#### Abstract

The main purpose of this planned manuscript is to establish an algorithm for the solution of multiattribute decision-making (MADM) issues, where the experts utilizing linguistic variables provide the information about attributes in the form of picture hesitant fuzzy numbers. So, for the solution of these kinds of issues, we develop the TOPSIS algorithm under picture hesitant fuzzy environment using linguistic variables, which plays a vital role in practical applications, notably MADM issues, where the decision information is arranged by the decision-makers (DMs) in the form of picture hesitant fuzzy numbers. Finally, a sample example is given as an application and appropriateness of the planned method. At the end, we conduct comparison analysis of the planned method with picture fuzzy TOPSIS method and intuitionistic fuzzy TOPSIS method.

#### 1. Introduction

There are two types of mathematical logic, true (T) and false (F), having notation of 1 and 0. This concept has been changed by Zadeh. In 1965, Zadeh coined his remarkable theory of fuzzy sets (FSs)[1] to handle the uncertainty and unpredictability known as “fuzziness,” which is by the cause of partial membership of an a member to a set. Many other hybrid concepts were established after the invention of fuzzy set (FS) theory. In the situation of decision-making (DM) problems [2], FS is now considered to be a good appliance. The problem is what ways to combine various data into a single output [3–5]. The idea of fuzzy set theory utilized by Bellman and Zadeh [6] in DM for the solution of unpredictability data became from human choice. For decision investigation, Dubois compares the old and the new procedures [7].

The idea of intuitionistic fuzzy set (IFS) was first introduced by Atanassov, which is the generalization of FS and is denoted by the degree of membership and degree of nonmembership [8, 9], under the limitation that the sum of its membership degrees and nonmembership degrees is 1. IFS can better handle fuzziness compared to FS. The concept of intuitionistic fuzzy set theory is widely applied in DM problems [10–13].

Torra [14] coined the opinion of hesitant fuzzy sets (HFSs), which are the extension of fuzzy sets. HFS is represented by the degree of membership function having set of viable values between 0 and 1. Many researchers utilized the idea of HFS to solve group DM (GDM) problems with aggregation operators in [3–5, 15, 16].

There are many problems in real life, which must not be shown in IFS theory, for example, in the issue of voting system human notions which include other answers, for example, yes, no, abstinence, and refusal. Then Coung covered these gaps by adding neutral-membership function in IFS theory. Coung [17] introduced the idea of picture fuzzy set (PFS) model, which is the expansion of IF model. In picture fuzzy set theory, basically, he combined the neutral terms in IFS theory. The only restraint in PFS theory is that the sum of positive-membership, neutral-membership, and negative-membership functions is less than or equal to 1. Some composition of PF relations was developed by Phong et al. [18]. Singh developed correlation coefficients for PFS theory in 2015. Coung and Van Hai [19] discussed the basic notion about the few fuzzy logics operations for PFS. Son introduced the generalized picture distance measure and also its uses [20]. For multiattribute decision-making (MADM) problem, Wei [21] coined the picture fuzzy cross entropy. Wei coined the picture fuzzy aggregation operator and also its uses [22]. Wei et al. [23] presented the projection model for MADM under picture fuzzy environment. Zeb et al. [24] presented the notion of extended Pythagorean fuzzy set and applied this concept to solve preference risk decision-making problem. Ullah et al. [25] initiated the concept of GRA method using picture hesitant fuzzy numbers with incomplete weight information. Some scholars are working in the field of PFS theory and introduced different type of DM approaches (Wang et al. [26] and Wang et al. [27]).

Wang and Li [28] introduced the concept of picture hesitant fuzzy set (PHFS) theory based on picture fuzzy set and hesitant fuzzy set and operations of picture hesitant fuzzy elements (PHFEs) according to the operation of intuitionistic fuzzy numbers (IFNs) [29].

