INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL ISSN 1841-9836, 13(5), 792-807, October 2018. Generalized Ordered Propositions Fusion Based on Belief Entropy Y. Li, Y. Deng Yangxue Li, Yong Deng* Institute of Fundamental and Frontier Science University of Electronic Science and Technology of China, Chengdu *Corresponding author: dengentropy@uestc.edu.cn Abstract: A set of ordered propositions describe the di�erent intensities of a char- acteristic of an object, the intensities increase or decrease gradually. A basic support function is a set of truth-values of ordered propositions, it includes the determinate part and indeterminate part. The indeterminate part of a basic support function indicates uncertainty about all ordered propositions. In this paper, we propose gen- eralized ordered propositions by extending the basic support function for power set of ordered propositions. We also present the entropy which is a measure of uncertainty of a basic support function based on belief entropy. The fusion method of general- ized ordered proposition also be presented. The generalized ordered propositions will be degenerated as the classical ordered propositions in that when the truth-values of non-single subsets of ordered propositions are zero. Some numerical examples are used to illustrate the e�ciency of generalized ordered propositions and their fusion. Keywords: Dempster-Shafer evidence theory, ordered proposition, uncertainty mea- sure, belief entropy, information fusion. 1 Introduction In resent year, with the intensi�cation of competition in the modern information war, infor- mation technology has developed rapidly, and the amount of information has increased explo- sively. Thus, as the critical technologies for information collection, storage and processing, the essentiality of information modeling and fusion has gradually increased. There are many methods to model information, such as probability theory [10], Dempster- Shafer evidence theory [7,23], rough sets [40], fuzzy sets [6,7,9,19,19,22�24,24], Z-numbers [17, 37,37], D numbers [1,5,76] and as so on. A specialized fusion algorithm is used for each method. Ordered proposition is a new approach to model information which is proposed by Liu et al. [35]. A set of ordered propositions describe the di�erent intensities of a characteristic of a objects, the intensities increase or decrease gradually. For example, consumers evaluate the quality of a product on a rank of "Wonderful, Good, Indi�erent, Weak". A set of ordered propositions can be expressed as a basic support function (similar to belief function in Dempster-Shafer evidence theory), whose elements represent the truth-value (belief value) of each proposition. The truth- values of a basic support function must be convex, because a subject cannot be two degrees in same characteristic. Such as, we cannot say the quality of a product is both wonderful and indi�erence simultaneously. A basic support function is divided into determinate part and indeterminate part [35]. The determinate part is the sum of truth-values of each ordered proposition. The sum of indeter- minate part and determinate part is one. In the ordered propositions fusion, the indeterminate part is prorated to each