Abstract
The concept of Pythagorean fuzzy sets (PFSs) was initially developed by Yager in 2013, which provides a novel way to model uncertainty and vagueness with high precision and accuracy compared to intuitionistic fuzzy sets (IFSs). The concept was concretely designed to represent uncertainty and vagueness in mathematical way and to furnish a formalized tool for tackling imprecision to real problems. In the present paper, we have used both probabilistic and nonprobabilistic types to calculate fuzzy entropy of PFSs. Firstly, a probabilistic-type entropy measure for PFSs is proposed and then axiomatic definitions and properties are established. Secondly, we utilize a nonprobabilistic-type with distances to construct new entropy measures for PFSs. Then a min–max operation to calculate entropy measures for PFSs is suggested. Some examples are also used to demonstrate suitability and reliability of the proposed methods, especially for choosing the best one/ones in structured linguistic variables. Furthermore, a new method based on the chosen entropies is presented for Pythagorean fuzzy multicriterion decision making to compute criteria weights with ranking of alternatives. A comparison analysis with the most recent and relevant Pythagorean fuzzy entropy is conducted to reveal the advantages of our developed methods. Finally, this method is applied for ranking China-Pakistan Economic Corridor (CPEC) projects. These examples with applications demonstrate practical effectiveness of the proposed entropy measures.
1. Introduction
The concept of fuzzy sets was first proposed by Zadeh [1] in 1965. With a widely spread use in various fields, fuzzy sets not only provide broad opportunity to measure uncertainties in more powerful and logical way, but also give us a meaningful way to represent vague concepts in natural language. It is known that most systems based on ‘crisp set theory’ or ‘two-valued logics’ are somehow difficult for handling imprecise and vague information. In this sense, fuzzy sets can be used to provide better solutions for more real world problems. Moreover, to treat more imprecise and vague information in daily life, various extensions of fuzzy sets are suggested by researchers, such as interval-valued fuzzy set [2], type-2 fuzzy sets [3], fuzzy multiset [4], intuitionistic fuzzy sets [5], hesitant fuzzy sets [6, 7], and Pythagorean fuzzy sets [8, 9].
Since fuzzy sets were based on membership values or degrees between 0 and 1, in real life setting it may not be always true that nonmembership degree is equal to (1- membership). Therefore, to get more purposeful reliability and applicability, Atanassov [5] generalized the concept of ‘fuzzy set theory’ and proposed Intuitionistic fuzzy sets (IFSs) which include both membership degree and nonmembership degree and degree of nondeterminacy or uncertainty where degree of uncertainty = (1- (degree of membership + nonmembership degree)). In IFSs, the pair of membership grades is denoted by satisfying the condition of . Recently, Yager and Abbasov [8] and Yager [9] extended the condition to and then introduced a class of Pythagorean fuzzy sets (PFSs) whose membership values are ordered pairs that fulfills the required condition of with different aggregation operations and applications in multicriterion decision making. According to Yager and Abbasov [8] and Yager [9], the space of all intuitionistic membership values (IMVs) is also Pythagorean membership values (PMVs), but PMVs are not necessary to be IMVs. For instance, for the situation when the numbers and , we can use PFSs, but IFSs cannot be used since , but . PFSs are wider than IFSs so that they can tackle more daily life problems under imprecision and uncertainty cases.
More researchers are actively engaged in the development of PFSs properties. For example, Yager [10] gave Pythagorean membership grades in multicriterion decision making. Extensions of technique for order preference by similarity to an ideal solution (TOPSIS) to multiple criteria decision making with Pythagorean and hesitant fuzzy sets were proposed by Zhang and Xu [11]. Zhang [12] considered a novel approach based on similarity measure for Pythagorean fuzzy multicriteria group decision making. Pythagorean fuzzy TODIM approach to multicriterion decision making was given by Ren et al. [13]. Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making was developed by Peng and Yang [14]. Zhang [15] gave a hierarchical QUALIFLEX approach. Peng et al. [16] investigated Pythagorean fuzzy information measures. Zhang et al. [17] proposed generalized Pythagorean fuzzy Bonferroni mean aggregation operators. Liang and Xu [18] extended TOPSIS to hesitant Pythagorean fuzzy sets. Pérez-Domínguez et al. [19] gave MOORA under Pythagorean fuzzy sets. Recently, Pythagorean fuzzy LINMAP method based on the entropy for railway project investment decision making was proposed by Xue et al. [20]. Zhang and Meng [21] proposed an approach to interval-valued hesitant fuzzy multiattribute group decision making based on the generalized Shapley-Choquet integral. Pythagorean fuzzy information measure for multicriteria decision making problem was presented by Guleria and Bajaj [22]. Furthermore, Yang and Hussain [23] proposed distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multicriteria decision making and clustering. Hussain and Yang [24] gave entropy for hesitant fuzzy sets based on Hausdorff metric with construction of hesitant fuzzy TOPSIS.
