INAk-out-of-n: G system, there are totally n components,

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1 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 5, OCTOBER Relablty Analyss of Uncertan Weghted k-out-of-n Systems Jnwu Gao,KaYao, Jan Zhou,andHuaKe Abstract The uncertan varable s used to model a quantty under human uncertanty, and the weghted k-out-of-n system s used to model a system of n components, whch functons f and only f the total weghts of functonng components s greater than k. Consderng the human uncertanty n operatng the system, ths paper ntroduces the uncertan varable to the weghted k-out-of-n system, and proposes a concept of uncertan weghted k-out-of-n system. Some formulas are derved to calculate the relablty ndex of such a system. As a generalzaton, ths paper also studes an uncertan weghted k-out-of-n system whose weghts are estmated by experts and modeled by uncertan varables nstead of crsp numbers. In addton, ths paper analyzes the mportance measure of the components n the uncertan weghted k-out-of-n systems. Index Terms Importance measure, k-out-of-n system,relablty analyss, uncertan varable. I. INTRODUCTION INAk-out-of-n: G system, there are totally n components, and the system functons f and only f there are at least k functonng components of those n components. Consderng the dfferent utltes of those components n the system, Wu and Chen [1] generalzed the k-out-of-n: G systems nto the weghted k-out-of-n: G systems n In a weghted k-out-ofn system, each component s assgned a real number to ndcate ts weght, and the system functons f and only f the total weght of functonng components s greater than k. Snce the weghts may be greater than 1, the number k mght be greater than the number n n a weghted k-out-of-n system. In order to calculate the relablty ndex of the weghted k-out-of-n system, Wu and Chen [1] desgned a recursve algorthm, whch was smplfed by Hgashyama [2] later. In 2005, Chen and Yang [3] extended the one-stage weghted k-out-of-n systems to the two-stage weghted k-out-of-n systems. After that, nspred by Levtn et al. [4], L and Zuo [5] proposed a Manuscrpt receved June 22, 2017; revsed December 19, 2017; accepted February 5, Date of publcaton February 15, 2018; date of current verson October 4, Ths work was ported n part by the Shuguang Program by Shangha Educaton Development Foundaton and Shangha Muncpal Educaton Commsson under Grant 15SG36 and n part by the Natonal Natural Scence Foundaton of Chna under Grant (Correspondng author: Jan Zhou) J. Gao s wth the School of Mathematcs, Renmn Unversty of Chna, Bejng , Chna (e-mal: jgao@ruc.edu.cn). K. Yao s wth the School of Economcs and Management, Unversty of Chnese Academy of Scences, Bejng , Chna (e-mal: yaoka@ucas.ac.cn). J. Zhou s wth the School of Management, Shangha Unversty, Shangha , Chna (e-mal: zhou_jan@shu.edu.cn). H. Ke s wth the School of Economcs and Management, Tongj Unversty, Shangha , Chna (e-mal: hke@tongj.edu.cn). Dgtal Object Identfer /TFUZZ type of multstate weghted k-out-of-n systems, and desgned a method of unversal generatng functon to evaluate the relablty of the systems. Recently, Zhu et al. [6] proposed a new approach for analyzng the Brnbaum mportance patterns of k-out-of-n systems. Erylmaz [7] presented a type of weghted k-out-of-n systems whose weghts are regarded as random varables. Burkschat et al. [8] studed a type of censored sequental k-out-of-n systems, and derved the maxmum lkelhood estmator. Snce the concept of fuzzy set was proposed by Zadeh [9] n 1965, relablty analyss of fuzzy systems has been wdely studed. In 1994, Utkn [10] studed relablty ndex of many types of fuzzy reparable systems, ncludng the seres, parallel, k-out-of-n systems. Then Cheng [11] proposed a new method to analyze the relablty of fuzzy consecutve-k-out-of-n systems; Wu [12] dscussed the fuzzy Bayes pont estmators of system relablty for the fuzzy k-out-of-n system; Guan and Wu [13] extended the fuzzy k-out-of-n systems nto the consecutve-kout-of-n systems wth fuzzy states. In 2011, Ebrahmpur et al. [14] ntroduced an optmzaton problem of multstate weghted fuzzy k-out-of-n systems, and solved the programmng models va the genetc algorthm. Except for randomness and fuzzness, there are some other types of ndetermnacy, for example, the human uncertanty. In order to model the human uncertanty arsng from human s belef degree on the chance that some event may occur, Lu [15] proposed an uncertan measure to ndcate the belef degree based on normalty, dualty, and subaddtvty axoms. Then an uncertan varable s used to model a quantty under human uncertanty, and an uncertanty dstrbuton s used to model the uncertan varable. So far, many works have been done n ths area. For example, Lu and Ha [16] and Yao [17] proposed some formulas to calculate the expected value and varance of an uncertan varable, respectvely. Da and Chen [18] verfed the lnearty of the entropy operator for uncertan varables. Gao and Qn [19] presented some formulas to calculate the edgeconnectvty degree of an uncertan graph. Yang and Gao [20] appled the uncertan lnear-quadratc dfferental game to the resource extracton problems. Yao and Zhou [21] nvestgated the run tme of an nsurance company by modelng the clams va an uncertan renewal reward process. In 2010, Lu [22] proposed the concept of uncertan system based on uncertan varables, and he calculated the relablty ndex of the uncertan seres, parallel and k-out-of-n systems. In the uncertan k-out-of-n system by Lu [22], all the components are assgned the same weghts, whch means they are IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See standards/publcatons/rghts/ndex.html for more nformaton.