proposition and itself. Therefore, the indeterminate part can express the uncertainty for all ordered propositions. In this paper, we de�ne the generalized ordered propositions, they extend the indeterminate part to all non-single subsets of ordered proposi- tions. The truth-value of a non-single subset expresses the uncertainty of the propositions in Copyright ©2018 CC BY-NC Generalized Ordered Propositions Fusion Based on Belief Entropy 793 it. For example, the "Wonderful, Good" express "the quality of this product is wonderful" or "the quality of this product is good". In order to ensure the convex property of a basic support function, the indeterminate part is listed separately. The generalized ordered propositions will be degenerated as the classical ordered propositions in that when the truth-values of non-single subsets of ordered propositions are zero. The ordered propositions fusion is an important and extensive problem [35]. Previously, a fusion algorithm based on centroid is proposed [42], which fuse the basic support functions of two sets of ordered propositions and ensure the convexity. However, this approach has a few shortages [35]. In order to address these shortages, a new fusion method based on consistency and uncertainty measurements was presented by Liu et al. for the fusion of ordered proposition [35]. They also introduced entropy to measure the uncertainty of the basic support function based on Shannon entropy [35]. But this entropy only considered the determinate part of a basic support function, the indeterminate part is ignored. In Dempster-Shafer evidence theory, an entropy is presented to measure the uncertainty of a belief function, named Deng entropy [6]. When we add the groups of propositions in ordered propositions, the basic support function is more similar with the belief function. In this paper, we introduce a new entropy to measure the uncertainty of a basic support function based on belief entropy. It will be degenerated as the entropy which is proposed by Liu et al. in that when the indeterminate part of a basic support function is zero. Additionally, the fusion method of generalized ordered propositions based on consistency and uncertainty measurements is introduced. When the truth-values of non-single subsets of ordered propositions are zero, the fusion result is same as the fusion result of Liu et al.'s method. The rest part of this paper is organized as follows. Section 2 brie�y discusses the de�nitions and properties of ordered propositions, Dempster-Shafer evidence theory and belief entropy. Section 3 introduces the de�nition and properties of generalized ordered propositions. Section 4 discusses the proposed method for measuring uncertainty of a basic support function. The fusion method of generalized ordered proposition is described in Section 5. Section 6 presents some numerical examples. Finally, this paper is concluded in Section 7. 