Abstract
Since real-world data are often inaccurate and working with fuzzy data and Z-numbers are very important and necessary, and in the real world, we need to rank and compare data, in this article, we introduce a new method for ranking Z-numbers. This ranking algorithm is based on centroid point. We evaluate the distance between the centroid points, and based on this distance, we rank the Z-numbers. The algorithm of this method has simple calculations and it provides satisfactory results for data in different conditions; we use this method in two practical examples. First, by ranking the return on assets of Tehran stock exchanges, with this method, we can choose the most risky stock and the least risky stock among the available stocks. The advantage of this method over conventional fuzzy methods is considering uncertainty and allocating credit in the opinion of experts to estimate fuzzy parameters, and in the second example, we ranked the factors which affect the productivity of tourism security. Then, by using the proposed algorithm, we introduce the most important and the least important factors for tourist security budgeting.
1. Introduction
In the real world, new computational methods are needed to cope with the increasing complexity of research, studies, and modeling and to solve new issues in various sciences. Fuzzy logic is one of them, but fuzzy logic does not consider the reliability of the information. To eliminate this restriction, Z-number was introduced by Zadeh in 2011 [1].
A Z-number which describes the restriction and the reliability is displayed by an ordered pair of fuzzy numbers . The first component shows fuzzy restriction and the second component is a reliability of the [2].
As a new concept, discussion about Z-numbers and their basic theories such as arithmetic and ranking are few. Various models have been presented to rank these numbers, each of them is based on criteria and specific characteristics of these numbers, but the reliability of the provided data is not considered.
For example, Kang et al. [3] introduced a method for converting Z-numbers to fuzzy numbers on the basis of a fuzzy expectation of a fuzzy set. By this method, the researchers first convert Z-numbers to fuzzy numbers, and then by using the theories and the methods of fuzzy numbers, they solve the problems in the uncertain situations.
Bakar and Gegov [4] suggested a method that initially turned the Z-numbers into a fuzzy numbers and then ranked the generalized fuzzy numbers.
According to the previous methods, it is evident that these methods depend on fuzzy number ranking methods, so before ranking Z-numbers, a new method for ranking generalized fuzzy numbers must be introduced. Chen et al. [5] introduced a method based on defuzzified values, the highest, and the spreads of generalized fuzzy numbers.
As can be seen, converting Z-numbers to generalized fuzzy numbers will lead to the loss of original information [3].
Over the last two decades, researchers have given attention to investigating the centroid point of fuzzy numbers and used it for comparing fuzzy numbers. Yager [6] introduced the concept of a centroid-based ranking method for comparing fuzzy numbers, but he used only the X coordinate of the centroid point. Murakami et al. [7] proposed a centroid-based ranking approach that calculated the centroid point of fuzzy numbers and ranked them according to the maximum of X coordinate and Y coordinate. Cheng [8] proposed a ranking method by using the distance between the original and centroid points. Chu and Tsao [9] proposed a ranking method of fuzzy numbers with an area between the centroid point and the original point. However, Wang et al. [10] demonstrated that the centroid formula which defined in [8, 9] is incorrect and they provided the correct centroid formula. Dat et al. [11] proposed an improved ranking method based on the centroid point for ranking various types of fuzzy numbers. In addition, centroid points have also been analyzed for other generalizations of fuzzy sets.
Therefore, as mentioned, many models have been presented with fuzzy ranking that all of them were based on certain criteria and characteristics of these numbers, but the reliability of the data provided has not been considered in many of them.
In this article, we have suggested a method that is based on centroid points, and we use them directly for ranking Z-numbers. The proposed method has some advantages as it does not convert to a fuzzy number and does not have a complicated algorithm, and it also provides acceptable results in different conditions.
As a practical example, this method was used in ranking the return on assets of the Tehran stock exchange.
As we know due to the nature of financial markets and the uncertainty in the available information in these markets, it is possible to use Z-numbers to describe future returns in the ranking and selection of investment portfolios.