2 2664 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 5, OCTOBER 2018 assumed to have the same utltes. Ths assumpton restrcts the wde applcatons of uncertan k-out-of-n systems as the utltes of the components may vary from each other n the real cases. That motvates us to nvestgate the uncertan weghted k-out-of-n systems as a generalzaton. In ths paper, we wll analyze the relablty of the uncertan weghted k-out-of-n systems, and calculate the mportance measure of the components n such systems. The rest of ths paper s organzed as follows. In Secton II, we wll ntroduce some basc concepts and formulas about uncertan varables. Then, we wll ntroduce the uncertan weghted k-out-of-n system and derve some formulas to calculate ts relablty ndex n Secton III. Based on these formulas, we wll also desgn an algorthm to calculate the relablty ndex. Then n Secton IV, we wll extend the weghts of the uncertan weghted k-out-of-n system from crsp numbers to uncertan varables, and study the k-out-of-n system wth uncertan weghts. In Secton V, we wll derve some formulas and desgn some algorthms to calculate the mportance measure of the components n uncertan weghted k-out-of-n systems. Fnally, some remarks wll be made n Secton VI. II. PRELIMINARY In ths secton, we ntroduce some basc concepts and formulas about uncertan varables. The uncertan varable, whch s used to model a quantty wth human uncertanty, s a measurable functon on an uncertanty space. Defnton 1: (Lu [15], [23]) Let L be a σ-algebra on a nonempty set Γ. A set functon M : L [0, 1] s called an uncertan measure f t satsfes the followng axoms: Axom 1 (Normalty Axom): MΓ =1for the unversal set Γ. Axom 2 (Dualty Axom): MΛ + MΛ c =1for any event Λ. Axom 3 (Subaddtvty Axom): For every countable sequence of events Λ 1, Λ 2,..., M Λ MΛ. Axom 4 (Product Axom): Let (Γ k, L k, M k ) be uncertanty spaces for k =1, 2,...Then the product uncertan measure M s an uncertan measure satsfyng M Λ k = mn M k Λ k k 1 k=1 where Λ k are arbtrarly chosen events from L k for k =1, 2,, respectvely. Defnton 2: (Lu [15]) An uncertan varable ξ s a measurable functon from the uncertanty space (Γ, L, M) to the set of real numbers,.e., for any Borel set B of real numbers, the set ξ B = γ Γ ξ(γ) B s an event. An uncertan varable ξ s called a Boolean uncertan varable f t can only take the values ether 0 or 1. Defnton 3: (Lu [23]) The uncertan varables ξ 1,ξ 2,...,ξ n are sad to be ndependent f n M (ξ B ) = mn Mξ B for any Borel sets B 1,B 2,...,B n of real numbers. Defnton 4: (Lu [15]) The uncertanty dstrbuton of an uncertan varable ξ s defned by Φ(x) =Mξ x for any real number x. An uncertan varable s sad to be contnuous f t has a contnuous uncertanty dstrbuton. An uncertan varable ξ s sad to be lnear f t has an uncertanty dstrbuton 0, f x<a Φ(x) = (x a)/(b a), f a x b 1, f x>b denoted by L(a, b). An uncertan varable ξ s sad to be lognormal f t has an uncertanty dstrbuton ( ( )) 1 π(e ln x) Ψ(x) = 1 + exp, x R 3σ denoted by LOGN (e, σ). It s easy to verfy that lnear uncertan varables and lognormal uncertan varables are both contnuous uncertan varables. For an uncertan varable ξ, the functon (α) =nfx R Mξ x α α (0, 1) s called the nverse uncertanty dstrbuton of ξ. Its essentally the generalzed nverse functon of the uncertanty dstrbuton of ξ. The nverse uncertanty dstrbutons of lnear uncertan varable L(a, b) and lognormal uncertan varable LOGN (e, σ) are and (α) =a +(b a)α ( Ψ 1 (α) = exp e + ) 3σ π ln α, 1 α respectvely. Theorem 1: (Lu [24]) Let ξ 1,ξ 2,...,ξ n be some ndependent uncertan varables. Then for a strctly ncreasng functon f(x 1,x 2,...,x n ),wehave Mf(ξ 1,ξ 2,...,ξ n ) x= f (x 1,x 2,,x n ) x mn Mξ x. Let ξ and η be two postve uncertan varables. Then accordng to Theorem 1, and Mξ + η x = mn (Mξ y, Mη z) y +z x Mξ η x = mn (Mξ y, Mη z). y z x

