INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 18, Issue: 2, Month: April, Year: 2023 Article Number: 5299, https://doi.org/10.15837/ijccc.2023.2.5299 CCC Publications A new belief entropy and its application in software risk analysis X.Y. Chen, Y. Deng Xingyuan Chen* School of Information Engineer Kunming University, Kunming, China *Corresponding author: chenxingyuan@kmu.edu.cn; stargarden.chen@hotmail.com Yong Deng** Institute of Fundamental and Frontier Science University of Electronic Science and Technology of China, Chengdu, China **Corresponding author: dengentropy@uestc.edu.cn; prof.deng@hotmail.com Abstract The measurement of uncertainty has been an important topic of research. In Dempster’s frame- work, Deng entropy serves as a reliable tool for such measurements. However, it fails to consider more comprehensive information, resulting in the loss of critical data. An improved belief entropy is proposed in this paper, which preserves all the merits of Deng entropy. When there is only a single element, it can be degraded to Shannon entropy. When dealing with multiple elements, the partitioning method employed for mass functions makes it more responsive and efficient than alternative measures of uncertainty. Some numerical examples are given to further illustrate the effectiveness and applicability of the proposed entropy measure. Additionally, a case study is con- ducted on software risk analysis, demonstrating the practical value and relevance of the proposed method in real-world scenarios. Keywords: Dempster-Shafer evidence theory, Deng entropy, Uncertainty measure, Belief en- tropy, Software risk analysis 1 Introduction Uncertainty exists everywhere in our daily life. Due to the large amount of useful messages con- tained in uncertainty information, uncertainty information processing is widely used in many fields. It plays an important role in the target recognition [13, 16], complex network [31], artificial intelli- gence [17, 23], fault diagnosis [40], decision-making [51] and risk analysis [22], etc. In uncertainty information processing, it is necessary to measure uncertainty effectively and reasonably. Therefore, uncertainty measurement (UM) is widely studied as a hot topic. Several methodologies have been proposed, such as Dempster-Shafer evidence theory [5, 33], Bayesian probability theory [27], possi- bility theory [11], rough sets [30], Shannon entropy [34], fuzzy set [6, 50], and so on [29]. https://doi.org/10.15837/ijccc.2023.2.5299 2 Within these approaches, Dempster-Shafer evidence theory (DST) is highly regarded because it needs fewer conditions and better deal with uncertainty than probability theory. It includes frame of discernment (FOD), basic probability assignment (BPA) and Dempster’s rules of combination for data fusion. It provides a powerful method for the expression and fusion of uncertainty information [10, 36, 42]. DST is therefore widely applied in many fields [7, 43, 52]. Shannon entropy occupies a central position in information theory. It solves the problem of quantitative measurement of information and has received a lot of attention in practical applications [38, 41, 48]. Nevertheless, Shannon entropy is only useful in the probabilistic framework, which is why many scholars have made a lot of attempts to introduce entropy into evidence theory. Such as confusion measure [15], Hartley entropy [12], discord measure [21], strife measure [20], total conflict measure [14], dissonance measure [46] and so on. However, these methods are not ideal for mea- suring the uncertainty of BPA in some situations. For measuring uncertainty, dissonance and non- specificity are the two main aspects. These methods simply take either discord or non-specificity into account. In terms of discord, confusion measure, and strife measure were employed, while in terms of non-specificity, Dubois and Prade¡¯s weighted Hartley entropy was used. Therefore, how to combine both discordance and non-specificity measurement uncertainty of BPA is a question worth investigating. Deng entropy [8] was proposed, which is more effective in some cases. First, both dissonance and non-specificity were considered by Deng entropy. Secondly, as a promotion of Shannon entropy, Deng entropy extends the probability to the mass function. Third, it is more reasonable that the Deng entropy increases monotonically with the size of the proposition. The greater the