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1、 Fuzzy arithmetic based reliability allocation approach during early design and development 初期初期階段的階段的設(shè)計和發(fā)展中基于模糊算法的可靠設(shè)計和發(fā)展中基于模糊算法的可靠度度分配方法分配方法V. Sriramdas , S.K. Chaturvedi , H. Gargama 1.Introduction2.Factors based conventional reliability allocation method3.Fuzzy numbers and arithmetic4.The method

2、ology5.Illustrative example6.Conclusions1.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 con

3、sider in design process. During the 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 i

4、ts constituent subsystems proportionately.42.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 apportion

5、ed reliability of ith subsystem Ri （子系統(tǒng)可靠度）and target system reliability R*（總系統(tǒng)可靠度） is defined with a weightage factor wi. Ri= (R*)wi (1) where weightage factor wi （權(quán)重因子）can be expressed with proportionality factor Zi（比例因子）as：wi = Zi /Zi(2)52.1. Complexity（復(fù)雜度） The complexity factor varies from subs

6、ystem to subsystem within a system and is 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 failur

7、e rate of the subsystem with high complexity 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

8、components comprising subsystem.62.2. Cost（成本） For 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

9、. State-of-the-art（工藝狀態(tài)） When the component has been available 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.72.4. Critic

10、ality（臨界值） Criticality is another very important factor in 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.

11、 2.5. Time of operation（運行時間） There may be some subsystems which 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

12、of operation factor for theith subsystem2.6. Maintenance（維護） A component 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 fac

13、tor for the itssubsystem. The process of allocation of relative scales is 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

14、 (Zi) as:3. Fuzzy numbers and arithmetic Definition: A fuzzy number is a fuzzy 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

15、are defined below: Then the standard operations on trapezoidal fuzzy numbersare expressed 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 n

16、o universal consensus. Each methodincludes computinga crisp value, to be used for comparison.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(計算模糊重心).3.1. Fuzzy division by using

17、linear programming Let be two trapezoidal fuzzy numbers parameterizedby (l1, c1, c11, r1), and (l2, c2, 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 ca

18、n be written as follows: The constraints and objective function of the linear programmingproblem for the trapezoidal 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 divisio

19、n of the same ratios calculated for In a similar manner, the second constraint is constructed as follows: The third 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 expressio

20、n forthe third constraint is given below: The right end point of a fuzzy number should be greater than theright center 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

21、 evaluate allocationfactors K, Co, Cr, M, T and S in a precise manner. Similar kind ofproblem has been observed in Failure 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

22、study, similar effort has been made to evaluate theallocation factors in linguistic way Suppose there are n subsystems, Ui (i = 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

23、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 subjective opinions on allocationfactors for the ith subsystem for i = 1,. . . ,n, as:(b) Define and compute the fuzzy allocation proportionalityfa

24、ctor (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 using Eq. (6) and the fuzzy division has been performed by usingliner programming,which is explained in Section 3.1.(c) Defuzzify the FZ

25、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 reliabilities 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 allocate reliability to each subsystem in order to achieve target reliability. The tra