3 GAO et al.: RELIABILITY ANALYSIS OF UNCERTAIN WEIGHTED k-out-of-n SYSTEMS 2665 TABLE I RELIABILITY INDEXES AND WEIGHTS OF THE COMPONENTS Components RIs Weghts Theorem 2: (Lu [24]) Assume ξ 1,ξ 2,...,ξ n are ndependent Boolean uncertan varables such that Mξ =1 = α and Mξ =0 =1 α for =1, 2,...,n.If f s a Boolean functon, then ξ = f(ξ 1,ξ 2,...,ξ n ) s a Boolean uncertan varable such that f Mξ =1= 1 f (x 1,x 2,,x n )=1 f f (x 1,x 2,,x n )=1 f (x 1,x 2,,x n )=0 f (x 1,x 2,,x n )=1 mn Mξ = x mn Mξ = x < 0.5 mn Mξ = x mn Mξ = x 0.5 where x take values ether 0 or 1 for =1, 2,...,n, respectvely. III. k-out-of-n SYSTEMS WITH CRISP WEIGHTS In a k-out-of-n: G system, there are totally n components wth ndfferent utltes, and the system works f and only f at least k components work. In a weghted k-out-of-n: G system, these n components have dfferent utltes whch are represented by the weghts w s, and the system works f and only f the total weght of the workng components s greater than k. Please note that the weghts w s mght be greater than 1, so k mght be greater than n. Consderng the human uncertanty n the system, we assume the relablty of each component s descrbed by an uncertan varable, and we study the uncertan k-out-of-n system wth crsp weghts n ths secton. Let the Boolean uncertan varable ξ denote the th component wth a relablty ndex α, and let w denote the weght of the th component when t functons, =1, 2,...,n. Then the weghted k-out-of-n: G system works f and only f w ξ k and the relablty ndex of the uncertan weghted k-out-of-n: G system s defned by RI = M w ξ k. (1) Example 1: Consder an uncertan weghted k-out-of-n: G system wth k =5and n =6. The relablty ndexes and the weghts of ts components are gven n Table I. In order to calculate the relablty ndex of such a system by usng Theorem 2, we ntroduce a Boolean functon 1, f 6 f(x 1,x 2,...,x 6 )= w x 5 0, f 6 w x < 5 and model the system va a Boolean uncertan varable ξ = f(ξ 1,ξ 2,...,ξ n ). Snce there exts a combnaton (x 1,x 2,x 3,x 4,x 5,x 6 )=(1, 1, 0, 0, 0, 1) such that f(x 1,x 2,...,x 6 )=1and Then, Notng that f (x 1,x 2,,x 6 )=1 mn Mξ = x =0.5 mn Mξ = x 0.5. RI = Mf(ξ 1,ξ 2,...,ξ 6 )=1 =1 mn Mξ = x w 1 x 1 + +w 6 x 6 <5 f (x 1,x 2,,x 6 )=0 =1 w 1 x 1 + +w 6 x 6 <5 mn Mξ = x mn Mξ = x. = M(ξ 1,ξ 2,ξ 3,ξ 4,ξ 5,ξ 6 )=(1, 1, 0, 1, 0, 0) =0.5 RI =1 0.5 =0.5 accordng to Theorem 2. Theorem 3: Let ξ 1,ξ 2,...,ξ n be ndependent Boolean uncertan varables. Then, the relablty ndex of the uncertan weghted k-out-of-n: G system (1) s RI = mn Mξ x w 1 x 1 +w 2 x 2 + +w n x n k where x take values ether 0 or 1 for =1, 2,...,n, respectvely. Proof: Defne a functon f(x 1,x 2,...,x n )= w x. Then, the relablty ndex of the weghted k-out-of-n: G system (1) s RI = Mf(ξ 1,ξ 2,...,ξ n ) k. By usng Theorem 1, RI = f (x 1,x 2,,x n ) k = The theorem s proved. w 1 x 1 +w 2 x 2 + +w n x n k mn Mξ x mn Mξ x.