The entropy of fuzzy sets is a measure of fuzziness between fuzzy sets. De Luca and Termini [25] first introduced the axiom construction for entropy of fuzzy sets with reference to Shannon’s probability entropy. Yager [26] defined fuzziness measures of fuzzy sets in terms of a lack of distinction between the fuzzy set and its negation based on Lp norm. Kosko [27] provided a measure of fuzziness between fuzzy sets using a ratio of distance between the fuzzy set and its nearest set to the distance between the fuzzy set and its farthest set. Liu [28] gave some axiom definitions of entropy and also defined a -entropy. Pal and Pal [29] proposed exponential entropies. While Fan and Ma [30] gave some new fuzzy entropy formulas. Some extended entropy measures for IFS were proposed by Burillo and Bustince [31], Szmidt and Kacprzyk [32], Szmidt and Baldwin [33], and Hung and Yang [34].
In this paper, we propose new entropies of PFS based on probability-type, distance, Pythagorean index, and min–max operation. We also extend the concept to -entropy and then apply it to multicriteria decision making. This paper is organized as follows. In Section 2, we review some definitions of IFSs and PFSs. In Section 3, we propose several new entropies of PFSs and then construct an axiomatic definition of entropy for PFSs. Based on the definition of entropy for PFSs, we find that the proposed nonprobabilistic entropies of PFSs are -entropy. In Section 4, we exhibit some examples for comparisons and also use structured linguistic variables to validate our proposed methods. In Section 5, we construct a new Pythagorean fuzzy TOPSIS based on the proposed entropy measures. A comparison analysis of the proposed Pythagorean fuzzy TOPSIS with the recently developed entropy of PFS [20] is shown. We then apply the proposed method to multicriterion decision making for ranking China-Pakistan Economic Corridor projects. Finally, we state our conclusion in Section 6.
2. Intuitionistic and Pythagorean Fuzzy Sets
In this section, we give a brief review for intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs).
Definition 1. An intuitionistic fuzzy set (IFS) in is defined by Atanassov [5] with the following form: where , and the functions denotes the degree of membership of in and denotes the degree of nonmembership of in . The degree of uncertainty (or intuitionistic index, or indeterminacy) of to is represented by .
For modeling daily life problems carrying imprecision, uncertainty, and vagueness more precisely and with high accuracy than IFSs, Yager [9, 10] presented Pythagorean fuzzy sets (PFSs), where PFSs are the generalizations of IFSs. Yager [9, 10] also validated that IFSs are contained in PFSs. The concept of Pythagorean fuzzy set was originally developed by Yager [8, 9], but the general mathematical form of Pythagorean fuzzy set was developed by Zhang and Xu [11].
Definition 2 (Zhang and Xu [11]). A Pythagorean fuzzy set (PFS) in proposed by Yager [8, 9] is mathematically formed aswhere the functions represent the degree of membership of in and represent the degree of nonmembership of in . For every , the following condition should be satisfied:
Definition 3 (Zhang and Xu [11]). For any PFS in , the value is called Pythagorean index of the element in with
In general, is also called hesitancy (or indeterminacy) degree of the element in It is obvious that It is worthy to note, for a PFS , if then , and if then and Similarly, if then If then and . If then . If then For convenience, Zhang and Xu [11] denoted the pair as Pythagorean fuzzy number (PFN), which is represented by
Since PFSs are a generalized form of IFSs, we give the following definition for PFSs.
Definition 4. Let be a PFS in . is called a completely Pythagorean if
Peng et al. [16] suggested various mathematical operations for PFSs as follows:
Definition 5 (Peng et al. [16]). If and are two PFSs in , then(i) if and only if , and ;(ii) if and only if and ;(iii);(iv);(v).
We next define more operations of PFS in , especially about hedges of “very”, “highly”, “more or less”, “concentration”, “dilation”, and other terms that are needed to represent linguistic variables. We first define the n power (or exponent) of PFS as follows.
