4.3 Analytic Development of Availability and Maintainability in Engineering Design 473 Table 4.5 Process capacities per subgroup Sub-system group No. of subgroups Capacity per subgroup A 1 100% B 2 75% C 3 50% • Process limitations • Quality limitations, etc. Each state in the simple power plant example has only one subgroup that is the limiting factor, or bottleneck, fo r the plant’s power output capab ility in that state. This constraint is illustrated in Fig. 4.35 where the example plant is represented as a set of pipes and valves of varying capacities. Each section of pip e and valve corresponds to a subgroup in which the subgroup’s unavailability is analogous to a valve being closed: • The single A subgroup (consisting of two coal bin sub-systems) is wide enough to handle 100% of the flow; • Each of the two B subgroups (consisting of the two slurry mill sub-systems) is wide enough to handle 75% of the flow; • Each of the three C subgroups (consisting o f three gas turbines and three gener- ators) is wide enough to handle 50% of the flow. For example, if two C subgroups are unavailable, and one B subgroup is un- available, the C subgroup is the limiting factor because its remaining capacity is only 50%, whereas the remaining capacity in any one of the B subgroups is 75%. Furthermore, when two C subgroups are unavailable, there could be either no un- available B subgroups or one unavailable B subgroup, without further reducing the process flow from the resulting 50% output brought about by the one available C subgroup. f) Defining Different States (1) Table 4.5 shows the percentage of the plant’s pr ocess flow capability that each type of subgroup could support. (2) Table 4.6 shows the reduction in plant flow capacity as the numberof unavail- able subgroups in each sub-system group increases, given that all other subgroups are available. Where excess capacity beyond 100% exists in a subgroup, 100% is given as the throughput capacity. (3) Table 4.7 shows the flow capacities and state definitions. The flow capacities are taken from the previous table. Note that, although the 100% entry appears four times, it is entered only once in the table below. All flow capacities other than 100% are entered as many times as they appear in the previous table. Thus, the 0% flow capacity is entered three times. The capacities should be entered in decreasing order 474 4 Availability and Maintainability in Engineering Design Table 4.6 Remaining capacity versus unavailable subgroups Sub-system Number of Subgroup Remaining capacity as n subgroups become unavailable group subgroups capacity n = 0 n = 1 n = 2 n = 3 A 1 100% 100% 0% B 2 75% 100% 75% 0% C 3 50% 100% 100% 50% 0% Table 4.7 Flow capacities and state definitions of unavailable subgroups State number Flow capacity Unavailable subgroups ABC 1 100% 0 0 0 or 1 2 75% − 1 − 3 50% −−2 40%1−− 50%− 2 − 60%−−3 to simplify the state definition process. The entries under columns A, B and C in Table 4.7 mustbe the same as the entries under columns n = 0, n = 1,n = 2andn= 3 in Tab le 4.6. Process of entering the different state definitions i) Enter for each sub-system group the number of unavailable subgroups that would still allow 100% process flow. In the example, no unavailablesubgroups in sub-system group A would allow for 100% process flow. Similarly, no sin- gle subgroup in sub-system group B would allow for 100% process flow. In sub-system group C, either zero or one unavailable subgroup allows for 100% process flow. ii) Enter for each state the number of unavailable subgroups in the appropriate sub-system group that are responsible for that state’s capacity. For example, the 75% capacity of state 2 is the result of one of sub-system group B’s sub- groups being unavailable; the 50% capacity of state 3 is the result of two of sub-system group C’s subgroups being unavailable; the 0% capacity of states 4, 5 and 6 each is the respective result that one of A’s subgroups, or two of B’s subgroups, or three of C’s subgroups are unavailable. This is indi- cated in Table 4 .8. iii) For each state that has a non-zeroflow capacity, enter the subgroups in each re- maining sub-system group that could be unavailable without further decreas- ing the flow capacity of that state. For example, state 3 has a 50% flow ca- pacity because of unavailability of two of C’s subgroups. Zero subgroups of sub-system group A can be unavailable, and either zero or one of sub-system group B’s subgroups can be unavailable without decreasing the flow capacity of 50% for state 3. 