Maintenance scheduling and process optimization under uncertainty


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1 Compuers and Chemical Engineering 25 (2001) ainenance scheduling and process opimizaion under uncerainy C.G. Vassiliadis, E.N. Piikopoulos * Deparmen of Chemical Engineering, Cenre for Process Syems Engineering, Imperial College, Prince Consor Road, London SW 72BY, UK Acceped 20 Sepember 2000 Abrac In his paper, we describe an opimizaion framework for (i) deriving opimal mainenance policies in coninuous process operaions in he presence of parameric uncerainy and (ii) analyzing and quanifying he impac of uncerainy on opimal mainenance schedules. A syems effeciveness measure is inroduced which depends on expeced process profiabiliy and process and reliabiliy/mainenance characeriics. A mixed ineger nonlinear opimizaion model is proposed which aims a idenifying he number of mainenance (prevenive or correcive) acions required over a given ime horizon of inere, he ime inans and sequence of hese mainenance acions on he various componens of he process syem, so ha he syem effeciveness is maximized. By inroducing he concep of availabiliy hreshold values, i is shown ha an efficien soluion raegy can be eablished which requires he soluion of much smaller nonlinear opimizaion problems. The applicaion of he proposed framework o an example problem highlighs he imporan ineracions beween process operaion and mainenance scheduling in he presence of uncerainy Elsevier Science Ld. All righs reserved. Keywords: Process operaion; Reliabiliy; ainenance; Process uncerainy; Opimizaion; Syem effeciveness 1. Inroducion * Corresponding auhor. Tel.: ; fax: address: (E.N. Piikopoulos). Process produciviy and effeciveness depend on he efficien uilizaion of asses and resources, he proper allocaion of which is ypically decided by he process engineer mainly in he design and parly in he operaion phase (process rucure, equipmen volumes, recipes, producion schedules, conrols, ec). However, i is process availabiliy ha criically deermines wheher asses and resources are, indeed, available o be used as planned. Aailabiliy, in general, is defined as he abiliy of an iem o perform is required funcion a a aed inan of ime or over a aed period of ime (BS4778, 2000). In erms of a chemical process i corresponds o he fracional amoun of ime he process is able o perform is producion. Especially nowadays, when modern echnology and engineering is leading o more highly inegraed plans and a failure in one par of he process can decisively influence oal plan performance, availabiliy is widely recognized as one of he mo imporan operabiliy characeriics. On he oher hand, availabiliy is closely conneced o he performance of he plan in erms of safey and environmenal impac. Laely, new and igher regulaions are imposed on indury demanding for specific availabiliy requiremens of safey syems (ichelsen, 1998). To underline he consequences of loss of availabiliy we noe ha lo producion cos in a chemical plan can range from $500 o 100,000 per h (Tan & Kramer, 1997). For refineries, in paricular, oal lo producion cos soar o millions of dollars (Nahara, 1993). Chemical process availabiliy is a funcion of boh equipmen and process reliabiliy and mainainabiliy. Reliabiliy is he abiliy of an iem o perform a required funcion, under given environmenal and operaional condiions and for a aed period of ime (BS4778, 2000). On he oher hand, mainainabiliy is he abiliy of an iem, under aed condiions of use, o be reained in, or reored o a ae in which i can perform is required funcions when mainenance is performed under aed condiions and using prescribed procedures and resources (BS4778, 2000). Reliabiliy /01/$  see fron maer 2001 Elsevier Science Ld. All righs reserved. PII: S (00)
