Maintenance scheduling and process optimization under
uncertainty
C.G. Vassiliadis, E.N. Pistikopoulos *
Department of Chemical Engineering,Centre for Process Systems Engineering,Imperial College,Prince Consort Road,London SW7 2BY,UK Accepted 20 September 2000
Abstract
In this paper, we describe an optimization framework for (i) deriving optimal maintenance policies in continuous process operations in the presence of parametric uncertainty and (ii) analyzing and quantifying the impact of uncertainty on optimal maintenance schedules. A systems effectiveness measure is introduced which depends on expected process profitability and process and reliability/maintenance characteristics. A mixed integer nonlinear optimization model is proposed which aims at identifying the number of maintenance (preventive or corrective) actions required over a given time horizon of interest, the time instants and sequence of these maintenance actions on the various components of the process system, so that the system effectiveness is maximized. By introducing the concept of availability threshold values, it is shown that an efficient solution strategy can be established which requires the solution of much smaller nonlinear optimization problems. The application of the proposed framework to an example problem highlights the important interactions between process operation and maintenance scheduling in the presence of uncertainty. © 2001 Elsevier Science Ltd. All rights reserved.
Keywords:Process operation; Reliability; Maintenance; Process uncertainty; Optimization; System effectiveness
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1. Introduction
Process productivity and effectiveness depend on the efficient utilization of assets and resources, the proper allocation of which is typically decided by the process engineer mainly in the design and partly in the opera-tion phase (process structure, equipment volumes, recipes, production schedules, controls, etc). However, it is process availability that critically determines whether assets and resources are, indeed, available to be used as planned.A6ailability, in general, is defined as the ability of an item to perform its required function at a stated instant of time or over a stated period of time (BS4778, 2000). In terms of a chemical process it corre-sponds to the fractional amount of time the process is able to perform its production. Especially nowadays, when modern technology and engineering is leading to more highly integrated plants and a failure in one part of the process can decisively influence total plant
per-formance, availability is widely recognized as one of the most important operability characteristics. On the other hand, availability is closely connected to the perfor-mance of the plant in terms of safety and environmen-tal impact. Lately, new and tighter regulations are imposed on industry demanding for specific availability requirements of safety systems (Michelsen, 1998). To underline the consequences of loss of availability we note that lost production costs in a chemical plant can range from $500 to 100,000 per h (Tan & Kramer, 1997). For refineries, in particular, total lost production costs soar to millions of dollars (Nahara, 1993).
Chemical process availability is a function of both equipment and process reliability and maintainability. Reliability is the ability of an item to perform a re-quired function, under given environmental and opera-tional conditions and for a stated period of time (BS4778, 2000). On the other hand, maintainability is the ability of an item, under stated conditions of use, to be retained in, or restored to a state in which it can perform its required functions when maintenance is performed under stated conditions and using prescribed procedures and resources (BS4778, 2000). Reliability * Corresponding author. Tel.: +44-171-5946620; fax: +
44-171-5946606.
E-mail address:[email protected] (E.N. Pistikopoulos).
0098-1354/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 1 3 5 4 ( 0 0 ) 0 0 6 4 7 - 5
characteristics (e.g. equipment characteristics, system configuration, buffer storage inventories, redundancy) and certain maintainability characteristics (e.g. accessi-bility to components, spares, consumables, etc.) are inherent attributes of the process established during the design stage and cannot be altered without a design change (Kapur & Lamberson, 1977). As a result, in the operating state of a process the way to achieve high availability is through the derivation and execution of effective maintenance strategies. Maintenance is broadly classified to preventive and corrective, the for-mer comprising activities such as inspection and re-placement while the latter concerning fixing or replacing equipment in the event of failure. A classifica-tion of maintenance activities is depicted in Fig. 1. Obviously, through its impact upon process availability, maintenance affects both the production and the safety functions of a chemical process. In particular, effective maintenance contributes (VanRijn, 1987) to sustaining production volume at the desirable levels, reducing operating (spares and consumable items) and fixed manufacturing costs and deterring the system from moving to a hazardous state.
Despite its big benefits, maintenance is, in general, very expensive. VanRijn (1987) places the cost of maintenance in the order of 20 – 30% of the plant’s total operating costs. Although proper planning of mainte-nance activities increases the profitability of the process, there are obvious trade-offs between process profitabil-ity and maintenance costs. Therefore, the maintenance optimization problem is described as trying to identify maintenance strategies which yield process availability levels that maintain such a balance between process revenue and maintenance cost which maximizes process profitability. For this purpose, maintenance optimiza-tion models are derived to determine the optimum balance between the costs and benefits of maintenance, taking into consideration all kinds of specifications and constraints (Dekker, 1996). A number of papers (Dekker & Scarf, 1998; Valdez-Flores & Feldman, 1989; Pierskalla & Voelker, 1976) present overviews of the research work in the field of maintenance optimiza-tion models, assessing their impact and applicaoptimiza-tions. In reference to the chemical process industry, in particular, it is admitted (Grievink, Smit, Dekker & VanRijn, 1993) that modeling the balance between maintenance
benefits and costs is quite a complicated task. Two of the main difficulties recognized are the following:
1.1. Quantification of the benefits of reliability and maintenance to process profitability
In the effort to achieve high availability there is a point at which this becomes economically unattractive. Woodhouse (1986) acknowledges that ‘‘although it may be worth spending $1000 to reduce a change of a bearing failure by 10%, it is unlikely that an extra $1,000,000 would be spent to reduce the probability of failure by a further 0.10%’’. Process profitability de-pends on production volume which, in the case of failure-free operation, is determined by the process mathematical model (production schedules, recipes, op-eration, etc.) and its feasibility. Traditional availability assessment and sensitivity analysis tools (fault trees, FMECA techniques), on the other hand, share the common limitation of not explicitly taking into account process models and process interactions and hence not fully capturing the physicochemical characteristics of the process; this may overestimate system efficiency and provide misleading information regarding critical com-ponents. The need to overcome these limitations is discussed both in the reliability and the process engi-neering community. Brunelle and Kapur (1997), for example, suggest that ‘‘reliability should be defined from the viewpoint of the customer’’ (i.e. process) and that reliability measures should be introduced to cap-ture the dynamic performance of the system. At the same time, process engineers suggest that (Grievink et al., 1993; Lamb, 1996; Pujadas & Chen, 1996) it is essential to introduce system effecti6eness measures to simultaneously take into consideration process, reliabil-ity and maintenance characteristics.
1.2. Process uncertainty
Uncertainty and variability is a common characteris-tic of all process systems and, based on the nature of its source, can be broadly classified as follows (Pistikopou-los, 1995):
model-inherent uncertainty, including kinetic con-stants, physical properties, transfer coefficients, etc. process-inherent uncertainty, including flowrate and
temperature variations, stream quality fluctuations, etc.
external uncertainty, including feedstream availabil-ity, product demands, prices and environmental conditions.
All types of uncertainties can be mathematically de-scribed either through ranges of possible realizations or with probability distribution functions using informa-tion obtained from experimental and pilot plant data, on-line measurements, historical data, customer orders, Fig. 1. Different types of maintenance.
market indicators and so on. Assessing and quantifying the impact of uncertainty,process flexibility(Swaney & Grossmann, 1985; Pistikopoulos & Mazzuchi, 1990; Straub & Grossmann, 1990) is a vital component of the operability of a chemical process and is defined as the probability that a system operates feasibly under con-tinuous uncertainties described by probability distribu-tion funcdistribu-tions. This suggests that the identificadistribu-tion of the optimal balance between process profitability — as a result of high availability — and maintenance costs, which is the core of the maintenance optimization problem, becomes a very complicated task in the pres-ence of uncertainty, since the operating pattern of the production process as well as the profile of process profitability may change (Pistikopoulos & Ierapetritou, 1995).
