+w3j′+w4j′4+−w1j′−w2j′+w3j′+w4j′2.To get normalized weights, we use the following formula:(4)∑i=1nWi=ε1∑k=1mε,ε2∑k=1mε,⋯,εk∑k=1mε.The sum of the normalization must be equal to 1.(5)∑i=1nWi=1.


### 3.2. Bipolar Fuzzy TOPSIS Method

The technique for order preference by similarity to ideal solution gives a framework to handle this type of problem. This model is based on the modified bipolar fuzzy TOPSIS method. The main steps include the construction of a decision matrix, weighted standardized decision matrix, determination of bipolar fuzzy positive ideal solution and negative ideal solution, calculation of the Euclidean distance, and computation of the score of alternatives and rank the preference order. The multicriteria decision-making problem is defined by set of alternatives and parameters. Let the alternatives beE1,E2,…,En, and H1,H2,…,Hm are the parameters, respectively. The crisp information based on different criteria is then converted into bipolar fuzzy numbers, and then it is represented in the bipolar decision matrix. The bipolar decision matrix is calculated as(6)M=mij,nijm×n,i=1,2,…,n;j=1,2,…,m,where mij and nij represent the bipolar information based on the set of alternatives and evaluation criteria. We can also express as(7)M=m11,n11m11,n11⋯m1n,n1nm21,n21m22,n22⋯m2n,n2n⋮⋮⋱⋮mm1,nm1mm1,nm1⋯mmn,nmnm×n,in which possible alternatives are E11,E12,…,Enm and evaluation criteria are presented as H1,H2,…,Hm. The weights are then assigned by the decision makers, and the weight of attributes is represented by(8)W=w1,w2,…,wnT.These weights are assigned by the faculty on the basis of set rules defined by the institute and∑i=1nWi=1. The standardized matrix of criterion weights is given as follows:(9)X=xijm×n,i=1,2,…,n;j=1,2,…,m,where xij are the weights of m criteria assigned by n decision makers. The standardized matrix is obtained by assigning linguistic scales to the criteria by the decision makers which is then transformed into trapezoidal bipolar fuzzy numbers. The weighted standardized decision matrix is represented as(10)X=wi⊗M,where wi represents the weights matrix and M represents the bipolar fuzzy decision matrix based on bipolar fuzzy weighted positive value and bipolar fuzzy weighted negative value, respectively. The bipolar fuzzy weighted positive value and negative value is calculated as(11)sij=wi⊗mij,uij=wi⊗nij.Here,sij and uij show the bipolar weighted positive value and negative value. The bipolar fuzzy positive and negative ideal solutions are calculated as follows:(12)t∗=x1∗,x2∗,…,xn∗=maxxij\j∈B,minxij\j∈C\i=1,2,…,m,t−=x1−,x2−,…,xn−=minxij\j∈B,maxxij\j∈C\i=1,2,…,m.The distance of each alternative from bipolar fuzzy positive ideal solution and negative ideal solution is calculated as follows:(13)△+=12∑i=1msij−maxxij2+uij−minxij2,△−=12∑i=1msij−minxij2+uij−maxxij2.The final score for the selection of the best course of action is obtained by utilizing(14)final scores=△−△++△−,where △+ is the distance from positive ideal solution and △− is the distance from negative ideal solution.The flow chart of the proposed methodology defined in Section3.2 is presented in Figure 1.Figure 1
Flow chart of the proposed bipolar algorithm using fuzzy TOPSIS.


### 3.3. Proposed Algorithm in Multicriteria Decision-Making Using Soft Expert Sets

The concept of soft expert sets is defined by Alkhazaleh and Salleh [24], and it has many uses in economics, decision making, social sciences, mathematics, and so on. In this study, we have proposed a bipolar fuzzy multicriteria decision-making method. We proposed an algorithm in which we compress the parameters to obtain a precise result by developing a preference level of the criteria. The following steps are involved in soft set technique:(1)
Input a bipolar fuzzy soft expert setF,U.(2)
Find the agree-bipolar fuzzy soft expert set and disagree-bipolar fuzzy soft expert set by using(15)μ+uj−v−uj,whereμ+uj is the positive response and v−uj is the negative response about each ujεU.(3)
Find the highest value for agree-BF soft expert set and disagree-BF soft expert set by using(16)maxei=Highestvalue,ei=Alternativese1,e2,e3.(4)
Find scores for agree-bipolar fuzzy soft expert set by using(17)Aj=∑ieij.(5)
Find scores for disagree-bipolar fuzzy soft expert set by using(18)Dj=∑ieij.(6)
By using agree-bipolar fuzzy soft expert set and disagree-bipolar fuzzy soft expert set, find the final scores using(19)sj=Aj−Dj,wheresj represent the final scores, Aj is agree-bipolar fuzzy soft expert set, and Dj is disagree-bipolar fuzzy soft expert set.(7)
Determining the value of the highest score max ofsj, then ei would be the optimal choice with the highest score.(8)
If there are more than one value, then any ofek can be chosen.The flow chart of the proposed algorithm as defined in Section3.3 is represented in Figure 2.Figure 2
The proposed algorithm using bipolar fuzzy soft sets.