For example, suppose a capital market expert believes that the return on a particular stock in the next year follows a fuzzy number , but the validity of this prediction can be shown by another fuzzy number such as . Therefore, the return of this share can be shown by the number .
This article is organized as follows: We bring the subject literature in Section 2. Section 3 contains some definitions and concepts of fuzzy numbers and Z-numbers. In Section 4, the proposed method for ranking Z-numbers is discussed, and then the method is illustrated by a numerical example. In Section 5, there are two practical examples that include a ranking of return on assets of the stock exchange and a ranking of factors affecting the productivity of tourism security by the proposed method, and interesting and desired results are shown at the end of the section, and the conclusion is presented in Section 6.
2. Subject Literature
The Z-number ranking literature is very extensive; although many research studies have been carried out in this field, still many existing methods face many shortcomings or complications. In this section, a number of new ranking methods are presented.
In Alam’s article [12], a new model was introduced using a ranking function based on central points that the main purpose of the model is to illustrate the use of IZN (intuitionist Z-number) in characterizing the decision maker’s preferences for the decision-making process.
In another article, Cheng et al. [13] proposed a new representative value for fuzzy numbers. Then, they applied it to the ranking of Z-number. For this purpose, Takagi–Sugeno–Kang (TSK) fuzzy model is used. The results of the Rep function are used to construct a new representative fuzzy subset of the Z-numbers and associating this new fuzzy subset with a scalar value. This fuzzy subset not only implies the fuzzy aspect of the Z-number but also contains the information that was contained in the hidden probability distribution. The scalar value is considered as the representative value of the Z-number to participate in the ranking. The proposed method largely preserves the original information of the Z-number and can overcome the shortcomings of the existing methods.
The method proposed by Hadaegh et al. [14] is based on calculating the center of gravity, normal distribution function, and confidence interval. In the first step, the value of the center of gravity of the confidence interval of the numbers is calculated and is considered as the confidence level. In the second step, the membership degree function of part A is determined using the normal distribution function. In the third step, the statistical confidence interval formula is obtained. The last step in bandwidth calculation is the ranking of Z-numbers.
Among the existing methods, the algorithm introduced in this article, unlike other methods, preserves the original nature of the z-number and can be used for numbers in different states and conditions, and it also has a simple algorithm.
3. Preliminaries
In this section, we explain some definitions that are used in this article.
Definition 1. A fuzzy subset of the real line with membership function , where , is a fuzzy number if it satisfies in the following properties:(1) is a continuous function from to (2) for all and (3) is strictly increasing on (4) for (5) is strictly decreasing on
Definition 2. A fuzzy subset is said to be normal if .
Definition 3. A fuzzy number is said a crisp number.
Definition 4. A fuzzy number is said a trapezoidal fuzzy number if it’s membership function is as follows:If we have , then is the triangular fuzzy number.
As we know, we can approximate each fuzzy number by a trapezoidal number in different approximations. Therefore, in this article, all of the fuzzy numbers are considered trapezoidal.
Definition 5. A Z-number is an ordered pair of fuzzy numbers that is shown as . The first component is a restriction on the values which can take and the second component is a measure of the reliability of the first component by the following membership functions:andwhere and are the maximum degrees of membership functions.
For ,(1) convert the (reliability) into a crisp number by using where denotes an algebraic integration.(2) Also, add the weight of the to the (restriction). The weighted Z-number is denoted as follows:Consider a Z-number as so that and that their membership functions have been defined in equations (2) and (3).
Definition 6. The centroid point of is shown by and is defined as follows:andSimilarly, the centroid point of as is defined in the previous equation.
Before introducing the algorithm we must note that if the Z-numbers are displayed with which , then it is easily understood that is the most important part of , because the final opportunity of Z-number is to describe the uncertain variable, while , as only a measure of the reliability of , can influence the ranking result but cannot decide the ranking result, so is more important and is the main part of Z-number. This point should be considered when we want to rank Z-numbers.