Nowadays, the need for integrating different oper-ability criteria (flexibility, controlloper-ability, relioper-ability and maintainability) in the study of chemical process opera-tion is widely recognized (VanRijn, 1987; Grievink et al., 1993). Obviously, qualitative techniques such as reliability centered maintenance (Anderson & Neri, 1990) or total productive maintenance (Nakajima, 1988), although very popular among maintenance prac-titioners, are not based on sound analytical process models to accommodate such an integration. On the other hand, traditional availability assessment of pro-cess systems, provides quantitative information on the characteristics of the maintenance policy to be fol-lowed, based on the system reliability model but with-out taking into account detailed process interactions, the process model complexity, and the uncertainty which may be involved in a number of model and process parameters. Typical studies on maintenance optimization (Alkhamis & Yellen, 1995; Vatn, Hokstad & Bodsberg, 1996; Tan & Kramer, 1997) deal with minimizing the cost of maintenance or maximizing an availability related performance measure without tak-ing into consideration the existtak-ing uncertainty in the process and its interactions with the optimal mainte-nance policy. Gradually, general mathematical frame-works have started to appear (Straub & Grossmann, 1993; Thomaidis & Pistikopoulos, 1994, 1995; Thomaidis, 1995; Dedopoulos & Shah, 1996) assessing the process both from a flexibility and a reliability point of view, without addressing, however, detailed mainte-nance considerations. In this context, the purpose of our work is to substantially expand and complete these initial works by proposing a rigorous mathematical framework for identifying optimal detailed maintenance and operating policies for processes operating in the presence of uncertainty, having accounted for their interactions and the impact of uncertainty. To achieve this, the proposed framework links process models to detailed reliability and maintenance models.
2. System characterization
2.1. Process model
Steady state continuous process operation in the presence of uncertainty, can be described by a set of process model equality and inequality constraints as follows (Biegler, Grossmann & Westerberg, 1997): h(z,x,q)=0
g(z,x,q)50 (1)
where
zZ is the vector of degrees of freedom (e.g. inlet streams, temperatures, split ratios, etc.) manipulated to achieve maximum process performance,
xX is the vector of process variables (e.g. flowrates),
q[ is the vector of continuous uncertain parame-ters (e.g. supply, demand, etc.) assumed to follows a continuous probability distribution function jpdf(q) and [={ql5q5qu} is the uncertain parameter
space.
The vector of equalities (h) corresponds to process equations (heat and mass balances, equilibria relations) while the vector of inequalities (g) denotes process specifications and logical constraints. The process model in Eq. (1) defines a feasible operating region (FOR) in the space of the uncertain parameters (e.g. supply and demand)
FOR=
!
q×(z,x):!
h(z,x,q)=0g(z,x,q)50
""
(2) The feasible operating region contains all the realiza-tions of the uncertain parameters for which feasible operation can be guaranteed by properly adjusting the values of the degrees of freedom. The revenue gener-ated per time unit is a function of the degrees of freedom (z), the process variable (x) and the parameter (q). Denoting this function by r(z, x, q), the revenue per time unit corresponds to the expected maximum attainable value of rachieved by proper adjustment of the degrees of freedom. The overall expected revenue rate can then be determined by the solution of the following optimization problem (Eq. (P1)).ERR=
&
q max x,z r(x,z,q)j(q)d(q) (P1) s.t. h(x,z,q)=0 g(x,z,q)50The evaluation of the expected revenue rate ERR is a challenging problem which, among others, is discussed in detail in Straub and Grossmann (1993), Pistikopou-los and Ierapetritou (1995), Acevedo and PistikopouPistikopou-los (1998).
Fig. 2. Feasible operating region of the four-component process system.
whose components are all functioning properly. Ac-cording to the operating program, expected process revenue can be evaluated using Eqs. (1) and (2) and problem (Eq. (P1)). If a failure occurs, the process model determining the operation changes, according to which components have failed, to a new (but also known) set of equations similar to Eq. (1) and, as a result, expected revenue changes. In discrete event sys-tem theory, the state of a syssys-tem at timetis defined as (Cassandras, 1993) the information required at t, such as the output of the system is uniquely determined by this information and the input. Since the output of the system is the expected revenue rate and the informa-tion required to determine it is which components are working and which are not at time t, the state of the process system is defined as a vector, the elements of which correspond to the operating status of each com-ponent. Failures can be represented by a transition from one system state to another and all the possible states form the state-space of the system. To illustrate these concepts, consider the chemical complex in Fig. 2a, comprising four units, for the production of chem-ical C either through the intermediate product B or by direct conversion of A to C, with uncertainty in sup-ply and demand (taken from Straub & Grossmann, 1990). Such a process can be described by a mathe-matical model similar to (Eq. (1)) and the expected revenue rate can be determined solving an optimiza-tion problem similar to (Eq. (P1)). In the time horizon of operation, a component can be either failed or functioning. The status of the component is described by a binary parameter jst, j=1, …, 4 where
jst=
!
1 if component j is functioning 0 if component j has failed and st=( 1 st, 2 st, 3 st, 4
st) is the state vector of this
four-component system. Each realization of the state vector describes a possible system state and hence a system with four components involves a total of 24=
16 states in which the system may reside with possible degradation and failure of equipment. The operable states of the process can be determined either by the structure function of the system or its minimal cuts (see, for example, Hoyland & Rausand, 1994). The structure function of the chemical complex in Fig. 2a is ( st) = 4st+ 1st· 3st+ 2st· 3st− 1st· 2st· 3st− 1st· 3st · 4st+ − 2 st· 3 st· 4 st+ 1 st· 2 st· 3 st· 4 st
and the minimal cuts are {3,4} and {1,2,4}. Subse-quently, the states of the system are classified as struc-turally operable or inoperable as shown in Table 1. Table 1
Structural classification of system states
( 1st, 2st, 3st, 4st) ( st) System statek (1,1,1,1) 1 Operable 1 2 (0,1,1,1) 1 Operable 3 (1,0,1,1) 1 Operable 1 Operable 4 (1,1,0,1) (1,1,1,0) 5 1 Operable (0,0,1,1) 1 Operable 6 (0,1,0,1) 1 Operable 7 (1,0,0,1) 1 Operable 8 1 Operable (1,0,1,0) 9 (0,1,1,0) 1 Operable 10 11 (0,0,0,1) 1 Operable (1,1,0,0) 0 Inoperable 12 (1,0,0,0) 0 Inoperable 13 (0,1,0,0) 14 0 Inoperable (0,0,1,0) 0 Inoperable 15 0 Inoperable (0,0,0,0) 16
2.2. State space representation
Due to the change in system configuration because of failure, each operable system state k is described by a different set of equality (hk) and inequality constraints (gk). For this particular process, the model given in Table 2a (Straub & Grossmann, 1990) describes all the 16 states in which the system may reside by adjusting the values of the binary parameters jst according to
whether unitjis functioning or not at system statek.F represents the flowrates through the process units and hj,dj, are the conversion rates and equipment volumes of each unit, respectively. Sup and Dem are the uncer-tain parameters corresponding to supply and demand, respectively.