## 3.1. Bipolar Fuzzy Analytical Hierarchical Process

Analytical hierarchy processAHP is used to estimate criterion weights through pairwise comparison tables. Through this comparison, the decision makers are allowed to assign weight to the parameters, and thus alternatives are compared without any difficulty. We convert crisp value into trapezoidal bipolar fuzzy numbers. By using these numbers, we construct a pairwise decision matrix defined as follows:(1)B=M,N=m1,m2,m3,m4,n1,n2,n3,n4,which are the trapezoidal bipolar fuzzy numbers. Here, M and N represent the positive and negative responses of the bipolar fuzzy number.Then, aggregated bipolar fuzzy weights are assigned with respect to the positive and negative responses of each criterion. These are represented as follows:(2)μi=w1j,w2j,w3j,w4j,w1j′,w2j′,w3j′,w4j′,where j=1,2,…,m and μi is calculated by taking the average of the weights with respect to positive and negative responses.The defuzzification process to obtain crisp values of weights using the ranking function is as follows:(3)εij=w1j+w2j+w3j+w4j4+−w1j−w2j−w3j−w4j2−w1j′+w2j′+w3j′+w4j′4+−w1j′−w2j′+w3j′+w4j′2.To get normalized weights, we use the following formula:(4)∑i=1nWi=ε1∑k=1mε,ε2∑k=1mε,⋯,εk∑k=1mε.The sum of the normalization must be equal to 1.(5)∑i=1nWi=1.


## 3.2. Bipolar Fuzzy TOPSIS Method

The technique for order preference by similarity to ideal solution gives a framework to handle this type of problem. This model is based on the modified bipolar fuzzy TOPSIS method. The main steps include the construction of a decision matrix, weighted standardized decision matrix, determination of bipolar fuzzy positive ideal solution and negative ideal solution, calculation of the Euclidean distance, and computation of the score of alternatives and rank the preference order. The multicriteria decision-making problem is defined by set of alternatives and parameters. Let the alternatives beE1,E2,…,En, and H1,H2,…,Hm are the parameters, respectively. The crisp information based on different criteria is then converted into bipolar fuzzy numbers, and then it is represented in the bipolar decision matrix. The bipolar decision matrix is calculated as(6)M=mij,nijm×n,i=1,2,…,n;j=1,2,…,m,where mij and nij represent the bipolar information based on the set of alternatives and evaluation criteria. We can also express as(7)M=m11,n11m11,n11⋯m1n,n1nm21,n21m22,n22⋯m2n,n2n⋮⋮⋱⋮mm1,nm1mm1,nm1⋯mmn,nmnm×n,in which possible alternatives are E11,E12,…,Enm and evaluation criteria are presented as H1,H2,…,Hm. The weights are then assigned by the decision makers, and the weight of attributes is represented by(8)W=w1,w2,…,wnT.These weights are assigned by the faculty on the basis of set rules defined by the institute and∑i=1nWi=1. The standardized matrix of criterion weights is given as follows:(9)X=xijm×n,i=1,2,…,n;j=1,2,…,m,where xij are the weights of m criteria assigned by n decision makers. The standardized matrix is obtained by assigning linguistic scales to the criteria by the decision makers which is then transformed into trapezoidal bipolar fuzzy numbers. The weighted standardized decision matrix is represented as(10)X=wi⊗M,where wi represents the weights matrix and M represents the bipolar fuzzy decision matrix based on bipolar fuzzy weighted positive value and bipolar fuzzy weighted negative value, respectively. The bipolar fuzzy weighted positive value and negative value is calculated as(11)sij=wi⊗mij,uij=wi⊗nij.Here,sij and uij show the bipolar weighted positive value and negative value. The bipolar fuzzy positive and negative ideal solutions are calculated as follows:(12)t∗=x1∗,x2∗,…,xn∗=maxxij\j∈B,minxij\j∈C\i=1,2,…,m,t−=x1−,x2−,…,xn−=minxij\j∈B,maxxij\j∈C\i=1,2,…,m.The distance of each alternative from bipolar fuzzy positive ideal solution and negative ideal solution is calculated as follows:(13)△+=12∑i=1msij−maxxij2+uij−minxij2,△−=12∑i=1msij−minxij2+uij−maxxij2.The final score for the selection of the best course of action is obtained by utilizing(14)final scores=△−△++△−,where △+ is the distance from positive ideal solution and △− is the distance from negative ideal solution.The flow chart of the proposed methodology defined in Section3.2 is presented in Figure 1.Figure 1
Flow chart of the proposed bipolar algorithm using fuzzy TOPSIS.


