INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 17, Issue: 4, Month: August, Year: 2022 Article Number: 4862, https://doi.org/10.15837/ijccc.2022.4.4862 CCC Publications A Modified Uncertainty Measure of Z-numbers Yangxue Li, Francisco Javier Cabrerizo, Enrique Herrera-Viedma, Juan Antonio Morente-Molinera Yangxue Li Andalusian Research Institute in Data Science and Computational Intelligence, Dept. of Computer Science and AI, University of Granada, 18071 Granada, Spain. liyangxue@correo.ugr.es Francisco Javier Cabrerizo Andalusian Research Institute in Data Science and Computational Intelligence, Dept. of Computer Science and AI, University of Granada, 18071 Granada, Spain. cabrerizo@decsai.ugr.es Enrique Herrera-Viedma* Andalusian Research Institute in Data Science and Computational Intelligence, Dept. of Computer Science and AI, University of Granada, 18071 Granada, Spain. *Corresponding author: viedma@decsai.ugr.es Juan Antonio Morente-Molinera* Andalusian Research Institute in Data Science and Computational Intelligence, Dept. of Computer Science and AI, University of Granada, 18071 Granada, Spain. *Corresponding author: jamoren@decsai.ugr.es Abstract The Z-number is a more adequate construct for describing real-life information. While consider- ing the uncertainty of the information, it also models the partial reliability of the information. It is a combination of probabilistric restriction and possibilistric restriction. In this paper, we modified the uncertainty measurement of the discrete Z-number and proposed the uncertainty measurement of the continuous Z-number. Some numerical examples are used to illustrate the calculation pro- cesses and advantages of the proposed method. An application of journey vehicle selection shows the effectiveness of the proposed uncertainty measurement in determining the weights of criteria. Keywords: Uncertainty, discrete Z-numbers, continuous Z-numbers, fuzzy numbers, decision- making. https://doi.org/10.15837/ijccc.2022.4.4862 2 1 Instruction Crisp numbers often cannot describe real-life information, it is insufficient. As the information is often uncertain, there are many mathematical models to handle such uncertain information, prob- ability theory [18], fuzzy sets theory [29], Dempster-Shafer evidence theory [11, 22] and other soft constraints. However, the information also is partially reliable. In the above soft constraints, they default the handled information is completely reliable. Unfortunately, the information in practice is not so. The Z-number, as a more sufficient formal construct for describing real-life information, models the uncertainty and partial reliability of the information together. A Z-number is an ordered pair of fuzzy numbers, Z = (A,B), where A is a fuzzy constrict on the values of the variable, X, can take, B is also a fuzzy constrict on the reliability value of A. The reliability value is a probability of the event ’X is A’. Thus, a Z-number combines the possibilistic restriction and the probabilistric restriction. Since the thought of Z-numbers was put forward in 2011 to now, the mathematical system and application of Z-numbers have become more and more complete. In applications of Z-number, The most widespread application areas are decision-making, such as selection problems [2, 10], fault diag- nosis [13], pattern recognition [25], risk assessment [1]. The Z-number has been especially useful for describing calculations in natural language. For instance, Ref. [20] analyzed