4.3 Analytic Development of Availability and Maintainability in Engineering Design 475 Table 4.8 Flow capacities of unavailable sub-systems per sub-system group State number Flow capacity Unavailable subgroups ABC 1 100% 0 0 0 or 1 2 75% 0 1 0 or 1 3 50% 0 0 or 1 2 40%1−− 50%− 2 − 60%−−3 Table 4.9 Unavailable sub-systems and flow capacities per sub-system group State number Flow capacity Unavailable subgroups AB C 1 100% 0 0 0 or 1 2 75% 0 1 0 or 1 3 50% 0 0 or 1 2 4 0% 1 0 or 1 or 2 0, 1, 2 or 3 50%020,1,2or3 60%00or13 Table 4.10 Unavailable sub-systems and flow capacities per sub-system group: final summary State number Flow capacity Unavailable subgroups ABC 1 100% 0 0 < 2 2 75% 0 1 < 2 3 50% 0 < 22 40%1< 3 < 4 50%02< 4 60%0< 23 iv) The remaining entries to be made are in the 0% capacity states. These remain- ing entries indicate the number of subgroups that can be unavailable in each sub-system group in conjunction with other sub-system groups, where a 0% capacity state can be defined. This is indicated in Table 4.9. The final summary is indicated in Table 4.10. g) Evaluating Complexity of the Different State Definitions One of the more significant challenges of engineering design is to provide a rational account of the uncertainty surrounding the state events of unavailable systems that could be responsible for diminishinga design’s capacity and/or performance.Classi- cal probability theory offe rs a feasible approach but it is burdened with well-known 476 4 Availability and Maintainability in Engineering Design epistemological flaws, consideredin Sect. 3.3.2(Zadeh1995; Laviolette et al. 1995). Theories of fuzzy sets and possibility represent attempts to rectify some of the de- ficiencies in classical probability theory (Dubois et al. 1993). However, all of these theories fundamentallyacceptthe basic fact that randomvariables form a significant part of uncertainty. Consider the state events of unavailable systems that diminish the overall ca- pacity of the example power plant: Let x i represent the sub-system states listed in Table 4.10 where i = states 1,2, 3, ,6. Furthermore, let y θ j represent the state events of unavailable sub-system groups that could affect the overall capacity of the example power plant, where the subscripts θ =sub-system group A, B or C, and j = subgroup 1, 2, 3. Individual elements of x i can then be combined into a primary set of state events of unavailable sub-systems, denoted by X,andthe elements y θ j can be combined into a secondary set of state events denoted by Y. A graphical representation of these elements is called a complex, whereby each x i element is taken as the vertex of a surface formed by connected points representing the possible state events of each related subgroup y θ j , the state event elements of Y, which are called simplices (Casti 1994). Thus, the system states represented by x i are: X = {x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 } (4.181) and the possible state events represented by y θ j are: Y = {y A0 ,y A1 ,y B0 ,y B1 ,y B2 ,y C0 ,y C1 ,y C2 ,y C3 } (4.182) The outcomes of the compound events resulting from the integration of systems forming each subgroup (as depicted in the availability block diagram of Fig. 4.27) are given by the values (expressed as percentages of the overall capacity of the example power plant) of the system states represented by x i , and are called