2 218 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) characeriics (e.g. equipmen characeriics, syem configuraion, buffer orage invenories, redundancy) and cerain mainainabiliy characeriics (e.g. accessibiliy o componens, spares, consumables, ec.) are inheren aribues of he process eablished during he design age and canno be alered wihou a design change (Kapur & Lamberson, 1977). As a resul, in he operaing ae of a process he way o achieve high availabiliy is hrough he derivaion and execuion of effecive mainenance raegies. ainenance is broadly classified o prevenive and correcive, he former comprising aciviies such as inspecion and replacemen while he laer concerning fixing or replacing equipmen in he even of failure. A classificaion of mainenance aciviies is depiced in Fig. 1. Obviously, hrough is impac upon process availabiliy, mainenance affecs boh he producion and he safey funcions of a chemical process. In paricular, effecive mainenance conribues (VanRijn, 1987) o suaining producion volume a he desirable levels, reducing operaing (spares and consumable iems) and fixed manufacuring cos and deerring he syem from moving o a hazardous ae. Despie is big benefis, mainenance is, in general, very expensive. VanRijn (1987) places he co of mainenance in he order of 20 30% of he plan s oal operaing cos. Alhough proper planning of mainenance aciviies increases he profiabiliy of he process, here are obvious radeoffs beween process profiabiliy and mainenance cos. Therefore, he mainenance opimizaion problem is described as rying o idenify mainenance raegies which yield process availabiliy levels ha mainain such a balance beween process revenue and mainenance co which maximizes process profiabiliy. For his purpose, mainenance opimizaion models are derived o deermine he opimum balance beween he cos and benefis of mainenance, aking ino consideraion all kinds of specificaions and conrains (Dekker, 1996). A number of papers (Dekker & Scarf, 1998; ValdezFlores & Feldman, 1989; Pierskalla & Voelker, 1976) presen overviews of he research work in he field of mainenance opimizaion models, assessing heir impac and applicaions. In reference o he chemical process indury, in paricular, i is admied (Grievink, Smi, Dekker & VanRijn, 1993) ha modeling he balance beween mainenance Fig. 1. Differen ypes of mainenance. benefis and cos is quie a complicaed ask. Two of he main difficulies recognized are he following: 1.1. Quanificaion of he benefis of reliabiliy and mainenance o process profiabiliy In he effor o achieve high availabiliy here is a poin a which his becomes economically unaracive. Woodhouse (1986) acknowledges ha alhough i may be worh spending $1000 o reduce a change of a bearing failure by 10%, i is unlikely ha an exra $1,000,000 would be spen o reduce he probabiliy of failure by a furher 0.10%. Process profiabiliy depends on producion volume which, in he case of failurefree operaion, is deermined by he process mahemaical model (producion schedules, recipes, operaion, ec.) and is feasibiliy. Tradiional availabiliy assessmen and sensiiviy analysis ools (faul rees, FECA echniques), on he oher hand, share he common limiaion of no explicily aking ino accoun process models and process ineracions and hence no fully capuring he physicochemical characeriics of he process; his may overeimae syem efficiency and provide misleading informaion regarding criical componens. The need o overcome hese limiaions is discussed boh in he reliabiliy and he process engineering communiy. Brunelle and Kapur (1997), for example, sugge ha reliabiliy should be defined from he viewpoin of he cuomer (i.e. process) and ha reliabiliy measures should be inroduced o capure he dynamic performance of he syem. A he same ime, process engineers sugge ha (Grievink e al., 1993; Lamb, 1996; Pujadas & Chen, 1996) i is essenial o inroduce syem effecieness measures o simulaneously ake ino consideraion process, reliabiliy and mainenance characeriics Process uncerainy Uncerainy and variabiliy is a common characeriic of all process syems and, based on he naure of is source, can be broadly classified as follows (Piikopoulos, 1995): modelinheren uncerainy, including kineic conans, physical properies, ransfer coefficiens, ec. processinheren uncerainy, including flowrae and emperaure variaions, ream qualiy flucuaions, ec. exernal uncerainy, including feedream availabiliy, produc demands, prices and environmenal condiions. All ypes of uncerainies can be mahemaically described eiher hrough ranges of possible realizaions or wih probabiliy diribuion funcions using informaion obained from experimenal and pilo plan daa, online measuremens, hiorical daa, cuomer orders,
