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1、Fuzzy arithmetic based reliability allocation approach during early design and development 初期阶段的设计和发展中基于模糊算法的可靠度分配方法,V.Sriramdas,S.K.Chaturvedi,H.Gargama,1.Introduction2.Factors based conventional reliability allocation method3.Fuzzy numbers and arithmetic4.The methodology5.Illustrative example6.Con
2、clusions,1.Introduction Reliability allocation is an important and iterative task during the design and development activities of any engineering system.It is also difficult task because of the obscured and incomplete design details and a number of factors have to consider in design process.During t
3、he design phase of a system with a specified target reliability level,the reliability levels of the subsystems affect the overall system reliability.Therefore,a proper reliability allocation method needs to be adopted to allocate the target system reliability to its constituent subsystems proportion
4、ately.,4,2.Factors based conventional reliability allocation method,In this allocation method,the target reliability based on the reliabilityfactors is apportioned to subsystems for which no predicted reliability values are known.The relationship between apportioned reliability of ith subsystem Ri（子
5、系统可靠度）and target system reliability R*（总系统可靠度）is defined with a weightage factor wi.,Ri=(R*)wi,(1),where weightage factor wi（权重因子）can be expressed with proportionality factor Zi（比例因子）as：,wi=Zi/Zi,(2),5,2.1.Complexity（复杂度）,The complexity factor varies from subsystem to subsystem within a system and i
6、s measured in terms of number of active components that a subsystem is composed of.The number of components in a subsystem has a direct bearing on the reliability of the subsystem.Thus,complexity has a strong impact on the reliability allocation.The failure rate of the subsystem with high complexity
7、 is generally going to be high.So,the failure rate is allocated proportional to the complexity of the subsystem.Hence,Zi Ki,where Ki is the complexity factor for the itssubsystem.,1.Multiple functional relationships with the other groups.2.Number of components comprising subsystem.,6,2.2.Cost（成本）,Fo
8、r a large system,the cost increment for reliability improvement is relatively high.The demonstration of a high reliability value for a costly system may be extremely uneconomical.Hence,Zi Coi,where Coi is the cost factor for the its subsystem.,2.3.State-of-the-art（工艺状态）,When the component has been a
9、vailable for a long time,it is quite difficult to further improve the reliability of a component evenif the reliability is considerably lower than desired.Hence,Zi 1/Si,where Si is the state-of-the-art factor for the itssubsystem.,7,2.4.Criticality（临界值）,Criticality is another very important factor i
10、n reliability allocation.It is logical,higher reliability target should be allocated tothe functionally critical sub-systems and thus Zi is proportionalto criticality.Hence,Zi 1/Cri,where Cri is the criticality factor for the itssubsystem.,2.5.Time of operation（运行时间）,There may be some subsystems whi
11、ch are required to be operatedfor a period less than the mission time.So,for the subsystems with operating time lessthan the mission time,it is only logical to allocate relativelylower reliability.Hence,Zi 1/Ti,where Ti is the time of operation factor for theith subsystem,2.6.Maintenance（维护）,A compo
12、nent which is periodically maintained or one which isregularly monitored or checked and repaired as necessary willhave,on an average higher availability than one which is not maintained Hence,Zi Mi,where Mi is the maintenance factor for the itssubsystem.The process of allocation of relative scales i
13、s carried out as ateam exercise,comprising of experienced members from the eachof the subsystem identified.From the previous discussions in thissection,after consideration of various factors,formula for proportionalityfactor(Zi)as:,3.Fuzzy numbers and arithmetic,Definition:A fuzzy number is a fuzzy
14、subset that is both convex,andnormal.The most commonly used fuzzy numbers are triangular and trapezoidal fuzzy numbers,parameterized by(a,b,c),and(a,b,c,d),respectively,the membership functions of these numbers are defined below:,Then the standard operations on trapezoidal fuzzy numbersare expressed
15、 as:,Addition:,Subtraction:,Multiplication:,Division:,Defuzzification（解模糊化）is the underlying reason that one cannot comparefuzzy numbers directly.Although many authors proposedtheir favorite methods,there is no universal consensus.Each methodincludes computinga crisp value,to be used for comparison.