MADM problems have been broadly applied in the fields of management [30, 31], engineering [32, 33], economy [34, 35], and so forth. The researchers have initiated different approaches to handle MADM issues, for example, TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) [36], ELECTRE (ELimination and Choice Expressing the REality) [37], and PROMETHEE (Preference Ranking Organization METHod for Enrichment of Evaluations) [38]. The assessments of alternatives are surely known in MADM [39, 40]. Due to addition of uncertainty in MADM issues, the decision-makers (DMs) are laborious to provide the correct evaluation for alternatives. To solve this problem, fuzzy set theory [1] has been applied to MADM [6, 41], which gives an essential means of representing the complicated information. In 2020, Ali et al. [42] introduced the concept of TOPSIS method under complex spherical fuzzy sets utilizing Bonferroni mean operators. Tahir and Zeeshan utilized complex q-rung orthopair fuzzy set (CQROFS) and uncertain linguistic variable set (ULVS) and explored the new idea of fuzzy set called complex q-rung orthopair uncertain linguistic set (CQROULS) [43]. Also, in 2020, [44], Mahmood and Ali introduced TOPSIS algorithm based on q-rung orthopair fuzzy set (q-ROFS). In 2019, Jan et al. [45] introduced some distance measures of picture hesitant fuzzy set and conducted comparison analysis of proposed distance measures with other existing distance measures. Picture hesitant fuzzy weighted averaging (PHFWA) operator, picture hesitant fuzzy ordered weighted averaging (PHFOWA) operator, and picture hesitant fuzzy hybrid averaging (PHFHA) operator were established by Ullah et al. [46]. The fuzzy cross-entropy for picture hesitant fuzzy sets was established by Mahmood and Ali in [47]. In 2020, Wang et al. [48] introduced interactive Hamacher power aggregation operators for Pythagorean fuzzy sets.

Based on what was discussed with regard to the above-mentioned studies, the contributions of this manuscript are given as follows:(1)Proposing the extended TOPSIS method under PHF environment using linguistics variables.(2)Solving numerical problem based on the planned TOPSIS algorithm.(3)To show the effectiveness and validity of the planned TOPSIS algorithm, a comparative study with other existing methods is discussed.

There are some situations in real-life MADM problems where the decision-makers provide the membership degree, neutral degree, and nonmembership degree represented by many viable values in [0, 1]. Under these circumstances, in this study, we will introduce the concept of TOPSIS method for supplier selection under picture hesitant fuzzy environment using linguistic variable and an illustrative example is given as an application and appropriateness of the proposed method.At the end, we conduct comparison analysis of the planned algorithm with preexisting algorithm.

So, the remainder of this manuscript is organized as follows: In the 2nd section, we give a short illustration of picture hesitant fuzzy sets. In the 3rd section, we develop a picture hesitant fuzzy TOPSIS method. In the 4th section, a numerical example is established. In the 5th section, we conduct comparison analysis. The 6th section presents our conclusion.

#### 2. Preliminaries

##### 2.1. Picture Fuzzy Set (PFS)

In this section, we recall some basic definitions and properties.

*Definition 1. *[49] A picture fuzzy set on a nonempty set is defined aswhere are called positive-membership, neutral membership, and negative-membership degrees of the function, respectively, satisfying the condition . Furthermore, is said to be the refusal-membership degree of the function. Note that every IFS can be defined asIf we put in equation (2), then we get PFS. Basically, PFSs models are used in those cases when we face human notions involving more answers, that is, “yes,” “no,” “ abstinence,” and “refusal.” We can elaborate this concept by an example. Suppose that there are students who have to choose places for study tour between Islamabad and Lahore. There are few students who want to visit Islamabad (positive-membership), not Lahore (negative-membership), and others who want to visit Lahore (positive-membership), not Islamabad (negative-membership). There are some students who want to visit both places, that is, neutral students. But there are also some students who do not want to visit either Islamabad or Lahore, that is, refusal.

*Definition 2. *[28] A picture fuzzy value (PFV) or picture fuzzy number (PFN) is represented bywhere , and .

