4.3 Analytic Development of Availability and Maintainability in Engineering Design 453 4.3.2.2 Designing for Availability Using Petri Net Modelling Returning to the initial quantitative example of designing for availability with the inclusion of preventivemaintenance, Fig. 4.24 illustrates the MRSPN representation of the system (Bobbio et al. 1997). The working state is modelled by place P up . The generally distributed transition t f models the failure distribution of which the firing results in the system moving to place P down . Upon system failure, the preventive maintenance activity is suspended; the inhibitor arc from place P down to transition t clock is used to model this fact. The deterministic transition t clock models the constant inspection interval. It is competitively enabled with t f , so that the one that fires first disables the other. Once t clock fires, a token moves in place P mai , as well as the activity related with the preventive maintenance (transition t mai starts). During the preventive m aintenance phase, the system is down and cannot fail by using the inhibitor arc from p lace P mai to transition t f . The completion of the maintenance (firing of t mai ) re-initialises the system in an as-good-as-new condition; hence, t f is assigned a prd policy. Since upon failure (and repair) a completed interval must elapse before the successive preventivemain- tenance takes place, t clock also must be assigned a prd policy. As can be observed from Fig. 4.24, t f and t clock are conflicting prd transitions. a) Numerical Computations fo r the Availability Petri Net Model Since there are no immediate transitions in the PN, all the markings are tangible. Starting from an initial marking m 1 , the token distribution of the reachable markings represented in Fig. 4.24 (assuming the following order for the places: P up , P clock , P down , P mai ) is given by: m 1 =(1,1,0,0) , m 2 =(0,1, 1,0), m 3 =(1,0, 0,1) . From marking m 1 , both t f and t clock may fire, leading to m 2 and m 3 respectively. From m 2 , only t down can fire, leading to m 1 and, finally, from m 3 only t mai can fire, p up p down p mai p clock t clock t f t mai t down Fig. 4.24 MRSPN model for availability with preventive maintenance (Bobbio et al. 1997) 454 4 Availability and Maintainability in Engineering Design leading to m 1 . As a consequence, the matrices E(t) and K(t) have the following structure: E(t)= ⎡ ⎣ E 11 (t) 00 0 E 22 (t) 0 00E 33 (t) ⎤ ⎦ (4.172) K(t)= ⎡ ⎣ 0 K 12 (t) K 13 (t) K 21 (t) 00 K 31 (t) 00 ⎤ ⎦ (4.173) Since E(t) is a diagonal matrix, the marking process is an SMP. Let G f (t) be the c.d.f. of the firing time associated with transition t f ,andd be the deterministic main- tenance interval associated with t clock . Furthermore, let λ 1 and λ 2 be the firing rates associated with the transitions t down and also t mai respectively. The non-zero matrix entries are: K 12 (t)= G f (t) 0 ≤ t < d G f (d) t ≥d (4.174) K 13 (t)= 00≤t < d 1−G f (d) t ≥d (4.175) K 21 (t)=1− e − λ 1 t (4.176) K 31 (t)=1− e − λ 2 t (4.177) E 11 (t)= 1−G f (I) 0 ≤t < d 0 t ≥ d (4.178) E 22 (t)=e − λ 1 t (4.179) E 33 (t)=e − λ 2 t (4.180) b) Steady-State Solution to the Availability Petr i Net Model To obtain the steady-state solution, the following procedure is given: STEP 1: α = ⎡ ⎢ ⎣ α 11 = [1−G f (t)]dt 00 0 α 22 = 1/ λ 1 0 00 α 33 = 1/ λ 2 ⎤ ⎥ ⎦ ϕ = ⎡ ⎣ 0 G f (d) 1−G f (d) 10 0 10 0 ⎤ ⎦ 4.3 Analytic Development of Availability and Maintainability in Engineering Design 455 Fig. 4.25 MRSPN model results for availability with preventive maintenance 500040003000200010000 0.9992 0.9993 0.9994 0.9995 0.9996 0.9997 0.9998 A vailability Optimum Optimal maintenance interval STEP 2: D =[1/2 , 1/2[G f (d)] , 1/2[1 −G f (d)]] STEP 3: ν =[1/A 2 · α 11 1/A 2 · α 22 G f (d) , 1/A 2 · α 33 [1−G f (d)]] A = 1/2 α 11 + 1/2 α 22 G f (d)+1/2 α 33 [1−G f (d)] The steady-state availability is given by the probability of being in state m 1 (entry v 1 in Step 3). The effect of the length of the preventive maintenance in terval d on system availability can now be determined. The numerical computations are made assuming the following values: i) Transition t f is distributed according to the Weibull cumulative distribution function G f (t)=1 − e −ct β , with β the shape parameter and c the scale pa- rameter. Assume β = 2.0 for an increasing failure rate (i.e. wear-out). ii) Let λ 1 = 0.1h −1 and λ 2 = 1.0h −1 for the firing rates of the transitions t down and t mai respectively. iii) The pr eventive maintenance interval d varies from 0 to 5,000 h. Figure 4.25 is a representative plo t of system availability t 1 versus the maintenance interval d.Ifd = 0, the system is always under maintenance and is completely un- available. As d increases, the steady-state availability increases as well. For large d, however, the effect of the preventive maintenance is nullified by the downtime due to failure and, in the limit d → ∞, the availability appr oaches the value when there is no p reventive maintenance. The optimal ma intenance interval d can now be com- puted at which the availability achieves its maximum value v t → maximum. 