3 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) marke indicaors and so on. Assessing and quanifying he impac of uncerainy, process flexibiliy (Swaney & Grossmann, 1985; Piikopoulos & azzuchi, 1990; Sraub & Grossmann, 1990) is a vial componen of he operabiliy of a chemical process and is defined as he probabiliy ha a syem operaes feasibly under coninuous uncerainies described by probabiliy diribuion funcions. This sugges ha he idenificaion of he opimal balance beween process profiabiliy as a resul of high availabiliy and mainenance cos, which is he core of he mainenance opimizaion problem, becomes a very complicaed ask in he presence of uncerainy, since he operaing paern of he producion process as well as he profile of process profiabiliy may change (Piikopoulos & Ieraperiou, 1995). Nowadays, he need for inegraing differen operabiliy crieria (flexibiliy, conrollabiliy, reliabiliy and mainainabiliy) in he udy of chemical process operaion is widely recognized (VanRijn, 1987; Grievink e al., 1993). Obviously, qualiaive echniques such as reliabiliy cenered mainenance (Anderson & Neri, 1990) or oal producive mainenance (Nakajima, 1988), alhough very popular among mainenance praciioners, are no based on sound analyical process models o accommodae such an inegraion. On he oher hand, radiional availabiliy assessmen of process syems, provides quaniaive informaion on he characeriics of he mainenance policy o be followed, based on he syem reliabiliy model bu wihou aking ino accoun deailed process ineracions, he process model complexiy, and he uncerainy which may be involved in a number of model and process parameers. Typical udies on mainenance opimizaion (Alkhamis & Yellen, 1995; Van, Hokad & Bodsberg, 1996; Tan & Kramer, 1997) deal wih minimizing he co of mainenance or maximizing an availabiliy relaed performance measure wihou aking ino consideraion he exiing uncerainy in he process and is ineracions wih he opimal mainenance policy. Gradually, general mahemaical frameworks have ared o appear (Sraub & Grossmann, 1993; Thomaidis & Piikopoulos, 1994, 1995; Thomaidis, 1995; Dedopoulos & Shah, 1996) assessing he process boh from a flexibiliy and a reliabiliy poin of view, wihou addressing, however, deailed mainenance consideraions. In his conex, he purpose of our work is o subanially expand and complee hese iniial works by proposing a rigorous mahemaical framework for idenifying opimal deailed mainenance and operaing policies for processes operaing in he presence of uncerainy, having accouned for heir ineracions and he impac of uncerainy. To achieve his, he proposed framework links process models o deailed reliabiliy and mainenance models. 2. Syem characerizaion 2.1. Process model Seady ae coninuous process operaion in he presence of uncerainy, can be described by a se of process model equaliy and inequaliy conrains as follows (Biegler, Grossmann & Weerberg, 1997): h(z, x, )=0 g(z, x, )0 (1) where zz is he vecor of degrees of freedom (e.g. inle reams, emperaures, spli raios, ec.) manipulaed o achieve maximum process performance, xx is he vecor of process variables (e.g. flowraes), is he vecor of coninuous uncerain parameers (e.g. supply, demand, ec.) assumed o follows a coninuous probabiliy diribuion funcion jpdf() and ={ l u } is he uncerain parameer space. The vecor of equaliies (h) corresponds o process equaions (hea and mass balances, equilibria relaions) while he vecor of inequaliies (g) denoes process specificaions and logical conrains. The process model in Eq. (1) defines a feasible operaing region (FOR) in he space of he uncerain parameers (e.g. supply and demand) h(z, x, )=0 FOR= (z, x): (2) g(z, x, )0 The feasible operaing region conains all he realizaions of he uncerain parameers for which feasible operaion can be guaraneed by properly adjuing he values of he degrees of freedom. The revenue generaed per ime uni is a funcion of he degrees of freedom (z), he process variable (x) and he parameer (). Denoing his funcion by r(z, x, ), he revenue per ime uni corresponds o he expeced maximum aainable value of r achieved by proper adjumen of he degrees of freedom. The overall expeced revenue rae can hen be deermined by he soluion of he following opimizaion problem (Eq. (P1)). ERR= maxr(x, z, )j()d() x,z s.. h(x, z, )=0 g(x, z, )0 (P1) The evaluaion of he expeced revenue rae ERR is a challenging problem which, among ohers, is discussed in deail in Sraub and Grossmann (1993), Piikopoulos and Ieraperiou (1995), Acevedo and Piikopoulos (1998).