*Definition 3. *[22] Let be three PFVs, , and is the complementary set of ; then

##### 2.2. Hesitant Fuzzy Set (HFS)

*Definition 4. *[14] Let be the set of all subsets of the unitary interval and . Let , then hesitant fuzzy set (HFS) in can be defined as follows:where denotes the set of possible values between 0 and 1.

*Definition 5. *[3] A hesitant fuzzy number (HFN) or hesitant fuzzy value (HFV) is a nonempty and finite subset of . For convenience, HFN is denoted by and HFNs is the set of all hesitant fuzzy numbers.

*Definition 6. *[14] Let be three hesitant fuzzy numbers. Then some basic operations on HFNs can defined as follows:(i)(ii)(iii)

##### 2.3. Picture Hesitant Fuzzy Set (PHFS)

*Definition 7. *[28] A picture hesitant fuzzy set on is defined aswhere , , and are three sets of some values belonging to , representing the potential positive-membership, neutral-membership, and negative-membership degrees. The above degrees satisfy the condition , where , , and . The pair is the picture hesitant fuzzy number (PHFN) or picture hesitant fuzzy value (PHFV). For convenience, is a PHFN, denoted by .

During the process of using the PHFNs in the MCDM problems, it is necessary to rank the PHFNs; thus, we develop the score and accuracy functions of PHFNs.

*Definition 8. *Let be a PHFN; the numbers of elements in are , respectively. Then the score function of is defined as follows:The accuracy function is defined as

*Definition 9. *[28] Let , , and be three PHFNs of . Then,Obviously, the following theorem can be obtained by using Definition 9.

Theorem 1. *Let , , and be three PHFNs of . Then,*(1)*(ii)**(iii)**(iv)**(v)**(vi)**(vii)*

*Definition 10. *Let be a family of picture hesitant fuzzy numbers (PHFNs), where , and let be the weight vector of with , where and . Then picture hesitant fuzzy weighted average (PHFWA) operator is a mapping , which can be defined as

.

*Definition 11. *Let and be two picture hesitant fuzzy sets on a set . A distance measure between and is a mapping from up to unit closed interval , satisfying the following conditions:(i)(ii)(iii)

*Definition 12. *Let and be two picture hesitant fuzzy values on a set . Then, we define the distance measure between and as follows:where , , and represent the numbers of elements in , respectively. Since the number of members for different picture hesitant fuzzy values (PHFVs) could be different, we can make those different PHFVs equivalent by adding members to the PHFV that has a less number of members. We can add the smallest member in terms of pessimistic principle, while the opposite case will be adopted in optimistic principle and add maximum value. Typically, the values are out of order; for easiness, we may set them out in any sequence. Assume that we set them out in a descending sequence.

*Example 1. *Let and be two PHFVs. Then, .

##### 2.4. TOPSIS Method and Linguistics Variables

Here, we briefly describe the TOPSIS method and its applications. After that, we explain the use of TOPSIS method in solving MADM problems. We present the relationship among linguistic variables and picture hesitant fuzzy numbers (PHFNs). TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is a very convenient practical method for selecting of suitable alternative and also for ranking of alternatives with respect to their distance from the positive-ideal solution and negative-ideal solution. Hwang and Yoon developed TOPSIS to MCDM problems. In TOPSIS method, the chosen alternative should have the shortest distance from the positive-ideal solution and the farthest distance from the negative-ideal solution. Dubois discussed few new methods of decision analysis in [7]. The TOPSIS method is extended for triangular fuzzy numbers in [50, 51]. A linguistic variable is a variable whose values are represented with words or sentence instead of numbers in a natural language. The concept of linguistic variable plays an important role in solving DM problems with complex content. For example, we can represent the performance ratings of alternatives on qualitative criteria by linguistic variables such as very important, important, and medium. Such linguistic values can be represented by PHFNs.