The process model (equality and inequality con-straintshkandgk) in each statekdefines a correspond-ing feasible operatcorrespond-ing region FORkin the space of the uncertain parameters (i.e. supply and demand):
FORk=
!
q×(zk,xk):!
hk(zk,xk,qk)=0gk(zk,xk,qk)50
""
, Ök (3) The feasible operating region contains all the realiza-tions of uncertain parameters for which feasible opera-tion of the system can be guaranteed by properly adjusting the values of the degrees of freedom. If the system is in the fully operable state (1,1,1,1), i.e. all theequipment components are functioning, this gives rise to a feasible operating region in the space of the uncertain parameters, as shown in Fig. 2b. On the other hand, if one or more units fail (e.g. if unit 4 fails) the system may go on operating in one of the degraded states, as shown in Fig. 2c. Note that, in principle, the corresponding feasible operating region is reduced, reflecting the fact that when equipment fails a smaller portion of demand and less throughput can be accom-modated through the process (see also, Thomaidis & Pistikopoulos, 1995).
As a result, for each stateka corresponding expected revenue rate ERRkcan be defined which can be evalu-ated by the solution of the following optimization problem (Eq. (P2)), similar to (Eq. (P1)):
ERRk=max xk,zk
&
q rk(xk,zk,qk)j(qk)d(q),Ök (P2) s.t. hk(xk,zk,qk)=0, ÖkS gk(xk,zk,qk)50, ÖkSA set of independent and identically distributed set of parameters (q1,q2, … qk) is used to denote the
realiza-tion of uncertain parameters in each state k.
Furthermore, since the transitions between the states are related to equipment failure and repair, each state has a probability of occurrence, Prk(t) which depends on the time-varying availability of each component determined by the reliability and maintainability char-acteristics of the equipment. This is described next.
2.3. Reliability/a6ailability modeling considerations The probability of the system being in statekat time tis a function of the degrading reliability characteristics of the equipment and the implemented maintenance policy. The defining elements of any maintenance policy are the maintenance policy assumptions, regarding the type of maintenance that can be performed on equip-ment components and the maintenance optimization variables, regarding the characteristics of optimal maintenance strategies to be determined.
The selection of the type of maintenance depends on the equipment attributes and specifications as well as the available maintenance facilities and capabilities. Different types of maintenance are defined both at a system level and at a component level. In the case of complex systems, for example, groups of components with similar operating conditions may be identified and treated uniformly during maintenance (e.g. group pre-ventive maintenance policies). Furthermore, at a com-ponent level, assumptions are made regarding the effectiveness of maintenance in restoring the component to a good condition. As-good as-new (AGAN) policies, Table 2a
Chemical complex process
Specifications Mass balance F1=F2+F3 F55d1· 1st F3=F4+F5 F45d2· 2st (F6+F7)5d3· 3 st F10=F8+F9 F7=h1F5 F25d4· 4st F6=h2F4 F15Sup F8=h3(F6+F7) F10]Dem F9=h4F2 Table 2b 2st 1st State 3st 4st ERRk 1 1 1 1 1 1147 1 1 2 0 1 1125 1 1 1133 0 1 3 4 1 1 0 1 343 5 1 1 1 0 114 343 1 1 6 0 0 1 7 0 1 0 343 1 0 8 0 1 343 0 0 0 9 1 343 SN=12;|
S=1; cost of A=40;DN=7;|D=1; revenue from C=200
for example, restore the component to its original con-dition at the beginning of operation, while as-good-as-old (AGAO) policies bring it back to where it was immediately before the maintenance task started.
Maintenance optimization variables correspond to the elements of the maintenance policy that can be treated as decision variables to optimize a maintenance or a performance related criterion. These variables usu-ally involve the number of maintenance actions to be performed, the length of maintenance intervals, the allocation of maintenance crews, etc.
Depending on the nature and complexity of the assumptions and the desired level of detail and depth, different mathematical modeling tools can be employed in maintenance optimization frameworks, such as ana-lytical techniques and Markov’s models (Tan & Kramer, 1997).
Markov’s models and decision processes are powerful modeling tools having the flexibility to model a wide variety of cases such as the existence of redundancy and spares, dependent failures and sequence dependent be-havior. The key elements behind Markov’s models are states and transitions. Transitions occur as a result of events or actions (e.g. failure, utilization of spares, repair, preventive maintenance, sequence of mainte-nance activities, etc.) and the different states represent the behavior of the system as a result of the transitions. According to whether a discrete or a continuous time representation is adopted, Markov’s models and deci-sion processes can lead to either dynamic programming or optimal control problems (Gertsbakh, 1977).
In an analytical approach, on the other hand, the objective is to express every term in the model (e.g. probability of occurrence of each state, equipment availability, maintenance costs, etc.) as a function of the maintenance optimization variables. Then, standard optimization techniques can be used to obtain the maintenance policy that optimizes a maintenance-re-lated criterion (see, for example, Vatn et al., 1996). In this work, an analytical approach will be followed. Extensions towards maintenance models described by continuous time Markov’s chains are discussed in Vas-siliadis and Pistikopoulos (1999b).
2.4. System effecti6eness measure
Having evaluated the expected revenue under uncer-tainty characteristics ERRk of each state k, as deter-mined by the solution of Eq. (P2), the expected revenue of the process is defined (Eq. (4)) as the weighted sum of the expected revenue of each system state using as weights the probability of the system being in each particular state ERH=
&
H % kS Prk(t)·ERRkdt (4)where Prk(t) is the probability of occurrence of state k as a function of time.
The expected revenue, as defined in Eq. (4), is a system effectiveness measure taking into account pro-cess (including the existing uncertainty), reliability and maintainability characteristics. The expected revenue rate ERRk of each state is determined by the process model, the feasible operating region in that state and the uncertainty. On the other hand, the state probabili-ties Prk(t) are a function of the reliability and maintain-ability characteristics of the process. The implemented maintenance policy depends on the maintenance policy assumptions (e.g. age replacement, block replacement, etc.) and the maintenance optimization variables (time of maintenance, number of maintenance activities, se-quence of actions, etc). The above suggests that in order to maximize system effectiveness, as measured by the expected revenue in Eq. (4), both process operation and maintenance have to be optimized accordingly.
3. Modelling and optimization framework
3.1. General description
Maintenance contributes to the profitability of the process mainly by keeping the plant functioning and capable of fulfilling production needs for longer periods of time (i.e. by providing higher plant availability). As already mentioned, determining the optimal mainte-nance policy corresponds to identifying the process availability level which sustains a balance between long-term maintenance costs and production level that max-imizes plant profitability. Traditional availability analysis considers functioning as a discrete event de-scribed by a zero-one variable, i.e. the system either functions or not; this simplification however cannot readily capture the physicochemical characteristics of the process. Therefore, the use of criteria that do not explicitly relate to process profitability (e.g. availability or reliability importance indices, etc.) as optimization objectives does not allow for the quantification of the balance between maintenance benefits and costs. To overcome these problems, the expected revenue — as a system effectiveness metric — is used as part of the maintenance optimization objective. Therefore, the simultaneous process and maintenance optimization problem can be conceptually posed as follows:
s.t.
max{expected revenue−maintenance costs} process model
maintenance model
(P3) The intention is to manipulate the process related and the maintenance related optimization variables so
as to maximize the balance between process revenue and maintenance costs, which are partly conflicting objectives. This optimization is subject to the rigorous process and maintenance models. Note that all process, reliability and maintenance characteristics are incorpo-rated in problem (Eq. (P3)), the solution of which simultaneously determines the optimal operating policy and the optimal maintenance policy. More specifically (i) the maintenance model and the implemented mainte-nance policy determine the cost of maintemainte-nance and the probability of occurrence of each state, Prk(t), and (ii) the process characteristics and the existing uncertainty, on the other hand, determine the expected revenue rate of each state, ERRk.