456 4 Availability and Maintainability in Engineering Design 4.3.3 Analytic Development of Availability and Maintainability Evaluation in Detail Design Appropriate methods for further development as tools for availability and main- tainability evaluation in de termining the integrity of engineering design during the detail design phase are: i. The application of systems engineering in engineering design, particularly sys- tems engineering analysis (SEA). ii. The evaluation of complexity in integrated systems through complex systems theory (CST). 4.3.3.1 Systems Engineering and Complex Systems Theory Systems engineering is a discipline that establishes a structured analysis approach to evaluate complex engineering design problems. Because systems engineering fo- cuses in this case on the methodology of analysis and synthesis for determining the overall integrity of complex integrated systems in engineering design, rather than its execution, describing it precisely is more difficult than for other engineering disciplines. Furthermore, its description varies considerably, particularly between industrial and research applications. Industrial demand for systems engineering is so pervasive that the approach is highly focused on methods for problem-solving in the operation of eng ineered installations as well as in their engineering design, while systems engineering in research concentrates mainly on mathematical methods and algorithms needed for evaluating the co mplexity of these designs. In the development of tools for availability and maintainability evaluation in de- termining the integrity of engineering design during the detail design phase, key characteristics of systems engineering are considered in industrial applications. Be- cause the initial engineering approach must be quantitative, systems engineering relies on mathematics, both for the representation of the real world as models and simulations, and for analysis and synthesis in mathematical methods or algorithms. This focus on mathematical methods and modelling translates the discipline of sys- tems engineering into systems engineering analysis (SEA). Systems engineering analysis is embodied in computer-based analysis of com- plexity in engineering design, as well as in software programming. With the grow- ing emphasis on computer aided design (CAD), systems engineering is increasingly providing a central and complementary role as an integrating factor in collaborative engineering design. Complex systems theory (CST) cuts across the boundaries between conventional scientific disciplines. It makes use of methods and examples from many disparate fields, and its results are widely applicable to a great variety of scientific and engi- neering problems (Wolfram 1988). Engineering systems, particularly industrial pro- cess systems, are often described as being complex (Boullart et al. 1988; Pritsker 1990). The dynamic nature o f process systems as well as the complex integration 4.3 Analytic Development of Availability and Maintainability in Engineering Design 457 of these systems make it difficult to predict the effect of design decisions on future system performance. Many integrated systems, which are designed to be flexible, are constrained by their complexity in being inflexible. An understanding of the effects of systems integration on system complexity is essential for realising the full potential of process systems, their successful deploy- ment in the process industry, and the economic justification of new process tech- nologies. However, literature on system complexity, specifically in the process and manufacturing context, is sparse (Ayres 1988; Deshmukh 1993). The notion of complex systems has been thoroughly considered in systems engi- neering analysis litera ture. Extrapolating fr om various different contexts in which the idea of complexity is used, a complex system may be defined as one having a static structure or dynamic behaviour that is counterintuitive or unpredictable (Casti 1979).A complex system may also be referred to as a system that has patterns of connections among sub-systems, such that any prediction of system behaviour is difficult without substantial analysis or computation; or one in which decision- making of alternative options in engineering design makes the effects of individual choices difficult to evaluate (Simon 1981). Computational or algorithmic complexity is often used for classifying process control problems (Garey et al. 1979). However, computational complexity does not capture all the aspects o f complexity in engineering systems. Also, computational complexity does not always relate to the performance of the system, since compu- tational comp lexity is an algorithm-related measure. The complexity of a physical system can be characterised in terms of its static structure or time-dependent behaviour. Static complexity can be viewed as a func- tion of the structure of the system, connective patterns, variety of components, and the strengths of