## 3.3. Proposed Algorithm in Multicriteria Decision-Making Using Soft Expert Sets

The concept of soft expert sets is defined by Alkhazaleh and Salleh [24], and it has many uses in economics, decision making, social sciences, mathematics, and so on. In this study, we have proposed a bipolar fuzzy multicriteria decision-making method. We proposed an algorithm in which we compress the parameters to obtain a precise result by developing a preference level of the criteria. The following steps are involved in soft set technique:(1)
Input a bipolar fuzzy soft expert setF,U.(2)
Find the agree-bipolar fuzzy soft expert set and disagree-bipolar fuzzy soft expert set by using(15)μ+uj−v−uj,whereμ+uj is the positive response and v−uj is the negative response about each ujεU.(3)
Find the highest value for agree-BF soft expert set and disagree-BF soft expert set by using(16)maxei=Highestvalue,ei=Alternativese1,e2,e3.(4)
Find scores for agree-bipolar fuzzy soft expert set by using(17)Aj=∑ieij.(5)
Find scores for disagree-bipolar fuzzy soft expert set by using(18)Dj=∑ieij.(6)
By using agree-bipolar fuzzy soft expert set and disagree-bipolar fuzzy soft expert set, find the final scores using(19)sj=Aj−Dj,wheresj represent the final scores, Aj is agree-bipolar fuzzy soft expert set, and Dj is disagree-bipolar fuzzy soft expert set.(7)
Determining the value of the highest score max ofsj, then ei would be the optimal choice with the highest score.(8)
If there are more than one value, then any ofek can be chosen.The flow chart of the proposed algorithm as defined in Section3.3 is represented in Figure 2.Figure 2
The proposed algorithm using bipolar fuzzy soft sets.


## 4. Numerical Computation

We have used the proposed method in education sector to deal with an issue of selecting a best university while getting admission. We have used a soft set technique to make the result more precise and obtain better results. The present research work depends on primary data collection. Data were collected from the decision makers who work in education sector. Different parameters have been developed to study the performance of alternatives. The selection of university depends on eleven parameters such as HEC ranking, location, environment, fee structure, faculty, infrastructure, library, research quality, merit, cocurricular activities, and career guidance. These linguistic terms are then changed into trapezoidal bipolar fuzzy numbers chosen by the decision makers. The three alternatives of the study were basically three universities, namely, National University of Science and Technology (NUST), Quaid-e-Azam University (QAU), and International Islamic University (IIU).


### 4.1. Parameters of Study

Nowadays, it has become very difficult for the students to take right decision in choosing a university while getting admission. Students need to focus on every aspect of the university that they want to get facilitated with while studying there. For the evaluation of the best university, the following parameters need to be compared and analysed. Table1 shows the parameters to choose the best university. The source of Table 1 is the decision makers referred as the faculty members who work in the respective universities as mentioned above. Table 1 presents the different criteria gathered from the decision makers.Table 1
Parameters of study.
SymbolsParametersH1HEC rankingH2LocationH3EnvironmentH4Fee structureH5FacultyH6InfrastructureH7LibraryH8Research qualityH9MeritH10Cocurricular activitiesH11Career guidanceThe hierarchical structure model of selected universities is represented by the flow chart in Figure3.Figure 3
Hierarchical structure model of universities flow chart.We provide weightage among different alternatives on the basis of the selected criteria; the following five sets of linguistic scales are used for the denied alternatives: very bad (VB), bad (B), neither good nor bad (NGNB), good (G), very good (VG), no (N), partially no (PN), neutral (Ne), partially yes (PY), yes (Y), very low (VL), low (L), neutral (Ne), high (H), and very high (VH), respectively. The bipolar fuzzy numbers are presented in Table2.Table 2
Linguistic terms for weight of parameters.
Linguistic variablesBipolar fuzzy numbersVery low0.0,0.0,0.1,0.2,0.7,0.8,0.9,1Low0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.8Neutral0.4,0.5,0.5,0.6,0.5,0.5,0.6,0.7High0.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3Very high0.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2The linguistic terms are assigned weights with respect to bipolar fuzzy numbers as shown in Tables3 and 4.Table 3
Linguistic terms for weight of parameters.
Linguistic variablesBipolar fuzzy numbersVery bad0.0,0.0,0.1,0.2,0.7,0.8,0.9,1Bad0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.8Neither good nor bad0.4,0.5,0.5,0.6,0.5,0.5,0.6,0.7Good0.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3Very good0.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2Table 4
Linguistic terms for weight of parameters.
Linguistic variablesBipolar fuzzy numbersNo0.0,0.0,0.1,0.2,0.7,0.8,0.9,1Partially no0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.8Neutral0.4,0.5,0.5,0.6,0.5,0.5,0.6,0.7Partially yes0.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3Yes0.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2The data are collected through questionnaires from the decision makers, and weights are assigned in linguistic terms for each parameter. The bipolar fuzzy numbers are used to compute the weights. It is to be noted that the sum of weights must always be equivalent to 1.