In this work, an analytical approach is developed to describe the maintenance model in (Eq. (P3)). In partic-ular, the probability of occurrence of each system state k (Prk(t)) and the maintenance costs — appearing in the objective function in (Eq. (P3)) comprising the terms that are determined by the maintenance strategy — are analytically expressed as a function of the maintenance optimization variables. This is described next.
3.2. Maintenance optimization model
Consider a system withMcomponents. LetA1,j(t) be
the initial (before any maintenance is performed) availability (i.e. reliability) function of component j, 15j5M as a function of time. Let, also, t1 and tN denote the beginning and the end, respectively, of the time horizon of interest and tu, 25u5N−1 the time instants at which we perform maintenance to any of the components according to a maintenance schedule. The number of maintenance actionsN−2, the maintenance time instants and the detailed maintenance schedule are optimization variables to be determined.
The execution of the maintenance activities depends on the following assumptions dictated by the reliability and maintainability characteristics of the system and the individual components:
at each maintenance time instant tu, maintenance is performed to only one of the units in the following way:
1. correcti6e, if the unit is down.
2. pre6enti6e, if it is operable.This assumption is common in the literature (see, for example, Tseng, 1996) and valid in many real cases. all maintenance is of an AGAN type, i.e. the
compo-nent is restored to its initial condition at the begin-ning of operation.
all failures are independent.
To determine the optimal maintenance policy three types of maintenance optimization variables are introduced:
N, which is an integer variable denoting the number of maintenance actions (N−2) performed to the equipment components within the time horizon of operation.
tu, which is a continuous variable representing the maintenance time instant of maintenance action (u−1).
uu,j, which are 0-1 variables defined as:
uu,j=
!
1 if maintenance actionu−1 is performed on component j 0 elseBased on the above maintenance policy assumptions, analytical expression can be recursively derived (see A.1) describing the availability of each component at each time instant in the time horizon as a function of the maintenance optimization variables (N, uu,j, tu).
Therefore, the availability of component j, 15j5M after (u−1)th and before theuth maintenance action is given by Au,j(t)= % u k=1
5u i=k+1 (1−ui,j)·uk,j·A1,j(t−t%k)n
, 25u5N−1 (5)and the availability of component j, 15j5M during the (u−1)th maintenance action is given by
A(u,j(t)=(1−uu,j) · % u−1 k=0
u5−1 i=k+1 (1−ui,j)·uk,j·A1,j(t−t%k)n
, 25u5N−1 (6)Note that during a maintenance action certain com-ponents may still be operable and therefore, the process may go on operating in one of the degraded states of the state-space. Taking this into consideration, the availability of each component has been expressed both during and after maintenance actions. Similarly, the probability of the system being in state k after the (u−1)th and before the uth maintenance action is given by (see A.2)
Prk,u(t)= 5 jOPk Au,j(t) · 5 jO(Pk (1−Au,j(t)), t%u5t5tu+1 (7) where OPk(O( Pk) is the set of operable (inoperable) components in system state k and Au,j(t) is the
availability of component j after the (u−1)th and before the uth maintenance action given by Eq. (5). During the (u−1)th maintenance action, on the other hand, the probability of occurrence of state k will be given by
P( rk,u(t)= 5 jOPk A( u,j(t) · 5 jO(Pk (1−A(u,j(t)), tu5t5t%k (8) where A( u,j(t) is the availability of component j during
the (u−1)th maintenance action. The expected dura-tion of the (u−1)th maintenance task as a function of the maintenance optimization variables is given by (see A.3) ~u= % M j=1 uu,j
1−%ku=−115iu=−k1+1(1−ui,j) ·uk,j·A1,j(tu−t%k)n
·~corr,j+%ku−=115iu=−k1+1 (1−ui,j)·uk,j·A1,j(tu−t%k)n
·~prev,j (9)where ~corr,j and ~prev,j are the fixed durations of cor-rective and preventive maintenance tasks for unit j, respectively. Then, the process revenue over the whole time horizon H, for a particular realization of process variables and uncertain parameters (zk, xk, qk) can be expressed as (see A.4)
RH= % N−1 u=1
&
tu+1 tu+~u % kS Pu,k(t) ·r(xk,zk,qk)dt + % N−1 u=2&
tu+~u tu % kS P(u,k(t) ·r(xk,zk,qk)dt (10)Eq. (10) comprises two terms. The first term corre-sponds to the revenue generated from the process be-tween theN−2 maintenance actions, while the second corresponds to the revenue generated during the N−
2 maintenance actions.
The total maintenance cost in the time horizon H is given by (see A.5)
m.cost= % N−1 u=2 %M j=1 uu,j
1−%ku=−11[(Piu=−k1+1(1−ui,j)) ·uk,j·A1,j(tu−t%k)]·Ccorr,j+%ku=−115iu=−k1+1 (1−ui,j)·uk,j·A1,j(tu−t%k)n
·Cprev,j (11)where Ccorr,j and Cprev,j are the fixed costs of correc-tive and prevencorrec-tive maintenance, respeccorrec-tively, for unit j.
Note that Eqs. (5) – (11) explicitly associate all maintenance terms with the set of maintenance opti-mization variables (N, uu,j, tu).
3.3. Process/maintenance optimization model
By collecting all the terms and equations together, problem (Eq. (P3)) is analytically written in the form
of problem (Eq. (P4)). Problem (Eq. (P4)) incorpo-rates all processes (including the existing uncertainty), reliability and maintenance characteristics of the sys-tem and aims at the optimization of the balance be-tween process revenue and maintenance costs.
&
q1&
q2 ···&
qk max N,uu,j,tu,zk,xk N%−1 u=1&
tu+1 tu+~u % ks Pu,k(t) ×·rk(zk,xk,qk) + % N−1 u=2&
tu+~u tu % ks P(u,k(t) ·rk(zk,xk,qk)dt − % N−1 u=2 %M j=1 uu,j[(1−Au−1,j(tu)) ·Ccorr,j +Au−1,j(tu) ·Cprev,j]jpdf(q1q2…qk)dq1dq2…dqkn
(P4) s.t. hk(xk,zk,qk)=0, ÖkS gk(xk,zkqk),50, ÖkS (12) %M j=1 uu,j=1, Öu[2 …N−1] (13) tu+~u5tu+1, 15u5N−1] (14)The objective function is a function of the mainte-nance optimization variables (N, uu,j, tu), the process
variables (xk, zk) and the uncertain parameters (qk). The first two terms in the objective correspond to the expected process revenue. Since all operable process states contribute to the process revenue the expec-tancy due to uncertainty is defined over the feasible operating regions of all operable states, as determined by the process model in each state. Therefore, a set of independent and identically distributed set of parame-ters (q1, q2, …, qk) is used to denote the realization
of uncertain parameters in each state k. Their joint probability distribution function is denoted by jpdf(q1,
q2, …, qk). The third term of the objective function is
the maintenance cost which is not associated with process uncertainty.