features and challenges of the Z-number approach to Computing With Words (CWW). In Ref. [9], Z-number is used as a tool for CWW and consequently subjective natural language understanding. In the mathematical system of Z-number, the basic computations were summarized in Ref. [8], including addition, subtraction, mul- tiplication, division, square, square root, absolute value, ranking and distance of Z-numbers. On this basis, more advanced arithmetic is proposed, including Z-numbers based Linear Program (Z-LP) [4], Z-numbers function [6], Approximate Reasoning [7], the parametric form [21], the negation operator [17], total utility [14], Z-Differential equations [19], multidimensional Z-numbers [23], Z-number if- then rules [5], soft likelihood function [24], Z-mixture-numbers [27]. Ref. [3, 26] defined the concepts of specificity and horizontal membership functions of Z-numbers. Besides, Li et. al. proposed an uncertainty measure of discrete Z-number [16]. However, we find there is an unintuitive point in this uncertainty measurement. When the second fuzzy number of the Z-number becomes sharper, the reliability value of the first fuzzy number is more certain, the uncertainty of the Z-number should reduce. However, in the method of Li et. al. [16], the uncertainty increases slowly. In addition, there is no approach to measure the uncertainty of continuous Z-number. In this paper, we modified the uncertainty measure of the discrete Z-number. The proposed measurement conforms to three intuitions about the uncertainty of Z-numbers: (1) When the first fuzzy number becomes shaper, the value of the variable is more accurate, the uncertainty of Z-number should be smaller. (2) When the second fuzzy number becomes sharper, the reliability value of the first fuzzy number is more certain, the uncertainty of Z-number should be smaller. (3) When the first fuzzy number is more reliable, the uncertainty of Z-number should be smaller. We also proposed an analytical solution of the uncertainty of continuous Z-number. If the analytical solution is difficult to obtain, we provided an appropriate method and set the precision to control the calculation consumption and accuracy of the result. This paper is structured as follows. Section 2 lists the concepts of Z-numbers and other related backgrounds. In section 3, we amend the measuring the uncertainty of discrete Z-numbers. And the uncertainty measurement of continuous Z-numbers is proposed. Some numerical examples are used to demonstrate the advantages of the proposed methodology in Section 4. In Section 5, an application of journey vehicle selection shows the usefulness of the proposed uncertainty measurement in determining the weights of criteria. Finally, this paper is concluded in Section 5. https://doi.org/10.15837/ijccc.2022.4.4862 3 2 Preliminaries In this section, some preliminaries about definitions of possibilistric restriction, probabilistric re- striction, Z-numbers and Z-restriction are discussed. Definition 1. A restriction Rprob(X) on real-valued uncertain variable X is a probabilistric restriction if Rprob(X) : X is P, (1) if X is a continuous variable, P is called the probability density function of X, Prob(u ≤ X ≤ u + du) = ∫ P(u)du. (2) If X is a discrete variable, P is called the probability distribution of X. Prob(X = u) = P(X = u). (3) Definition 2. A possibilistric restriction R)poss(X) on variable X is denoted as Rposs(X) : X is A, (4) where A is a fuzzy number in universe U with membership