random variables. According to Table 4.10, outcomes of the compound events are: x 1 =(y A0 + y B0 + y C0 ,y C1 ,y C2 ,y C3 ) = 100% x 2 =(y B1 ,y B2 ,y B1 ,+y C1 ,y B1 + y C2 ,y B1 + y C3 , y B2 + y C1 ,y B2 + y C2 ,y B2 + y C3 ) = 75% x 3 =(y B1 + y C1 + y C2 ,y B1 + y C1 + y C3 ,y B1 + y C2 + y C3 , y C1 + y C2 ,y C1 + y C3 ,y C2 + y C3 ) = 50% x 4 =(y A1 ) = 0% 4.3 Analytic Development of Availability and Maintainability in Engineering Design 477 Table 4.11 Unavailable subgroups and flow capacities incidence matrix State number Flow capacity Unavailable subgroups 1 100% (y A0 + y B0 + y C0 ,y C1 ,y C2 ,y C3 ) 2 75% (y B1 ,y B2 ,y B1 ,+y C1 ,y B1 + y C2 ,y B1 + y C3 ,y B2 + y C1 , y B2 + y C2 ,y B2 + y C3 ) 3 50% (y B1 + y C1 + y C2 ,y B1 + y C1 + y C3 ,y B1 + y C2 + y C3 ,y C1 + y C2 , y C1 + y C3 ,y C2 + y C3 ) 40%(y A1 ) 50%(y B1 + y B2 ) 60%(y C1 + y C2 + y C3 ) Table 4.12 Probability of incidence of unavailable systems and flow capacities State number Flow capacity Unavailable subgroups Probability of incidence AB C 1 100% 0 0 0 or 1 0.100 2 75% 0 1 0 or 1 0.200 3 50% 0 0 or 1 2 0.533 4 0% 1 0 or 1 or 2 0, 1, 2 or 3 0.017 50%020,1,2or30.017 60%00or130.133 x 5 =(y B1 + y B2 ) = 0% x 6 =(y C1 + y C2 + y C3 ) = 0% Taking the elements of X to be the vertices of the unavailability complex of the power plant, and denoting the elements of Y to be simplices formed from these vertices, the relation R Y linking the two sets can be established, such that the pairs of elements (y θ j ,x i ) are in the relation R Y if, and only if, the possible state events of unavailable subgroups, y θ j , form part of the elementary system states x i . Thus, (y C1 ,x 1 ) is in R Y ;however,(y A1 ,x 1 ) and (y B1 ,x 1 ) are not. Computing all the chains of connections in this complex enables the formation of an incidence matrix for R Y . This matrix is the kind of incidence structure for which classical probability theory works well to express the concept of uncertainty in evaluating the integrity of the design. The complex of which the simplices are the state event elements of Y represents the sample space of the various unavailability states, expressed as percentages of the overall capacities, as indicated in Table 4 .11. The probability of system unavail- ability incidence is given in Table 4.12. 478 4 Availability and Maintainability in Engineering Design h) Evaluation of Alternatives At this point in systems engineering analysis, alternative design solutions that sat- isfy system constraints are developed.Effectiveness measu resare initially quantified for each solution without serious consideration of cost. Later, both effectiveness and costs are evaluated. After alternative system configurations have been synthesised and the effectiveness requirements have been established for each alternative, they can be compared. A typical trade-off matrix technique is appropriate. In most stud- ies, the analysis is restricted to an evaluation of cost and to some physical attributes of the system such as reliability, availability, maintainab ility or safety. It is, how- ever, necessary to analyse cost and effectiveness in monetary terms. An adequate analysis cannot be performed unless both parts of the relationship are evaluated in commensurate terms—i.e. when evaluating on the basis of costs, all comparisons must be kept in terms of costs. Prior to such a cost versus effectiveness compari- son, however, it is necessary to determine the physical attributes of the system (i.e. system integrity). The following example indicates how overall system integrity can be determined through systems engineering analysis to obtain the system’s sub-system and/or as- sembly attributes of mean times between failures and failure repair times. Figure 4.36 represents a process block diagram (i.e. a simplified process flow diagram) of a turbine/generator system. After the development of an availability block diagram (ABD), the overall in- tegrity of the system