Definition 6. Let be a PFS in . For any positive real number n, the n power (or exponent) of the PFS , denoted by , is defined asIt can be easily verified that, for any positive real number n,
By using Definition 6, the concentration and dilation of a PFS can be defined as follows.
Definition 7. The concentration of a PFS in is defined aswhere and
Definition 8. The dilation of a PFS in is defined aswhere and
In next section, we construct new entropy measures for PFSs based on probability-type, entropy induced by distance, Pythagorean index, and max-min operation. We also give an axiomatic definition of entropy for PFSs.
3. New Fuzzy Entropies for Pythagorean Fuzzy Sets
We first provide a definition of entropy for PFSs. De Luca and Termini [25] gave the axiomatic definition of entropy measure of fuzzy sets. Later on, Szmidt and Kacprzyk [32] extended it to entropy of IFS. Since PFSs developed by Yager [8, 9] are generalized forms of IFSs, we use similar notions as IFSs to give a definition of entropy for PFSs. Assume that represents the set of all PFSs in X.
Definition 9. A real function is called an entropy on if E satisfies the following axioms:;, ;, ;, is crisper than i.e., , and for and ;, where is the complement of ;
For probabilistic-type entropy, we need to omit the axiom (A0). On the other hand, because we take the three-parameter , and as a probability mass function , the probabilistic-type entropy should attain a unique maximum at Therefore, for probabilistic-type entropy, we replace the axiom with and the axiom with as follows: attains a unique maximum at , , if is crisper than , i.e.,, , and for and for .
In addition to the five axioms (A0)~(A4) in Definition 9, if we add the following axiom (A5), E is called - entropy:, and for or and for
We present a property for the axiom when is crisper than .
Property 10. If is crisper than in the axiom , then we have the following inequality:
Proof. If is crisper than , then , and for . Therefore, we have that and and so , i.e., and . Thus, we have , , and . This induces the inequality. Similarly, the part of and for also induces the inequality. Hence, we prove the property.
Since PFSs are generalized form of IFSs, the distances between PFSs need to be computed by considering all the three components and in PFSs. The well-known distance between PFSs is Euclidean distance. Therefore, the inequality in Property 10 indicates that the Euclidean distance between and is larger than the Euclidean distance between and . This manifests that is located more nearby to than that of . From a geometrical perspective, the axiom is reasonable and logical because the closer PFS to the unique point with maximum entropy reflects the greater entropy of that PFS.
We next construct entropies for PFSs based on a probability-type. To formulate the probability-type of entropy for PFSs, we use the idea of entropy of Havrda and Charavát [35] to a probability mass function withLet be a finite universe of discourses. Thus, for a PFS in , we propose the following probability-type entropy for the PFS with
Apparently, one may ask a question: “Are these proposed entropy measures for PFSs are suitable and acceptable?” To answer this question, we present the following theorem.
Theorem 11. Let be a finite universe of discourses. The proposed probabilistic-type entropy for a PFS satisfies the axioms , and in Definition 9.
To prove the axioms and for Theorem 11, we need Lemma 12.
Lemma 12. Let be defined asThen is a strictly concave function of
Proof. By twice differentiating , we get , for Then is a strictly concave function of Similarly, we can show that is also a strictly concave function of
Proof of Theorem 11. It is easy to check that satisfies the axioms and . We need only to prove that satisfies the axioms and . To prove for the case that satisfies the axiom , we use Lagrange multipliers for with = + . By taking the derivative of with respect to , and , we obtainFrom above PDEs, we get and then ; and ; and ; and then . Thus, we have . We obtain , and then We also get and That is, Similarly, we can show that the equation of for the case also obtains By Lemma 12, we learn that the function is a strictly concave function of We know that and so is also a strictly concave function. Therefore, it is proved that attains a unique maximum at We next prove that the probabilistic-type entropy satisfies the axiom . If is crisper than , we notice that is far away from compared to according to Property 10. However, is a strictly concave function and attains a unique maximum at From here, we obtain if is crisper than . Thus, we prove the axiom .