16、This assignment of areal value to a fuzzy number is called defuzzification.It can takemany forms,but the most standard defuzzification is through computing the centroid(计算模糊重心).,3.1.Fuzzy division by using linear programming,Let be two trapezoidal fuzzy numbers parameterizedby(l1,c1,c11,r1),and(l2,c
17、2,c22,r2),where l1 and l2,c1 and c2,c11 and c22,and r1 and r2 denotes left end points,left center points,right center points,and right end points,respectively.The resulting fuzzy numbers can be written as follows:,The constraints and objective function of the linear programmingproblem for the trapez
18、oidal fuzzy division are defined as follows:,The first constraint is constructed based on the left spreads.Theleft spread value over center value for should be equal to or lessthan division of the same ratios calculated for,In a similar manner,the second constraint is constructed as follows:,The thi
19、rd and fourth constraints can be defined based on the definitionof fuzzy numbers.The left end point of a fuzzy number shouldbe less than the left center value.The mathematical expression forthe third constraint is given below:,The right end point of a fuzzy number should be greater than theright cen
20、ter value.The mathematical expression for the fourth constraintis given below:,The objective function is given below:,4.The methodology,It can be very well argued that it is not easy to evaluate allocationfactors K,Co,Cr,M,T and S in a precise manner.Similar kind ofproblem has been observed in Failu
21、re Mode Effect and CriticalityAnalysis,while evaluating the risk factors.Significant efforts have beenmade to evaluate risk factors in a linguistic way using fuzzy logic.In this study,similar effort has been made to evaluate theallocation factors in linguistic way,Suppose there are n subsystems,Ui(i
22、=1,.,n)be evaluatedand allocated by a system design and analysis team consisting ofm members,TMj(j=1,.,m).,Let hj(j=1,.,m)be the relative importance weights of the m team members,satisfying,Based on these assumptions,the n subsystems can be allocated by following steps:,(a)Aggregate the team members
23、 subjective opinions on allocationfactors for the ith subsystem for i=1,.,n,as:,(b)Define and compute the fuzzy allocation proportionalityfactor(FZ)of each subsystem as,Now,FZ of each subsystem can be computed by combining fuzzy multiplication and division.Fuzzy multiplication has been performed by
24、using Eq.(6)and the fuzzy division has been performed by usingliner programming,which is explained in Section 3.1.,(c)Defuzzify the FZ by using Eq.(8).(d)Evaluate the weightage of each subsystem using the Eq.(2)by the solution of trapezoidal centroid defuzzification values.(e)Evaluate the allocate r
25、eliabilities of the each subsystem using the Eq.(1).,5.Illustrative example,In this section,we provide a numerical example to illustrate thepotential applications of the proposed fuzzy allocation method.A team consisting of the three members identifies four subsystems in a transceiver and needs to a
26、llocate reliability to each subsystem in order to achieve target reliability.The transceiver subsystems are power supply,transmitter,receiver,and antenna system.,require:the transceiver to operate 1000 h with a probability of 0.875.,Three team members TM1,TM2and TM3 are assigned with relative weight
27、s of 45%,25%and 30%,respectively.,6.Conclusions,Reliability allocation is very important task during the early designand development stages of the system.In this paper,a fuzzybased reliability allocation technique considering engineeringand design attributes has been proposed.Besides,a formulationfo
28、r decreasing the fuzziness in the division operation of trapezoidalfuzzy numbers using linear programming also been provided.Thenotion has been presented by an example,where the approach isemployed to allocate the reliability of each subsystem of transceiverto achieve the target reliability during early design anddevelopment stage.,