can be determined based on the ABD configuration and at- tributes of the system’s sub-systems and/or assemblies (Table 4.13). An ABD of the super-heated steam turbine/generator system illustrated in th e process block diagram of Fig. 4.36 is given in Fig. 4.37. Determining overall mean time to repair (MTTR system) From the integrity values given in Table 4.13: MTTR system = Σ ( λ R) Σ ( λ ) (4.183) where: λ = failure rate R = repair time (h) . MTTR system = 39,227/370.43 MTTR system = 105.9 Determining overall mean time between failures (MTBF system) From the in- tegrity values given in Table 4.13: MTBF system = 10 6 Σ ( λ ) (4.184) MTBF system = 2.699 4.3 Analytic Development of Availability and Maintainability in Engineering Design 479 Unit 1 boiler Feed water heater Boiler feed pump Hot well pump Cond. pump Unit 1 turbine Condenser De-aerator Super heater Unit 1 generator Fig. 4.36 Process block diagram of a turbine/generator system Power generating system A = 96.2 MTBF = 2.699 MTTR = 105.9 MTBF = 8.965 R = 124.5 MTBF = 20.653 R = 142.5 MTBF = 51.046 R = 148.6 MTBF = 62.344 R = 39.5 MTBF = 36.063 R = 48.3 MTBF = 85.616 R = 42.5 MTBF = 91.408 R = 96.3 MTBF = 112.306 R = 98.5 MTBF = 8.652 R = 96.3 Generator Turbine Steam condenserCondenser pumps Hot well pump Boiler feed pumpDe-aerator Feed water heater Boiler Fig. 4.37 Availability block diagram of a turbine/generator system, where A = availability, MTBF = mean time between failure (h), MTTR = mean time to repair (h) 480 4 Availability and Maintainability in Engineering Design Table 4.13 Sub-system/assembly integrity values of a turbine/generator system Power system Failure rate MTBF Repair rate λ ×R items ( λ fail/10 6 h) (10 6 / λ h) (R,h) 1. Generator 111.55 8.965 124.513.888 2. Turbine 48.42 20.653 142.56.900 3. Hot pump 27.73 36.062 48.31.339 4. Condenser 19.59 51.046 148.62.911 5. Cond. pump 16.04 62.344 39.50.634 6. De-aerator 10.94 91.408 96.31.053 7. Feed pump 11.68 85.616 42.50.496 8. Feed heater 8.90 112.306 98.50.876 9. Boiler 115.58 8.652 96.311.130 370.43 39.227 Determining overall availability (A system) From the integrity values given in Table 4.13, and from the formula for steady-state availability, we get: A = MTBF MTBF+ MTTR (4.185) = 2.699 2.699+ 105.9 A = 96.2% where, in Eqs. (4.184) and (4.185): λ = failure rate A = availability MTBF = mean time between failure (h) MTTR = mean time to repair (h). 4.3.3.4 Evaluating Complexity in Engineering Design With the phenomenal advancement in p rocess technology, there has been an almost similar increase in the complexity of engineered installations, p articularly large in- tegrated systems. Much engineering effort has gone into analysing and understand- ing systems complexity in an attempt to try and manage or reduce it at the design stage. Relatively recent research has shown, however, that the real issue is not so much reducing systems complexity but, rather, reducing complicatedness.Thisis an important distinction because complexity can, in fact, be a desirable property of integrated systems, provided it is specifically engineered complexity that reduces complicatedness (Tang et al. 2001). Complexity and complicatedness are not synonymous. Complexity is an inher- ent property of systems and the integration of systems; complicatedness is a derived 4.3 Analytic Development of Availability and Maintainability in Engineering Design 481 function of complexity, introduced in the notion of complicatedness of complex sys- tems. Equations for each can be developed showing that they are separate and dis- tinct properties that not only reflect the fundamental behaviour of complex systems but that also p rovide a design methodology whereby complicatedness can be evalu- ated. The implications for systems design engineers are enormous, especially con- cerning complex systems analysis in engineering design. The difference b etween complexity and complicatedness can be illustrated by the following example (Tang et