number of focus elements of the FOD, the higher the entropy. Due to its advantages, it is widely used in several fields [4, 18, 25]. However, Deng entropy did not consider the most possible states when splitting the mass function. There are limitations when it comes to dealing with certain issues. If all possible states are not taken into account, then the information is incomplete and lost. A novel measure of belief entropy founded on Deng entropy is proposed. The proposed entropy inherits all the benefits of Deng entropy. At the same time, the way the mass function splits makes the proposed entropy more sensitive and effective than other uncertainty measures. Some examples and a risk assessment case demonstrate the effectiveness of the new entropy for uncertainty mea- surement. The structure of this paper is as follows: An summary of the Dempster-Shafer evidence theory, Shannon entropy, and Deng entropy principles is given in 2. Section 2 provides a brief overview of the concepts of Dempster-Shafer evidence theory, Shannon entropy, and Deng entropy. Section 3 proposes a new measure of belief entropy based on Deng en- tropy and provides several numerical examples. Section 4 illustrates an application of the proposed belief entropy to demonstrate its validity. Finally, the paper concludes in Section 5. 2 Preliminaries In this section, some essential basics are briefly introduced. 2.1 Dempster-Shafer Evidence Theory D-S evidence theory, also called as the dempster-Shafer evidence theory (DST) [5, 33], is an exten- sion of Bayesian theory and is a potent tool for handling uncertain information. Some fundamental concepts are outlined below. The set X represents the framework of discernment (FOD), which contains all possible mutually exclusive and exhaustive hypotheses [33], defined as follows: X = {θ1, θ2, ..., θn} (1) The power set 2X contains all possible subsets of X, as defined below: 2X = {∅, {θ1}, {θ2}, ..., {θn}, {θ1, θ2}, ..., {θ1, θn}, ..., {θ1, ..., θn}} (2) https://doi.org/10.15837/ijccc.2023.2.5299 3 where ∅ is an empty set. For FOD X, the mass function can also be referred to as Basic Belief Assignment (BBA) or Basic Probability Assignment (BPA). It is mapping from 2X to [0, 1], and description is as follows: m : 2X → [0, 1] (3) constrained conditions as follows, { ∑ A∈2X m(A) = 1 m(∅) = 0 (4) A is called the focal element when m(A) > 0. m(A) indicates the belief value that supports the proposition A. There are some studies about mass function [9, 44, 47], which has been used in many fields [2, 24]. For A ⊆ X, the belief function Bel : 2X → [0, 1] is following: Bel(A) = ∑ B⊆A m(B) (5) The plausibility function Pl : 2X → [0, 1] is defined as, Pl(A) = 1 − Bel(Ā) = ∑ B ⋂ A ̸=∅ m(B) (6) Apparently, ∀A ⊆ X, Bel(A) ≤ m(A) ≤ Pl(A). Where Bel(A)is the lower bound of proposition A and Pl(A) is the upper bound of Proposition A. [Bel(A), Pl(A)] indicates the confidence interval of A. Considering two bpa, the Dempster combination rule is denoted as[33]: m(A) = { 1 1−k ∑ B ⋂ C=A m1(B)m2(C), A ̸= ∅ 0, A = ∅ (7) where A, B, C ∈ 2X , k is a normalization factor, k = ∑ B ⋂ C=∅ m1(B)m2(C) (8) The conflict coefficient, denoted by k, is a measure of the degree of conflict between two basic belief assignments (BPAs) m1 and m2. A value of k = 0 indicates that m1 is consistent with m2, while a value of k = 1 signifies that m1 and m2 are in total contradiction. However, when combining highly contradictory evidence, a counterintuitive conclusion may arise from Dempster’s combination rule [45, 49]. Numerous research are devoted to solving the problem [39]. 2.2 Shannon Entropy Shannon entropy, also called information entropy, holds significant role in the field of information theory. Shannon entropy is defined as [34], Es(m) = − N ∑ i=1 Pi logb Pi (9) where N represents the number of basic states, while Pi is the probability associated with state i. The probabilities Pi must satisfy the condition N ∑ i=1 Pi = 1. When the unit of information is the bit, then b = 2. Shannon entropy can be defined as: Es(m) = − N ∑ i=1 Pi log2 Pi (10) Within the framework of probability theory, Shannon entropy has proven to be a successful ap- proach for measuring information uncertainty. Nevertheless, the measurement of information uncer- tainty within the DST framework is also a subject worth studying. https://doi.org/10.15837/ijccc.2023.2.5299 4 2.3 Uncertainty Measure in the DST Framework