function µA : U → [0, 1]. The membership degree of a base value u ∈ U can be seen as the possibility value of ’X = u’. More explicitly, Rposs(X) : Poss(X = u) = µA(u) (5) Definition 3. A Z-number is an ordered pair of fuzzy numbers, Z = (A,B). Where A is a possibilistric restriction on values of variable X can take with membership function µA(u) : X → [0, 1], B is another possibilistric restriction on reliability of A with membership function µB(v) : [0, 1] → [0, 1] [28]. v is a base value of B and can be seen as the reliability value. Definition 4. A Z-restriction combine a possibilistric restriction and a series of probabilistric restric- tions [28], RZ(X) : X is Z → Prob(X is A) is B, (6) where Prob(X is A) is the probability of the event ’X is A’ occur and can be expressed as: Prob(X is A) = ∫ R µA(u)pX(u)du. (7) where pX is the underlying probability distribution of X and is unknown. If A and pX are matching, then the centroids of µA is same with the expected value of pX,∫ upX(u)du = ∫ uµA(u)du∫ µA(u)du (8) 3 Uncertainty measure of Z-numbers In this section, we propose a novel uncertainty measure of discrete and continuous Z-numbers. We try to find the underlying probability distribution for reliability value, and then combine the underlying probability distributions with their corresponding possibilities. 3.1 Uncertainty measure of discrete Z-numbers For a discrete Z-number Z = (A,B), A =< u,µA(u) >, B =< v,µB(v) >, the underlying probability distribution PvX(u) of reliability v satisfy the following three conditions: (1) ∑n i=1 P v X(ui) = 1, https://doi.org/10.15837/ijccc.2022.4.4862 4 (2) ∑n i=1 P v X(ui)µA(ui) = v, (3) ∑n i=1 uiP v X(ui) = ∑n i=1 uiµA(ui)∑n i=1 µA(ui) , where {u1,u2, · · · ,un} is the base values of fuzzy number A, v ∈ {v1,v2, · · · ,vm} is a base value of fuzzy number B. Most of time, we may obtain more than one underlying probability distribution according to three conditions. The maximum entropy method [12] is used to select one with the maximum Shannon entropy. Then the calculation of underlying probability distribution is an optimization problem: max : H(PvX) = − n∑ k=1 PvX log(P v X), (9) subject to: n∑ i=1 PvX(ui) = 1, n∑ i=1 PvX(ui)µA(ui) = v, n∑ i=1 uiP v X(ui) = ∑n i=1 uiµA(ui)∑n i=1 µA(ui) We obtained all underlying probability distributions of reliabilities {v1,v2, · · · ,vm}, the uncertainty of Z is calculated by HZ(Z) = m∑ i=1 H(PviX )µB(vi) (10) 3.2 Uncertainty measure of continuous Z-numbers For a continuous Z-number Z = (A,B), A =< u,µA(u) >, B =< v,µB(v) >, we assume the underlying probability density function is a normal distribution: PvX(u) = N(α,β 2) = 1 √ 2πσ2 exp(− (u−α)2 2β2 ), (11) subject to: ∫ PvX(u)du = 1,∫ PvX(u)µA(u)du = v,∫ uPvX(u)du = α = ∫ uµA(u)du∫ µA(u)du According to Equation (11), the obtained underlying probability is a function about variable u and v. More precisely, its β is a function about variable v. Then calculate the uncertainty of this normal probability density function by: H(PvX(u)) = − ∫ PvX(u)log(P v X(u))du = 1 2 (log(2πβ2) + 1) (12) where β is a function about variable v. The uncertainty of Z is HZ(Z) = ∫ H(PvX(u))µB(v)dv = ∫ 1 2 (log(2πβ2) + 1)µB(v)dv (13) https://doi.org/10.15837/ijccc.2022.4.4862 5 4 Numerical example In this section, some numerical examples are used to show the calculation process and the advan- tages of the proposed method. In which, Example 1 and Example 2 show the uncertainty calculation processes of discrete Z-numbers and continuous Z-numbers in