In addition, the information of Z-numbers should be retained as much as we deal with them, so ranking Z-numbers should satisfy in two principles as follows:(1) is the main part of the Z-number and is the subsidiary factor of the Z-number. So, the weight of should be larger than when Z-numbers are ranked.(2)In the ranking process, the information of the Z-number must be kept intact and protected without defects, and the conversion of and to a crisp number must be avoided.So, in this algorithm, we try to save most information of Z-numbers by keeping the nature of the Z-numbers.
4. The Proposed Method for Ranking Z-Numbers Based on Centroid Point
Due to the importance of ranking Z-numbers, many methods have been introduced, but the methods that have satisfactory results for all data are less. The presented method in this section has a very simple algorithm and does not require complex calculations and provides acceptable results for different values and conditions.
The correct operation of this method is shown with a numerical example and some special cases at the end of this section.
This section presents a new method that is based on centroid point. Let us consider a set of Z-numbers where . We compute the centroid points of and , then we use their convex combination for the definition centroid point of , and finally, we rank Z-numbers by using the proposed algorithm. This ranking process can be carried out according to the following steps:
Step 1. For each such that and , compute the centroid point for and by equations (6) and (7).
Step 2. Suppose thatwhere and are the mean values of and which are defined as follows:
Step 3. Suppose that such thatwhere is a parameter and , and it opposed to if two fuzzy numbers and in a Z-number are symmetrical.
Step 4. and are defined as follows:where and are the reference values according to the meters that will be defined in the next step, all the fuzzy quantities are compared by using these reference values.
Step 5. Evaluation of the distance between and for by using the Minkowski metric as follows:Similarly, the distance between and for is measured as follows:So, if the centroid points are equal for two Z-numbers such as and , we put and otherwise , and is a distance parameter.
Step 6. Compute the ranking index that is shown by as follows:where the parameter is referred to as the decision maker’s preference attitude and .
Step 7. Rank the Z-numbers by using values of obtained in the previous step as follows:(1)We say that if ; in the other words, if we have hence So, we must have the following equation:(2)We say that if ; in other words, if we have the following equation, we must have the following equation:
Step 8. If one of the components of Z-number is crisp, for example, , then we let
Theorem 1. This algorithm satisfies in following axioms of ranking: . For an arbitrary Z-number as , we have . For arbitrary Z-numbers as and , if we have and , then we should have . If , , and are arbitrary Z-numbers and we have and , then we have . For three arbitrary Z-numbers , , and , if we have , then we should have . For arbitrary Z-numbers as and , if and is a crisp number such that , then we have
Proof. . If we put in (17), then for an arbitrary Z-number as , we have the following: It is clear that and so . . Suppose that and are two Z-numbers such that and , so we have the following equation: Hence, Moreover, by equation (17), we have . . Suppose that , , and are three Z-numbers and we have and , so by (14), we should have the following equation: So, we must have the following equation: and then we have . . For three Z-numbers , , and , suppose that , then Moreover, we have the following equation: So, it is clear that . . For arbitrary Z-numbers as and , if and is a crisp number such that , then we have the following equation: Hence, .We now use the proven theorems and results from this article in a numerical example and compare the results with the ranking methods found in previous articles.
Example 1. In this example, we use 6 sets of Z-numbers to compare the ranking results of the proposed ranking method for the Z-number of existing ranking methods.
For all of the examples, let be the of in the 6 sets. On the other hand, all the Z-numbers have the same part . The is shown in Figure 1 and all the of Z-numbers in the 6 sets are shown in Figure 2. Also, in all of the examples, we suppose that , , and .
In Table 1, we have some existed methods, and we can see that these methods cannot distinguish between Z-numbers.
The ’s in the 6 sets are different, but they have a same ; hence, the different Z-numbers in the 6 sets have the same ranking scores by existing methods.
Table 2 clearly shows the desired results of comparing our proposed method with Jiang’s method [17]. We have ranked 6 different sets by two methods and listed the result.