Table 1 lists several measures of uncertainty in the DST framework. Table 1: Some uncertainty measures Literature Expression [15] Ch(m) = − ∑ A⊆X m(A)log2 Bel(A) [46] Ey(m) = − ∑ A⊆X m(A)log2 Pl(A) [12] Ed p(m) = − ∑ A⊆X m(A)log2|A| [21] Dkr(m) = − ∑ A⊆X m(A)log2 ∑ B⊆X m(B) |A∩B||B| [20] Sk p(m) = − ∑ A⊆X m(A)log2 ∑ B⊆X m(B) |A∩B||A| [14] TCg p(m) = − ∑ A⊆X m(A)log2 ∑ B⊆X m(B)[1 − |A∩B||A∪B|] 2.4 Deng Entropy The Deng entropy is an appropriate metric for the BPA in the DST. It can be described as [8], Ed(m) = − ∑ A⊆X m(A)log2 m(A) 2|A| − 1 (11) Equation (11) defines the Deng entropy, where m is a BPA defined on FOD, and A is a focal element of m, with |A| representing the cardinality of A. Through a simple transformation, the equation can be expressed by: Ed(m) = ∑ A⊆X m(A)log2(2 |A| − 1) − ∑ A⊆X m(A)log2(m(A)) (12) It can be divided into two components: ∑ A⊆X m(A)log(2|A| − 1), which measures nonspecificity, and − ∑ A⊆X m(A)logm(A), which measures discord. As an extension of Shannon entropy, Deng entropy is formally similar to it. The difference is that the belief of each focal element A is divided by the number of possible states in A, which is 2|A|−1. If BPA degenerates to probability, Deng entropy can deteriorate to the Shannon entropy. Ed(m) = − ∑ A⊆2X m(A)log2 m(A) 2|A| − 1 = − ∑ A⊆2X m(A)log2(m(A)) (13) Some of the uncertainty measures within the framework of Deng entropy can be found in [4, 26, 28, 53]. 3 The Proposed Belief Entropy This section presents an approach to measuring uncertainty in DST. Specifically, Section 3.1 in- troduces a novel belief entropy that builds upon Deng entropy. In Section 3.2, the efficacy of the proposed method is illustrated through numerical examples and a comprehensive discussion. https://doi.org/10.15837/ijccc.2023.2.5299 5 3.1 The Proposed Belief Entropy En(m) = − ∑ A⊆X m(A)log2 m(A) ∑i(2|Ai| − 1) (14) For each i = 1, 2, ..., 2X − 1, where X is FOD, and m is a BPA, A is a focal element, the belief value of A is normalized by dividing it with a term ∑i(2 |Ai| − 1), where |A| indicates the cardinality of A. This term represents the total number of possible states that A can assume.s Assuming there are two balls inside a box, all the possible state are displayed in Fig. 1. Let FOD X = {a, b}, the power set is A = {∅, {a}, {b}, {a, b}}. The most possible state in A is obtained by calculating ∑i(2 |Ai| − 1) = (20 − 1) + (21 − 1) + (21 − 1) + (22 − 1) = 0 + 1 + 1 + 3 = 5. The figure 3 represents the total number of states that comprise components belonging to both {a} and {b}, namely, {a}, {b}, and {a, b}. Alternatively, FOD = X = {a, b, c }, then ∑i(2 |Ai| − 1) = 19, as depicted in Fig. 2. Figure 1: All the possible states when there are two balls inside a box Figure 2: All the possible states when there are three balls inside a box One simple conversion is as follows: En(m) = ∑ A⊆X m(A)log2(∑ i 2|Ai| − 1) − ∑ A⊆X m(A)log2m(A) (15) where the first parameter measures the total non-specificity, whereas the second one measure of discord. An example is presented. Example 1. Suppose X = {A, B, C}, and the following are the two BPAs, m1 : m1({A}) = 0.6, m1({A, B}) = 0.4 m2 : m2({B}) = 0.6, m2({X}) = 0.4 According to Eq. (14), the calculations are as follows, https://doi.org/10.15837/ijccc.2023.2.5299 6 En(m1) = −0.6 × log2 0.6 21 − 1 − 0.4 × log2 0.4 (21 − 1) + (21 − 1) + (22 − 1) = 1.8997 En(m2) = −0.6 × log2 0.6 21 − 1 − 0.4 × log2 0.4 3 × (21 − 1) + 3 × (21 − 1) + (23 − 1) = 2.6701 It is shown that the entropy of m2 is larger than that of m1. which means the uncertainty of m2 is greater. 3.2 Numerical Examples and Discussion A. Probabilistic consistency When there is only a single element in a BPA or |A| ≡ 1, the proposed entropy can be calculated as follows, En(m) = − ∑ A⊆X m(A)log2 m(A) ∑i 2|Ai| − 1 = − ∑ A⊆X m(A)log2 m(A) 21 − 1 = − ∑ A⊆X m(A)log2 m(A) En(m) = Ed(m) = Es(m) Example 2. Suppose FOD X={A} and m3 (A)=1, Shannon entropy (Es), Deng entropy (Ed) and the new entropy (En) are calculated according to (10), Eq. (11) and Eq. (14), Es(m3) = −1 × log21 = 0 Ed(m3) = −1 × log2 1 21 − 1 = 0 En(m3) = −1 × log2 1 21 − 1 = 0 As shown in Example 2, it can be seen that m3 assigns a belief of one hundred percent on the proposition A, which means there is no uncertainty in an information system. It is clear from the results that all entropies are equal to 0, which is reasonable. Example 3. Suppose that in FOD X = A, B, C, D, the