## 4.1. Parameters of Study

Nowadays, it has become very difficult for the students to take right decision in choosing a university while getting admission. Students need to focus on every aspect of the university that they want to get facilitated with while studying there. For the evaluation of the best university, the following parameters need to be compared and analysed. Table1 shows the parameters to choose the best university. The source of Table 1 is the decision makers referred as the faculty members who work in the respective universities as mentioned above. Table 1 presents the different criteria gathered from the decision makers.Table 1
Parameters of study.
SymbolsParametersH1HEC rankingH2LocationH3EnvironmentH4Fee structureH5FacultyH6InfrastructureH7LibraryH8Research qualityH9MeritH10Cocurricular activitiesH11Career guidanceThe hierarchical structure model of selected universities is represented by the flow chart in Figure3.Figure 3
Hierarchical structure model of universities flow chart.We provide weightage among different alternatives on the basis of the selected criteria; the following five sets of linguistic scales are used for the denied alternatives: very bad (VB), bad (B), neither good nor bad (NGNB), good (G), very good (VG), no (N), partially no (PN), neutral (Ne), partially yes (PY), yes (Y), very low (VL), low (L), neutral (Ne), high (H), and very high (VH), respectively. The bipolar fuzzy numbers are presented in Table2.Table 2
Linguistic terms for weight of parameters.
Linguistic variablesBipolar fuzzy numbersVery low0.0,0.0,0.1,0.2,0.7,0.8,0.9,1Low0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.8Neutral0.4,0.5,0.5,0.6,0.5,0.5,0.6,0.7High0.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3Very high0.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2The linguistic terms are assigned weights with respect to bipolar fuzzy numbers as shown in Tables3 and 4.Table 3
Linguistic terms for weight of parameters.
Linguistic variablesBipolar fuzzy numbersVery bad0.0,0.0,0.1,0.2,0.7,0.8,0.9,1Bad0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.8Neither good nor bad0.4,0.5,0.5,0.6,0.5,0.5,0.6,0.7Good0.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3Very good0.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2Table 4
Linguistic terms for weight of parameters.
Linguistic variablesBipolar fuzzy numbersNo0.0,0.0,0.1,0.2,0.7,0.8,0.9,1Partially no0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.8Neutral0.4,0.5,0.5,0.6,0.5,0.5,0.6,0.7Partially yes0.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3Yes0.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2The data are collected through questionnaires from the decision makers, and weights are assigned in linguistic terms for each parameter. The bipolar fuzzy numbers are used to compute the weights. It is to be noted that the sum of weights must always be equivalent to 1.


## 5. Results and Discussion

### 5.1. BF-AHP Results along with Weight of Parameters

The weights gathered through AHP as shown in Table5 totally depend on the judgment of the decision makers. AHP has an advantage that it is easy to utilize and provide with pairwise comparison. The weights are normalized for multicriteria decision making.Table 5
Weights of criteria.
CriteriaWNH10.0964H20.0548H30.0262H40.0405H50.1393H60.1214H70.1107H80.0988H90.1059H100.1393H110.0667Total1It is clear from the beginning that the sum of the weights must be equivalent to 1. So, there is a need to normalize the weights to obtain sum of 1 which are shown in Table5.The bar chart shown in Figure4 depicts all the eleven criteria with their weights, respectively. The criterion weights can be further used for decision making as these weights are suitable for further process.Figure 4
Normalized weights of criteria.