4 220 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Fig. 2. Feasible operaing region of he fourcomponen process syem. Table 1 Srucural classificaion of syem aes Syem ae k ( 1, 2, 3, 4 ) ( ) 1 (1,1,1,1) 1 Operable 2 (0,1,1,1) 1 Operable 3 (1,0,1,1) 1 Operable 4 (1,1,0,1) 1 Operable 5 (1,1,1,0) 1 Operable 6 (0,0,1,1) 1 Operable 7 (0,1,0,1) 1 Operable 8 (1,0,0,1) 1 Operable 9 (1,0,1,0) 1 Operable 10 (0,1,1,0) 1 Operable 11 (0,0,0,1) 1 Operable 12 (1,1,0,0) 0 Inoperable 13 (1,0,0,0) 0 Inoperable 14 (0,1,0,0) 0 Inoperable 15 (0,0,1,0) 0 Inoperable 16 (0,0,0,0) 0 Inoperable 2.2. Sae space represenaion Consider a process syem in eadyae operaion whose componens are all funcioning properly. According o he operaing program, expeced process revenue can be evaluaed using Eqs. (1) and (2) and problem (Eq. (P1)). If a failure occurs, he process model deermining he operaion changes, according o which componens have failed, o a new (bu also known) se of equaions similar o Eq. (1) and, as a resul, expeced revenue changes. In discree even syem heory, he ae of a syem a ime is defined as (Cassandras, 1993) he informaion required a, such as he oupu of he syem is uniquely deermined by his informaion and he inpu. Since he oupu of he syem is he expeced revenue rae and he informaion required o deermine i is which componens are working and which are no a ime, he ae of he process syem is defined as a vecor, he elemens of which correspond o he operaing aus of each componen. Failures can be represened by a ransiion from one syem ae o anoher and all he possible aes form he aespace of he syem. To illurae hese conceps, consider he chemical complex in Fig. 2a, comprising four unis, for he producion of chemical C eiher hrough he inermediae produc B or by direc conversion of A o C, wih uncerainy in supply and demand (aken from Sraub & Grossmann, 1990). Such a process can be described by a mahemaical model similar o (Eq. (1)) and he expeced revenue rae can be deermined solving an opimizaion problem similar o (Eq. (P1)). In he ime horizon of operaion, a componen can be eiher failed or funcioning. The aus of he componen is described by a binary parameer j,,, 4 where 1 if componen j is funcioning j = 0 if componen j has failed and =( 1, 2, 3, 4 ) is he ae vecor of his fourcomponen syem. Each realizaion of he ae vecor describes a possible syem ae and hence a syem wih four componens involves a oal of 2 4 = 16 aes in which he syem may reside wih possible degradaion and failure of equipmen. The operable aes of he process can be deermined eiher by he rucure funcion of he syem or is minimal cus (see, for example, Hoyland & Rausand, 1994). The rucure funcion of he chemical complex in Fig. 2a is ( ) = and he minimal cus are {3,4} and {1,2,4}. Subsequenly, he aes of he syem are classified as rucurally operable or inoperable as shown in Table 1.
5 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Due o he change in syem configuraion because of failure, each operable syem ae k is described by a differen se of equaliy (h k ) and inequaliy conrains (g k ). For his paricular process, he model given in Table 2a (Sraub & Grossmann, 1990) describes all he 16 aes in which he syem may reside by adjuing he values of he binary parameers j according o wheher uni j is funcioning or no a syem ae k. F represens he flowraes hrough he process unis and j, d j, are he conversion raes and equipmen volumes of each uni, respecively. Sup and Dem are he uncerain parameers corresponding o supply and demand, respecively. The process model (equaliy and inequaliy conrains h k and g k ) in each ae k defines a corresponding feasible operaing region FOR k in he space of he uncerain parameers (i.e. supply and demand): FOR k = (z k,x k ): h k(z k, x k, k )=0 g k (z k, x k, k )0, k (3) The feasible operaing region conains all he realizaions of uncerain parameers for which feasible operaion of he syem can be guaraneed by properly adjuing he values of he degrees of freedom. If he syem is in he fully operable ae (1,1,1,1), i.e. all he Table 2a Chemical complex process ass balance F 1 =F 2 +F 3 F 3 =F 4 +F 5 F 10 =F 8 +F 9 F 7 = 1 F 5 F 6 = 2 F 4 F 8 = 3 (F 6 +F 7 ) F 9 = 4 F 2 Table 2b Sae Specificaions F 5 d 1 1 F 4 d 2 2 (F 6 +F 7 )d 3 3 F 2 d 4 4 F 1 Sup F 10 Dem 4 ERR k S N =12; S =1; co of A=40; D N =7; D =1; revenue from C=200 equipmen componens are funcioning, his gives rise o a feasible operaing region in he space of he uncerain parameers, as shown in Fig. 2b. On he oher hand, if one or more unis fail (e.g. if uni 4 fails) he syem may go on operaing in one of he degraded aes, as shown in Fig. 2c. Noe ha, in principle, he corresponding feasible operaing region is reduced, reflecing he fac ha when equipmen fails a smaller porion of demand and less hroughpu can be accommodaed hrough he process (see also, Thomaidis & Piikopoulos, 1995). As a resul, for each ae k a corresponding expeced revenue rae ERR k can be defined which can be evaluaed by he soluion of he following opimizaion problem (Eq. (P2)), similar o (Eq. (P1)): ERR k =max x k,z k s.. r k (x k, z k, k )j( k )d(),k h k (x k, z k, k )=0, ks g k (x k, z k, k )0, ks (P2) A se of independen and idenically diribued se of parameers ( 1, 2, k ) is used o denoe he realizaion of uncerain parameers in each ae k. Furhermore, since he ransiions beween he aes are relaed o equipmen failure and repair, each ae has a probabiliy of occurrence, Pr k () which depends on he imevarying availabiliy of each componen deermined by he reliabiliy and mainainabiliy characeriics of he equipmen. This is described nex Reliabiliy/aailabiliy modeling consideraions The probabiliy of he syem being in ae k a ime is a funcion of he degrading reliabiliy characeriics of he equipmen and he implemened mainenance policy. The defining elemens of any mainenance policy are he mainenance policy assumpions, regarding he ype of mainenance ha can be performed on equipmen componens and he mainenance opimizaion variables, regarding he characeriics of opimal mainenance raegies o be deermined. The selecion of he ype of mainenance depends on he equipmen aribues and specificaions as well as he available mainenance faciliies and capabiliies. Differen ypes of mainenance are defined boh a a syem level and a a componen level. In he case of complex syems, for example, groups of componens wih similar operaing condiions may be idenified and reaed uniformly during mainenance (e.g. group prevenive mainenance policies). Furhermore, a a componen level, assumpions are made regarding he effeciveness of mainenance in reoring he componen o a good condiion. Asgood asnew (AGAN) policies,
6 222 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) for example, reore he componen o is original condiion a he beginning of operaion, while asgoodasold (AGAO) policies bring i back o where i was immediaely before he mainenance ask ared. ainenance opimizaion variables correspond o he elemens of he mainenance policy ha can be reaed as decision variables o opimize a mainenance or a performance relaed crierion. These variables usually involve he number of mainenance acions o be performed, he lengh of mainenance inervals, he allocaion of mainenance crews, ec. Depending on he naure and complexiy of he assumpions and he desired level of deail and deph, differen mahemaical modeling ools can be employed in mainenance opimizaion frameworks, such as analyical echniques and arkov s models (Tan & Kramer, 1997). arkov s models and decision processes are powerful modeling ools having he flexibiliy o model a wide variey of cases such as he exience of redundancy and spares, dependen failures and sequence dependen behavior. The key elemens behind arkov s models are aes and ransiions. Transiions occur as a resul of evens or acions (e.g. failure, uilizaion of spares, repair, prevenive mainenance, sequence of mainenance aciviies, ec.) and he differen aes represen he behavior of he syem as a resul of he ransiions. According o wheher a discree or a coninuous ime represenaion is adoped, arkov s models and decision processes can lead o eiher dynamic programming or opimal conrol problems (Gersbakh, 1977). In an analyical approach, on he oher hand, he objecive is o express every erm in he model (e.g. probabiliy of occurrence of each ae, equipmen availabiliy, mainenance cos, ec.) as a funcion of he mainenance opimizaion variables. Then, andard opimizaion echniques can be used o obain he mainenance policy ha opimizes a mainenancerelaed crierion (see, for example, Van e al., 1996). In his work, an analyical approach will be followed. Exensions owards mainenance models described by coninuous ime arkov s chains are discussed in Vassiliadis and Piikopoulos (1999b) Syem effecieness measure Having evaluaed he expeced revenue under uncerainy characeriics ERR k of each ae k, as deermined by he soluion of Eq. (P2), he expeced revenue of he process is defined (Eq. (4)) as he weighed sum of he expeced revenue of each syem ae using as weighs he probabiliy of he syem being in each paricular ae ER H = H ks Pr k () ERR k d (4) where Pr k () is he probabiliy of occurrence of ae k as a funcion of ime. The expeced revenue, as defined in Eq. (4), is a syem effeciveness measure aking ino accoun process (including he exiing uncerainy), reliabiliy and mainainabiliy characeriics. The expeced revenue rae ERR k of each ae is deermined by he process model, he feasible operaing region in ha ae and he uncerainy. On he oher hand, he ae probabiliies Pr k () are a funcion of he reliabiliy and mainainabiliy characeriics of he process. The implemened mainenance policy depends on he mainenance policy assumpions (e.g. age replacemen, block replacemen, ec.) and he mainenance opimizaion variables (ime of mainenance, number of mainenance aciviies, sequence of acions, ec). The above sugges ha in order o maximize syem effeciveness, as measured by he expeced revenue in Eq. (4), boh process operaion and mainenance have o be opimized accordingly. 