5. Numerical Examples
Example 2. Ranking of return on assets of Tehran stock exchange.
In this section, using the theory of Z-numbers and considering the future return on assets in the form of z-numbers, we will rank the return on assets of the Tehran stock exchange. To do this, we have randomly selected ten assets from the Tehran stock exchange [18], the symbols of which are listed in Table 2.
For this purpose, the following three factors are considered:(1)Arithmetic Mean. Although the arithmetic mean of a return on an asset should not be directly expressed as a future return, it can be considered as a good approximation for its calculation.(2)Historical Efficiency Trend. If recent returns on securities are increasing, it can be believed that the expected return on securities based on historical data is higher than the average, and vice versa. This factor can have a significant impact on estimating future returns on assets.(3)Use of Financial Statements and Expert Opinion. The use of expert opinion depends on predictions based on financial reports and the personal experience of experts.The forecast of future return on assets (the first factor of the Z-number) is considered as a trapezoidal number. Also, the validity of these predictions (the second factor of the Z-number) according to the number of fluctuations per share during the last few years as well as the opinion of experts about the probability of predictions has been considered in the form of symmetrical triangular fuzzy numbers.
For example, suppose an expert believes that the return on a particular share over the next year can be expressed by a trapezoidal fuzzy number in the form of . This number indicates that the return on equity will not be less than 0.12 in the next year. On the other hand, the return on this share does not exceed 0.35. Also, the membership rate of share return in the range of 0.22 to 0.28 is equal to 1. Now, given the person’s background, as well as the predictability of the share over the past few years, we can show the confidence level in his prediction is equal to another fuzzy triangular number such as . In other words, we express the probability that this person’s prediction is correct through the fuzzy number .
Therefore, the Z-numbers of ten random assets from the Tehran stock exchange are summarized in Table 3.
The chart of each Z-number is shown in Figure 3.
Now, by using the proposed algorithm and supposing that , , we will get Table 4.
Moreover, by Table 4, we have the following equation:This means that BSwich and CShoaml stocks with the scores of 1.000 and have the lowest risk for investment and Khodro stock has the highest risk. So, BSwich is the best choice if you want to invest.
Example 3. Ranking of Factors Affecting the Productivity of Tourism Security
Over the past decades, tourism has continued to grow and diversify and become one of the largest growing economic sectors in the world. Increasing the number of tourists will boost business and increase the income of companies and institutions that operate in this field. The development of tourism is an effective factor in combating poverty and increasing the income of various groups. It reduces unemployment and economic prosperity, and thus improves the quality of people’s lives and increases social welfare.
The tourism industry today is more diverse and complex than in the past. At present, tourists have shorter stays in tourist destinations. Instead of long stays, they have increased the number of their trips. They place more emphasis on the destination environment, expect more value for money, and demand better services. As a result, in order to succeed in tourism, a region must be economically, socially, and environmentally secure, and in order to be secure, it must be carefully planned managed and taken various factors into consideration. In fact, improving security in tourist areas is an attractive key factor in encouraging tourists to travel safely. It can be an incentive for people and industries to provide services and facilities for the development of this industry.
In this section, we prioritize the factors affecting the productivity of tourism security and Isfahan (in Iran) has been selected as a case study. Based on the research background, a series of studies conducted on sustainable tourism security in Iran, as well as the views of professors and experts at various levels in the field of tourism in Isfahan, 15 factors affecting security efficiency in tourism supply, including attraction, transportation, information, advertising, and services were identified in the form of a questionnaire. [19].