BPAs are ascribed as below, m5 : m5(A) = m5(B) = m5(C) = m5(D) = 1 4 In this example, A, B, C, D have the same belief values with the same level of support. (Es), (Ed) and (En) can be calculated as below, Es(m5) = − 1 4 × log2 1 4 − 1 4 × log2 1 4 − 1 4 × log2 1 4 − 1 4 × log2 1 4 = 2 Ed(m5) = − 1 4 × log2 1 4 21 − 1 − 1 4 × log2 1 4 21 − 1 − 1 4 × log2 1 4 21 − 1 − 1 4 × log2 1 4 21 − 1 = 2 En(m5) = − 1 4 × log2 1 4 21 − 1 − 1 4 × log2 1 4 21 − 1 − 1 4 × log2 1 4 21 − 1 − 1 4 × log2 1 4 21 − 1 = 2 https://doi.org/10.15837/ijccc.2023.2.5299 7 From the above results in Example 3, the proposed belief entropy is the same with both Shannon entropy and Deng entropy if the mass value is only assigned one element. B. Superiority Example 4. Suppose X = {1, 2, ..., 14, 15} with 15 elements, and the following are the BPAs [8], m({3, 4, 5}) = 0.05, m(6) = 0.05, m(A) = 0.8, m(X) = 0.1 The range of propositions denoted by the symbol A spans from 1 to 14. Table 2 presents the results obtained from various uncertainty metrics applied to both new entropy and Deng entropy, while Fig. 3 provides a visual comparison of the findings. Notably, the new belief entropy grows monotonically with the increase in the size of A, which is the same as Deng entropy. Nevertheless, a key distinction arises in that the proposed entropy significantly surpasses Deng entropy. The reason is that more available information contained in BPA is taken into account in the proposed entropy. Table 2: Results of a variable number of elements in A Cases Deng entropy The new entropy A={1} 2.6623 3.6114 A={1, 2} 3.9303 5.3286 A={1, 2, 3} 4.9082 6.86941 A={1, ..., 4} 5.7878 8.2890 A={1, ..., 5} 6.6256 9.6479 A={1, ..., 6} 7.4441 10.9728 A={1, ..., 7} 8.2532 12.2773 A={1, ..., 8} 9.0578 13.5689 A={1, ..., 9} 9.8600 14.8524 A={1, ..., 10} 10.6612 16.1306 A={1, ..., 11} 11.4617 17.4053 A={1, ..., 12} 12.2620 18.6778 A={1, ..., 13} 13.0622 19.9487 A={1, ..., 14} 13.8622 21.2187 Figure 3: Comparison between the new entropy and Deng entropy Furthermore, to provide a more comprehensive comparison, six additional uncertainty measures are introduced and depicted in Fig. 4. With the increasing number of elements in proposition A, measures can be divided into three different categories. The first category entails a constant level of uncertainty, exemplified by Hohle’s confusion measure. The second category is characterized by a decreasing level of uncertainty, which includes Klir & Ramer’s discord measure, Yager’s dissonance, Klir & Parviz’s strife measure, and George & Pal’s conflict measure. Notably, both categories one and two appear counterintuitive as they disregard nonspecific uncertainty and only account for dis- cordant uncertainty. In contrast, the third category, which encompasses Dubois & Prade’s weighted https://doi.org/10.15837/ijccc.2023.2.5299 8 Hartley entropy, Deng entropy, and the proposed entropy, exhibits a corresponding increase in un- certainty as A expands. However, Dubois & Prade’s weighted Hartley entropy exclusively considers non-specificity while ignoring discord uncertainty, a perspective that is equally unsound. Ultimately, the proposed entropy outperforms other uncertainty measures by offering a more reasonable and ef- fective approach. Figure 4: Comparison between the new entropy and other uncertainty measures 4 Applicaiton Numerous techniques have been proposed to determine the weight of software risk [32, 35, 37]. This paper presents a straightforward risk evaluation model that utilizes the proposed entropy. As we know, it is crucial to establish a rational weight for risk evaluation. The proposed entropy is used to measure the uncertainty of risk. The uncertainty of risk increases with entropy, so the weight decreases with entropy. Whereas, the larger the weight, the smaller the risk uncertainty and the lower the entropy. 