2 Preliminaries 2.1 Ordered propositions In this section, some background knowledge about ordered propositions is brie�y intro- duced [35]. De�nition 1 (Ordered propositions). For a set of propositions p1,p2, · · · ,pn, the truth-value of pi is denoted as λ(pi). λ(pk) = max{λ(p1), · · · ,λ(pn)}. p1,p2, · · · ,pn are ordered propositions if [35] (1) ∀i = 1, 2, · · · ,n, all subjects described in pi are S; (2) ∀i = 1, 2, · · · ,n, si describes the same characteristics or features of S; (3) ∀i = 1, 2, · · · ,k − 1, λ(pi) ≤ λ(pi+1); and ∀i = k,k + 1, · · · ,n− 1, λ(pi) ≥ λ(pi+1). De�nition 2 (Basic support function). For a set of ordered propositions P = {p1,p2, · · · ,pn}, a function λ is called the basic support function of the ordered propositions if [35] (1) λ is de�ned on {P}∪{{pi}|1 ≤ i ≤ n}, where P indicates indeterminacy; (2) λ(pi) ≥ 0, 1 ≤ i ≤ n; 794 Y. Li, Y. Deng (3) ∑ 1≤i≤n λ(pi) ≤ 1; (4) λ(P) = 1 − ∑ 1≤i≤n λ(pi). De�nition 3 (Determinate part and indeterminate part). For a basic support function λ, the determinate part λ(P) and indeterminate part λ(P) are de�ned as [35] λ(S) = ∑ i=1,··· ,n λ(pi), λ(P) = 1 −λ(P). (1) De�nition 4 (Mean value). The mean value of a basic support function λ is de�ned as [35] λ = ∑n i=1 λ(pi) n . (2) De�nition 5 (Measure of convexity). The measure of convexity of a basic support function λ is de�ned as [35] convex(λ) = max{λ(p1),λ(p2), · · · ,λ(pn)}−λ. (3) It was clear that the maximum of the measure of convexity is 1 −λ. Thus, the normalized convex(λ) as follows: [35] NC(λ) = (max{λ(p1),λ(p2), · · · ,λ(pn)}−λ)/(1 −λ). (4) De�nition 6 (Center of a basic support function). For a basic support function λ = (λ(p1),λ(p2), · · · , λ(pn)), the center of λ is de�ned as [35] CI(λ) =   argmaxi=1,··· ,nλ(pi), NC(λ) ≥ θ∑ i=1,··· ,n∧λ(pi)≥τ·λ λ(pi) × i∑ i=1,··· ,n∧λ(pi)≥τ·λ λ(pi) , otherwise, (5) θ is set to 0.55 in [35], 1 < τ ≤ 1.5. In order to model the complex information of interaction, complex networks are proposed [4, 20,21,40,65,69]. The measure of consistency is essential to information, a�ected by the reliability of the information source [9, 11, 17, 32, 46, 64, 66, 74]. The reliability of obtaining data is very important for information fusion [41]. De�nition 7 (Measure of consistency). If CI(λ1) and CI(λ2) are the centers of the basic support functions λ1 and λ2.The consistency between λ1 and λ2 is de�ned as [35] ∆G(λ1,λ2) = |CI(λ1) −CI(λ2)|/(n− 1). (6) If ∆G = 1, then λ1 and λ2 are totally con�icting. If ∆G = 0, then λ1 and λ2 are consistent. Otherwise, if 0 < ∆G < 1, then λ1 and λ2 are partially con�icting. The consistency between λ1 and λ2 can be divided into 3 degrees [35]. 0 ≤ ∆G ≤ δ1 indicates the consistency between λ1 and λ2 is high. δ1 ≤ ∆G ≤ δ2 indicates the consistency between λ1 and λ2 is medium. δ2 ≤ ∆G ≤ 1 indicates the consistency between λ1 and λ2 is poor. Generalized Ordered Propositions Fusion Based on Belief Entropy 795 2.2 Dempster-Shafer evidence theory Evidence theory is widely used in many applications such as target recognition [29,30], deci- sion making [1,11], uncertain processing [3,13,16,16,20,21,26�28,31,35], risk management [18,36], fault diagnosis [4,15,25,56,60] and as so on. The frame of discernment Θ is the exhaustive hy- potheses of variable, X. Θ = {x1,x2, · · · ,xi, · · · ,xn}. The power set of Θ is 2Θ = {∅,{x1}, · · · ,{xn},{x1,x2}, · · · , {x1,x2, · · · ,xi}, · · · , Θ}, where ∅ is an empty set [7,23]. De�nition 8 (Basic probability assignment (BPA)). A basic probability assignment function m : 2Θ → [0, 1], which satis�es [7,23]: m(Θ) = 0 ∑ A∈2Θ m(A) = 