4 2666 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 5, OCTOBER 2018 Example 2: Consder the weghted k-out-of-n: G system n Example 1. There are 40 combnatons for (x 1,x 2,x 3,x 4, x 5,x 6 ) such that w 1 x 1 + w 2 x w 6 x 6 5. Comparng these combnatons, we fnd that (1, 1, 0, 1, 0, 1) s the optmal combnaton, and RI = w 1 x 1 +w 2 x 2 + +w 6 x 6 5 mn Mξ x 1 5 = M(ξ 1,ξ 2,ξ 3,ξ 4,ξ 5,ξ 6 )=(1, 1, 0, 1, 0, 1) =0.5. Based on Theorem 3, a numercal method could be desgned to calculate the relablty ndex of the uncertan weghted k-outof-n: G system. Frst, we enumerate all the possble states of the components. Second, we fnd out those states of the elements that satsfy the constrants. Fnally, we calculate the relablty ndex of the system under each state, and fnd the optmal one whch s just the relablty ndex of the system. However, ths numercal method s of hgh computatonal complexty. In order to desgn an effcent algorthm, we derve a formula to calculate the relablty ndex of the k-out-of-n: G system va the nverse uncertanty dstrbuton. Theorem 4: Let ξ 1,ξ 2,...,ξ n be ndependent Boolean uncertan varables. Then, the relablty ndex of the uncertan weghted k-out-of-n: G system (1) s where RI = α [0, 1] w 1 (α) k 1, f α α 1 (α) = 0, f α>α are some real functons for =1, 2,...,n, respectvely. Proof: For smplcty, wrte α 0 = α [0, 1] w 1 (α) k. On the one hand, for any gven α l <α 0,letξ l1,ξ l2,...,ξ lp denote those components whose relablty ndexes α l1,α l2,...,α lp are greater than α l. Snce w 1 (α l ) k w ξ k p ξ l =1. Then by usng the monotoncty of uncertan measure and the ndependence of uncertan varables, we further have p M w ξ k M (ξ l =1) = mn Mξ l =1 1 p = mn α l α l. 1 p The above nequalty holds for any α l <α 0,so M w ξ k α l = α 0. (2) α l α 0 On the one hand, for any gven α u >α 0,letξ u 1,ξ u 2,...,ξ u q denote those components whose relablty ndexes α u 1,α u 2,...,α u q are less than α u. Snce w 1 (α u ) <k w ξ <k q ξ u =0. Then by usng the monotoncty of uncertan measure and the ndependence of uncertan varables, we further have q M w ξ <k M (ξ u =0) = mn Mξ u =0 1 q = mn (1 α u ) 1 α u. 1 q Accordng to the dualty of uncertan measure, M w ξ k α u. The above nequalty holds for any α u >α 0,so M w ξ k nf α u = α 0. (3) α u >α 0 It follows mmedately from Inequaltes (2) and (3) that M w ξ k = α 0. The theorem s proved. Example 3: Consder the uncertan weghted k-out-of-n: G system n Example 1. Snce 6 6 w 1 (0.8) = 1.8 < 5, w 1 (0.7) = 3.3 < 5 6 w 1 (0.6) = 3.8 < 5, 6 w 1 (0.5) = 6.8 5

5 GAO et al.: RELIABILITY ANALYSIS OF UNCERTAIN WEIGHTED k-out-of-n SYSTEMS 2667 TABLE II RELIABILITY INDEXES AND WEIGHTS OF THE COMPONENTS Components RIs Weghts Components RIs Weghts t follows from Theorem 4 that the relablty ndex of the system s RI =0.5. Now, we present an algorthm to calculate the relablty ndex of the uncertan weghted k-out-of-n: G system (1) based on Theorem 4. Step 1: Set j =0, and resort the relablty ndexes of the components n the descendng order whch are denoted by β 1,β 2,...,β n. Step 2: Set j j +1. Step 3: If w 1 (β j ) <k then go back to Step 2. Step 4: The relablty ndex of the system s β j. Example 4: Consder an uncertan weghted k-out-of-n: G system wth k =12and n =10. The relablty ndexes and the weghts of the components are gven n Table II. By usng the algorthm ntroduced above, the relablty ndex of the system s RI =0.6. IV. k-out-of-n SYSTEMS WITH UNCERTAIN WEIGHTS In a complex weghted k-out-of-n system, the weghts of the components may not be precsely obtaned. In ths case, we have to nvte the experts to estmate the weghts of these components, so we could model the weghts va uncertan varables. Ths secton wll study the weghted k-out-of-n system wth uncertan weghts. We denote the state of the components by Boolean uncertan varables ξ s, and denote the weghts of the components by contnuous uncertan varables