The solution of Eq. (P4) provides the operating policy and the maintenance schedule that maximizes the balance between process revenue and maintenance costs, for a system operating under process uncertainty and under the maintenance policy assumptions stated in Section 3.2. In particular, the optimal number of maintenance actions required for the time horizon of operation (N), the optimal maintenance schedules (uu,j)
and the exact optimal maintenance time instants (tu) are identified in conjunction with the optimal values for the process variables (zk,xk). Eq. (13) suggests that one maintenance task is performed at a time while Eq. (14)
Fig. 3. Availability threshold values.
Problem (Eq. (P5)) corresponds to a mixed-integer nonlinear programming formulation involving discrete decisions (number and sequence of maintenance ac-tions) and integral terms with integrands defined implic-itly (as a function of 0-1 variables). The main difficulties in handling a problem such as the above are the following:
the presence of highly nonlinear terms in terms of the 0-1 variables (e.g. the terms Piu=−k1+1(1−
ui,j)uk,j included in the availability expressions in-corporated in the terms Prk(t) in the expected revenue terms of the objective function). Simplifica-tions and linearizaSimplifica-tions are not straightforward and may lead to a dramatic increase of the size of the problem.
it is highly combinatorial. An increase in the num-ber of components and maintenance actions leads to a large number of possible maintenance sched-ules.
The direct solution of (Eq. (P5)) is therefore very complex. To overcome this, an effective two step tion strategy is proposed which avoids the direct solu-tion of (Eq. (P5)) by an approximate reformulasolu-tion based on the concept of availability threshold values. This solution strategy is described next.
4. Solution strategy
4.1. A6ailability threshold 6alues
Consider the case when maintenance actions are per-formed periodically to each component j every tj time units. The availability function of component j before the first maintenance action onjis equal to the reliabil-ity function of j,
Aj(t)=Rj(t), 05t5tj, 15j5M (15) Since maintenance restores the component to an AGAN condition, the availability of j after the first maintenance action will be given by
Aj(t)=Rj(t−tj), tj5t52 ·tj, 15j5M
and, therefore, the availability of componentjafter the nth maintenance action on j is given by
Aj(t)=Rj(t−n·tj), n·tj5t5(n+1) ·tj, 15j5M This is depicted in Fig. 3a.
Due to the monotonicity of the reliability function and the periodicity of maintenance, these maintenance time instants (tj, 2tj, …, ntj) for unit j correspond to a specific (same for each unit) availability value Ath,jfor that unit. This one-to-one relationship is determined by the reliability function of the unit
ensures that the time instant of the uth maintenance action follows that of the (u−1)th.
Taking advantage of the mathematical structure of problem (Eq. (P4)) (see Vassiliadis, 2000 for the mathe-matical proof), the process optimization part of the problem can be isolated and performed separately. Therefore, (Eq. (P4)) can be recasted as follows:
max N,uu,j,tu % N−1 u=1
&
tu+1 tu+~u % ks Pu,k(t) · ERRkdt + % N−1 u=2&
tu+~u tu % ks P(u,k(t) · ERRkdt − % N−1 u=2 %M j=1 uu,j[(1−Au−1,j(tu)) ·Ccorr,j +Au−1,j(tu) ·Cprev,j] (P5) s.t % M j=1 uu,j=1, Öu[2…N−1] tu+~u5tu+1, 15u5N−1where ERRkis the expected revenue rate for each state k of the system obtained from the solution of the process optimization problem (Eq. (P2)).
Ath,j(t)=Rj(tj), 15j5M (16)
This is also depicted in Fig. 3a.
These maintenance time instants are unknown; for their determination it would suffice to determine the component availability value, Ath,j, they correspond to. Therefore, an alternative way to identify the opti-mal maintenance time instants for each component and, hence, the sequence and number of maintenance actions, is to determine a set of optimal a6ailability threshold 6alues Ath,j. This implies that the time
in-stant at which a maintenance action will be performed for component j will correspond to the time instant at which the availability of the component falls to the threshold valueAth,j.
This is schematically shown in Fig. 3b, for a two-component system. Whenever the availability function of the first unit falls to its threshold value, Ath,1 maintenance is performed on unit 1, while whenever the availability function of the second unit falls to Ath,2 maintenance is performed on unit 2. The number
of maintenance actions for each component and their sequence is automatically determined by representing the maintenance time instants on the time axis. In the case of the example depicted in Fig. 3b, nine mainte-nance actions take place (three for component 1 and six for component 2) and their sequence is 212212212. The significance of the availability threshold values is attributed to the fact that it is feasible to derive good approximations of all the terms and expressions in problem (Eq. (P4)) as a function of the availability threshold values (see Appendix B). This property is used for the construction of the first step of the pro-posed solution strategy.
4.2. Algorithm
The solution of the proposed formulation (Eq. (P5)) seeks to determine the number and sequence of maintenance actions, i.e. optimal values forN and uu,j and the exact maintenance time instants, i.e. optimal values for tu. In this section, we propose a two-step solution strategy according to which in the first step the number and an initial sequence of maintenance actions is obtained while in the second step this infor-mation is used to determine the optimal maintenance sequence and the optimal exact maintenance time in-stants.
4.2.1. Step 1:determination of the a6ailability threshold
6alues — number and initial sequence of maintenance actions
Using the concept of availability threshold values, a nonlinear mathematical programming problem (Eq. (P6)) approximating (Eq. (P5)) is proposed in which all the terms in the constraints and the objective
func-tion are expressed as funcfunc-tions of the threshold values Ath,j max Ath,j % kS Hk(Ath,j) · Prk(A*j) · ERRk−m.cost(Ath,j) (P6) s.t. 05Ath,j51, Öj where
Hk(Ath,j) is the maximum amount of time that the
system can spend in state k as a function of the availability threshold values Ath,j. This is equal to the time horizon H subtracting the expected dura-tion of the maintenance acdura-tions in this state. Aj* is the average availability of component j in the
time period to the first maintenance action tj. Since on the first step of the algorithm we have assumed periodic maintenance, A* is the average availabilityj of component j for the whole time horizon except the time that component j is actually maintained. The average availability of component j over the time interval tj is evaluated from (Lewis, 1994) A*j= 1 tj
&
tj 0 Aj(t) dtwhere Aj(t) is the availability function of compo-nent j given by Eq. (15).
Prk(Aj*) is the probability of the system being in state k as a function of the average availability of each component j, Aj*. Since independent failures have been assumed, these probabilities can be eval-uated from Prk(A*j)= 5 iOPk A*j · 5 jO(Pk (1−A*)j
Obviously, an approximation of the expected rev-enue generated by the process in the time horizon H is given by multiplying the probability of the system being in state k by the maximum amount of time the system can spend in state k.
m.cost(Ath,j) is the expected maintenance cost
ex-pressed as a function of the availability thresholds. Note that in problem (Eq. (P6)), the average availability levels, Aj*, for each component j are opti-mized so as to maximize the expected profit of the process and balance the trade-offs between process revenue and maintenance costs.
The formulation in problem (Eq. (P6)) is general and not based on any assumptions regarding the probability distribution functions describing the reli-ability characteristics of the components. In this work, we assume that equipment failure rate is linearly in-creasing, approximating the wear-out phase of a unit the reliability of which is described by a Weibull probability distribution function.