Concept to determine uncertainty from a fuzzy set and its complement was first given by Yager [26]. In this section, we first use the similar idea to measure uncertainty of PFSs in terms of the amount of distinction between a PFS and its complement . However, various distance measures are made to express numerically the difference between two objects with high accuracy. Therefore, the distance between two fuzzy sets plays a vital role in theoretical and practical issues. Let be a finite universe of discourses; we first define a Pythagorean normalized Euclidean (PNE) distance between two Pythagorean fuzzy sets asWe next propose fuzzy entropy induced by the PNE distance between the PFS and its complement . Let be any Pythagorean fuzzy set on the universe of discourse with its complement . The PNE distance between PFSs and will be Thus, we define a new entropy for the PFS as
Theorem 13. Let be a finite universe of discourse. The proposed entropy for a PFS satisfies the axioms (A0)~(A5) in Definition 9, and so is a -entropy.
Proof. We first prove the axiom (A0). Since the distance is between 0 and 1, , and then The axiom (A0) is satisfied. For the axiom (A1), if is crisp, i.e., or , , then . Thus, we obtain Conversely, if , then , i.e., . Thus, or and so is crisp. Thus, the axiom (A1) is satisfied. Now, we prove the axiom (A2). , implies that Conversely, implies . Thus, we have that For the axiom (A3), since and for imply that , then we have the distance Again from the axiom (A3) of Definition 9, we have and for implies that , and then the distance From inequalities and , we have , and then This concludes that In this way the axiom (A3) is proved. Next, we prove the axiom (A4). Since = Hence, the axiom (A4) is satisfied. Finally, for proving the axiom (A5), let and be two PFSs. Then, we have(i) and for , , or(ii) and for , . From (i), we have , then and . That is, which implies that . Also, , and , then implies Hence, Again, from (ii), we have , . Then, and , and so which implies that Also, and , then implies that Hence, . Thus, we complete the proof of Theorem 13.
Burillo and Bustince [31] gave fuzzy entropy of intuitionistic fuzzy sets by using intuitionistic index. Now, we modify and extend the similar concept to construct the new entropy measure of PFSs by using Pythagorean index as follows.
Let be a PFS on We define an entropy of using Pythagorean index as
Theorem 14. Let be a finite universe of discourse. The proposed entropy for the PFS using Pythagorean index satisfies the axioms (A0)~(A4) and (A5) in Definition 9, and so it is a - entropy.
Proof. Similar to Theorem 13.
We next propose a new and simple method to calculate fuzzy entropy of PFSs by using the ratio of min and max operations. All three components of a PFS are given equal importance to make the results more authentic and reliable. The new entropy is easy to be computed. Let be a PFS on , and we define a new entropy for the PFS as
Theorem 15. Let be a finite universe of discourse. The proposed entropy for a PFS satisfies the axioms (A0)~(A4) and (A5) in Definition 9, and so it is a -entropy.
Proof. Similar to Theorem 13.
Recently, Xue at el. [20] developed the entropy of Pythagorean fuzzy sets based on the similarity part and the hesitancy part that reflect fuzziness and uncertainty features, respectively. They defined the following Pythagorean fuzzy entropy for a PFS in a finite universe of discourses . Let be a PFS in . The Pythagorean fuzzy entropy, , proposed by Xue at el. [20], is defined asThe entropy will be compared and exhibited in next section.
4. Examples and Comparisons
In this section, we present simple examples to observe behaviors of our proposed fuzzy entropies for PFSs. To make it mathematically sound and practically acceptable as well as choose better entropy by comparative analysis, we give an example involving linguistic hedges. By considering linguistic example, we use various linguistic hedges like “more or less large”, “quite large” “very large”, “very very large”, etc. in the problems under Pythagorean fuzzy environment to select better entropy. We check the performance and behaviors of the proposed entropy measures in the environment of PFSs by exhibiting its simple intuition as follows.
Example 1. Let , and be singleton element PFSs in the universe of discourse defined as , , and The numerical simulation results of entropy measures , and are shown in Table 1 for the purpose of numerical comparison. From Table 1, we can see the entropy measures of almost have larger entropy than and without any conflict, except That is, the degree of uncertainty of is greater than that of and Furthermore, the behavior and performance of all entropies are analogous to each other, except . Apparently, all entropies , , , and behave well, except .
However, in Example 1, it seems to be difficult for choosing appropriate entropy that may provide a better way to decide the fuzziness of PFSs. To overcome it, we give an example with structured linguistic variables to further analyze and compare these entropy measures in Pythagorean fuzzy environment. Thus, the following example with linguistic hedges is presented to further check behaviors and performance of the proposed entropy measures.