4.1 A Simple Software Risk Assessment Framework Based On The New Belief Entropy Software risk encompasses two important aspects. The probability of risk occurrence and the severity of loss are affected. In accordance with Boehm’s seminal work on software engineering [1], risk may be explicated as follows: Risk = Probability(P) × Severity(S) (16) As shown in the Fig. 5, software risk assessment can include the following steps, Step 1. Expression of risk assessment. Step 1.1. Establishing assessment criteria. Since most experts and decision-makers (DMs) usually use linguistic information such as low, high, very high, etc to assess risk. In this paper, we provided the following evaluation criteria, as shown in Table 3, and some explanations are given in Table 4 and 5. Notably, the criteria are not fixed, and can be adjusted according to the actual situation. Step 1.2. Assign and translate assessments into BPA. https://doi.org/10.15837/ijccc.2023.2.5299 9 Figure 5: software risk assessment framework Table 3: Linguistic terms Level Linguistic terms −2 Very low −1 Low 0 Medium 1 High 2 Very high Table 4: The probability of risk Linguistic terms Possibility Very low The consequences are negligible Low The consequences are not overly severe Medium The consequences are moderate High The consequences are serious Very high The consequences are extremely severe Table 5: The severity of risk Linguistic terms Severity Very low The consequences are negligible. Low The consequences are not too severe. Medium The consequences are moderate High The consequences are serious Very high The consequences are extremely severe https://doi.org/10.15837/ijccc.2023.2.5299 10 N experts are invited to assess both the probability and severity of risk using the criteria pre- sented in Table 3. For example {p−1(0.6), p−1 p0(0.4)} means the probability of "low" is 60% and the probability of hesitation between "low" and "medium" is 40%. Then the evaluation results should be translate into BPA. {p−1(0.6), p−1 p0(0.4)} is equivalent to m1({p−1}) = 0.6, m2({p−1, p0}) = 0.4. If {s−1(0.5), s0(0.5)} means the severity of "low" is 50% and "medium" is also 50%. Translating the evaluation results into BPA is m1({s−1}) = 0.5, m2({s0}) = 0.5. Step 2. Integrating Risk Information. For each risk, the assessment values given by all experts are fused using the Dempster’s combi- nation rule. Step 3. Calculating risk weights WR. Step 3.1. Calculating the uncertainty of the probability and severity. According to Eq. (17), use the new entropy to calculate the uncertainty of the probability En(P) and the uncertainty of the severity En(S). En(R) = En(P) + En(S) (17) Step 3.2. Calculating the credit value for each risk. According to Eq. (18), CredR are calculated as following, CredR = En(R) 1 + En(R) (18) Step 3.3. Calculating the weight for each risk According to Eq. (19), we are calculated as following, wR = CredR ∑ny=1 CredR (19) Step 4. Ranking of the risk. Step 4.1. Calculate the value the probability and the severity. Table 6: Values of score BPA score BPA score m({p−2})/m({s−2}) 09 m({p−2, p−1})/m({s−2, s−1}) 1 9 m({p−1})/m({s−1}) 29 m({p−1, p0})/m({s−1, s0}) 3 9 m({p0})/m({s0}) 49 m({p0, p1})/m({s0, s1}) 5 9 m({p1})/m({s1}) 69 m({p1, p2})/m({s1, s2}) 7 9 m({p2})/m({s2}) 89 m({pX}/m({sX}) 9 9 For example, a severity of risk value after fusion is as follows: (m{s0}) = 0.2079, m({s0, s1}) = 0.1386, m({s1}) = 0.5941, m({s1, s2}) = 0.0594. According to Table 6, the value of S is calculated as: S = 4 9 × m({s0}) + 5 9 × m({s0, s1}) + 6 9 × m({s1}) + 7 9 × m({s1, s2}) = 4 9 × 0.2079 + 5 9 × 0.1386 + 6 9 × 0.5941 + 7 9 × 0.0594 = 0.6117 Step 4.2. Calculate the risk using the Eq. (20) and proceed with ranking the results. Risk = wR × P × S (20) https://doi.org/10.15837/ijccc.2023.2.5299 11 4.2 A Case Study In this section, we will give a case study. There are six risk factors in a software project. And three experts are invented to the risks. Step 1. Based on the assessment criteria in Table 3, three experts gave their assessment and converted into BPAs. The results are shown in Table 7. Table 7: Experts assignment Risks Experts P S RS1 m1(.) {p−1 p0(0.6), p0 p1(0.4)} {s−1(0.5), s0(0.5)} m2(.) {p−1(0.4), p0 p1(0.25)} {s0s1(0.8)} m3(.) {p−1(0.65), p1(0.35)} {s0(0.7)} RS2 m1(.) {p−1(0.7), p0(0.2)} {s0(0.4), s1s2(0.6)} m2(.) {p−1 p0(0.6), p0 p1(0.4)} {s1(0.65)} m3(.) {p0(0.5)} {s0(0.6)} RS3 m1(.) {p−2(0.3), p−2 p−1(0.3), p−1(0.4)} {s0s1(0.7), s1s2(0.3)} m2(.) {p−1(0.6), p−1 p0(0.3)} {s0(0.6)} m3(.) {p−1 p0(0.7)} {s1(0.75)} RS4 m1(.) {p−1(0.7)} {s1s2(0.8)} m2(.) {p0(0.9)} {s1(0.75)} m3(.) {p−1(0.4), p0 p1(0.25)} {s0s1(0.6), s1s2(0.4)} RS5 m1(.) {p0(0.5), p0 p1(0.4)} {s−1(0.5), s0(0.5)} m2(.) {p0(0.5)} {s0s1(0.8)} m3(.) {p−1 p0(0.7), p0 p1(0.3)} {s−1s0(0.6), s1(0.4)} RS6 m1(.) {p1(0.5), p1 p2(0.4)} {s−1s0(0.8), s1(0.2)} m2(.) {p0 p1(0.8)} {s0(0.65)} m3(.) {p0(0.6), p1(0.25)} {s0s1(0.9)} Step 2. After using Dempster’s combination rule, the fusion results of P and S are