interactions, whereas dynamic complexity is concerned with unpre- dictability in the b ehaviour of the system over a time period (Deshmukh 1993). The process environment consists of physical systems in which concurrent and sequential p rocesses take place in order to produce an output. The nature of these processes is dependent not only upon system capabilities but also on the p rocess characteristics (inputs, throughputs and outputs) being produced in the system. Hence, any measure of system complexity should be dependent on both the system and process characteristics, particularly with integrated systems that result in a mul- tiplicity of characteristic-related event criteria. However, the complexity principle states that as the complexity and uncertainty of an engineering system increases, our ability to predict its behaviour diminishes, until a threshold is reached beyond which accuracy and significance become almost mutually exclusive. This is often termed the threshold of chaos. Phenomena that are chaotic are unpredictable (non- repeatable) and, hence, cannot be optimised. The main reason for this is an extreme sensitivity to initial conditions. Most complex systems contain some chaos. All that can be done with chaotic phenomena is to increase the analysis of their p roperties, patterns, structure and occurrence (Zadeh 1979). The difficulty in making design decisions with complex systems arises from the number of choices available at each decision point relating to each event in the range of the event-criteria possibilities, and the unpredictability of the effects of 458 4 Availability and Maintainability in Engineering Design these events on system performance. Computational complexity can be considered to be the algorith mic effort required to evaluate these choices. In addition to systems complexity, there is a further aspect that relates to the control of process systems: static and dynamic complexities are usually considered assuming constant control schemes. However, different process systems require varying levels of control, fur- ther complicating the difficulty of regulatingcomplexprocesses. The notion of com- plexity needs to be qualified and quantified in order to compare different system alternatives. The general lack of understanding in this area has hindered d esigners in deciding to what extent systems integration is beneficial, and beyond which point integration is actually detrimental to system performance, since correct decisions are difficult to make due to high system complexity. Another important consequence of developing an analytical framework for com- plexity would be to assist designers in managing desired levels of complexity in the system because, realistically, this cannot be eliminated due to the unavoidability of systems integration in large engineered installations, resulting in unpredictable changes in operating conditions. A fundamental problem of defining the notion of complexity beyond simply a term for a phenomenon that appears to be counter- intuitive or unpredictable, into a more formalised language (i.e. mathematically) whereby the differences between the complex and commonplace can be better un- derstood, is that it involves making that which is fuzzy, precise (Casti 1994). In an effort to understand what is involved with complex systems, it would be intuitive to first consider some of the properties associated with simple systems, before attempting to express complexity in mathematical terms. • Predictable behaviour: simple systems give rise to behaviour that is easy to de- duce if the system’s process characteristics (i.e. inputs, throughputs and outputs) can be defined. Such predictable behaviour is one of the principal characteristics of simple systems. • Interaction and feedback loops: simple systems generally involve a small num- ber of components with interactions that dominate the linkages among the pro- cess characteristic variables. In addition to having only a few variables, simple systems generally consist of a few feedback/feed-forwardloops that enable mod- ification or regulation of interactions among the process characteristic variables. • Centralised control: in simple systems, control is centred with very little, if any, independent interactive control between lower-tiered components. Such systems tend to be more robust, as they are better able to absorb process fluctuations. • Decomposable and reducible: a sim ple system involves relatively weak interac- tions among its various components that, if disconnected or degraded, would not result in a total loss of process control or unstable behaviour. By establishing a workable complex systems theory, a framework can be structured within which complex systems can be better understood from the perspective of en- gineering design, process control and operational stability. More importantly, CST can provide a means of determining the limits of reduction of complex systems for systems engineering analysis. 