1 0 ≤ m(A) ≤ 1, (7) the mass m(A) indicates how strongly the evidence supports A. 2.3 Belief entropy Shannon entropy is widely used to measure the uncertainty of a probability. In addition, a belief entropy named Deng entropy is proposed to measure the uncertainty of a BPA [6]. De�nition 9 (Belief entropy). For a BPA, m, de�ned on the frame of discernment Θ, it's belief entropy is de�ned as [6] Ed(m) = − ∑ A⊆Θ m(A) ln m(A) 2|A| − 1 , (8) where A is the focal element of m, |A| is the cardinality of A. 3 Generalized ordered propositions 3.1 De�nitions De�nition 10 (Generalized ordered propositions). For a set of propositions p1,p2, · · · ,pn, it's power set, {∅,{p1},{p2}, · · · ,{pn},{p1,p2}, · · · ,{p1, · · · ,pn}}, let λ(pi,pj · · ·) represent the truth-value of {pi,pj, · · ·} and λ(pk) = max{λ(p1), · · · ,λ(pn)}. p1,p2, · · · ,pn are generalized ordered propositions, if (1)∀i = 1, 2, · · · ,n, all subjects described in pi are S; (2)∀i = 1, 2, · · · ,n, pi describes the same characteristics or features of S; (3)∀i = 1, 2, · · · ,m− 1, λ(pi) ≤ λ(pi+1); and ∀i = m,m + 1, · · · ,n− 1, λ(pi) ≥ λ(pi+1). De�nition 11 (Basic support function of the generalized ordered propositions). For a set of gen- eralized ordered propositions P = {p1,p2, · · · ,pn}, it's power set 2P = {∅,{p1},{p2}, · · · ,{pn}, {p1,p2},{p1,p3}, · · · ,{p1,p2, · · · ,pn}} a function λ is called a basic support function of the gen- eralized ordered propositions if (1) λ is de�ned on 2P ; (2) λ(A) ≥ 0,A ⊆ P ; (3) λ(∅) = 0; (4) ∑ 1≤i≤n λ(A) = 1, where A ⊆ P ; 796 Y. Li, Y. Deng Take the example of "the quality of a product", the basic support function is{(0.1, 0.3, 0.2, 0.0), (λ(p1,p2) = 0.2,λ(p2,p3) = 0.2)}. λ(p1) = 0.1 means the truth-value of 1st proposition "the quality of a product is wonderful" is 0.1; λ(p2) = 0.3 means the truth-value of 2nd proposition "the quality of a product is good" is 0.3; λ(p3) = 0.2 means the truth-value of 3rd proposition "the quality of a product is indi�erence" is 0.2; λ(p4) = 0.0 means the truth-value of 4rd proposition "the quality of a product is weak" is 0.0; λ(p1,p2) = 0.2 means the uncertain truth-value of 1st proposition and 2nd proposition is 0.2; λ(p2,p3) = 0.2 means the uncertain truth-value of 2st proposition and 3rd proposition is 0.2. 3.2 Properties De�nition 12 (Determinate part and indeterminate part). For a basic support function of generalized ordered proposition, the determinate part and indeterminate part is λ(P) = n∑ i=1 λ(pi), λ(P) = ∑ A⊆P∧A 6={q1},··· ,{qn} λ(A) = 1 −λ(P). (9) De�nition 13 (Mean value). The mean value of a basic support function λ of generalized ordered propositions is λ = ∑n i=1 λ(pi)(1 + ∑ pi⊂A λ(A)) n , (10) where A ( {p1,p2, · · · ,pn}. De�nition 14 (Degree of convexity). The degree of convexity of a basic support function λ of generalized ordered propositions is: convex(λ) = maxi=1,··· ,n{λ(pi)(1 + ∑ pi⊂A λ(A))}−λ,i = 1, 2, · · · ,n, (11) where A ( {p1,p2, · · · ,pn}. The normalized convex(λ) is NC(λ) = (maxi=1,··· ,n{λ(pi)(1 + ∑ pi⊂A λ(A))}−λ)/(1 −λ), i = 1, 2, · · · ,n. (12) De�nition 15 (Center of a basic support function). A basic support function of generalized or- dered propositions λ = {(λ(p1),λ(p2), · · · ,λ(pn)), (λ(p1,p2), · · · ,λ(p1,p2, · · · ,pn))}, the