η s. Then the weghted k-out-of-n: G system works f and only f η ξ k and the relablty ndex of the uncertan weghted k-out-of-n: G system s defned by RI = M η ξ k. (4) Theorem 5: Assume ξ 1,η 1,...,ξ n,η n are ndependent uncertan varables. Then the relablty ndex of the uncertan weghted k-out-of-n: G system (4) s RI = w 1 x 1 + +w n x n k mn mn (Mη w, Mξ x ), where w take any real numbers and x take values ether 0 or 1 for =1, 2,..., n, respectvely. Proof: Defne a functon f(w 1,...,w n,x 1,...,x n )= w x. Then, the relablty ndex of the weghted k-out-of-n: G system (4) s RI = Mf(η 1,...,η n,ξ 1,...,ξ n ) k. By usng Theorem 1, RI = f (w 1,,w n,x 1,,x n ) k = w 1 x 1 + +w n x n k mn Mη ξ w x mn mn (Mη w, Mξ x ). The theorem s proved. Theorem 6: Assume ξ 1,η 1,...,ξ n,η n are ndependent uncertan varables. Then, the relablty ndex of the uncertan weghted k-out-of-n: G system (4) s RI = α (0, 1) (1 α)1 (α) k where Φ are the uncertanty dstrbutons of η, and 1, f α α 1 (α) = 0, f α>α are some real functons for =1, 2,...,n, respectvely. Proof: For smplcty, wrte α 0 = α [0, 1] (1 α)1 (α) k. On the one hand, for any gven α l <α 0,letξ l1,ξ l2,...,ξ lp denote those components whose relablty ndexes α l1,α l2,...,α lp are greater than α l. Snce η ξ k (1 α l )1 (α l ) k p ( ηl l (1 α l ) ) (ξ l =1).

6 2668 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 5, OCTOBER 2018 Then by usng the monotoncty of uncertan measure and the ndependence of uncertan varables, we further have M η ξ k M p ( ηl l (1 α l ) ) (ξ l =1) = mn mn ( M η l 1 p l (1 α l ), Mξ l =1 ) ( ) = mn α l, mn α l = α l. 1 p The above nequalty holds for any α l <α 0,so M η ξ k α l = α 0. (5) α l α 0 On the one hand, for any gven α u >α 0,letξ u 1,ξ u 2,,ξ u q denote those components whose relablty ndexes α u 1, α u 2,,α u q are less than α u. Snce (1 α u )1 (α u ) <k η ξ <k q ( ξu u (1 α u ) ) (ξ u =0). Then by usng the monotoncty of uncertan measure and the ndependence of uncertan varables, we further have M η ξ <k M q ( ξu u (1 α u ) ) (ξ u =0) = mn 1 q mn ( M ξ u = mn ( (1 α u ), mn 1 q (1 α u ) u (1 α u ), Mξ u =0 ) =1 α u. Accordng to the dualty of uncertan measure, M η ξ k α u. The above nequalty holds for any α u >α 0,so M η ξ k nf α u = α 0. (6) α u >α 0 It follows mmedately from Inequaltes (5) and (6) that M η ξ k = α 0. The theorem s proved. ) TABLE III RELIABILITY INDEXES AND WEIGHTS OF THE COMPONENTS Components RIs Weghts L(1, 2) L(2, 2.8) L(2.4, 3.2) L(3, 3.6) L(3, 4) LOGN (3, 0.2) LOGN (2, 0.1) LOGN (3.5, 0.3) LOGN (2.5, 0.2) LOGN (3, 0.3) Based on Theorem 6, we desgn an algorthm to calculate the relablty ndex of the uncertan weghted k-out-of-n: G system (4) as follows. Step 0: Set ε as the degree of the precson. Step 1: Set β 1 =1.If (1 β 1 )1 (β 1 ) k then the relablty ndex of the system s 1, and exst. Step 2: Set β 2 =0.If (1 β 2 )1 (β 2 ) <k then the relablty ndex of the system s 0, and exst. Step 3: Set β =(β 1 + β 2 )/2.If β 1 β 2 <ε,then the relablty ndex of the system s β, and exst. Step 4: If (1 β)1 (β) <k then set β 1 = β; otherwse set β 2 = β. Go back to Step 3. Example 5: Consder an uncertan weghted k-out-of-n: G system wth k =14and n =10. The relablty ndexes and the weghts of the components are gven n Table III. By usng the algorthm ntroduced above, the relablty ndex of the system s RI = V. IMPORTANCE MEASURE OF WEIGHTED k-out-of-n SYSTEMS The mportance measure s used to evaluate the relatve mportance of the components n a system. In ths secton, we gve some formulas to calculate the mportance measure of the components n the uncertan weghted k-out-of-n: G systems. Based on the concept of mportance measure by Brnbaum [25] for stochastc system, the mportance measure of the jth