3. odelling and opimizaion framework 3.1. General descripion ainenance conribues o he profiabiliy of he process mainly by keeping he plan funcioning and capable of fulfilling producion needs for longer periods of ime (i.e. by providing higher plan availabiliy). As already menioned, deermining he opimal mainenance policy corresponds o idenifying he process availabiliy level which suains a balance beween longerm mainenance cos and producion level ha maximizes plan profiabiliy. Tradiional availabiliy analysis considers funcioning as a discree even described by a zeroone variable, i.e. he syem eiher funcions or no; his simplificaion however canno readily capure he physicochemical characeriics of he process. Therefore, he use of crieria ha do no explicily relae o process profiabiliy (e.g. availabiliy or reliabiliy imporance indices, ec.) as opimizaion objecives does no allow for he quanificaion of he balance beween mainenance benefis and cos. To overcome hese problems, he expeced revenue as a syem effeciveness meric is used as par of he mainenance opimizaion objecive. Therefore, he simulaneous process and mainenance opimizaion problem can be concepually posed as follows: s.. max{expeced revenue mainenance cos} process model mainenance model (P3) The inenion is o manipulae he process relaed and he mainenance relaed opimizaion variables so
7 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) as o maximize he balance beween process revenue and mainenance cos, which are parly conflicing objecives. This opimizaion is subjec o he rigorous process and mainenance models. Noe ha all process, reliabiliy and mainenance characeriics are incorporaed in problem (Eq. (P3)), he soluion of which simulaneously deermines he opimal operaing policy and he opimal mainenance policy. ore specifically (i) he mainenance model and he implemened mainenance policy deermine he co of mainenance and he probabiliy of occurrence of each ae, Pr k (), and (ii) he process characeriics and he exiing uncerainy, on he oher hand, deermine he expeced revenue rae of each ae, ERR k. In his work, an analyical approach is developed o describe he mainenance model in (Eq. (P3)). In paricular, he probabiliy of occurrence of each syem ae k (Pr k ()) and he mainenance cos appearing in he objecive funcion in (Eq. (P3)) comprising he erms ha are deermined by he mainenance raegy are analyically expressed as a funcion of he mainenance opimizaion variables. This is described nex ainenance opimizaion model Consider a syem wih componens. Le A 1,j () be he iniial (before any mainenance is performed) availabiliy (i.e. reliabiliy) funcion of componen j, 1j as a funcion of ime. Le, also, 1 and N denoe he beginning and he end, respecively, of he ime horizon of inere and,2 he ime inans a which we perform mainenance o any of he componens according o a mainenance schedule. The number of mainenance acions N 2, he mainenance ime inans and he deailed mainenance schedule are opimizaion variables o be deermined. The execuion of he mainenance aciviies depends on he following assumpions dicaed by he reliabiliy and mainainabiliy characeriics of he syem and he individual componens: a each mainenance ime inan, mainenance is performed o only one of he unis in he following way: 1. correcie, if he uni is down. 2. preenie, if i is operable.this assumpion is common in he lieraure (see, for example, Tseng, 1996) and valid in many real cases. all mainenance is of an AGAN ype, i.e. he componen is reored o is iniial condiion a he beginning of operaion. all failures are independen. To deermine he opimal mainenance policy hree ypes of mainenance opimizaion variables are inroduced: N, which is an ineger variable denoing he number of mainenance acions (N 2) performed o he equipmen componens wihin he ime horizon of operaion., which is a coninuous variable represening he mainenance ime inan of mainenance acion ( 1). u, j, which are 01 variables defined as: u, j = 1 if mainenance acion 1 is performed on componen j 0 else Based on he above mainenance policy assumpions, analyical expression can be recursively derived (see A.1) describing he availabiliy of each componen a each ime inan in he ime horizon as a funcion of he mainenance opimizaion variables (N, u,j, ). Therefore, he availabiliy of componen j, 1j afer ( 1)h and before he h mainenance acion is given by n A, j ()= (1 u i, j ) u k, j A 1, j ( k ), k=1 i=k+1 2 (5) and he availabiliy of componen j, 1j during he ( 1)h