#### 3. Picture Hesitant Fuzzy TOPSIS

In this section, we define the TOPSIS method under picture hesitant fuzzy environment using linguistic variables. Let be a set of alternatives and let be a set of attributes. We create the plane of picture hesitant fuzzy TOPSIS method, which is as follows.

##### 3.1. Steps of Picture Hesitant Fuzzy TOPSIS Method

*Step 1. *Determine the weight of decision-makers.

Suppose that our decision group contains “n” decision-makers. The importance of decision-makers is expressed as a linguistic term represented by picture hesitant fuzzy numbers. Let be a PHFN for rating of *r*th decision makers. Thus, the weight of *r*th decision-makers can be obtained asWe have that and is the refusal membership degree.

*Step 2. *Calculate aggregated picture hesitant fuzzy decision matrix under the notions of the decision-makers.

Let be a picture fuzzy decision matrix of each decision-maker. is the weight of each decision-maker and , . We need to obtain picture hesitant fuzzy decision S. For this, we utilized the PHFWA operator as follows:wherewith denoting a picture hesitant fuzzy number.

The aggregated picture hesitant fuzzy decision matrix of decision-maker can be defined as follows:

*Step 3. *Calculate the weights of attributes.In the decision-making method, each attribute as reported by decision-makers may have different importance. By combining the weight values and the attributes values of decision-makers for the importance of each attribute, we can obtain the weights of the attributes. Suppose that the weight of attributes is denoted by , where represents the relative importance of attributes . Let be a picture hesitant fuzzy number expressing the attributes by the *r*th decision-maker. The weights of attributes are calculated by using the PHFWA operator as follows:

*Step 4. *Calculate aggregated weighted picture hesitant fuzzy decision matrix with respect to attributes.

After finding the aggregated weighted picture hesitant fuzzy decision matrix with respect to decision-makers and determining the weights of attributes , we obtain the aggregated weighted picture hesitant fuzzy decision matrix with respect to criteria by using the aggregated weighted picture hesitant fuzzy decision matrix and the weights of attributes. Then, it can be defined as follows:wherewith denoting a picture fuzzy number.

The aggregated weighted picture hesitant fuzzy decision matrix of attributes can be defined as follows:

*Step 5. *Calculation of picture hesitant fuzzy positive-ideal solution (PF-PIS) and picture hesitant fuzzy negative-ideal solution (PF-NIS).

In TOPSIS method, the evaluation of attributes can be classified into two groups, benefit and cost. Let and be benefit attributes and cost attributes, respectively. is picture hesitant fuzzy positive-ideal solution and is picture hesitant fuzzy negative-ideal solution. Then and are defined aswhere

*Step 6. *Calculate the separation distance of each alternative from picture hesitant fuzzy positive-ideal solution (PHF-PIS) and picture hesitant fuzzy negative-ideal solution (PHF-NIS).

To measure distance of each alternative from PHF-PIS and PHF-NIS, we use the distance measure given by equation (10).

*Step 7. *Measure the closeness coefficient (CC).

Finally, we calculate the relative closeness coefficient of each alternative with respect to picture hesitant fuzzy positive-ideal solution (PF-PIS) , which is defined as follows:

*Step 8. *Measure the rank of alternatives.

After finding the relative closeness coefficient of each alternative we can rank all the alternatives in a descending order according to relative closeness coefficient.

#### 4. An Illustrative Example

In this part, we adopt a numerical example of MADM problem from the study of [52] to describe the application of the planned approach and conduct a comparison analysis with PF-TOPSIS and IF-TOPSIS [11].

##### 4.1. Implementation

*Example 2. *Suppose that there is a production industry; for supplier selection, four decision-makers have been appointed to evaluate 5-supplier alternatives with respect to four performance attributes. We have the following notations:(i): delivery performance(ii): product quality (manufacture quality)(iii): service(iv): priceThe importance weights from PHFNs of linguistic terms are listed in Table 1.