detail. Example 3, Example 4 and Example 5 verify the proposed measurement satisfies the three intuitions and compare the proposed method and the measurement in [16]. Example 1. Given a discrete Z-number Z = (A,B), the membership functions of A and B are A = µA u = 0 1 + 0.3 2 + 0.5 3 + 0.7 4 + 1 5 + 0.7 6 + 0.5 7 + 0.3 8 + 0 9 , B = µB v = 0 0.1 + 0.3 0.2 + 0.5 0.3 + 0.7 0.4 + 1 0.5 + 0.7 0.6 + 0.5 0.7 + 0.3 0.8 + 0 0.9 . Probability distributions corresponding to different reliability value for Z and their entropy are list as Table 1. The uncertainty of Z is v pX(u) u1 = 1 u2 = 2 u3 = 3 u4 = 4 u5 = 5 u6 = 6 u7 = 7 u8 = 8 u9 = 9 Entropy v1 = 0.1 0.3724 0.0838 0.0310 0.0115 0.0026 0.0115 0.0310 0.0838 0.3724 1.4849 v2 = 0.2 0.2759 0.1156 0.0647 0.0362 0.0152 0.0362 0.0647 0.1156 0.2759 1.8678 v3 = 0.3 0.1978 0.1240 0.0908 0.0665 0.0417 0.0665 0.0908 0.1240 0.1978 2.0875 v4 = 0.4 0.1348 0.1175 0.1072 0.0978 0.0853 0.0978 0.1072 0.1175 0.1348 2.1871 v5 = 0.5 0.0851 0.1008 0.1129 0.1264 0.1497 0.1264 0.1129 0.1008 0.0851 2.1815 v6 = 0.6 0.0478 0.0776 0.1071 0.1478 0.2395 0.1478 0.1071 0.0776 0.0478 2.0733 v7 = 0.7 0.0223 0.0514 0.0896 0.1564 0.3605 0.1564 0.0896 0.0514 0.0223 1.8552 v8 = 0.8 0.0073 0.0263 0.0617 0.1447 0.5199 0.1447 0.0617 0.0263 0.0073 1.5067 v9 = 0.9 0.0010 0.0073 0.0271 0.1011 0.7270 0.1011 0.0271 0.0073 0.0010 0.9765 Table 1: Probability distributions corresponding to different reliability value for Z. HZ(Z) = 1.4849 × 0 + 1.8678 × 0.3 + 2.0875 × 0.5 + 2.1871 × 0.7 + 2.1815 × 1 + 2.0733 × 0.7 + 1.8552 × 0.5 + 1.5067 × 0.3 + 0.9765 × 0 = 8.1475 Example 2. Given a continuous Z-number A = (A,B), where A and B are two triangular fuzzy number A = (2, 5, 8), B = (0.2, 0.5, 0.8). The membership functions of A and B are: µA(u) =   u− 2 3 2 < u ≤ 5, 8 −u 3 5 < u ≤ 8, 0, otherwise. µB(v) =   10u− 2 3 0.2 < u ≤ 0.5, 8 − 10u 3 0.5 < u ≤ 0.8, 0, otherwise. According to Equation (11), The analytical solutions of underlying probability density functions is very hard to obtain in practice. Thus, we use the approximate method. First set a precision pre ∈ [0.0001, 0.1], we assume the underlying probability density function is same in interval [v,v + pre]. Then calculate the underlying probability density functions at points · · · ,v,v + pre,v + 2pre, · · ·. The results as shown in Table 2. The precision is pre = 0.1. The uncertainty of Z is https://doi.org/10.15837/ijccc.2022.4.4862 6 v1 = 0.2 v2 = 0.3 v3 = 0.4 v4 = 0.5 v5 = 0.6 v6 = 0.7 v7 = 0.8 β 3.0079 2.6319 2.2559 1.8799 1.5039 1.1279 0.7519 H(pX) 2.5201 2.3866 2.2324 2.0501 1.8270 1.5392 1.1337 Table 2: The value of β and entropy under different reliability. HZ(Z) = m−1∑ i=1 ∫ vi+pre vi H(pviX) + H(p vi+pre X ) 2 µB(v)dv = 0.0408 + 0.1154 + 0.1784 + 0.1615 + 0.0842 + 0.0223 = 0.6026 Example 3. The first intuition is: when the membership function of A is sharper, the uncertainty of Z should be smaller. In Example 1, when the membership degrees of u2 and u8 reduce to 0, and then the membership degrees of u3 and u7 reduce to 0, and then the membership degrees of u4 and u6 reduce to 0. The graphical representations of changes are shown in Figure 1. The changes of the uncertainty of Z computed by the proposed approach and the measurement in [16] are shown in Figure 2. 