### 5.2. BF-TOPSIS Results and Scoring of Alternatives

TOPSIS is the frequently used among many decision-making techniques due to its simplicity of steps and precision. Its ultimate goal is to identify an alternative with distance closest to the positive ideal solution and farthest to negative ideal solution. In this study, we have highlighted one of the socialistic issue and applied TOPSIS on that for decision making. We have gathered information from three different universities and want to decide that which university is the best for getting admission in all the three alternatives. These alternatives are studied on basis of fifteen criteria designed by the expert opinion. Table6 shows the rating of alternative 1 in linguistic scale.Table 6
Linguistic rating of alternative 1.
CriteriaDecision maker 1Decision maker 2Decision maker 3H1VGVGVGH2NeYNeH3YYYH4YPYPYH5YYYH6VGVGVGH7YYYH8VGVGGH9VHVHVHH10YPYPYH11PYPYYWe repeat the same process as defined in Table6, on the rating of alternative 2 and alternative 3. After converting all the linguistic terms into bipolar fuzzy numbers, the next step is to find the aggregated rating of alternatives. The aggregated decision matrix of alternatives is represented in Table 7.Table 7
Aggregated bipolar fuzzy rating of alternatives.
CriteriaAlternativesE1E2E3H10.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3H20.5,0.6,0.6,0.7,0.3,0.3,0.4,0.50.5,0.6,0.6,0.7,0.2,0.6,0.4,0.40.4,0.5,0.6,0.6,0.2,0.3,0.4,0.5H30.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H40.6,0.7,0.8,0.8,0.0,0.1,0.2,0.20.7,0.8,0.9,0.9,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.3,0.3,0.4,0.5H50.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.6,0.7,0.8,0.8,0.0,0.1,0.2,0.2H60.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.6,0.7,0.7,0.8,0.1,0.2,0.3,0.30.6,0.7,0.8,0.8,0.0,0.1,0.2,0.2H70.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.6,0.4,0.40.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2H80.7,0.8,0.9,0.9,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H90.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.7,0.8,0.9,0.9,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H100.6,0.7,0.8,0.8,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H110.6,0.7,0.7,0.8,0.1,0.2,0.3,0.30.5,0.6,0.6,0.7,0.3,0.3,0.4,0.50.4,0.5,0.6,0.6,0.3,0.4,0.5,0.5,Next, we need to find the weighted normalized decision matrix which is obtained by multiplying the aggregated matrix with the weighted matrix. The weights were first obtained using analytical hierarchical process. The obtained weighted normalized decision matrix is shown in Table8.Table 8
Weighted normalized decision matrix.
CriteriaAlternativesE1E2E3H10.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.05,0.06,0.06,0.0,0.00,0.01,0.02,0.02H20.02,0.03,0.03,0.04,0.01,0.02,0.02,0.020.02,0.03,0.03,0.04,0.01,0.03,0.02,0.020.02,0.02,0.03,0.03,0.01,0.02,0.02,0.02H30.02,0.02,0.02,0.02,0.0,0.00,0.0,0.00.02,0.02,0.02,0.02,0.0,0.00,0.0,0.00.01,0.01,0.01,0.01,0.00,0.00,0.01,0.01H40.02,0.03,0.03,0.0,0.00,0.007,0.01,0.010.02,0.03,0.03,0.03,0.00,0.00,0.00,0.000.02,0.02,0.02,0.02,0.01,0.01,0.0,0.0H50.11,0.12,0.13,0.13,0.0,0.13,0.02,0.020.11,0.12,0.13,0.13,0.0,0.13,0.02,0.020.0,0.10,0.11,0.12,0.00,0.02,0.03,0.03H60.097,0.109,0.12,0.12,0.0,0.01,0.02,0.020.07,0.08,0.08,0.09,0.01,0.02,0.03,0.030.08,0.09,0.09,0.10,0.00,0.02,0.03,0.03H70.08,0.09,0.1,0.1,0.0,0.01,0.02,0.020.05,0.06,0.07,0.08,0.02,0.07,0.04,0.040.0,0.09,0.11,0.11,0.0,0.01,0.02,0.02H80.07,0.08,0.08,0.09,0.00,0.01,0.02,0.020.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.053,0.06,0.06,0.07,0.02,0.02,0.03,0.04H90.08,0.09,0.10,0.10,0.0,0.01,0.02,0.020.07,0.08,0.09,0.09,0.00,0.01,0.02,0.020.05,0.06,0.06,0.07,0.02,0.03,0.04,0.04H100.09,0.107,0.11,0.121,0.00,0.02,0.03,0.030.11,0.12,0.13,0.13,0.0,0.01,0.02,0.020.07,0.08,0.08,0.10,0.03,0.04,0.05,0.06H110.04,0.04,0.04,0.05,0.00,0.01,0.02,0.020.03,0.04,0.04,0.04,0.022,0.02,0.03,0.030.02,0.03,0.04,0.04,0.025,0.02,0.03,0.03The maximum and minimum values are highlighted in Table8. The next step is to find bipolar fuzzy positive and negative ideal solutions. The calculated bipolar fuzzy positive and negative ideal solutions are shown in Table 9.Table 9
Bipolar fuzzy positive and negative ideal solutions.
CriteriaPISNISH10.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.05,0.06,0.06,0.07,0.00,0.01,0.02,0.02H20.02,0.03,0.03,0.04,0.01,0.03,0.02,0.020.02,0.02,0.03,0.03,0.01,0.02,0.02,0.02H30.02,0.02,0.02,0.02,0.0,0.00,0.00,0.000.01,0.01,0.01,0.01,0.00,0.00,0.01,0.01H40.02,0.03,0.03,0.03,0.00,0.00,0.00,0.000.02,0.02,0.02,0.02,0.01,0.01,0.01,0.02H50.11,0.12,0.13,0.13,0.0,0.01,0.02,0.020.09,0.10,0.11,0.12,0.00,0.02,0.03,0.03H60.09,0.10,0.12,0.12,0.0,0.12,0.02,0.020.07,0.08,0.08,0.09,0.01,0.02,0.03,0.03H70.08,0.09,0.11,0.11,0.0,0.01,0.02,0.020.05,0.06,0.07,0.08,0.02,0.07,0.04,0.04H80.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.05,0.06,0.06,0.07,0.02,0.02,0.03,0.04H90.08,0.09,0.10,0.10,0.0,0.01,0.02,0.020.05,0.06,0.06,0.07,0.02,0.03,0.04,0.04H100.11,0.12,0.13,0.13,0.0,0.01,0.02,0.020.07,0.08,0.08,0.10,0.03,0.04,0.05,0.06H110.04,0.04,0.04,0.05,0.00,0.01,0.02,0.020.02,0.03,0.04,0.04,0.02,0.02,0.03,0.03The final scores of the alternatives are computed. Results are displayed in Table10 which clearly shows that alternative 1 which is National University of Science and Technology has got the highest score 0.7774 and is ranked first, second is Quaid-e-Azam University with score 0.602, and third is International Islamic University getting the score of 0.3241. The ranking of the alternatives is shown in Table 10 by which we have concluded that the best university is NUST.Table 10
Ranking and scores of alternatives.
AlternativesScoresRankingE10.7759311E20.5677482E30.28323In Figure5, the highest score is alternative 1, i.e., E1 (NUST university) and is the best university and alternative 2 which is E2 (Quaid-e-Azam University) has got the second highest score and is ranked number 2, and E3 (International Islamic University) has obtained lowest scores and is ranked number 3.Figure 5
Scores of alternatives.