shown in Table 8. Step 3. Based on Eq. (17), Eq. (18) and Eq. (19) the weights of risks are w1 = 0.1704, w2 = 0.1870, w3 = 0.1613, w4 = 0.1879, w5 = 0.1403, w6 = 0.1531. Step 4. Based on Eq. (20), the risk values and rankings are shown in Table 9. We can easily obtain that RS4 > RS6 > RS2 > RS1 > RS5 > RS3. The RS4 is the highest risk and it should be focused on. While RS6 is the second highest one, and also needs to be given attention. Figure 6: Comparison of risk values In Table (9) and Fig 6, we can see that the results will be different by using the proposed method. And if we just use P*S, the risk values and rankings are different. This is because the uncertainty of https://doi.org/10.15837/ijccc.2023.2.5299 12 Ta b le 8: F u si on R es u lt s R is k s P S R S1 {p − 1 (0 .5 82 1) ,p 0 (0 .2 50 7) ,p 0 p 1 (0 .1 67 2) } {s 0 (0 .7 69 2) ,s 1 (0 .2 30 9) } R S2 {p − 1 (0 .4 11 8) ,p − 1 p 0 (0 .0 58 8) ,p 0 (0 .4 90 2) ,p 0 p 1 (0 .0 39 2) } {s 0 (0 .3 68 4) ,s 1 (0 .4 10 5) ,s 1 s 2 (0 .2 21 1) } R S3 {p − 2 (0 .0 12 7) ,p − 2 p − 1 (0 .0 12 7) ,p − 1 (0 .9 74 6) } {s 0 s 1 (0 .2 07 9) ,s 0 s 1 (0 .1 38 6) ,s 1 (0 .5 94 1) ,s 1 s 2 (0 .0 59 4) } R S4 {p − 1 (0 .2 63 8) ,p 0 (0 .6 62 6) ,p 0 p 1 (0 .0 30 7) ,p x (0 .0 42 9) } {s 0 (0 .3 75 ), s 0 s 1 (0 .0 93 8) ,s 1 (0 .4 68 8) ,s 1 s 2 (0 .0 62 5) } R S5 {p − 1 p 0 (0 .0 35 ), p 0 (0 .8 9) ,p 0 p 1 (0 .0 75 )} {s − 1 (0 .1 66 7) ,s 0 (0 .8 33 3) } R S6 {p 0 (0 .1 30 4) ,p 0 p 1 (0 .0 26 1) ,p 1 (0 .8 10 9) ,p 1 p 2 (0 .0 26 1) ,p x (0 .0 06 5) } {s − 1 s 0 (0 .0 32 2) ,s 0 (0 .8 87 4) ,s 1 (0 .0 80 5) } https://doi.org/10.15837/ijccc.2023.2.5299 13 Table 9: Risk values and rankings Risks P*S Proposed method P*S Rating Proposed method Rating RS1 0.1654 0.0282 5 4 RS2 0.2137 0.0400 3 3 RS3 0.1333 0.0215 6 6 RS4 0.2395 0.0450 2 1 RS5 0.1829 0.0257 4 5 RS6 0.2936 0.0449 1 2 the expert assessment value is measured using the proposed entropy. More information is consid- ered. Also, avoiding subjective bias of experts and assigning weights from an objective point of view, it has a more fair and reasonable character. To prove the validity of the proposed entropy, we also made some other comparisons. We anal- ysed the example in Ref. [3]. We replaced the Deng entropy with the proposed entropy and calculated the uncertainty. The final rankings of the risks are the same as in Ref. [3]. Therefore, the proposed entropy is effective. 5 Conclusion Measure uncertainty in the DST framework is still an open issue. In this paper, a new belief entropy based Deng entropy has been proposed. In the DST framework, the performance of uncer- tainty measures can be improved when using the new entropy. The new entropy can identify the uncertainty more effectively and provides a new way of the splitting of the mass function thinking about measuring uncertainty. The experimental results illustrate that the new entropy we proposed is better than other methods. A case study on risk assessment further highlights the effectiveness of the proposed entropy in practical applications. However, it is worth noting that the new entropy may not fully satisfy all the characteristics proposed by Klir and Wierman [19]. Whether or not it is necessary in Dempster-Shafer theory is still a topic for discussion. In the future, based on the new belief entropy, we can do more research. According to the main ideas of BPA divisions in this paper, more divisions methods can be designed. Besides more applica- tion could be use this entropy. Acknowledgment The work is partially supported by National Natural Science Foundation of China (Grant No. 61973332), JSPS Invitational Fellowships for Research in Japan (Short-term). References [1] Boehm, B.W., 1991. Software risk management: principles and practices. IEEE software 8, 32–41. [2] Chen, L., Deng, Y., Cheong, K.H., 2021. Probability transformation of mass function: A weighted network method based on the ordered visibility graph. Engineering Applications of Artificial Intelligence 105, 104438. doi:https://doi.org/10.1016/j.engappai.2021.104438. [3] Chen, X., Deng, Y., 2022. An evidential software risk evaluation model. Mathematics 10, 2325. [4] Cui, H., Liu, Q., Zhang, J., Kang, B., 2019. An improved deng entropy and its application in pattern recognition. IEEE Access 7, 18284–18292. [5] Dempster, A.P., 2008. Upper and lower probabilities induced by a multivalued mapping, in: Classic works of the Dempster-Shafer