4.2.1.1. Linearly increasing failure rate. Suppose that the system components operate in the wear-out period. To approximate this situation, we assume that compo-nent failure rates are not constant, but linearly increas-ing with time
hj(t)=bj·t+ej
In this case it is shown (see Appendix B) that prob-lem (Eq. (P6)) can be rewritten as
max Ath,j % kS Hk(Ath,j) · Prk(A*j) · ERRk−m.cost(Ath,j) (P7) s.t. 05Ath,j51, Öj m.cost(Ath,j) = −% j H [−ej+ej2−2 ·bj· ln(A th,j)]/bj ((1−Ath,j) ·Ccorr,j+Ath,j·Cprev,j), Öj Hk(Ath,j) =H+ % jOP H [−ejej2−2 ·bj· ln(A th,j)]/bj .~j, Öj, k ~j=(1−Ath,j) ·~corr,j+Ath,j·~prev,j, Öj Prk(A*j)= 5 iOP A*j · 5 jO(P (1−A*),j Öj, k Aj*=0.5 · % q i=0 q· e−bj/2 · [(zq·tj+tj)/2]2−ej·[ (zq·tj+tj)/2], Öj Problem (Eq. (P3)) is a nonlinear mathematical pro-gramming formulation, the solution of which provides the availability thresholds for each component in order to maintain average component availability to a level that maximizes the expected profitability of the process, given that maintenance is performed periodically. From the values of the availability thresholds we can deter-mine the number of maintenance actions Nj required per unit and also an initial sequence of maintenance actions.
The assumption of periodic maintenance, (which ex-plains why there is only one availability threshold value for each component), will be relaxed in the second step of the algorithm, where the initial maintenance se-quence will be corrected, if necessary, and the exact optimal maintenance time instants will be obtained.
4.2.2. Step 2:determining the optimal maintenance sequence and the optimal maintenance time instants
The solution of the first step is used as an initial point to obtain the optimal maintenance sequence and the optimal exact maintenance time instants. For this purpose, a series of NLPs solved iteratively in the following way:
1. with fixed number and sequence of maintenance actions N( and u¯u,j, as they are provided by step1,
problem (Eq. (P5)) reduces to a standard nonlinear optimization problem, where the optimization vari-ables are the maintenance time instants tu.
max tu % N(−1 u=1
&
tu+1 tu+~u % kS Pu,k(t)ERRk dt + % N( −1 u=2&
tu+~u tu % kS P(u,k(t) · ERRk dt − % N( −1 u=2 %M j=1 u¯u,j[(1−Au−1,j(tu)) ·Ccorr,j +Au−1,j(tu) ·Cprev,j] (P8) s.t. tu+~u5tu+1, 25u5N−1 (17)The solution of (Eq. (P8)) provides the maintenance time sequencetu,u[2, …,N( −1], for the particular maintenance sequence u¯u,j, which is imposed by constraint (Eq. (17))
2. if some of the inequality constraints in (Eq. (P4)) are active, i.e.
tu+~u5tu+1, 25u5N−1
we go back to step 2(a) and check whether changing the sequence of the maintenance actions involved in this constraint will improve the solution of (Eq. (P8)).
3. if there are no active constraints or no better solu-tion is obtained, stop.
It should be noted that the two-step strategy does not theoretically guarantee to identify the optimal solution of problem (Eq. (P1)) in a strict mathematical sense but only a lower bound solution (valid if each NLP is solved to global optimality). If the approximation, however, used in the first step is good, then the solution is often very close to the optimal solution as will be discussed in the following section.
Next, the optimal inspection policy will be identified in the case of equipment units with linearly increasing failure rates; subcases are examined to illustrate the impact of process uncertainty, maintenance costs and maintenance action duration.
5. Numerical example
The chemical complex shown in Fig. 2a converts species A to C either through the production of the intermediate product B or by direct conversion of A to C. The supply of raw material A and the demand of product C are considered to be continuous uncertain parameters, described by normal probability distribu-tion funcdistribu-tions. The process model, the mathematical
Table 3a
Case 1, failure rate characteristics and results of steps 1 and 2 e b Unit 1 2×10−4 5×10−7 5×10−7 2 2×10−4 5×10−7 2×10−4 3 2×10−4 4 5×10−7 Table 3c
Item Maintenance sequence Profit 4,3,4,1,3,2,4,3,4,1,3,2,4,3,4 1.98×106 1 4,3,4,1,3,2,4,1,3,4,3,2,4,3,4 1.99×106 2 4,3,4,1,3,2,4,1,3,4,2,3,4,3,4 3 2.00×106 4 4,3,4,1,2,3,4,1,3,4,2,3,4,3,4 2.01×106
description of the uncertainty, the nine structurally operable states and their expected revenue rates are presented in Tables 2a and 2b. The conversions fac-tors hj for each unit j are 0.92, 0.9, 0.85, 0.75 and the equipment volumes dj are 5, 5, 7, 9, respectively. The revenue rate for each system state is evaluated by integrating the probability distribution function of the revenue rate function over the feasible operating re-gion of each state (problem Eq. (P2)). The solution to this problem is obtained using a Gaussian integration scheme with ten quadrature points for both continu-ous uncertainties, similar to the one presented in Acevedo and Pistikopoulos (1998). The results are summarized in Table 2b. All the NLP formulations were solved using the GAMS modeling package (Brooke, Kendrick & Meeraus, 1988). In this case, there are nine states with a non-zero revenue rate.
5.1. Increasing failure rate
5.1.1. Case 1:finding the optimal maintenance policy The data required to describe the component fail-ure rates is given in Table 3a. The cost and the dura-tion of preventive and corrective maintenance acdura-tions are the same as in the previous case. In the first step of the algorithm, the optimal availability threshold and the optimal average availability for each compo-nent are obtained by solving problem (Eq. (P7)). These values yield a number of maintenance actions per component and an initial maintenance sequence. The results of the first step are shown in Table 3b. In the second step of the algorithm, the maintenance sequence and the number of maintenance actions per unit obtained from the first step are used to solve problem (Eq. (P8)). By reversing the order of the active constraints (see Vassilaidis, 2000) the optimal
solution is obtained in four iterations (Table 3c). Fi-nally, the optimal maintenance schedule obtained is shown in Fig. 4a.
5.1.2. Case2:analyzing the impact of uncertainty Suppose there is no uncertainty in the process. The values of supply and demand are considered constant and equal to the nominal values (SN=12, DN=7) of normal probability distribution functions that de-scribed them when they were uncertain. The new op-erable process states and their corresponding maximum revenue rates are summarized in Table 4a. Implementing the first step of the maintenance opti-mization algorithm, the optimal availability threshold values and average availabilities for each component are obtained (see Table 4b). Then, the number of maintenance actions and an initial maintenance se-quence can be determined. Note that due to the fact that there are only three operable states, the average availability levels required per unit are higher. As a result, in order to achieve these levels, the number of maintenance actions is bigger. Note, also, that the threshold values for units 3 and 4 are identical. This is expected since neither in the process (both compo-nents participate in all the operable states), nor in the reliability and maintenance data, any distinguishing elements between these two units can be found. On the other hand, the availability threshold value for unit 1 is larger than that for unit 2, Ath,1\Ath,2, since unit 1 participates in the most profitable states of the process. In the second step of the algorithm an opti-mal solution is obtained, by reversing the order of the active constraints, in nine iterations. The results are summarized in Table 4c. The optimal mainte-nance schedule is shown in Fig. 4b.
Table 3b
Ath A* Maintenance instants Maintenance actions
Unit 0.782 669, 1338 1 0.903 2 2 0.771 0.899 695, 1390 2 0.957 342, 684, 1026, 1368, 1710 0.907 5 3 315, 630, 945, 1260, 1575, 1890 4 0.916 0.961 6
Fig. 4. Cases 1 – 4, optimal maintenance schedules.