Moreover, in Table 2, we give the set of linguistic terms to rate the importance of alternatives as reported by decision-makers.

To find the performance of each attribute, the decision-maker utilizes a linguistic set of weights. The information of weights given to four attributes by four decision-makers is listed in Table 3.

We assume that the decision-makers utilize the linguistic variables and ratings to describe the appropriateness of the supplier alternatives with respect to each of the individual attributes. The results are listed in Tables 4–7.

Next, we apply the procedure of picture hesitant fuzzy TOPSIS method, which is as follows. Step 1: determine the weights of decision-makers. By utilizing equation (11), we get the weights of decision-makers, which are listed in Table 8. Then, we denote the weight vector of the decision-makers by . Step 2: calculation of aggregated picture hesitant fuzzy decision matrix under the notions of decision-makers. The ratings selected by the decision-makers to all alternatives were given in Table 4–7. Then the aggregated picture hesitant fuzzy decision matrix is obtained by utilizing equation (13) and the result is given in Table 9. We have the following: Step 3: calculate the weight of each attribute. We find the weight of each attribute by utilizing equation (14) and we use the information from Table 1 and present it in Table 10. Step 4: calculation of aggregated weighted picture hesitant fuzzy decision matrix with respect to attributes. To calculate the aggregated weighted picture fuzzy decision matrix, we utilize equation (15) and give it in Table 11. The details of Table 11 are presented in Appendix A. Step 5: Calculation of picture hesitant fuzzy positive-ideal solution and picture hesitant fuzzy negative-ideal solution. Delivery performance, product quality, and service are benefit attributes and price is cost attribute . To calculate picture hesitant fuzzy positive-ideal solution PHF-PIS and picture hesitant fuzzy negative ideal solution PHF-NIS , we use equation 17 and equation (18), respectively. For the details of the calculation of PHF-PIS and PHF-NIS, see Appendix B. Step 6: calculation of the separation measures. To calculate the separation measure of each alternative from the picture hesitant fuzzy positive-ideal solution and picture hesitant fuzzy negative-ideal solution, we use equation (19) and equation (20), respectively, and the calculation is given in Table 12. Step 7: calculate relative closeness coefficient (CC). We find the closeness coefficient of each alternative by using equation (20). The 4th column of Table 12 presents the result. Step 8: rank the alternatives. After finding the relative closeness coefficients, five alternatives are ranked in a descending order according to relative closeness coefficient. The alternatives are ranked as , chosen as suitable supplier among the alternatives. So the most suitable is .

##### 4.2. Comparison Analysis with Picture Fuzzy TOPSIS Method

Picture fuzzy numbers can be considered as a special case of picture hesitant fuzzy numbers when there is only one element in membership, neutral, and nonmembership degrees. For comparison, the picture hesitant fuzzy numbers can be converted into picture fuzzy numbers by finding the average values of membership, neutral, and nonmembership degrees; after calculation, the picture information is listed in Table 13 and Table 14.

Now we calculate the weight of decision-makers using the picture fuzzy TOPSIS method. The weights of decision-makers are given in Table 15.

The construction of aggregated picture fuzzy decision matrix with respect to decision-makers is given in Table 16.

Table 17 gives the weights of attributes.

The construction of aggregated weighted picture fuzzy decision matrix with respect to attributes is given in Table 18.

The information of PF-PIS and PF-NIS is given in Table 19.

The separation measures of each alternative from PF-PIS and PF-NIS, respectively, are listed in Table 20.

The relative closeness coefficient of each alternative from PF-PIS and ranking of the alternatives are given in Table 21.

Rank the alternatives with respect to the relative closeness coefficients :

So, the most suitable alternative is .