## 5.1. BF-AHP Results along with Weight of Parameters

The weights gathered through AHP as shown in Table5 totally depend on the judgment of the decision makers. AHP has an advantage that it is easy to utilize and provide with pairwise comparison. The weights are normalized for multicriteria decision making.Table 5
Weights of criteria.
CriteriaWNH10.0964H20.0548H30.0262H40.0405H50.1393H60.1214H70.1107H80.0988H90.1059H100.1393H110.0667Total1It is clear from the beginning that the sum of the weights must be equivalent to 1. So, there is a need to normalize the weights to obtain sum of 1 which are shown in Table5.The bar chart shown in Figure4 depicts all the eleven criteria with their weights, respectively. The criterion weights can be further used for decision making as these weights are suitable for further process.Figure 4
Normalized weights of criteria.


## 5.2. BF-TOPSIS Results and Scoring of Alternatives

TOPSIS is the frequently used among many decision-making techniques due to its simplicity of steps and precision. Its ultimate goal is to identify an alternative with distance closest to the positive ideal solution and farthest to negative ideal solution. In this study, we have highlighted one of the socialistic issue and applied TOPSIS on that for decision making. We have gathered information from three different universities and want to decide that which university is the best for getting admission in all the three alternatives. These alternatives are studied on basis of fifteen criteria designed by the expert opinion. Table6 shows the rating of alternative 1 in linguistic scale.Table 6
Linguistic rating of alternative 1.
CriteriaDecision maker 1Decision maker 2Decision maker 3H1VGVGVGH2NeYNeH3YYYH4YPYPYH5YYYH6VGVGVGH7YYYH8VGVGGH9VHVHVHH10YPYPYH11PYPYYWe repeat the same process as defined in Table6, on the rating of alternative 2 and alternative 3. After converting all the linguistic terms into bipolar fuzzy numbers, the next step is to find the aggregated rating of alternatives. The aggregated decision matrix of alternatives is represented in Table 7.Table 7
Aggregated bipolar fuzzy rating of alternatives.
CriteriaAlternativesE1E2E3H10.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.6,0.7,0.7,0.8,0.1,0.2,0.3,0.3H20.5,0.6,0.6,0.7,0.3,0.3,0.4,0.50.5,0.6,0.6,0.7,0.2,0.6,0.4,0.40.4,0.5,0.6,0.6,0.2,0.3,0.4,0.5H30.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H40.6,0.7,0.8,0.8,0.0,0.1,0.2,0.20.7,0.8,0.9,0.9,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.3,0.3,0.4,0.5H50.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.6,0.7,0.8,0.8,0.0,0.1,0.2,0.2H60.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.6,0.7,0.7,0.8,0.1,0.2,0.3,0.30.6,0.7,0.8,0.8,0.0,0.1,0.2,0.2H70.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.6,0.4,0.40.8,0.9,1.0,1.0,0.0,0.1,0.2,0.2H80.7,0.8,0.9,0.9,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H90.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.7,0.8,0.9,0.9,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H100.6,0.7,0.8,0.8,0.0,0.1,0.2,0.20.8,0.9,1.0,1.0,0.0,0.1,0.2,0.20.5,0.6,0.6,0.7,0.2,0.3,0.4,0.4H110.6,0.7,0.7,0.8,0.1,0.2,0.3,0.30.5,0.6,0.6,0.7,0.3,0.3,0.4,0.50.4,0.5,0.6,0.6,0.3,0.4,0.5,0.5,Next, we need to find the weighted normalized decision matrix which is obtained by multiplying the aggregated matrix with the weighted matrix. The weights were first obtained using analytical hierarchical process. The obtained weighted normalized decision matrix is shown in Table8.Table 8