Comparing cases 1 and 2, we observe that the opti-mal preventive maintenance schedule for the same system is different in the presence of uncertainty. This should not be surprising despite the fact that uncer-tainty is process related and not reliability or mainte-nance related. If there is uncertainty in the process, then process revenue changes. Therefore since mainte-nance optimization defined as identifying the balance between process revenue and maintenance costs that maximizes profitability, the optimal maintenance pol-icy has to change in the presence of uncertainty. The error induced if uncertainty is not taken into account when trying to find the optimal preventive mainte-nance policy is estimated in Table 4d. If uncertainty had been ignored, the optimal maintenance policy along with the values of expected profit and mainte-nance costs would have been as shown in the first column (‘imaginary’ case) of Table 4d. However, the expected profit would have been much smaller if the policy depicted as optimal in the ‘imaginary’ case was implemented for the same system in the presence of uncertainty (second column), because it does not ac-count for it. On the other hand, if uncertainty is taken into account, the optimal maintenance policy is different leading to a larger expected profit and smaller maintenance costs (third column).
5.1.3. Case 3:effect of maintenance task durations In this case, the process characteristics remain the same while the duration of preventive and corrective maintenance tasks for the first unit is increased to
~prev,1=100 and ~corr,1=20, respectively. The
im-pact of this increase to the maintenance schedule is captured quantitatively in the first step of the al-gorithm where the optimal availability threshold val-ues and average availabilities are determined. Note that the availability threshold value of the first unit is dramatically decreased compared with case 1 while there is a big increase in the availability threshold value of the second unit, which operates in parallel with the first. These changes suggest that unit one should be maintained less frequently since mainte-nance activities are economically unattractive due to the potential loss of production from the large repair times of this unit. On the other hand, unit 2 should be maintained more frequently to compensate for the reduction in the availability of the first unit. As a result the number of maintenance actions for the first unit is reduced to one and the number of mainte-nance actions for the second unit is increased to three. The results are depicted in Table 5a. In the
Table 4a
Case 2, revenue rate characteristics and results of steps 1 and 2
3st 1st 2st ERR k State 4st 1 1 1 1200 1 1 2 0 1 1 1 1177 1 3 1 0 1 1193
Table 4b
Ath
Unit A* Maintenance interval Maintenance actions
0.913 1 0.805 615 3 0.906 655 0.788 2 2 0.919 3 0.962 310 6 0.962 4 0.919 310 6 4,3,4,1,3,2,4,3,1,4,3,2,4,3,1,4,3 Initial maintenance sequence
second step of the algorithm, the information derived from the first step is used in order to obtain the optimal maintenance sequence and the optimal exact mainte-nance time instants in three iterations. The results of the second step are depicted in Table 5b. The optimal maintenance schedule is shown in Fig. 4c.
5.1.4. Case 4:effect of maintenance costs
In this case, the costs of preventive and corrective maintenance of units 3 and 4 are increased tocprev,3=
cprev,4=5000 and ccorr,3=ccorr,4=25 000, respectively.
This increase of costs results in a significant decrease of the optimal availability thresholds and average levels of availability of units 3 and 4. Subsequently, the optimal number of maintenance actions for units 3 and 4 is smaller, since the big increase in maintenance costs renders maintenance activities economically unattrac-tive. The results of the first step are shown in Table 6a. In the second step, solving the NLP formulation in Eq. (P8) for the resulting initial maintenance sequence (4, 3, 1, 2, 4, 3, 4, 3), the solution depicted in Table 6b is obtained which corresponds to the maintenance sched-ule depicted in Fig. 4d. Because of the small number of maintenance actions, units are more likely to fail and the system is more likely to operate in less profitable states. Furthermore, expected maintenance costs are high. As a result, the expected profit of the system is significantly reduced compared with case 1.
6. Concluding remarks
In this work, a rigorous optimization framework for the optimization of preventive maintenance in process systems is presented. The proposed methodology differs from traditional approaches in that:
the objective is to identify the maintenance policy that optimized process profitability instead of an availability-related criterion;
in addition to the maintenance model, the interac-tions of maintenance, process characteristics and the process uncertainty are taken into consideration by including the full process model.
Therefore, the optimal maintenance policy and the optimal process-operating pattern are identified via the
solution of the same optimization problem, having accounted for their interactions. This approach facili-tates the quantification of the impact of process charac-teristics, including the existing uncertainty, upon the optimal maintenance policy. It is shown that although uncertainty is related to process and not reliability-maintenance parameters, it strongly influences the de-Table 4c
Maintenance sequence Profit Item 1 4,3,4,1,3,2,4,3,1,4,3,2,4,3,1,4,3 2.054×106 2.0551×106 4,3,4,1,3,2,4,3,1,4,2,3,4,3,1,4,3 2 3 4,3,4,1,3,2,4,3,1,2,4,3,4,3,1,4,3 2.0554×106 4,3,4,1,3,2,4,3,2,1,4,3,4,3,1,4,3 4 2.0556×106 2.0558×106 4,3,4,1,2,3,4,3,2,1,4,3,4,3,1,4,3 5 4,3,4,2,1,3,4,3,2,1,4,3,4,3,1,4,3 6 2.056×106 4,3,4,2,1,3,4,2,3,1,4,3,4,3,1,4,3 7 2.0562×106 4,3,2,4,1,3,4,2,3,1,4,3,4,3,1,4,3 8 2.0564×106 4,3,2,4,1,3,4,2,3,4,1,3,4,3,1,4,3 9 2.0565×106 Table 4d No uncertainty Uncertainty ‘Would’ve’
(‘Imaginary’ case) ‘Should’ve’
2 056 425 1 962 047 1 981 433 Expected profit 15 (2,2,5,6) 17 (3,2,6,6) Maintenance 17 (3,2,6,6) actions (N) 210 430 Maintenance 229 544 229 544 cost Table 5a
Case 3, results of steps 1 and 2
Unit Ath A* Maintenance actions 1 0.860 1 0.682 0.832 2 0.925 3 0.907 0.957 5 3 0.916 0.916 6 4 Initial maintenance 4,3,2,4,3,1,4,3,2,4,3,4,2,3,4 sequence
Table 5b
Item Maintenance sequence Profit 1 4,3,2,4,3,1,4,3,2,4,3,4,2,3,4 1977×106 1.978×106 4,3,4,2,3,1,4,3,2,4,3,4,2,3,4 2 4,3,4,2,3,1,4,3,2,4,3,4,2,3,4 1.98×106 3
poration of Markov’s maintenance models, which have the flexibility to describe a wide variety of maintenance policies (Vassiliadis & Pistikopoulos, 1999b).
Finally, in terms of the required computational ef-fort, an advantageous feature of the proposed solution strategy is the approximation of the original MINLP formulation by a series of NLPs. This greatly increases the number of the maintenance decision (in particular the 0-1) variables that can be handled to address bigger problems.