4.3 Analytic Development of Availability and Maintainability in Engineering Design 459 4.3.3.2 Systems Engineering in Engineering Design Like many other engineering disciplines, systems engineering: • involves central concepts; • uses specific methodologies; • includes both analysis and synthesis for evaluating engineering design; • relies on mathematics to express knowledge; • bearsan interdependent relationship to other engineering disciplines (since many design problems are cross-disciplinary); • provides profound benefit to engineering and industry in particular; • stimulates research for further engineering benefit. Many of the key thrusts of systems engineering are found within the other engineer- ing disciplines. However, systems engineering is qualitatively different. Systems engineering differs from the basic engineering disciplines in that these disciplines concentrate on using knowledge of the real world for systems construction, e.g. ma- terials, structures, electrical circuits, r obotics, whereas systems engineering finds its focus in constructs of analysis and synthesis for problems involving multiple as- pects of complex real-world systems. The effectiveness o f systems engineering in analysing complex systems is determined by methodologies, algorithms and tools available for advanced systems engineering analysis, such as performance metrics, optimisation methods in the presence of various kinds of constraints, marginal and sensitivity analysis, linear/non-linear programming, dynamic programming, utility theory, decision analysis, mathematical modelling and simulation modelling (IN- COSE 2002). Systems engineering in engineering design involves several distinguishing char- acteristics, such as: • Design problems are highly inter-disciplinary: systems engineering in engineer- ing design typically involves a spectrum of conventional engineering and science disciplines. • Design problems require high-levelmetrics: systems engineeringproblems place a high priority on measuring and optimising values at higher levels of systems integration. • Design problems are hierarchical: as a result of integrating various factors into high-level metrics, systems engineering structures large-scale systems into a ver- tical hierarchy. • Multiple metrics and optimisation are crucial: integrating a plurality of system performance metrics leads to difficult challeng es in multivariate optimisation of design input variables to achieve reasonable optimisations of the various output metrics. • Thereis additional h eterogeneity: the behaviour of systems brings additional het- erogeneity into systems engineering problems that add more diversity to com- plexity considerations. 460 4 Availability and Maintainability in Engineering Design • The problems are dynamic: systems engineering places emphasis on dynamic variations in time, necessitating design-for-integrity through a concurrent engi- neering approach. • Methodologies for process life cycle are central: because systems engineering emphasises a structured approach to the analysis of design, analytic methodolo- gies are central. • Systems definition and development: systems engineering methods such as anal- ysis of systems complexity, hierarchical modelling and concurrent engineering design provide for a more comprehensive approach to process engineering de- sign. • Integrity of design: uncertainties in the development process underscore the im- portance of systems engineering approaches to the integrity of process engineer- ing design, together with system performance and life-cycle costs. • Non-technical components and metrics: while cost and human resource factors are normally considered intrinsic factors in conventional engineering disciplines, systems engineering places an explicit, high priority on these factors. • Other non-technical disciplines: human factors play a crucial role in systems engineering, such as the disciplinary requirements in computer systems. • Government regulatory policy and decision-making: systems engineering appli- cation in many large-scale process projects involve not only technical compo- nents but political, economic and sociological factors as well. 4.3.3.3 Complexity in Engineering Design and Systems Engineering Systems engineering analysis (SEA) brings out clearly a systematic reasoning pro- cess in which all the un certainties associated with complex integrations of multiple systems that may have impeded d ecision-making during the early phases o f engi- neering design are properly considered. Systems engineering analysis examines un- certainties and assumptions made in the conceptual and preliminary design phases, to determine the end-result integrity of the engineered installation as a whole. It is a study of total systems performance, rather than a study of its elements. It stems from the recognition th a t, even if each element of a sy stem is optimised fro m a de- sign point of view, the total systems performance may be less than optimal, owing to complex interactions between the elemen ts. All complex systems have certain characteristics encountered not only in their design but also in their application that cause many of the