To rank the factors, 15 factors were selected as follows: . Cooperation of travel agencies and tour guides with the security police . Cooperation of hotel officials with security police . Establishment of parking lots in and out of tourist places for security and environmental protection . Using of public tourist transport fleet equipped with GPS and approved by the police . Modification of roads and installation of traffic signs on tourist routes . Using a comprehensive database to obtain visas and tourist information . Creating a comprehensive security control system, and intelligent law enforcement to know the status of tourists . Developing virtual tourism to combat malicious advertising . Developing ethical standards for tourism police personnel in dealing with tourists . Informing tourists of local values and laws and the need to respect these values . Providing timely services for renewal and obtaining visas and accommodation for foreign tourists . The seriousness of the police in pursuing the complaints of tourists . Strengthening the expert force in all areas of tourism . Existence of 24-hour police patrols in seasons when there are many tourists .Periodic police visits of railways, bus terminals, hotels, and campsThe Z-numbers of these factors are summarized in Table 5.
The charts of Z-numbers are shown in Figure 4.
Now using the proposed algorithm, we have the following ranking Table 6.
So, we have the following equation:It means that the presence of 24-hour police patrols in seasons, when there are many tourists, can have the greatest impact on the security and productivity of tourism. Therefore, in allocating the tourism budget, more attention should be paid to this part and more manpower should be considered for it.
Moreover, the creating a comprehensive security control system, intelligent law enforcement to know the status of tourists will have less impact, so the lowest budget belongs to this item.
6. Conclusion
When we have a problem of the linguistic with Z-numbers data, the question arises, which of these numbers is bigger and which is smaller? To answer this question, ranking methods are introduced.
In this article, we presented a new method for ranking Z-numbers. In comparison with the existing methods for ranking, the proposed method has some advantages. We considered a Z-number as , then we obtained the centroid points of and , and with a new ranking algorithm, based on the calculation the distance between centroid points, we ranked Z-numbers.
The most important advantage of the method is not to convert the Z-number into a fuzzy number, and therefore there is no need to use complex fuzzy ranking algorithms and it is carried out directly with a simple algorithm on Z-numbers. This method also provides acceptable results in different conditions and for different numbers.
Finally, two numerical examples for this ranking method were presented. First, by selecting several assets from the Tehran Stock Exchange, the ranking of these assets was carried out.
As the results have been shown, the use of Z-numbers in the problem of investment ranking causes that in addition to allocating uncertainty on the return on assets, the amount of credit for the forecast of experts is also taken into account. In fact, using the Z-numbers theory, uncertainty in estimating fuzzy parameters can also be considered.
In the second example, we used a descriptive method to rank the factors affecting the productivity of tourism security and tried to use the knowledge of experts and specialists in this industry to rank the indicators for evaluating tourist attractions.
In problem modeling, when a cycle is completed, there will be a favorable opportunity to improve productivity or avoid wasting resources in the concerned sector. Therefore, this research is a step towards prioritizing needs and using resources in the investment and planning of managers, which is an inevitable necessity.
The correct performance of the method and its advantages were shown with numerical and practical examples. As seen from the results of solving the examples, the method is accurate and it corresponds to objective observations from intuitive comparisons and it has the ability to discriminate z-numbers.
Considering the great importance of ranking Z-numbers in today’s world, if a method is presented to help the decision makes for ranking in the shortest possible time with the most accuracy, also considering the special situations of z-numbers, if that method can simultaneously support all modes, a big step will be taken to improve ranking methods.
In this article, we mentioned two examples out of thousands of existing problems. In future articles, this method can be used for different problems that need ranking. For example, the proposed method can be used in “CN-q-ROFS”: connection number-based q-rung orthopair fuzzy set that Garg introduced in [20] and also expanding the proposed method to intuitionistic fuzzy set that Atanassov introduced in [21] will be valuable.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this article.
Authors’ Contributions
M. Afshar kermani, T. Allahviranloo, and A.Aminataei devised the project, provided the main conceptual ideas and proof outline, and verified the analytical methods. M. Farzam developed the theory and performed the computations and wrote the manuscript with support from M. Afshar kermani and T. Allahviranloo. M. Afshar kermani encouraged M. Farzam to investigate and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript. A preprint has previously been published (Maryam Farzam et al., 2021) by Research Square [22].