7 GAO et al.: RELIABILITY ANALYSIS OF UNCERTAIN WEIGHTED k-out-of-n SYSTEMS 2669 component n the uncertan weghted k-out-of-n: G system s IM j =M w ξ k, ξ j =1 M w ξ k, ξ j =0. (7) Theorem 7: Let ξ 1,ξ 2,...,ξ n be ndependent Boolean uncertan varables. Then the mportance measure of the jth component n the weghted k-out-of-n: G system (1) s IM j = α [0, 1] w 1 (α)+w j k α [0, 1] w 1 (α) k. Proof: Accordng to Theorem 4, the relablty ndex of the system when the jth component functons s M w ξ k, ξ j =1 = α [0, 1] w 1 (α)+w j k and the relablty ndex of the system when the jth component fals s M w ξ k, ξ j =0 = α [0, 1] w 1 (α) k. Then the theorem follows mmedately from (7). Based on Theorem 7, an algorthm to calculate the mportance measure of the components n the uncertan k-out-of-n system wth crsp weghts s stated as follows. Step 1: Set j 1 =0and j 2 =0, and resort the relablty ndexes of the components n the descendng order whch are denoted by β 1,β 2,...,β n. Step 2: Set j 1 j Step 3: If w 1 (β j1 )+w j <k then go back to Step 2. Step 4: The relablty ndex of the system when the jth component functons s β j1. Step 5: Set j 2 j Step 6: If w 1 (β j2 ) <k then go back to Step 5. Step 7: The relablty ndex of the system when the jth component fals s β j2. Step 8: The mportance measure of the jth component n the system s β j1 β j2. The algorthm s of low computatonal complexty. To fnd the exact value of the mportance measure of a component, we need at most O(n) steps, so t works well for an uncertan weghted k-out-of-n system wth large values k and n. Example 6: Consder the uncertan weghted k-out-of-n: G system wth k =12and n =10n Example 4. By usng the algorthm above, the mportance measure of the second component s IM 2 =0.1. In fact, when the second component functons, the relablty ndex of the system s 0.6; when the second component fals, the relablty ndex of the system s 0.5. Hence, the mportance measure of the second component s IM 2 = =0.1. When the weghts of the components are uncertan varables n the weghted k-out-of-n: G system, the mportance measure of the jth component s IM j =M η ξ k, ξ j =1 M η ξ k, ξ j =0. (8) Theorem 8: Assume ξ 1,η 1,...,ξ n,η n are ndependent uncertan varables. Then the mportance measure of the jth component n the weghted k-out-of-n: G system (4) s α [0, 1] α [0, 1] (1 α)1 (α)+ j (1 α) k (1 α)1 (α) k. Proof: Accordng to Theorem 6, the relablty ndex of the system when the jth component functons s M η ξ k, ξ j =1 = α [0, 1] (1 α)1 (α)+ j (1 α) k and the relablty ndex of the system when the jth component fals s M η ξ k, ξ j =0 = α [0, 1] (1 α)1 (α) k. Then, the theorem follows mmedately from (8). Based on Theorem 8, an algorthm to calculate the mportance measure of the components n the uncertan k-out-of-n system wth uncertan weghts s stated as follows. Step 0: Set ε as the degree of the precson.

8 2670 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 5, OCTOBER 2018 Step 1: Set β 11 =1.If (1 β 11 )1 (β 11 )+ j (1 β 11 ) k then the relablty ndex of the system when the jth component functons s β 1 =1, and go to Step 5. Step 2: Set β 12 =0.If (1 β 12 )1 (β 12 )+ j (1 β 12 ) <k then the relablty ndex of the system when the jth component functons s β 1 =0, and go to Step 5. Step 3: Set β 1 =(β 11 + β 12 )/2.If β 11 β 12 <ε,then the relablty ndex of the system when the jth component functons s β 1, and go to Step 5. Step 4: If (1 β 1 )1 (β 1 )+ j (1 β 1 ) <k then set β 11 = β 1 ; otherwse set β 12 = β 1. Go back to Step 3. Step 5: Set β 21 =1.If (1 β 21 )1 (β 21 ) k then the relablty ndex of the system when the jth component fals s β 2 =1, and go to Step 9. Step 6: Set β 22 =0.If (1 β 22 )1 (β 22 ) <k then the relablty ndex of the system when the jth component fals s β 2 =0, and go to Step 9. Step 7: Set β 2 =(β 21 + β 22 )/2.If β 21 β 22 <ε,then the relablty ndex of the system when the jth component fals s β 2, and go to Step 9. Step 8: If (1 β 2 )1 (β 2 ) <k then set β 21 = β 2 ; otherwse set β 22 = β 2. Go back to Step 7. Step 9: The mportance measure