al. 2001). Relative to a manual transmission, a motor vehicle’s automatic transmission has more parts and more intricate linkages, making it more complex. To the vehicle driver (operator), it is unquestionably less complicated but to the mechanic (main- tainer), who has to repair it, it is more complicated. This illustrates a fundamental fact about systems: operational control has an important role on systems to manage their behaviour. Complexity, therefore, is an inherent property of systems. Compli- catedness is a derived property that characterises the a bility to control a complex system. A system of complexity level C a may present different degrees of compli- catedness K to distinct control units E and F, where: K E = K E (C a ) K F = K F (C a ) (4.186) and: K E ,K F = complicatedness of systems E and F . a) Complexity in Systems There is hardly any research on complicatedness and complexity as distinct prop- erties of systems. The focus is on modularisation and integrated interactions with a bias to linear systems and qualitative metrics. Overwhelmingly, the literature considers systems with a large number of elements as complex (Suh 1999). Very few studies address integrated linkages among the elements (Warfield 2000), and at least one considers their bandwidth (Tang et al. 2001). All these factors are in- herent characteristics of systems; the number of elements, the number of interac- tions among these, and the bandwidth of the interactions determine the complex- ity of the system. As these increase, system complexity is expected to increase. For example, consider the system N = {n i } i = 1,2, , p with binar y interactions among the elements. Complexity C N of this system does not exceed p 2 ,whichis denoted by: C N = O p 2 482 4 Availability and Maintainability in Engineering Design Thus, the system M = {m j } j = 1,2, , p can have complexity: C M = O(p k ) where k > 2 . (4.187) Thus, when M has {m j × m r } jr and {m j × m r × m s } jrs interactions, then C M = O(p 3 ). Furthermore, when M has {m j × m r × m s × m t } jrst interactions, then C M = O(p 4 ). This characterisation of complex systems considers systems with feedback loops of arbitrary nesting (i.e. arbitrary loops within loops), and high bandwidth (i.e. vol- ume or number) of interactions among system elements. Complexity is a monoton- ically increasing function, as the size of the system and the number of interactions, as well as the bandwidth of interactions increase (Tang et al. 2001). In the limit, complexity → ∞. Complexity is thus defined by: C = X n ∑ b B b (4.188) where: X is an integer denoting the number of elements {x e }e = 1,2, ,p n is the integer indicated in the relation O(p n ) and: B 1 = ∑ ij λ ij β ij (4.189) B 2 = ∑ k λ k ij β k ij (4.190) where: λ ij = the number of linkages between x i and x j β ij = the bandwidth of linkages between x i and x j λ k ij = the number of linkages between x k and (x i ,x j ) β k ij = the bandwidth of linkages between x k and (x i ,x j ). In general: B n = ∑ n λ p ijk n−1 β n ijk n−1 (4.191) where: λ p ijk n−1 is the number of linkages among x k and (x i ,x j ),(x i ,x j ,x k ), , (x i ,x j ,x k , ,x n−1 ) β n ijk n−1 is the bandwidth of linkages for x k and (x i ,x j ),(x i ,x j ,x k ), , (x i ,x j ,x k , ,x n−1 ) B n is a measure of the capacity among the n elements of the system. . percentages of the overall capacities, as indicated in Table 4 .11. The probability of system unavail- ability incidence is given in Table 4.12. 478 4 Availability and Maintainability in Engineering Design h). derived 4.3 Analytic Development of Availability and Maintainability in Engineering Design 481 function of complexity, introduced in the notion of complicatedness of complex sys- tems. Equations for each. flow capacity of 50% for state 3. 4.3 Analytic Development of Availability and Maintainability in Engineering Design 475 Table 4.8 Flow capacities of unavailable sub-systems per sub-system group State