Nomenclature
vector of process equality constraints hk
at state k
gk vector of process inequality con-straints at statek
vector of degrees of freedom at state zk
k
xk vector of process variables at statek q vector of uncertain process
parameters
rk revenue rate function at state k FORk feasible operating region at state k
expected revenue rate at statek ERRk
jst binary parameter denoting whether unitj is up or down
( st) structure function of the system
hj Conversion factor for unit j equipment volume for unit j dj
Prk(t) probability of being at state kas a function of time t
N integer variable denoting the number of maintenance actions for all units of the system
uu,j 0-1 variable denoting whether the (u−1)th maintenance action is going to be performed on unitj or not tu continuous variable denoting the (u−1)th maintenance time instant Au,j(t) availability of unit j after the (u−
1)th and before the uth maintenance action as a function of the mainte-nance optimization variables (N, uu,j, tu) and time t
availability of unit j during the (u−
A(u,j(t)
1)th maintenance action as a func-tion of the maintenance optimizafunc-tion variables (N, uu,j, tu) and time t
Prk,u(t) probability of being at state kafter
the (u−1)th and before the uth maintenance action as a function of the maintenance optimization vari-ables (N, uu,j, tu) and time t
Table 6a
Case 4, results of steps 1 and2
Unit Ath A* Maintenance actions 0.779 1 0.902 2 2 0.769 0.897 2 0.817 3 0.918 3 4 0.839 0.928 3
Initial maintenance sequence 4,3,1,2,4,3,4,3 Table 6b
Maintenance sequence Profit Item
1 4,3,1,2,4,3,4,3 1.635×106
termination of the optimal maintenance policy. This is attributed to the fact that, in the presence of uncer-tainty, process profitability changes and, as a result, the balance between maintenance benefits and costs, which determines the solution of the maintenance optimiza-tion problem, also changes.
The proposed methodology allows for the study of the interactions between various reliability-maintenance characteristics and process operation in the determina-tion of the optimal maintenance policy. This is illus-trated in cases 3 and 4 of the numerical example for increasing failure rate, where changes in maintenance costs and maintenance task durations are taken into account. This significantly affects expected process profitability and the maintenance policy is modified accordingly.
Another important issue concerns the interactions of design and maintainability, since the selection of pro-cess design critically determines future reliability char-acteristics as well as preventive maintenance policies and maintenance costs. Extensions of this work toward incorporating maintainability issues at the design phase of a process are discussed in Pistikopoulos and Vassil-iadis (1998) and VassilVassil-iadis and Pistikopoulos (1999a). Depending on the complexity of the maintenance procedures and maintenance policy assumptions, it is not always possible to express the maintenance model in an analytical form. The structure of the proposed optimization framework, however, allows for the
incor-probability of being at state k during P( rk,u(t)
the (u−1)th maintenance action as a function of the maintenance op-timization variables (N, uu,j, tu)
and time t
expected duration of the (u−1)th ~u
maintenance action as a function of the maintenance optimization variables (N, uu,j,tu)
reliability function of unit j Rj(t)
Ath,j availability threshold value for unit j Aj* average availability for unit in the
time horizon j
mean to time to failure of unit j in vj*
the case of exponentially dis-tributed reliability functions bj, ej linear failure rate parameters for
unit jin the case of units that are in the wear-out phase
cost of a preventive maintenance cprev,j
task for unit j
cost of a corrective maintenance task ccorr,j
for unit j
duration of a preventive maintenance ~prev,j
task for unit j
~corr,j duration of a corrective maintenance
task for unit j
H time horizon of operation S set of operable system states
set of(in)operable components in OPk(O( Pk)
state k
Appendix A. Analytical expressions for problem (Eq. (P3))
A.1. Component a6ailability
Suppose that the maintenance action (u−1) takes place at time instanttu and is performed on one of the equipment components (both the time and the unit of maintenance are unknown to be determined).
The availability of each component j, 15j5M of the system after this maintenance action (u−1) and before the next one (u) can be expressed as follows: Au,j(t)=(1−uu,j) ·Au−1,j(t)+uu,j·A1,j(t−t%u), t%u
5t5tu+1 (18)
where tu%=tu+~u, ~u being the expected duration of
maintenance action (u−1) (see A.3). Eq. (18) describes the following two cases:
uu,j=0: then maintenance at tu has been planned to
take place on a component other than j. Hence,
componentjretains the same availability characteris-tics as in the previous maintenance interval
Au,j(t)=Au−1,j(t), t%u5t5tu+1
uu,j=1: then maintenance action (corrective or pre-ventive) attuis performed on unitjwhich, according to our assumptions, becomes AGAN
Au,j(t)=A1,j(t−tu%), t%u5t5tu+1
FunctionA1,j(t−tu%) is simply function A1,j(t) shifted
in time by tu% units.
With the use of Eq. (18) we can, recursively, establish analytical expressions for the availability functions of all units in the time intervals after each maintenance action and before the next one. Therefore, the availabil-ity of equipment unit j after the first maintenance action and before the second will be
A2,j(t)=(1−u2,j) ·A1,j(t)+u2,j·A1,j(t−t%2), t%25t
5t3
where t2 and t3 are maintenance optimization
corre-sponding to the first and the second maintenance time instants, respectively. Similarly, the availability of unitj after the second and before the third maintenance action would be A3,j(t)=(1−u3,j) · [(1−u2,j) ·A1,j(t) +u2,j·A1,j(t−t%2)]+u3,j·A1,j(t−t%3) =(1−u3,j) · (1−u2,j) ·A1,j(t) +(1−u3,j) ·u2,j·A1,j(t−t%2) +u3,j·A1,j(t−t%3)t%35t5t4
In this manner, we can recursively establish analyti-cal expressions for the availability of unit j in the time interval after the (u−1)th and before the uth mainte-nance action: Au,j(t)= % u k=1
Pu i=k+1 (1−ui,j)·uk,j·A1,j(t−t%k)n
, 2 5u5N−1 (19) where, by definition, u1,j=1.During the maintenance action (u−1), on the other hand, the availability of each componentjof the system can be expressed as follows:
Eq. (20) describes the following two cases:
If maintenance is performed (uu,j=1) then the com-ponent is unavailable
A( u,j(t)=0, tu5t5t%u
if no maintenance action takes place (uu,j=0), then componentjhas the same availability function as in interval (u−1)
A( u,j(t)=Au−1(t), tu5t5t%u
This means that the component is up with a proba-bility of Au−1,j(t) or down with a probability of
1−Au−1,j(t), tu5t5tu
By combining Eqs. (19) and (20) the availability of equipment unit j during the (u−1)th maintenance ac-tion is given by A( u,j(t)=(1−uu,j) · % u−1 k=0
uP−1 i=k+1(1−ui,j) ·uk,j·A1,j(t−t%k)n
, 25u5N−1 (21) Eqs. (19) and (21) describe equipment availability within the time horizon of operation as a function of the initial reliability characteristics of the equipment (A1,j(t)) and the maintenance optimization variables (tu,uu,j)
A.2. Probability of being at each state k of the system Each operable system state k has a probability of occurrence which depends on the reliability characteris-tics of the equipment, their degradation with time and the maintenance action taken to increase the availabil-ity of the components. In our model, in particular, since we have assumed independent failures and repairs, the probability of the system being in state k after the (u−1)th and before the uth maintenance action is given by
Prk,u(t)= P
jOPAu,j(t) ·jPO(P
(1−Au,j(t)), t%u5t5tu+1
(22) where OP(O(P) is the set of operable (inoperable) com-ponents in system statekandAu−j(t) is the availability of component j after the (u−1)th and before the uth maintenance action given by Eq. (19). During the (u−
1)th maintenance action, on the other hand, the proba-bility of occurrence of state k will be given by
P( rk,u(t)= P
jOPA(u,j(t) ·jPO(P(1−A(u,j(t)), tu5t5t%u
(23) where A( u,j(t) is the availability of component j during
the (u−1)th maintenance action. Note that Eqs. (22) and (23) describe the probability of occurrence of each
system statekas a function of time, equipment reliabil-ity and maintenance optimization variables (tu, uu,j).
A.3.Duration of maintenance ta