center of λ is CI(λ) =   argmaxi=1,··· ,nλ(pi)(1 + ∑ pi⊂A λ(A)), NC(λ) ≥ θ ∑ i=1,··· ,n∧λ(pi)≥τ·λ λ(pi)(1 + ∑ pi⊂A λ(A)) × i∑ i=1,··· ,n∧λ(pi)(1+ ∑ pi⊂A λ(A))≥τ·λ λ(pi)(1 + ∑ pi⊂A λ(A)) , otherwise, (13) where A ( {p1,p2, · · · ,pn}. Generalized Ordered Propositions Fusion Based on Belief Entropy 797 4 Uncertainty measure Uncertainty can evaluate the quality of information [2,3,13,15,16,31,32,39,39,47,48,50,61,71]. The more uncertainty, the less information [7,8]. A method to measure the uncertainty of a basic support function of ordered propositions based on Shannon entropy is proposed by Liu et al. [35]. De�nition 16 (Liu et al.'s entropy). For a basic support function λ = (λ(p1),λ(p2), · · · ,λ(pn)), λ 6= (λ(p1) = 0,λ(p2) = 0, · · · ,λ(pn) = 0) and n ≥ 2. Let λ(pk) = max{λ(p1),λ(p2), · · · ,λ(pn)}, 1 ≤ k ≤ n. If βλ(pk) ≤ λ(pj) ≤ λ(pk),β ≥ 0.9 and 1 ≤ j ≤ n, then λ(pj) is quasi-maxima. Let n′ is the total number of maxima and quasi-maxima. The Liu et al.'s entropy of λ is de�ned as: [35] E(λ) =   − n∑ i=1 λ(pi) ln λ(pi), n ′ = 1, − n∑ i=1 λ(pi) ln λ(pi) + (ln n + n∑ i=1 λ(pi) ln λ(pi)))(n ′/n)α, 2 ≤ n′ ≤ n, (14) where α = 0.1. When indeterminate part of a basic support function is equal to zero, this entropy can accu- rately measure the uncertainty of a basic support function. For example, given two basic sup- port functions µ1 = (0.005, 0.99, 0.005, 0.0, 0.0), µ2 = (0.0049995, 0.990001, 0.0049995, 0.0, 0.0), we can given E(µ1) = 0.062933 and E(µ2) = 0.062928 using Eq. (14). E(µ1) is greater than E(µ2), this means that the uncertainty of µ1 is higher than the uncertainty of µ2. The result is reasonable. When there are multiple maxima of a basic support function, Liu et al.'s method can also measure uncertainty accurately. Take two basic support functions µ3 = (0.5, 0.5, 0.0, 0.0), µ4 = (0.15, 0.7, 0.1, 0.05), then E(µ3) = 1.34 and E(µ4) = 0.914. It is reasonable that E(µ3) > E(µ4). However, when indetermination part of a basic support function is not equal to zero, this entropy doesn't apply to measure uncertainty of a basic support function. For example, for two basic support functions µ5 = (0.2, 0.3, 0.0, 0.0) and µ6 = (0.7, 0.1, 0.1, 0), then E(µ5) = 0.6831, E(µ6) = 0.7103. E(µ5) < E(µ6), this means that the degree of uncertainty of µ6 is higher. It is obviously counterintuitive. In order to take into considered not only the determinate part but also indeterminate part, we present the a new method to measure uncertainty of a basic support function of generalized ordered proposition based on belief entropy [1,6]. De�nition 17 (The entropy based on belief entropy). For a basic support function of generalized ordered propositions λ = {(λ(p1),λ(p2), · · · ,λ(pn)), (λ(p1,p2),λ(p1,p3), · · · ,λ(p1,p2, · · · ,pn))}, λ 6= (λ(p1) = 0,λ(p2) = 0, · · · ,λ(pn) = 0) and n ≥ 2. Let λ(pk) = max{λ(p1),λ(p2), · · · ,λ(pn)}, 1 ≤ k ≤ n. If βλ(pk) ≤ λ(pj) ≤ λ(pk),β ≥ 0.9 and 1 ≤ j ≤ n, then λ(qj) is quasi-maxima. Let n′ is the total number of maxima and quasi-maxima. The entropy of λ is de�ned as: Ed(λ) =   − n∑ i=1 λ(A) ln( λ(A) 2|A|−1 ), n′ = 1, − n∑ i=1 λ(A) ln( λ(A) 2|A|−1 ) + (ln n + λ(A) ln( λ(A) 2|A|−1 ))(n′/n)α, 2 ≤ n′ ≤ n, (15) where A ⊆{q1, 12, · · · ,qn}, |A| is the number of elements of A, α = 0.1. Using Eq.