Example 2. Let be a PFS in a universe of discourse defined as , . By Definitions 6, 7, and 8 where the concentration and dilation of are defined as Concentration: , Dilation: . By considering the characterization of linguistic variables, we use the PFS to define the strength of the structural linguistic variable in . Using above defined operators, we consider the following: is regarded as “More or less LARGE”; is regarded as “LARGE”; is regarded as “Quite LARGE”; is regarded as “Very LARGE”; is regarded as “Quite very LARGE”; is regarded as “Very very LARGE”. We use above linguistic hedges for PFSs to compare the entropy measures , and , respectively. From intuitive point of view, the following requirement of (18) for a good entropy measure should be followed:After calculating these entropy measures , and for these PFSs, the results are shown in Table 2. From Table 2, it can be seen that these entropy measures , and satisfy the requirement of (18), but fails to satisfy (18) that has . Therefore, we say that the behaviors of , and are good, but is not.
In order to make more comparisons of entropy measures, we shake the degree of uncertainty of the middle value “5” in . We decrease the degree of uncertainty for the middle point in X, and then we observe the amount of changes and also the impact of entropy measures when the degree of uncertainty of the middle value in X is decreasing. To observe how different PFS “LARGE” in X affects entropy measures, we modify asAgain, we use PFSs , and with linguistic hedges to compare and observe behaviors of entropy measures. The results of degree of fuzziness for different PFSs from entropy measures are shown in Table 3. From Table 3, we can see that these entropies , and satisfy the requirement of (18), but and could not fulfill the requirement of (18) with and Therefore, the performance of , and is good, and is not good, but presents very poor. We see that a little change in uncertainty for the middle value in X did not affect entropies , and , and it brings a slight change in , but it gives an absolute big effect for entropy .
In viewing the results from Tables 1, 2, and 3, we may say that entropy measures , and present better performance. On the other hand, from the viewpoint of structured linguistic variables, we see that entropy measures , and are more suitable, reliable, and well suited in Pythagorean fuzzy environment for exhibiting the degree of fuzziness of PFS. We, therefore, recommend these entropies , and in a subsequent application involving multicriteria decision making.
In the following example, we conduct the comparison analysis of proposed entropies , and with the entropy , developed by Xue at el. [20], to demonstrate the advantages of our developed entropies , and .
Example 3. Let , and be PFSs in the singleton universe set as The degrees of entropy for different PFSs between the proposed entropies and the entropy by Xue at el. [20] are shown in Table 4. As can be seen from Table 4, we find that despite having three different PFSs, the entropy measure could not distinguish the PFSs , and . However, the proposed entropy measures , and can actually differentiate these different PFSs , and .
5. Pythagorean Fuzzy Multicriterion Decision Making Based on New Entropies
In this section, we construct a new multicriterion decision making method. Specifically, we extend the technique for order preference by similarity to an ideal solution (TOPSIS) to multicriterion decision making, based on the proposed entropy measures for PFSs. Impreciseness and vagueness is a reality of daily life which requires close attentions in the matters of management and decision. In real life setting with decision making process, information available is often uncertain, vague, or imprecise. PFSs are found to be a powerful tool to solve decision making problems involving uncertain, vague, or imprecise information with high precision. To display practical reasonability and validity, we apply our proposed new entropies , and in a multicriteria decision making problem, involving unknown information about criteria weights for alternatives in Pythagorean fuzzy environment.
We formalize the problem in the form of decision matrix in which it lists various project alternatives. We assume that there are m project alternatives and we want to compare them on n various criteria . Suppose, for each criterion, we have an evaluation value. For instance, the first project on the first criterion has an evaluation The first project on the second criterion has an evaluation , and the first project on nth criterion has an evaluation . Our objective is to have these evaluations on individual criteria and come up with a consolidated value for the project 1 and do something similar to the project 2 and so on. We then ultimately obtain a value for each of the projects. Finally, we can rank the projects with selecting the best one among all projects.
Let be the set of alternatives, and let the set of criteria for the alternatives be represented by . The aim is to choose the best alternative out of the n alternatives. The construction steps for the new Pythagorean fuzzy TOPSIS based on the proposed entropy measures are as follows.