theory of belief functions. Springer, pp. 57–72. http://dx.doi.org/https://doi.org/10.1016/j.engappai.2021.104438 https://doi.org/10.15837/ijccc.2023.2.5299 14 [6] Deng, J., Deng, Y., 2021. Information volume of fuzzy membership function. International Journal of Computers Communications & Control 16, 4106. doi:https://doi.org/10.15837/ ijccc.2021.1.4106. [7] Deng, X., Jiang, W., 2020. On the negation of a dempster–shafer belief structure based on maxi- mum uncertainty allocation. Information Sciences 516, 346–352. [8] Deng, Y., 2016. Deng entropy. Chaos, Solitons & Fractals 91, 549–553. [9] Deng, Y., 2020a. Information volume of mass function. International Journal of Computers Communications & Control 15, 3983. doi:https://doi.org/10.15837/ijccc.2020.6.3983. [10] Deng, Y., 2020b. Uncertainty measure in evidence theory. SCIENCE CHINA Information Sci- ences 63, 210201. [11] Dubois, D., 2006. Possibility theory and statistical reasoning. Computational statistics & data analysis 51, 47–69. [12] Dubois, D., Prade, H., 1985. A note on measures of specificity for fuzzy sets. International Journal of General System 10, 279–283. [13] Gao, X., Deng, Y., 2019. The generalization negation of probability distribution and its appli- cation in target recognition based on sensor fusion. International Journal of Distributed Sensor Networks 15, 1550147719849381. [14] George, T., Pal, N.R., 1996. Quantification of conflict in dempster-shafer framework: a new approach. International Journal Of General System 24, 407–423. [15] Hohle, U., 1982. Entropy with respect to plausibility measures, in: Proc. of 12th IEEE Int. Symp. on Multiple Valued Logic, Paris, 1982. [16] Jishuang, Q., Chao, W., Zhengzhi, W., 2003. Structure-context based fuzzy neural network ap- proach for automatic target detection, in: IGARSS 2003. 2003 IEEE International Geoscience and Remote Sensing Symposium. Proceedings (IEEE Cat. No. 03CH37477), Ieee. pp. 767–769. [17] Kanal, L.N., Lemmer, J.F., 2014. Uncertainty in artificial intelligence. Elsevier. [18] Kazemi, M.R., Tahmasebi, S., Buono, F., Longobardi, M., 2021. Fractional deng entropy and extropy and some applications. Entropy 23, 623. [19] Klir, G., Wierman, M., 1999. Uncertainty-based information: elements of generalized informa- tion theory. volume 15. Springer Science & Business Media. [20] Klir, G.J., Parviz, B., 1992. A note on the measure of discord, in: Uncertainty in Artificial Intelli- gence, Elsevier. pp. 138–141. [21] Klir, G.J., Ramer, A., 1990. Uncertainty in the dempster-shafer theory: a critical re-examination. International Journal of General System 18, 155–166. [22] Levkina, R., Kravchuk, I., Sakhno, I., Kramarenko, K., Shevchenko, A., et al., 2019. The economic-mathematical model of risk analysis in agriculture in conditions of uncertainty. Fi- nancial and credit activity problems of theory and practice 3, 248–255. [23] Li, D., Du, Y., 2017. Artificial intelligence with uncertainty. CRC press. [24] Liang, H., Cai, R., 2021. A new correlation coefficient of bpa based on generalized information quality. International Journal of Intelligent Systems . [25] Liao, H., Ren, Z., Fang, R., 2020. A deng-entropy-based evidential reasoning approach for multi-expert multi-criterion decision-making with uncertainty. International Journal of Com- putational Intelligence Systems 13, 1281–1294. http://dx.doi.org/https://doi.org/10.15837/ijccc.2021.1.4106 http://dx.doi.org/https://doi.org/10.15837/ijccc.2021.1.4106 http://dx.doi.org/https://doi.org/10.15837/ijccc.2020.6.3983 https://doi.org/10.15837/ijccc.2023.2.5299 15 [26] Mambe, M.D., Tchimou N’Takp’e, T., Nogbou, G.A., Oumtanaga, S., et al., 2018. A new un- certainty measure in belief entropy framework. International Journal of Advanced Computer Science and Applications 9. [27] Olshausen, B.A., 2004. Bayesian probability theory. The Redwood Center for Theoretical Neuro- science, Helen Wills Neuroscience Institute at the University of California at Berkeley, Berkeley, CA . [28] Pan, L., Deng, Y., 2018. A new belief entropy to measure uncertainty of basic probability assign- ments based on belief function and plausibility function. Entropy 20, 842. [29] Pan, Y., Zhang, L., Li, Z., Ding, L., 2019. Improved fuzzy bayesian network-based risk analysis with interval-valued fuzzy sets and d–s evidence theory. IEEE Transactions on Fuzzy Systems 28, 2063–2077. [30] Pawlak, Z., 1982. Rough