of the jth component n the system s β 1 β 2. The algorthm s of low computatonal complexty. To fnd an approxmate value wth an error less than ε for the mportance measure of a component, we need at most O(n) ( log ε) steps, so t works well for an uncertan weghted k-out-of-n system wth large values k and n. Example 7: Consder the uncertan weghted k-out-of-n: G system wth k = 14 and n = 10 n Example 5. By usng the algorthm above, the mportance measure of the sxth component s IM 6 = In fact, when the sxth component functons, the relablty ndex of the system s ; when the sxth component fals, the relablty ndex of the system s Hence, the mportance measure of the sxth component s IM 6 = = VI. CONCLUSION Ths paper proposed an uncertan weghted k-out-of-n system, and derved a formula to calculate ts relablty ndex. As a generalzaton, t also ntroduced a k-out-of-n system whose components have both uncertan relablty ndexes and the uncertan weghts. Besdes, ths paper studed the mportance measure of the components n the uncertan weghted k-out-of-n systems. In the future research, we wll study consecutve uncertan k-out-of-n systems. REFERENCES [1] J. S. Wu and R. J. Chen, An algorthm for computng the relablty of weghted-k-out-of-n systems, IEEE Trans. Rel., vol. 43, no. 2, pp , Jun [2] Y. Hgashyama, A factored relablty formula for weghted-k-out-of-n system, Asa Pac. J. Oper. Res., vol. 18, no. 1, pp , May [3] Y. Chen and Q. Yang, Relablty of two-stage weghted-k-out-of-n systems wth components n common, IEEE Trans. Rel., vol. 54, no. 3, pp , Sep [4] G. Levtn, A. Lsnansk, H. Ben-Ham, and D. Elmaks, Redundancy optmzaton for seres-parallel mult-state systems, IEEE Trans. Rel., vol. 47, no. 2, pp , Jun [5] W. L and M. J. Zuo, Relablty evaluaton of mult-state weghted k-outof-n systems, Rel. Eng. Syst. Safe.,vol.93,no.1,pp ,Jan [6] X. Zhu, Q. Yao, and W. Kuo, Patterns of the Brnbaum mportance n lnear consecutve-k-out-of-n systems, IIE Trans.,vol.44,no.4,pp , Apr [7] S. Erylmaz, On relablty analyss of a k-out-of-n system wth components havng random weghts, Rel. Eng. Syst. Safe., vol. 109, pp , Jan [8] M. Burkschat, E. Cramer, and J. Górny, Type-I censored sequental k-outof-n systems, Appl. Math. Model., vol. 40, nos , pp , Oct [9] L. A. Zadeh, Fuzzy sets, Inf. Control, vol. 8, no. 3, pp , Jun [10] L. V. Utkn, Fuzzy relablty of reparable systems n the possblty context, Mcroelectron. Rel., vol. 34, no. 12, pp , Dec [11] C. H. Cheng, Fuzzy consecutve-k-out-of-n: F system relablty, Mcroelectron. Rel., vol. 34, no. 12, pp , Dec [12] H. C. Wu, Bayesan system relablty assessment under fuzzy envronments, Rel. Eng. Syst. Safe., vol. 83, no. 3, pp , Mar [13] J. Guan and Y. Wu, Reparable consecutve-k-out-of-n: F system wth fuzzy states, Fuzzy Set. Syst., vol. 157, no. 1, pp , Jan [14] V. Ebrahmpur, S. F. Quraysh, A. Shaban, and B. Malek-Shoja, Relablty optmzaton of mult-state weghted k-out-of-n systems by fuzzy mathematcal programmng and genetc algorthm, Int. J. Syst. Assur. Eng. Manag., vol. 2, no. 4, pp , Dec [15] B. Lu, Uncertanty Theory, 2nd ed. Berln, Germany: Sprnger-Verlag, [16] Y. H. Lu and M. Ha, Expected value of functon of uncertan varables, J. Uncertan Syst., vol. 4, no. 3, pp , Aug [17] K. Yao, A formula to calculate the varance of uncertan varable, Soft Comput., vol. 19, no. 10, pp , Oct [18] W. Da and X. Chen, Entropy of functon of uncertan varables, Math. Comput. Model., vol. 55 no. 3, pp , Feb [19] Y. Gao and Z. Qn, On computng the edge-connectvty of an uncertan graph, IEEE Trans. Fuzzy Syst., vol. 24, no. 4, pp , Aug [20] X. F. Yang and J. Gao, Lnear quadratc uncertan dfferental game wth applcaton to resource extracton problem, IEEE Trans. Fuzzy Syst., vol. 24, no. 4, pp , Aug [21] K. Yao and J. Zhou, Run tme of uncertan nsurance rsk process, IEEE Trans. Fuzzy Syst., vol. 26, no. 1, pp , Feb. 2018, do: /TFUZZ , [22] B. Lu, Uncertan rsk analyss and uncertan relablty analyss, J. Uncertan Syst., vol. 4, no. 3, pp , Aug [23] B. Lu, Some research problems n uncertanty theory, J. Uncertan Syst., vol. 3, no. 1, pp. 3 10, Feb

