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.Introduction 2.Factors based conventional reliability allocation method 3.Fuzzy numbers and arithmetic 4.The methodology 5.Illustrative example 6.Conclusions,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 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 its constituent subsystems proportionately.,4,2.Factors based conventional reliability allocation method,In this allocation method, the target reliability based on the reliability factors is apportioned to subsystems for which no predicted reliability values are known. The relationship between apportioned reliability of ith subsystem Ri （子系统可靠度）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 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 failure 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 its subsystem.,1. Multiple functional relationships with the other groups. 2. Number of components comprising subsystem.,6,2.2. 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. State-of-the-art（工艺状态）,When the component has been available for a long time, it is quite difficult to further improve the reliability of a component even if the reliability is considerably lower than desired. Hence, Zi 1/Si, where Si is the state-of-the-art factor for the its subsystem.,7,2.4. Criticality（临界值）,Criticality is another very important factor in reliability allocation. It is logical, higher reliability target should be allocated to the functionally critical sub-systems and thus Zi is proportional to criticality. Hence, Zi 1/Cri, where Cri is the criticality factor for the its subsystem.,2.5. Time of operation（运行时间）,There may be some subsystems which are required to be operated for 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 the ith subsystem,2.6. Maintenance（维护）,A component which is periodically maintained or one which is regularly monitored or checked and repaired as necessary will have, on an average higher availability than one which is not maintained Hence, Zi Mi, where Mi is the maintenance factor for the its subsystem. The process of allocation of relative scales is carried out as a team exercise, comprising of experienced members from the each of the subsystem identified. From the previous discussions in this section, after consideration of various factors, formula for proportionality factor (Zi) as:,3. Fuzzy numbers and arithmetic,Definition: A fuzzy number is a fuzzy subset that is both convex,and normal. 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 numbers are expressed as:,Addition :,Subtraction :,Multiplication :,Division :,Defuzzification（解模糊化）is the underlying reason that one cannot compare fuzzy numbers directly. Although many authors proposedtheir favorite methods, there is no universal consensus. Each methodincludes computing a 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(计算模糊重心).,3.1. Fuzzy division by using linear programming,Let be two trapezoidal fuzzy numbers parameterized by (l1, c1, c11, r1), and (l2, c2, c22, r2), where l1 and l2, c1 and c2, c11 and c22, an