Weighted normalized decision matrix.
CriteriaAlternativesE1E2E3H10.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.05,0.06,0.06,0.0,0.00,0.01,0.02,0.02H20.02,0.03,0.03,0.04,0.01,0.02,0.02,0.020.02,0.03,0.03,0.04,0.01,0.03,0.02,0.020.02,0.02,0.03,0.03,0.01,0.02,0.02,0.02H30.02,0.02,0.02,0.02,0.0,0.00,0.0,0.00.02,0.02,0.02,0.02,0.0,0.00,0.0,0.00.01,0.01,0.01,0.01,0.00,0.00,0.01,0.01H40.02,0.03,0.03,0.0,0.00,0.007,0.01,0.010.02,0.03,0.03,0.03,0.00,0.00,0.00,0.000.02,0.02,0.02,0.02,0.01,0.01,0.0,0.0H50.11,0.12,0.13,0.13,0.0,0.13,0.02,0.020.11,0.12,0.13,0.13,0.0,0.13,0.02,0.020.0,0.10,0.11,0.12,0.00,0.02,0.03,0.03H60.097,0.109,0.12,0.12,0.0,0.01,0.02,0.020.07,0.08,0.08,0.09,0.01,0.02,0.03,0.030.08,0.09,0.09,0.10,0.00,0.02,0.03,0.03H70.08,0.09,0.1,0.1,0.0,0.01,0.02,0.020.05,0.06,0.07,0.08,0.02,0.07,0.04,0.040.0,0.09,0.11,0.11,0.0,0.01,0.02,0.02H80.07,0.08,0.08,0.09,0.00,0.01,0.02,0.020.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.053,0.06,0.06,0.07,0.02,0.02,0.03,0.04H90.08,0.09,0.10,0.10,0.0,0.01,0.02,0.020.07,0.08,0.09,0.09,0.00,0.01,0.02,0.020.05,0.06,0.06,0.07,0.02,0.03,0.04,0.04H100.09,0.107,0.11,0.121,0.00,0.02,0.03,0.030.11,0.12,0.13,0.13,0.0,0.01,0.02,0.020.07,0.08,0.08,0.10,0.03,0.04,0.05,0.06H110.04,0.04,0.04,0.05,0.00,0.01,0.02,0.020.03,0.04,0.04,0.04,0.022,0.02,0.03,0.030.02,0.03,0.04,0.04,0.025,0.02,0.03,0.03The maximum and minimum values are highlighted in Table8. The next step is to find bipolar fuzzy positive and negative ideal solutions. The calculated bipolar fuzzy positive and negative ideal solutions are shown in Table 9.Table 9
Bipolar fuzzy positive and negative ideal solutions.
CriteriaPISNISH10.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.05,0.06,0.06,0.07,0.00,0.01,0.02,0.02H20.02,0.03,0.03,0.04,0.01,0.03,0.02,0.020.02,0.02,0.03,0.03,0.01,0.02,0.02,0.02H30.02,0.02,0.02,0.02,0.0,0.00,0.00,0.000.01,0.01,0.01,0.01,0.00,0.00,0.01,0.01H40.02,0.03,0.03,0.03,0.00,0.00,0.00,0.000.02,0.02,0.02,0.02,0.01,0.01,0.01,0.02H50.11,0.12,0.13,0.13,0.0,0.01,0.02,0.020.09,0.10,0.11,0.12,0.00,0.02,0.03,0.03H60.09,0.10,0.12,0.12,0.0,0.12,0.02,0.020.07,0.08,0.08,0.09,0.01,0.02,0.03,0.03H70.08,0.09,0.11,0.11,0.0,0.01,0.02,0.020.05,0.06,0.07,0.08,0.02,0.07,0.04,0.04H80.07,0.08,0.09,0.09,0.0,0.00,0.01,0.010.05,0.06,0.06,0.07,0.02,0.02,0.03,0.04H90.08,0.09,0.10,0.10,0.0,0.01,0.02,0.020.05,0.06,0.06,0.07,0.02,0.03,0.04,0.04H100.11,0.12,0.13,0.13,0.0,0.01,0.02,0.020.07,0.08,0.08,0.10,0.03,0.04,0.05,0.06H110.04,0.04,0.04,0.05,0.00,0.01,0.02,0.020.02,0.03,0.04,0.04,0.02,0.02,0.03,0.03The final scores of the alternatives are computed. Results are displayed in Table10 which clearly shows that alternative 1 which is National University of Science and Technology has got the highest score 0.7774 and is ranked first, second is Quaid-e-Azam University with score 0.602, and third is International Islamic University getting the score of 0.3241. The ranking of the alternatives is shown in Table 10 by which we have concluded that the best university is NUST.Table 10
Ranking and scores of alternatives.
AlternativesScoresRankingE10.7759311E20.5677482E30.28323In Figure5, the highest score is alternative 1, i.e., E1 (NUST university) and is the best university and alternative 2 which is E2 (Quaid-e-Azam University) has got the second highest score and is ranked number 2, and E3 (International Islamic University) has obtained lowest scores and is ranked number 3.Figure 5
Scores of alternatives.