critical malfunctions in industrial p lant and equipment. Among these characteristics are the following: • Change: the present state or condition of a system is the result of past per- formances or its engineering design. No real-world system remains static over a long period of time. The flow of the process enters and leaves the system either through a birth-and-death occurrence or by p assing through system boundaries. • Environment: each system has its own environment and is, in fact, a sub-system of some broader system. The environment of a system is a set of elements and 4.3 Analytic Development of Availability and Maintainability in Engineering Design 461 their relevant properties that, although not a part of the system, if modified can produce a change in the state of the system as a whole. • Counteraction results: examination of some systems might indicate the need for corrective action. This action can often be ineffective or even adverse in its re- sults. Corrective action in complex systems may intensify a problem, rather than solve it. • Drift to low performance: complex systems generally tend towards a condition of reduced performance with time. Components deteriorate and inefficiencies creep in, their counteraction nature causing detrimental d esign changes. • Interdependency: no activity in a complex system takes place in total isolation. Each event is influenced by its predecessor and affects its successors. I n addition, real-world activities are generally parallel and u ltimately in flu ence each o th e r. • Organisation: all complex systems consist of highly o rganised elements or com- ponents. These elements are combined into hierarchies of sub-systems, assem- blies, components and parts that interact to carry out the function of the system. • Variance: outputs from complex systems tend to have greater variances about a mean result, because of the individual variances in performance of the con- stituent elements. Open and closed systems A closed system is considered to be one in which only the components within the system are assumed to exist. All other influences or vari- ables from outside the system are considered to be non-existent, or to be insignifi- cant. It is a hypothetical assumptive system, as there probably never is a completely closed system. Components within a system are always subject to outside influ- ences. Closed systems are u sually adopted for initial analysis, as they are usually simple and each component in the system is m ore easily analysed with regard to its effect on the other components in the system. An open system is described by the basic properties of: • Inputs: inputs (exogenous variables) are the independent variables of the system model, and are pre-determined. Input variables can be classified as either con- trollable or non-controllable. Controllable input variables can be manipulated. Non-controllable input variables are generated by the environment in which the system exists, and not by the system. • Throughputs: throughputs (status variables) are indicative of the capability of the system to achieve the desired output. • Outputs: outputs (endogenousvariables) are the dependent or output variables of the system model, and are generated from the interaction of the system’s output and status variables, according to the system’s operating characteristics. Added to this are other attributes such as cyclic events, continuity, and differentia- tion of functions. An open system recognises and permits all interactions of its components to take place across the boundariesof the system. It is more realistic, though more complex, than closed systems and, therefore, more difficult to analyse or control. Diagrams of a model of a closed system and a model of an open system are given b elow (Fig. 4.26), together with the typical symbols used in such system models. 462 4 Availability and Maintainability in Engineering Design Closed system Decision Flow line Auxiliary operation Operation or end Input/output connection/ information Closed system from other systems to other systems Adj. system Fig. 4.26 Models of closed and open systems a) Functions of Systems Engineering Analysis Systems engineering analysis consists of the following functions: • Problem definition • System objectives • System boundaries • System components • Requirements analysis • Functional analysis • Effectiveness measures • Constraints evaluation • Choosing alternatives • Evaluating alter natives. . availability and main- tainability evaluation in de termining the integrity of engineering design during the detail design phase are: i. The application of systems engineering in engineering design, particularly. of sys- tems engineering into systems engineering analysis (SEA). Systems engineering analysis is embodied in computer-based analysis of com- plexity in engineering design, as well as in software. evaluating the co mplexity of these designs. In the development of tools for availability and maintainability evaluation in de- termining the integrity of engineering design during the detail design