9 GAO et al.: RELIABILITY ANALYSIS OF UNCERTAIN WEIGHTED k-out-of-n SYSTEMS 2671 [24] B. Lu, Uncertanty Theory: A Branch of Mathematcs for Modelng Human Uncertanty. Berln, Germany: Sprnger-Verlag, [25] Z. W. Brnbaum, On the mportance of dfferent components n a multcomponent system, n Multvarate Anal.,vol. 2, Krshnaah PR, Ed. New York, NY, USA: Academc Press, 1969, pp Jnwu Gao receved the B.S. degree n mathematcs from Shaanx Normal Unversty, Xan, Chna, n 1996, and the M.S. and Ph.D. degrees n mathematcs from Tsnghua Unversty, Bejng, Chna, n He s currently an Assocate Professor wth the School of Mathematcs, Renmn Unversty of Chna. He has authored or co-authored more than 50 papers that have appeared n IEEE TRANSACTIONS ON FUZZY SYSTEMS, Fuzzy Optmzaton and Decson Makng, Journal of Intellgent Manufacturng, Soft Computng, Internatonal Journal of Uncertanty, Fuzzness & Knowledge-Based Systems, Computer & Mathematcs wth Applcatons,and other publcatons. Hs current research nterests nclude fuzzy systems, uncertan systems and ther applcaton n optmzaton, game theory, and fnance. Dr. Gao s the Co-Edtor-n-Chef of the Journal of Uncertan Systems snce 2011, an Assocate Edtor of Soft Computng from 2017 and an edtoral board member of several other journals. He has been a Fellow of the Asa Pacfc Industral Engneerng and Management Socety snce 2015, the Presdent of Intellgent Computng Chapter of Operatons Socety of Chna snce 2015, and the Vce Presdent of Internatonal Consortum for Uncertanty Theory snce Ka Yao receved the B.S. degree n computatonal mathematcs from the Nanka Unversty, Tanjn, Chna, n 2009, and the Ph.D. degree n operatons research from Tsnghua Unversty, Bejng, Chna, n He s currently an Assocate Professor wth the School of Economcs and Management, Unversty of Chnese Academy of Scences, Bejng , Chna. He has authored or coauthored about 40 artcles on several journals ncludng IEEE TRANS- ACTIONS ON FUZZY SYSTEMS, Knowledge-Based Systems, Appled Soft Computng, Appled Mathematcal Modellng, Appled Mathematcs and Computaton, Fuzzy Optmzaton and Decson Makng, and Soft Computng. Hs current research nterests nclude uncertan renewal processes, uncertan systems, uncertan dfferental equatons and ther applcatons. Jan Zhou receved the B.S. degree n appled mathematcs and the M.S. and Ph.D. degrees n computatonal mathematcs from Tsnghua Unversty, Bejng, Chna, n 1998 and 2003, respectvely. She s currently a Professor wth the School of Management, Shangha Unversty, Shangha, Chna. She has authored or coauthored more than 50 artcles n natonal and nternatonal journals, ncludng IEEE TRANSACTIONS ON FUZZY SYSTEMS, Computers and Industral Engneerng, Neucomputng, Engneerng Applcatons of Artfcal Intellgence, Fuzzy Optmzaton and Decson Makng, and Soft Computng. Her research nterests nclude network optmzaton, fuzzy clusterng, uncertanty theory and ts applcatons. For the other nformaton about her please vst the followng lnk: Hua Ke receved the B.S. degree from X an Jaotong Unversty, X an, Chna, n 2001, and the Ph.D. degree from Tsnghua Unversty, Bejng, Chna, n He s currently an Assocate Professor wth the School of Economcs and Management, Tongj Unversty, Shangha, Chna. He has authored or coauthored more than 50 artcles n natonal and nternatonal journals, ncludng European Journal of Operatonal Research, Internatonal Journal of Producton Economcs, Relablty Engneerng & System Safety, Journal of Intellgent Manufacturng, Appled Mathematcal Modellng, Journal of the Operatonal Research Socety, Soft Computng, and Fuzzy Optmzaton and Decson Makng. Hs research nterests nclude project schedulng, ply chan optmzaton and coordnaton, uncertanty theory and ts applcatons.