( 15) to calculate the uncertainty of µ5 and µ6, the results are Ed(µ5) = 2.3837, Ed(µ6) = 1.2114. Ed(µ5) > Ed(µ6), it is reasonable. For two basic support functions of general- ized ordered propositions µ7 = {(0.2, 0.5, 0.1, 0.0), (µ7(p1,p2) = 0, 1,µ7(p2,p3) = 0.1)} and µ8 = 798 Y. Li, Y. Deng {(0.2, 0.6, 0.1, 0.0), (µ8(p1,p2) = 0.1)}. The results are Ed(µ7 = 1.5790 and Ed(µ8) = 1.1988 using Eq. (12). Ed(µ7) > Ed(µ8), this means that the degree of uncertainty of µ7 is higher than µ8. 5 Fusion of generalized ordered propositions For a set of generalized ordered propositions P = {p1,p2, · · · ,pn}, let λ1 and λ2 are two basic support functions of P . Denote the fusion result of λ1 and λ2 is ω. The processes of method for fusion of basic support functions of generalized ordered propositions is shown in Fig. 5. The steps of this method can be explained as follows: Start λ1, λ2, Ω1, Ω2 . 1 ={ 1(p1,p2, ,pnYes 2 ={ 2(p1,p2, ,pn No 2 ={ 2(p1,p2, ,pn No Yes No Calculate the initial fusion result ω ʹ. Calculate the center CI(ω ʹ ). Calculate the consistency ΔG( ). CI(ω ʹ ) is Integer? Yes End Calculate the final fusion result with method . Calculate the final fusion result with method . No ω = 2. ω = 1.Yes Figure 1: The processes of proposed method Step 1: Give two basic support functions λ1, λ2 of a set of generalized ordered propositions P = {p1,p2, · · · ,pn}, and the weights Ω1, Ω2 of two basic support functions respectively. Generalized Ordered Propositions Fusion Based on Belief Entropy 799 Table 1: Process of calculating ω′ by Eq. (16). λ1 to ω′ λ2 to ω′ ω′ Truth-value obtained by ω′(p1) 0.025 0.08625 0.11125 Truth-value obtained by ω′(p2) 0.345 0.345 0.69 Truth-value obtained by ω′(p3) 0.08625 0.025 0.11125 Truth-value obtained by ω′(p4) 0.025 0.025 0.05 Truth-value obtained by ω′(p1,p2) 0 0.01875 0.01875 Truth-value obtained by ω′(p2,p3) 0.01875 0 0.01875 Step 2: Determine whether λ1 is equal to {λ1(p1,p2, · · · ,pn) = 1} and if λ2 is equal to {λ2(p1,p2, · · · ,pn) = 1}. If λ1 = {λ1(p1,p2, · · · ,pn) = 1} and λ2 = {λ2(p1,p2, · · · ,pn) = 1}, the fusion result ω = (1/n, 1/n, · · · , 1/n). If λ1 = {λ1(p1,p2, · · · ,pn) = 1} but λ2 6= {λ2(p1,p2, · · · ,pn) = 1}, the fusion result ω = λ2. If λ2 = {λ2(p1,p2, · · · ,pn) = 1} but λ1 6= {λ1(p1,p2, · · · ,pn) = 1}, the fusion result ω = λ1. If λ1 6= {λ1(p1,p2, · · · ,pn) = 1} and λ2 6= {λ2(p1,p2, · · · ,pn) = 1}, take the next step. Step 3: Calculate the initial fusion result. ω′(A) =   Ω1 ·λ1(A)(1 + ∑ A⊂B λ1(B)) + Ω2 ·λ1(A)(1 + ∑ A⊂C λ1(C)), |A| = 1, Ω1 ·λ1(A)(1 − ∑ pi⊂A λ1(pi)) + Ω2 ·λ2(A)(1 − ∑ pi⊂A λ1(pi)), 1 < |A| ≤ n, (16) where A,B,C ⊆{p1,p2, · · · ,pn}, i = 1, 2, · · · ,n, Ω1 + Ω2 = 1. For example, there are two basic support functions λ1 = {(0.05, 0.6, 0.15, 0.05), (λ1(p2,p3) = 0.15} and λ2 = {(0.15, 0.6, 0.05, 0.05), (λ2(p1,p2) = 0.15)}. The weights are Ω1 = Ω2 = 0.5. The process of calculating initial fusion result ω′ by using Eq. (16) is illustrated in Table 1. Step 4: Calculate the center of initial fusion result ω′ with Eq. (13), CI(ω′). Step 5: Calculate the consistency between λ1 and λ2 with Eq. (6), ∆G(λ1,λ2). Step 6: Determine whether the center of initial fusion result CI(ω′) is Integer. If CI(ω′) is Integer, take the step 7, otherwise