1 2 3 4 5 6 7 8 9 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m e m b e rs h ip f u n c ti o n o f A Figure 1: The change of membership function of fuzzy number A. As shown in Figure 2, the results of the two methods are consistent with our intuition. Example 4. The second intuition is: when the membership function of B is sharper, the uncertainty of Z should be smaller. In Example 1, when the membership degrees of v2 and v8 reduce to 0, and then the membership degrees of v3 and v7 reduce to 0, and then the membership degrees of v4 and v6 reduce to 0. The graphical representations of changes are shown in Figure 3. The changes of the uncertainty of Z computed by the proposed approach and the measurement in [16] are shown in Figure 4. As shown in Figure 4, the uncertainty calculated by method in [16] increases slowly as the second fuzzy number becomes sharper. The uncertainty calculated by the proposed measurement reduce sharply. it is consistent with our intuition. Example 5. When the first fuzzy number is more reliable, the uncertainty of the Z-number is smaller. Given a discrete Z-number Z1 = (A1,B1), where A1 = A, the membership function B1 is B1 = µB v = 0 0.2 + 0.3 0.3 + 0.5 0.4 + 0.7 0.5 + 1 0.6 + 0.7 0.7 + 0.5 0.8 + 0.3 0.9 + 0 1 . HZ(Z1) = 7.6651 < HZ(Z). https://doi.org/10.15837/ijccc.2022.4.4862 7 u2,u8=0.2 u2,u8=0 u3,u7=0.3 u3,u7=0.1 u4,u6=0.6 u4,u6=0.4 u4,u6=0.2 u4,u6=0 1 2 3 4 5 6 7 8 9 U n c e rt a in ty o f Z The proposed method The method of Li et.al. [16] Figure 2: The change of uncertainty of Z as A changes. 1 2 3 4 5 6 7 8 9 v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m e m b e rs h ip f u n c ti o n o f B Figure 3: The change of membership function of fuzzy number B. 5 Application In this section, the weights of criteria in a decision-making problem are determined by the proposed uncertainty measurement. There are three different options of vehicle for journey, car (a1), taxi (a2) and train (a3). The set of alternatives is denoted as A = {a1,a2,a3}. Three criteria, expense (c1), time consumption (c2) and comfort (c3) are took into account. The set of criteria is denoted as C = {c1,c2,c3}. The Z-information decision matrix is shown Table 3. The uncertainty of Z-numbers in decision matrix are displayed in Table 4 (the precision is 0.01). https://doi.org/10.15837/ijccc.2022.4.4862 8 v2,v8=0.2 v2,v8=0 v3,v7=0.3 v3,v7=0.1 v4,v6=0.6 v4,v6=0.4 v4,v6=0.2 v4,v6=0 2 3 4 5 6 7 8 9 U n c e rt a in ty o f Z The proposed metho The method of Li et.al. [16] Figure 4: The change of uncertainty of Z as B changes. c1 c2 c3 a1 ((8, 10, 12), (0.83, 1, 1)) ((70, 100, 120), (0, 0.17, 0.33)) ((3, 5, 7), (0.33, 0.5, 0.67)) a2 ((20, 24, 25), (0, 0, 0.17)) ((40, 70, 100), ((0.33, 0.5, 0.67)) ((5, 8, 11), (0.67, 0.83, 1)) a3 ((14, 15, 16), (0.83, 1, 1)) ((60, 80, 100), (0.67, 0.83, 1)) ((1, 4, 7), (0.67, 0.83, 1)) Table 3: The decision matrix with Z-information. The sum of uncertainty of criteria ’Price’ is Hsum(c1) = 0.1378 + 0.2491 − 0.1967 = −0.0854. The sum of the uncertainty of criteria ’Journey time’ is Hsum(c2) = 0.7719 + 0.7408 + 0.4514 = 1.9641. The sum of the uncertainty of criteria ’Comfort’ is Hsum(c3) = 0.2788 + 0.1389 + 0.1389 = 0.5566. AS there are negative numbers, we add a constant: Hnor(cj) = Hsum(c1) + 0.5,j = 1, 2, 3. Then the weight of criteria ’Price’ is wc1 = 1/(−0.0854 + 0.5) 1/(−0.0854 + 0.5) + 1/(1.9641 + 0.5) + 1/(0.5566 + 0.5) = 0.6408. c1 c2 c3 a1 −0.1378 0.7719 0.2788 a2 0.2491 0.7408 0.1389 a3 −0.1967 0.4514 0.1389 Table 4: The uncertainty of Z-numbers. https://doi.org/10.15837/ijccc.2022.4.4862 9 c1 c2 c3 a1 ((0.6667, 0.8, 1), (0.83, 1, 1)) ((0.3333, 0.4, 0.5714), (0, 0.17, 0.33)) ((0.2727, 0.4545, 0.6363), (0.33, 0.5, 0.67)) a2 ((0.32, 