Step 1 (construction of Pythagorean fuzzy decision matrix). Consider that the alternative acting on the criteria is represented in terms of Pythagorean fuzzy value , where denotes the degree of fulfillment, represents the degree of not fulfillment, and represents the degree of hesitancy against the alternative to the criteria with the following conditions: , and . The decision matrix is constructed to handle the problems involving multicriterion decision making, where the decision matrix can be constructed as follows:
Step 2 (determination of the weights of criteria). In this step, the crux to the problem is that weights to criteria have to be identified. The weights or priorities can be obtained by different ways. Suppose that the criteria information weights are unknown and therefore, the weights of criteria for Pythagorean fuzzy entropy measures can be obtained by using , and , respectively. Suppose the weights of criteria are with and . Since weights of criteria are completely unknown, we propose a new entropy weighting method based on the proposed Pythagorean fuzzy entropy measures as follows:where the weights of the criteria is calculated with
Step 3 (Pythagorean fuzzy positive-ideal solution (PFPIS) and Pythagorean fuzzy negative-ideal solution (PFNIS)). In general, it is important to determine the positive-ideal solution (PIS) and negative-ideal solution (NIS) in a TOPSIS method. Since the evaluation criteria can be categorized into two categories, benefit and cost criteria in TOPSIS, let and be the sets of benefit criteria and cost criteria in criteria , respectively. According to Pythagorean fuzzy sets and the principle of a TOPSIS method, we define a Pythagorean fuzzy PIS (PFPIS) as follows:Similarly, a Pythagorean fuzzy NIS (PFNIS) is defined as
Step 4 (calculation of distance measures from PFPIS and PFNIS). In this step, we need to use a distance between two PFSs. Following the similar idea from the previously defined PNE distance between two PFSs, we define a Pythagorean weighted Euclidean (PWE) distance for any two PFSs asWe next use the PWE distance to calculate the distances and of each alternative from PFPIS and PFNIS, respectively, as follows:
Step 5 (calculation of relative closeness degree and ranking of alternatives). The relative closeness degree of each alternative with respect to PFPIS and PFNIS is obtained by using the following expression:
Finally, the alternatives are ordered according to the relative closeness degrees. The larger value of the relative closeness degrees reflects that an alternative is closer to PFPIS and farther from PFNIS, simultaneously. Therefore, the ranking order of all alternatives can be determined according to ascending order of the relative closeness degrees. The most preferred alternative is the one with the highest relative closeness degree.
In the next example, we present a comparison between the proposed entropies and with the entropy by Xue at el. [20] based on PFSs for multicriteria decision making problem. The prime objective of decision makers is to select a best alternative from a set of available alternatives according to some criteria in multicriteria decision making process. Corruption is the misuse and mishandle of public power and resources for private and individual interest and benefits, usually in the form of bribery and favouritism. In addition, corruption twists and manipulates the basis of competitions by misallocating resources and slowing economic activity (Wikipedia).
Example 4. In this example, a real world problem on selection of well renowned national or international audit company is taken into account to demonstrate the comparison analysis among the proposed probabilistic entropy and nonprobabilistic entropy with the entropy [20]. To ensure the transparency and accountability of state run intuitions, the ministry of finance of a developing country offers quotations to select a renowned audit company to get unbiased and fair audit report to keep on tract the economic development of a state. The quotations which are gone through scrutiny process and found successful by a committee are comprised of experts. The quotations which are found successfully by the committee are called eligible while the rest are rejected. A commission of experts is invited to rank the audit companies and to select the best one on the basis of set criteria . The descriptions about criteria are given in Table 5, and Pythagorean fuzzy decision matrices are presented in Table 6. The obtained weights of criteria from the entropies , , and are shown in Table 7. From Table 7, it is seen that the weights of criteria obtained by the entropy are always the same despite having different alternatives. However, the proposed entropies and correctly differentiate the weights of criteria for each alternative . The weights of criteria in Table 7 are also used to calculate the distances and of each alternative from PFPIS and PFNIS, respectively, where the results are shown in Table 8. Furthermore, the relative closeness degrees of each alternative to ideal solution are shown in Table 9. It can be seen that the relative closeness degrees obtained by the proposed entropies and are different for different alternatives, but the relative closeness degrees obtained by the entropy [20] could not differentiate different alternatives so that it gives bias ranking of alternatives. These ranking results of different alternatives by the entropies , and are shown in Table 10. As can be seen, the ranking of alternatives by the proposed entropies and is well; however, the entropy [20] could not rank the alternative and . It is found that there is no conflict in ranking alternatives by using the proposed Pythagorean fuzzy TOPSIS method based on the proposed entropies and . Totally, the comparative analysis shows that the best alternative is .