take the step 8. Step 7: Calculate the �nal fusion result ω with method I. Step 7.1: Positive regulation. ω(pi) =   i∑ k=1 ω′(pk)[1 + ϕ(i−k)]∑CI(ω′)−k j=0 (1 + jϕ) , if i < CI(ω′), CI(ω)∑ k=1 ω′(pk)[1 + ϕ(i−k)]∑CI(ω′)−k j=0 (1 + jϕ) + n∑ k=CI(ω′)+1 ω′(pk)[1 + ϕ(k −CI(ω′))]∑k−CI(ω′) j=0 (1 + jϕ) if i = CI(ω′), n∑ k=i ω′(pk)[1 + ϕ(k − i)]∑k−CI(ω′) j=0 (1 + jϕ) if i > CI(ω′), ω(A) = ω′(A), (17) where ϕ = 0.2, 0.1, 0 when the consistency between two basic support function is high, medium, poor respectively, A is the non-simple subset of P . Step 7.2: Negative regulation. 800 Y. Li, Y. Deng When the consistency between two basic support functions is poor, the measure of uncertainty is used to compress the curve of truth-value of ω vertically until the entropy of ω approximately equals the entropy of ω′, that is |E(ω) −E(ω′)| ≤ �. This process is called negative regulation and outlined in Algorithm 1. Algorithm 1 The procedure of negative regulation. Input: The initial fusion result ω′ and basic support function ω after positive regulation Output: The �nal fusion result ω 1: δ ← 1 2: while |E(ω) −E(ω′)| ≤ � do 3: I ← index of maximum truth-value of ω 4: k ← 1 5: for k = I to n− 1 do 6: if ω(pk) > ω(pk+1) then 7: ω(pk) = ω(pk) − δω(pk+1)(ω(pk)−ω(pk+1)) ω(pk)+ω(pk+1) 8: ω(pk+1) = ω(pk+1) + δω(pk+1)(ω(pk)−ω(pk+1)) ω(pk)+ω(pk+1) 9: end if 10: end for 11: for k = I; k > 1; k −− do 12: if ω(pk) > ω(pk−1) then 13: ω(pk) = ω(pk) − δω(pk−1)(ω(pk)−ω(pk−1)) ω(pk)+ω(pk−1) 14: ω(pk−1) = ω(pk−1) + δω(pk−1)(ω(pk)−ω(pk−1)) ω(pk)+ω(pk−1) 15: end if 16: end for 17: if E(ω) < E(ω′) − � then 18: δ ← 1 19: end if 20: if E(ω) > E(ω′) + � then 21: δ ← δ/2 22: end if 23: end while Step 8: Calculate the �nal fusion result ω with method II. Step 8.1: Positive regulation. Denote a = dCI(ω)e, b = bCI(ω′)c for convenience, thus ω(pi) =   i∑ k=1 ω′(pk)[1 + ϕ(i−k)] (( ∑a−k j=0 (1 + jϕ)) −ϕ) , if i < b, ω′(pb) + Γ(a−CI(ω′)), If i = b∧ω′(pb) 6= ω′(pa), ω′(pa) + Γ(CI(ω ′) − b), If i = a∧ω′(pb) 6= ω′(pa), ω′(pb) + Γ/2, If i = b∧ω′(pb) = ω′(pa), ω′(pa) + Γ/2, If i = a∧ω′(pb) = ω′(pa), n∑ k=i ω′(pk)[1 + ϕ(k − i)] (( ∑k−b j=0(1 + jϕ)) −ϕ) , if i > a, ω(A) = ω′(A), (18) Generalized Ordered Propositions Fusion Based on Belief Entropy 801 Table 2: The fusion process and result of example (1) variables values λ1 {(0.1, 0.4, 0.2, 0.1), (λ1(p1,p2) = 0.1,λ1(p2,p3) = 0.1)} λ2 {(0.1, 0.4, 0.2, 0.1), (λ2(p1,p2) = 0.1,λ2(p2,p3) = 0.1)} λ1 = λ2 0.2275 NC(λ1) = NC(λ2) 0.3269 CI(λ1) = CI(λ2) 2.4615 ∆G(λ1,λ2) 0 ω′ {(0.11, 0.48, 0.22, 0.1), (ω′(p1,p2) = 0.05,ω′(p2,p3) = 0.04)} ω′ 0.2419 NC(ω′) 0.3711 CI(ω′) 2.3407 ω {(0.0324, 0.5777, 0.2705, 0.1482), (ω(p1,p2) = 0.05,ω(p2,p3) = 0.04)} where Γ = Γ1 + Γ2, Γ1 = b−1∑ k=1 ω′(pk)[1 + ϕ(b−k)] ( ∑a−k j=0 (1 + jϕ)) −ϕ + n∑ a+1 ω′(pk)[1 + ϕ(k −a)] ( ∑k−b j=0(1 + ϕj)) −ϕ , Γ2 = b−1∑ k=1 ω′(pk)[1 + ϕ(a−k) −ϕ] ( ∑a−k j=0 (1 + jϕ)) −ϕ + n∑ a+1 ω′(pk)[1 + ϕ(k −a)] ( ∑k−b j=0(1 + ϕj)) −ϕ , ϕ = 0.2, 0.1, 0 when the consistency between two basic support function is high, medium, poor respectively, A is the non-simple subset of P . Step 8.2: Negative regulation. It is same as Step 7.2. 