Clearly, the result obtained from picture fuzzy TOPSIS method is different from the result of the proposed method (see Figure 1). The difference in the alternative ranking is due to the fact that, in our planned approach, we assume that the membership degree, neutral degree, and nonmembership degree in picture hesitant fuzzy information are represented by many viable values between [0, 1], and in PF set membership degree, neutral degree, and nonmembership degree, it is represented by only single value. Obviously, our proposed method is more accurate and more stable. . So our planned algorithm is most reliable for such types of problems. In particular, when we face some situations where the information is represented by many possible values here, our planned approach shows its superiority to handle those decision-making problems. Thus, our planned approach is more suitable than PF-TOPSIS because it assumes the situation where decision-makers would like to use many viable values to represent the membership, neutral, and nonmembership degrees.

##### 4.3. Comparison Analysis with Intuitionistic Fuzzy TOPSIS Method

In this subsection, we will compare our propose method with intuitionistic fuzzy TOPSIS method. The importance weight as linguistic variable is given in Table 22.

The linguistic terms for rating the importance of alternatives are listed in Table 23.

Now we calculate the weight of decision-makers using the intuitionistic fuzzy TOPSIS method. The weights of decision-makers are given in Table 24.

The construction of aggregated intuitionistic fuzzy decision matrix with respect to decision-makers is given in Table 25.

The weights of attributes are given in Table 26.

The construction of aggregated weighted intuitionistic fuzzy decision matrix with respect to attributes is given in Table 27.

The information of IF-PIS and IF-NIS is given in Table 28.

The separation measures of each alternative from IF-PIS and IF-NIS, respectively, are given in Table 29.

The relative closeness coefficient of each alternative from IF-PIS and ranking of the alternatives are given in Table 30.

Rank the alternatives with respect to the relative closeness coefficients :

So, the most suitable alternative is .

Clearly, the result obtained from intuitionistic fuzzy TOPSIS method is different from the result of the proposed method (see Figure 2). The difference in the alternative ranking is due to the fact that IFNs only consider positive-membership degrees and negative-membership degrees and ignore neutral-membership degrees, which may result in information loss, giving rise to a difference in the accuracy of data; it will have an effect on the final result. Obviously, our proposed approach is more suitable than IF-TOPSIS method because in PHFNs we consider neutral-membership degrees.

#### 5. Conclusion

The main goal of this manuscript is to present a real-world multiattribute decision-making problem for supplier selection utilizing PHF- TOPSIS approach. PHFTOPSIS methodis an appropriate way to deal with unreliable information, as solving real-world DM issue expressed by crisp data under uncertain environment.. We defined PHF distance formula between picture hesitant fuzzy values. In the rating procedure aggregation of attributes the weight of decision makers and the impact of alternatives on attributes under the notion of DMs is most necessary to suitably perform 4 evaluation process. In order to do that, the ratings of each alternative under each attribute and the weights of each attribute were given as a linguistic terms expressed by picture fuzzy numbers. Also, PHFWA operator is applied to aggregate notions of DMs. After picture hesitant fuzzy positive-ideal solution and picture hesitant fuzzy negative-ideal solution were obtained by using distance formula, the relative closeness coefficients of each alternative were calculated and then the alternatives were ranked utilizing the closeness coefficient. Finally, the appropriateness of the planned algorithm has been illuminated with an example. At the end, we compared the planned algorithm with picture fuzzy TOPSIS algorithm and intuitionistic fuzzy TOPSIS method and proved the appropriateness of the proposed method.

In the future, the proposed TOPSIS algorithm can be utilized for dealing with unpredictability in multiattribute decision-making issues such as selecting manufacturing system, selecting project, and many other areas of management decision issues and marketing research problems. Also, we used Maclaurin Symmetric Mean Operator under picture cubic fuzzy value and its application to multiattribute decision-making issue.

#### Appendix

#### A. Detail of Aggregated Weighted Picture Hesitant Fuzzy Decision Matrix with respect to Attributes for Example 2

Table 31. Step 1: Calculation of aggregated weighted picture hesitant fuzzy decision matrix with respect to attributes.

To calculate the aggregated weighted picture fuzzy decision matrix, we utilize equation (15) and give it in Table 31.

We have