mainenance acion is given by A, j ()=(1 u, j ) 1 1 k=0 i=k+1 (1 u i, j ) u k, j A 1, j ( k ) n, 2 (6) Noe ha during a mainenance acion cerain componens may ill be operable and herefore, he process may go on operaing in one of he degraded aes of he aespace. Taking his ino consideraion, he availabiliy of each componen has been expressed boh during and afer mainenance acions. Similarly, he probabiliy of he syem being in ae k afer he ( 1)h and before he h mainenance acion is given by (see A.2) Pr k, ()= jop k A, j () jo P k (1 A, j ()), +1 where OP k (O P k ) is he se of operable (inoperable) componens in syem ae k and A,j () is he availabiliy of componen j afer he ( 1)h and before he h mainenance acion given by Eq. (5). During he ( 1)h mainenance acion, on he oher hand, he probabiliy of occurrence of ae k will be given by (7)
8 224 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) P r k, ()= jop k A, j () jo P k (1 A, j ()), k (8) where A,j () is he availabiliy of componen j during he ( 1)h mainenance acion. The expeced duraion of he ( 1)h mainenance ask as a funcion of he mainenance opimizaion variables is given by (see A.3) 1 1 i=k+1 (1 u i, j ) = u, j 1 k=1 u k, j A 1, j ( k ) n corr,j + k=1 (1 u i, j ) u k, j A 1, j ( k ) n prev, j 1 1 i=k+1 (9) where corr,j and prev,j are he fixed duraions of correcive and prevenive mainenance asks for uni j, respecively. Then, he process revenue over he whole ime horizon H, for a paricular realizaion of process variables and uncerain parameers (z k, x k, k ) can be expressed as (see A.4) R H = = =2 P,k () r(x k, z k, k )d ks + ks P,k () r(x k, z k, k )d (10) Eq. (10) comprises wo erms. The fir erm corresponds o he revenue generaed from he process beween he N 2 mainenance acions, while he second corresponds o he revenue generaed during he N 2 mainenance acions. The oal mainenance co in he ime horizon H is given by (see A.5) m.co= =2 u, j k=1 [( i=k+1 (1 u i, j )) 1 i=k+1 1 u k, j A 1, j ( k )] C corr, j + k=1 (1 u i, j ) u k, j A 1, j ( k ) n C prev, j (11) where C corr,j and C prev,j are he fixed cos of correcive and prevenive mainenance, respecively, for uni j. Noe ha Eqs. (5) (11) explicily associae all mainenance erms wih he se of mainenance opimizaion variables (N, u, j, ) Process/mainenance opimizaion model By collecing all he erms and equaions ogeher, problem (Eq. (P3)) is analyically wrien in he form of problem (Eq. (P4)). Problem (Eq. (P4)) incorporaes all processes (including he exiing uncerainy), reliabiliy and mainenance characeriics of he syem and aims a he opimizaion of he balance beween process revenue and mainenance cos. 1 2 k max N,u, j,,z k,x k r k (z k, x k, k ) + =2 + ks =1 +1 P,k () + ks P,k () r k (z k, x k, k )d u, j [(1 A 1, j ( )) C corr, j =2 +A 1, j ( ) C prev, j ]jpdf( 1 2 k )d 1 d 2 d kn s.. h k (x k, z k, k )=0, ks g k (x k, z k k ),0, ks (P4) (12) u, j =1, [2 ] (13) + +1,1] (14) The objecive funcion is a funcion of he mainenance opimizaion variables (N, u, j, ), he process variables (x k, z k ) and he uncerain parameers ( k ). The fir wo erms in he objecive correspond o he expeced process revenue. Since all operable process aes conribue o he process revenue he expecancy due o uncerainy is defined over he feasible operaing regions of all operable aes, as deermined by he process model in each ae. Therefore, a se of independen and idenically diribued se of parameers ( 1, 2,, k ) is used o denoe he realizaion of uncerain parameers in each ae k. Their join probabiliy diribuion funcion is denoed by jpdf( 1, 2,, k ). The hird erm of he objecive funcion is he mainenance co which is no associaed wih process uncerainy. The soluion of Eq. (P4) provides he operaing policy and he mainenance schedule ha maximizes he balance beween process revenue and mainenance cos, for a syem operaing under process uncerainy and under he mainenance policy assumpions aed in Secion 3.2. In paricular, he opimal number of mainenance acions required for he ime horizon of operaion (N), he opimal mainenance schedules (u, j ) and he exac opimal mainenance ime inans ( ) are idenified in conjuncion wih he opimal values for he process variables (z k, x k ). Eq. (13) sugges ha one mainenance ask is performed a a ime while Eq. (14)