6 Numerical examples (1) Two basic support functions are λ1 = {(0.1, 0.4, 0.2, 0.1), (λ1(p1,p2) = 0.1,λ1(p2,p3) = 0.1)} and λ2 = {(0.1, 0.4, 0.2, 0.1), (λ2(p1,p2) = 0.1,λ2(p2,p3) = 0.1)}. The weights of λ1 and λ2 are Ω1 = Ω2 = 0.5. The fusion processes and results are shown in Table 2. λ1 and λ2 are consistent, and they all mean that the 2nd proposition is most likely to be correct. So the fusing basic support function should reach the maximum truth-value at the index 2. The results are reasonable. (2) Two basic support functions are λ1 = (0, 0.1, 0.2, 0.7) and λ2 = {(0.1, 0.1, 0.1, 0.6), (λ2(p3,p4) = 0.1)}. The weights of λ1 and λ2 are Ω1 = Ω2 = 0.5. The fusion result is ω = {(0.0096, 0.0394, 0.1172, 0.8188), (ω(p3,p4) = 0.015)}. λ1 and λ2 are not exactly the same, but NC(λ1) = 0.6 > 0.55 and NC(λ2) = 0.5512 > 0.55, so CI(λ1) = CI(λ2) = 4 and ∆G(λ1,λ2) = 0. Similar to the previous example, the fusing basic support function should reach the maximum truth-value at the index 4. So the results are reasonable. (3) Two basic support functions are λ1 = (0.7, 0.2, 0.1, 0) and λ2 = {(0.1, 0.1, 0.1, 0.6), (λ2(p3,p4) = 0.1)}. The weights of λ1 and λ2 are Ω1 = Ω2 = 0.5. The fusion results are shown in Table 4. λ1 and λ2 are totally con�icting and the fusion result is ω = {(0.1333, 0.4737, 0.2680, 0.11), (ω(p3,p4) = 0.015)}. The result shows that the 2nd proposition is most likely to be true, which is logical. It is reasonable that the uncertainty of the result is high. 802 Y. Li, Y. Deng Table 3: The fusion process and result of example (3) variables values λ1 (0.7, 0.2, 0.1, 0) λ2 {(0.1, 0.1, 0.1, 0.6), (λ2(p3,p4) = 0.1)} ω′ {(0.4, 0.15, 0.105, 0.33), (ω′(p3,p4) = 0.015} Ed(ω ′) 1.5185 ω {(0.1333, 0.4737, 0.2680, 0.11), (ω(p3,p4) = 0.015)} Ed(ω) 1.4832 7 Conclusion In order to better model the uncertain information of the characteristics of a subject, we proposed the generalized ordered propositions based on classical ordered propositions. The gen- eralized ordered propositions extended the indeterminate part of a basic support function to all groups of propositions, not just the universal set of propositions. Then we considered the determinate part, indeterminate part, mean, degree of convexity and center of a basic support function in the situation of generalized ordered propositions. These properties can also be applied in classical ordered propositions. Additionally, we found the existing entropy of a basic support function does not apply when the indeterminate part is not zero. To address this shortage, we presented a new entropy based on belief entropy. This entropy measures not only the uncertainty of the determinate part but also indeterminate part of a basic support function. When the inde- terminate part equals to zero, this entropy is degenerated into the existing entropy. Finally, we instructed the fusion method of basic support functions in generalized ordered propositions based on consistency and uncertainty. The experimental results show that the method is e�ective. Acknowledgment The authors greatly appreciate the reviews' suggestions and the editor's encouragement. The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61573290, 61503237). Con�ict of interest The authors declare that there is no con�ict of interests regarding the publication of this paper. Bibliography [1] Abelln, J. 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