We next apply the constructed Pythagorean fuzzy TOPSIS in multicriterion decision making for China-Pakistan Economic Corridor projects.
Example 5. A case study in ranking China-Pakistan Economic Corridor projects on priorities basis in the light of related experts’ opinions is used in order to demonstrate the efficiency of the proposed Pythagorean fuzzy TOPSIS being applied to multicriterion decision making. China-Pakistan Economic Corridor (CPEC) is a collection of infrastructure projects that are currently under construction throughout in Pakistan. Originally valued at $46 billion, the value of CPEC projects is now worth $62 billion. CPEC is intended to rapidly modernize Pakistani infrastructure and strengthen its economy by the construction of modern transportation networks, numerous energy projects, and special economic zones. CPEC became partly operational when Chinese cargo was transported overland to Gwadar Port for onward maritime shipment to Africa and West Asia (see Wikipedia). It is not only to benefit China and Pakistan, but also to have positive impact on other countries and regions. Under CPEC projects, it will have more frequent and free exchanges of growth, people to people contacts, and integrated region of shared destiny by enhancing understanding through academic, cultural, regional knowledge, and activity of higher volume of flows in trades and businesses. The enhancement of geographical linkages and cooperation by a win-win model will result in improving the life standard of people, road, rail, and air transportation systems and also sustainable and perpetual development in China and Pakistan.
Now suppose the concern and relevant experts are allowed to rank the CPEC projects according to needs of both countries on priorities basis. Assume that initially there are five mega projects which are Gwadar Port (A1), Infrastructure (A2), Economic Zones (A3), Transportation and Energy (A4), and Social Sector Development (A5), according to the following four criteria: time frame and infrastructural improvement (C1), maintenance and sustainability (C2), socioeconomic development (C3), and eco-friendly (C4). A detailed description of such criteria is displayed in Table 11. Consider a decision organization with the five concerns, where relevant experts are authorized to rank the satisfactory degree of an alternative with respect to the given criterion, which is represented by a Pythagorean fuzzy value (PFV). The evaluation values of the five alternatives with Pythagorean fuzzy decision matrix are given in Table 12.
We next use entropy measures , and based on better performance in numerical analysis to calculate the criteria weights using (22). These results are shown in Table 13. From Table 13, we can see that the criteria weights and ranking of weights obtained by each entropy measure are different. We find the distances and of each alternative from PFPIS and PFNIS using (23) and (24). The results are shown in Table 14. We also calculate the relative closeness degrees of alternatives using (27). The results are shown in Table 15. From Table 15, it can be clearly seen that, under different entropy measures, the relative closeness degrees of alternatives obtained are different, but the gap between these values are considerably small. Thus, the ranking of alternatives is almost the same. The final ranking results from different entropies are shown in Table 16. From Table 16, it can be identified that there is no conflict in selecting the best alternative among alternatives by using the proposed Pythagorean fuzzy TOPSIS method based on the entropies , and . There is only one conflict to be found in deciding the preference ordering of alternatives and in Hence, the results of ranking of alternatives according to the closeness degrees are made in an increasing order. Therefore, our analysis shows that the most feasible alternative is which is unanimously chosen by all proposed entropy measures.
6. Conclusions
In this paper, we have proposed new fuzzy entropy measures for PFSs based on probabilistic-type, distance, Pythagorean index, and min–max operator. We have also extended nonprobabilistic entropy to -entropy for PFSs. The entropy measures are considered especially for PFSs on finite universes of discourses. As these are not only used in purposes of computing environment, but also used in more general cases for large universal sets. Structured linguistic variables are used to analyze and compare behaviors and performance of the proposed entropies for PFSs in different Pythagorean fuzzy environments. We have examined and analyzed these comparison results obtained from these entropy measures and then selected appropriate entropies which can be useful and also be helpful to decide fuzziness of PFSs more clearly and efficiently. We have utilized our proposed methods to perform comparison analysis with the most recently developed entropy measure for PFSs. In this connection, we have demonstrated a simple example and a problem involving MCDM to show the advantages of our suggested methods. Finally, the proposed entropy measures of PFSs are applied in an application to multicriterion decision making for ranking China-Pakistan Economic Corridor projects. Based on obtained results, we conclude that the proposed entropy measures for PFSs are reasonable, intuitive, and well suited in handling different kinds of problems, involving linguistic variables and multicriterion decision making in Pythagorean fuzzy environment.
Data Availability
All data are included in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.