sets. International journal of computer & information sciences 11, 341–356. [31] Raimondo, S., De Domenico, M., 2021. Measuring topological descriptors of complex networks under uncertainty. Physical Review E 103, 022311. [32] Sangaiah, A.K., Samuel, O.W., Li, X., Abdel-Basset, M., Wang, H., 2018. Towards an efficient risk assessment in software projects–fuzzy reinforcement paradigm. Computers & Electrical Engineering 71, 833–846. [33] Shafer, G., 1976. A mathematical theory of evidence. Princeton university press. [34] Shannon, C.E., 1948. A mathematical theory of communication. The Bell system technical jour- nal 27, 379–423. [35] Song, H., Wu, D., Li, M., Cai, C., Li, J., 2010. An entropy based approach for software risk as- sessment: A perspective of trustworthiness enhancement, in: The 2nd International Conference on Software Engineering and Data Mining, IEEE. pp. 575–578. [36] Song, Y., Wang, X., Zhu, J., Lei, L., 2018. Sensor dynamic reliability evaluation based on evidence theory and intuitionistic fuzzy sets. Applied Intelligence 48, 3950–3962. [37] Suresh, K., Dillibabu, R., 2020. A novel fuzzy mechanism for risk assessment in software projects. Soft Computing 24, 1683–1705. [38] Vashishtha, G., Kumar, R., 2022. Pelton wheel bucket fault diagnosis using improved shan- non entropy and expectation maximization principal component analysis. Journal of Vibration Engineering & Technologies 10, 335–349. [39] Wang, L., Bao, Y., 2021. An improved method for multisensor high conflict data fusion. Journal of Sensors 2021. [40] Wang, T., Liu, W., Zhao, J., Guo, X., Terzija, V., 2020. A rough set-based bio-inspired fault diagnosis method for electrical substations. International Journal of Electrical Power & Energy Systems 119, 105961. [41] Wu, S., Fu, Y., Shen, H., Liu, F., 2018. Using ranked weights and shannon entropy to modify regional sustainable society index. Sustainable cities and society 41, 443–448. [42] Wu, X., Liao, H., 2021. An adaptive evidence combination method for decision analysis under uncertainty. Journal of the Operational Research Society , 1–15. [43] Xiao, F., 2020. Generalization of dempster–shafer theory: A complex mass function. Applied Intelligence 50, 3266–3275. https://doi.org/10.15837/ijccc.2023.2.5299 16 [44] Xiao, F., 2021. Ceqd: A complex mass function to predict interference effects. IEEE Transactions on Cybernetics . [45] Xiong, L., Su, X., Qian, H., 2021. Conflicting evidence combination from the perspective of networks. Information Sciences 580, 408–418. [46] Yager, R.R., 2008. Entropy and specificity in a mathematical theory of evidence, in: Classic works of the Dempster-Shafer theory of belief functions. Springer, pp. 291–310. [47] Yan, Z., Zhao, H., Mei, X., 2021. An improved conflicting-evidence combination method based on the redistribution of the basic probability assignment. Applied Intelligence , 1–27. [48] Yang, W., Xu, K., Lian, J., Ma, C., Bin, L., 2018. Integrated flood vulnerability assessment ap- proach based on topsis and shannon entropy methods. Ecological Indicators 89, 269–280. [49] Zadeh, L.A., 1979. On the validity of Dempster’s rule of combination of evidence. Infinite Study. [50] Zadeh, L.A., 1996. Fuzzy sets, in: Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh. World Scientific, pp. 394–432. [51] Zhang, S., Xiao, F., 2023. A tfn-based uncertainty modeling method in complex evidence theory for decision making. Information Sciences 619, 193–207. [52] Zhao, K., Li, L., Chen, Z., Sun, R., Yuan, G., Li, J., 2022. A survey: Optimization and applica- tions of evidence fusion algorithm based on dempster-shafer theory. Applied Soft Computing , 109075. [53] Zhou, D., Tang, Y., Jiang, W., 2017. An improved belief entropy and its application in decision- making. Complexity 2017, 1–15. Copyright ©2023 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 Cite this paper as: Chen, X.Y.; Deng, Y. (2023). A new belief entropy and its application in software risk analysis, International Journal of Computers Communications & Control, 18(2), 5299, 2023. https://doi.org/10.15837/ijccc.2023.2.5299 Introduction Preliminaries Dempster-Shafer Evidence Theory Shannon Entropy Uncertainty Measure in the DST Framework Deng Entropy The Proposed Belief Entropy The Proposed Belief Entropy Numerical Examples and Discussion Applicaiton A Simple Software Risk Assessment Framework Based On The New Belief Entropy A Case Study Conclusion