## 6. Proposed Bipolar Fuzzy Soft Expert Sets Algorithm

The next step is to compute soft sets of the criteria. We have used the soft expert set technique. On the basis of students priority, few of parameters are selected as soft sets and have questioned the decision makers for the agree and disagree of the selected soft sets.Input a weighted bipolar fuzzy soft expert setF,U, where alternatives are E=e1,e2,e3 which from the universe and the chosen parameter are H=h1,h2,h3, where hih=1,2,3,4 are “HEC Ranking,” “Environment,” “Fee structure,” and “Merit,” and Z=p,q,r be set of decision makers.(20)F,U=h1,p,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.6,0.7,0.7,0.8,h1,p,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.6,0.7,0.7,0.8,h1,q,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.6,0.7,0.7,0.8,h1,r,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.6,0.7,0.7,0.8,h2,p,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.6,0.7,0.7,0.8,h2,q,1,e10.8,0.9,1.0,1.0,e20.6,0.7,0.7,0.8,e30.6,0.7,0.7,0.8,h2,r,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.4,0.5,0.5,0.6,h3,p,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.8,0.9,1.0,1.0,h3,q,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.8,0.9,1.0,1.0,h3,r,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.8,0.9,1.0,1.0,h4,p,1,e10.8,0.9,1.0,1.0,e20.8,0.9,1.0,1.0,e30.4,0.5,0.5,0.6,h4,q,1,e10.6,0.7,0.7,0.8,e20.8,0.9,1.0,1.0,e30.4,0.5,0.5,0.6,h4,r,1,e10.6,0.7,0.7,0.8,e20.6,0.7,0.7,0.8,e30.8,0.9,1.0,1.0,(21)h1,p,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.1,0.2,0.3,0.3,h1,q,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.1,0.2,0.3,0.3,h1,r,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.1,0.2,0.3,0.3,h2,p,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.1,0.2,0.3,0.3,h2,q,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.1,0.2,0.3,0.3,h2,r,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.5,0.5,0.6,0.7,h3,p,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.0,0.1,0.2,0.2,h3,q,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.0,0.1,0.2,0.2,h3,r,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.0,0.1,0.2,0.2,h4,p,0,e10.0,0.1,0.2,0.2,e20.0,0.1,0.2,0.2,e30.5,0.5,0.6,0.7,h4,q,0,e10.1,0.2,0.3,0.3,e20.0,0.1,0.2,0.2,e30.5,0.5,0.6,0.7,h4,r,0,e10.1,0.2,0.3,0.3,e20.1,0.2,0.3,0.3,e30.0,0.1,0.2,0.2.Values of agree-BF soft expert set and disagree-BF soft expert set are found which are shown in Tables11 and 12.Table 11
Agree-bipolar fuzzy soft expert set.
Ee1e2e3h1,p,13.73.72.8h2,p,13.73.72.8h3,p,13.73.73.7h4,p,13.73.72h1,q,13.73.72.8h2,q,13.72.82.8h3,q,13.73.73.7h4,q,12.83.72h1,r,13.73.72.8h2,r,13.73.72h3,r,13.73.73.7h4,r,12.82.83.7Table 12
Disagree-bipolar fuzzy soft expert set.
Ee1e2e3h1,p,00.50.50.9h2,p,00.50.50.9h3,p,00.50.50.5h4,p,00.50.52.3h1,q,00.50.50.9h2,q,00.50.90.9h3,q,00.50.50.5h4,q,00.90.52.3h1,r,00.50.50.9h2,r,00.50.52.3h3,r,00.50.50.5h4,r,00.90.90.5Highest numerical value for agree-bipolar fuzzy expert set and disagree-bipolar fuzzy expert set is obtained using Tables11 and 12. Then, final scores are extracted which are shown in Table 13.Table 13
Final scores.
Agree scoresAiDisagree scoresDiSiScoree1=37Scoree1=2.434.6Scoree2=33.3Scoree2=3.330Scoree3=18.5Scoree3=12.95.6


## 7. Validation and Robustness

We check the accuracy of the proposed bipolar fuzzy TOPSIS modeling technique by validation and robustness. From the scores obtained in Table13, it can be seen that the highest score is 34.6 for alternative 1, that is, NUST, so the optimal decision is to choose alternative 1 as the best university. Also, it is clear that by applying soft expert set technique, optimal result remains same as obtained by the BF-TOPSIS method. This method is then applied on the same alternatives for university selection. At the end, we get the same optimal solution. We conclude from Table 13 that E1 has obtained the highest scores and is the best university among the other two universities.


## 8. Conclusion

Decision making in education-based problems is so common nowadays. Students also pay for counselors and advisors for the right advice and right decision to take. Bipolar fuzzy and soft sets are combined together in this study to solve the problems that occur while taking multicriteria decisions. Fuzzy covers the vagueness, and bipolar covers human dual behavior. Soft sets on the other side give more precise results by giving order of preference. Hence, we have demonstrated a modified bipolar fuzzy soft set decision-making method. We have used the TOPSIS method for decision making as the computational steps of TOPSIS are simple and results are easy to evaluate, which is named as the bipolar fuzzy method of order preference by similarity measure to ideal solution (BF-TOPSIS). The results which are based on the positive and negative aspects are covered by the bipolar fuzzy TOPSIS method. The comparison of the existing and modified methods is as follows.


### 8.1. Existing Method

The existing method deals directly with the alternatives which could be very time consuming and can cause errors. The existing method does not give precise results. The existing decision-making methods in [15, 16] based on fuzzy TOPSIS and fuzzy soft sets simply consider the best optimal solution by distance measure and preferences. The method fails when the decision maker wants to assess the variables based on judgmental thinking.


### 8.2. Proposed Method

The bipolar fuzzy AHP minimizes this issue by allowing pairwise comparison and by giving hierarchy. Using soft sets a preference level is made, and thus more compressed and precise results are obtained. Then, bipolar fuzzy TOPSIS is introduced to assess the linguistic variables based on dual thinking. Our proposed method can efficiently achieve the optimum solution with preferences and positive and negative aspects.


## 8.1. Existing Method

The existing method deals directly with the alternatives which could be very time consuming and can cause errors. The existing method does not give precise results. The existing decision-making methods in [15, 16] based on fuzzy TOPSIS and fuzzy soft sets simply consider the best optimal solution by distance measure and preferences. The method fails when the decision maker wants to assess the variables based on judgmental thinking.


## 8.2. Proposed Method

The bipolar fuzzy AHP minimizes this issue by allowing pairwise comparison and by giving hierarchy. Using soft sets a preference level is made, and thus more compressed and precise results are obtained. Then, bipolar fuzzy TOPSIS is introduced to assess the linguistic variables based on dual thinking. Our proposed method can efficiently achieve the optimum solution with preferences and positive and negative aspects.



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*Source: 5742288-2021-10-19.xml*