0.3333, 0.4), (0, 0, 0.17)) ((0.4, 0.5714, 1), ((0.33, 0.5, 0.67)) ((0.4545, 0.7272, 1), (0.67, 0.83, 1)) a3 ((0.5, 0.5333, 0.5714), (0.83, 1, 1)) ((0.4, 0.5, 0.6666), (0.67, 0.83, 1)) ((0.0909, 0.3636, 0.6363), (0.67, 0.83, 1)) Table 5: The normalized decision matrix. c1 c2 c3 a1 0.7985 0.1775 0.3214 a2 0.0835 0.4646 0.5311 a3 0.5195 0.4767 0.2655 Table 6: The crisp numbers of Z-numbers in decision matrix. The weight of criteria ’Journey time’ is wc1 = 1/(1.9641 + 0.5) 1/(−0.0854 + 0.5) + 1/(1.9641 + 0.5) + 1/(0.5566 + 0.5) = 0.1078. The weight of criteria ’Comfort’ is wc1 = 1/(0.5566 + 0.5) 1/(−0.0854 + 0.5) + 1/(1.9641 + 0.5) + 1/(0.5566 + 0.5) = 0.2514. The scales of criteria are different. In order to compare these Z-numbers, the first fuzzy numbers must be normolized. The results are summarized in Table 5. Then we used the converting methods to converse the Z-numbers to crisp numbers [15], the decision matrix with crisp numbers is shown in Table 6. The evalution value of alternative a1 is: <(a1) = 0.7985 ∗ 0.6408 + 0.1775 ∗ 0.1078 + 0.3214 ∗ 0.2514 = 0.6116. The evalution value of alternative a2 is: <(a2) = 0.0835 ∗ 0.6408 + 0.4646 ∗ 0.1078 + 0.5311 ∗ 0.2514 = 0.2371. And the evalution value of alternative a2 is: <(a3) = 0.5195 ∗ 0.6408 + 0.4767 ∗ 0.1078 + 0.2655 ∗ 0.2514 = 0.4510. The ranking of alternatives is a1 > a3 > a2. The best option of journey is a1, car. 6 Conclusion Z-number is a new framework to address uncertain and partial reliable information. This paper modifies the uncertainty measure of discrete and continuous Z-numbers, overcomes the shortcoming of the previous method. The proposed measurement satisfies all intuitions about the uncertainty of Z-numbers. Finally, some numerical examples and an application illustrate the advantages and effectiveness of the proposed method. Acknowledgements This work was supported by the project B-TIC-590-UGR20 co-funded by the Programa Operativo FEDER 2014-2020 and the Regional Ministry of Economy, Knowledge, Enterprise and Universities (CECEU) of Andalusia, by the Andalusian Government through the project P20_00673, by the China Scholarship Council (CSC), and by the project PID2019-103880RB-I00 funded by MCIN / AEI / 10.13039/501100011033. https://doi.org/10.15837/ijccc.2022.4.4862 10 References [1] Abbaspour Onari, Mohsen, Yousefi, Samuel, and Jahangoshai Rezaee, Mustafa (2021). 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Information Sciences, 181(14), 2923–2932, 2011. [29] Zimmermann, Hans-Jürgen (2011). Fuzzy set theory—and its applications. Springer Science & Business Media, 2011. Copyright ©2022 by the authors. Licensee Agora University, Oradea, Romania. This is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International License. Journal’s webpage: http://univagora.ro/jour/index.php/ijccc/ This journal is a member of, and subscribes to the principles of, the Committee on Publication Ethics (COPE). https://publicationethics.org/members/international-journal-computers-communications-and-control https://doi.org/10.15837/ijccc.2022.4.4862 12 Cite this paper as: Li Y.; Cabrerizo F.J.; Herrera-Viedma E.; Morente-Molinera J.A. (2022). A Modified Uncertainty Measure of Z-numbers, International Journal of Computers Communications & Control, 17(4), 4862, 2022. https://doi.org/10.15837/ijccc.2022.4.4862 Instruction Preliminaries Uncertainty measure of Z-numbers Uncertainty measure of discrete Z-numbers Uncertainty measure of continuous Z-numbers Numerical example Application Conclusion