9 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Problem (Eq. (P5)) corresponds o a mixedineger nonlinear programming formulaion involving discree decisions (number and sequence of mainenance acions) and inegral erms wih inegrands defined implicily (as a funcion of 01 variables). The main difficulies in handling a problem such as he above are he following: he presence of highly nonlinear erms in erms of 1 he 01 variables (e.g. he erms i=k+1 (1 u i, j )u k, j included in he availabiliy expressions incorporaed in he erms Pr k () in he expeced revenue erms of he objecive funcion). Simplificaions and linearizaions are no raighforward and may lead o a dramaic increase of he size of he problem. i is highly combinaorial. An increase in he number of componens and mainenance acions leads o a large number of possible mainenance schedules. The direc soluion of (Eq. (P5)) is herefore very complex. To overcome his, an effecive wo ep soluion raegy is proposed which avoids he direc soluion of (Eq. (P5)) by an approximae reformulaion based on he concep of availabiliy hreshold values. This soluion raegy is described nex. 4. Soluion raegy Fig. 3. Availabiliy hreshold values. ensures ha he ime inan of he h mainenance acion follows ha of he ( 1)h. Taking advanage of he mahemaical rucure of problem (Eq. (P4)) (see Vassiliadis, 2000 for he mahemaical proof), he process opimizaion par of he problem can be isolaed and performed separaely. Therefore, (Eq. (P4)) can be recaed as follows: max N,u, j, + =2 =2 =1 +1 P,k () ERR k d + ks + ks P,k () ERR k d u, j [(1 A 1, j ( )) C corr, j +A 1, j ( ) C prev, j ] (P5) s. u, j =1, [2 ] + +1,1 where ERR k is he expeced revenue rae for each ae k of he syem obained from he soluion of he process opimizaion problem (Eq. (P2)) Aailabiliy hreshold alues Consider he case when mainenance acions are performed periodically o each componen j every j ime unis. The availabiliy funcion of componen j before he fir mainenance acion on j is equal o he reliabiliy funcion of j, A j ()=R j (), 0 j,1j (15) Since mainenance reores he componen o an AGAN condiion, he availabiliy of j afer he fir mainenance acion will be given by A j ()=R j ( j ), j 2 j,1j and, herefore, he availabiliy of componen j afer he nh mainenance acion on j is given by A j ()=R j ( n j ), n j (n+1) j,1j This is depiced in Fig. 3a. Due o he monooniciy of he reliabiliy funcion and he periodiciy of mainenance, hese mainenance ime inans ( j,2 j,,n j ) for uni j correspond o a specific (same for each uni) availabiliy value A h, j for ha uni. This oneoone relaionship is deermined by he reliabiliy funcion of he uni
10 226 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) A h,j ()=R j ( j ), 1j (16) This is also depiced in Fig. 3a. These mainenance ime inans are unknown; for heir deerminaion i would suffice o deermine he componen availabiliy value, A h,j, hey correspond o. Therefore, an alernaive way o idenify he opimal mainenance ime inans for each componen and, hence, he sequence and number of mainenance acions, is o deermine a se of opimal aailabiliy hreshold alues A h,j. This implies ha he ime inan a which a mainenance acion will be performed for componen j will correspond o he ime inan a which he availabiliy of he componen falls o he hreshold value A h,j. This is schemaically shown in Fig. 3b, for a wocomponen syem. Whenever he availabiliy funcion of he fir uni falls o is hreshold value, A h,1 mainenance is performed on uni 1, while whenever he availabiliy funcion of he second uni falls o A h,2 mainenance is performed on uni 2. The number of mainenance acions for each componen and heir sequence is auomaically deermined by represening he mainenance ime inans on he ime axis. In he case of he example depiced in Fig. 3b, nine mainenance acions ake place (hree for componen 1 and six for componen 2) and heir sequence is The significance of he availabiliy hreshold values is aribued o he fac ha i is feasible o derive good approximaions of all he erms and expressions in problem (Eq. (P4)) as a funcion of he availabiliy hreshold values (see Appendix B). This propery is used for he conrucion of he fir ep of he proposed soluion raegy Algorihm The soluion of he proposed formulaion (Eq. (P5)) seeks o deermine he number and sequence of mainenance acions, i.e. opimal values for N and u,j and he exac mainenance ime inans, i.e. opimal values for. In his secion, we propose a woep soluion raegy according o which in he fir ep he number and an iniial sequence of mainenance acions is obained while in he second ep his informaion is used o deermine he opimal mainenance sequence and he opimal exac mainenance ime inans Sep 1: deerminaion of he aailabiliy hreshold alues number and iniial sequence of mainenance acions Using he concep of availabiliy hreshold values, a nonlinear mahemaical programming problem (Eq. (P6)) approximaing (Eq. (P5)) is proposed in which all he erms in he conrains and he objecive funcion are expressed as funcions of he hreshold values A h,j max A h, j s.. H k (A h, j ) Pr k (A*) j ERR k m.co(a h, j ) (P6) ks 0A h, j 1, j where H k (A h, j ) is he maximum amoun of ime ha he syem can spend in ae k as a funcion of he availabiliy hreshold values A h,j. This is equal o he ime horizon H subracing he expeced duraion of he mainenance acions in his ae. A j * is he average availabiliy of componen j in he ime period o he fir mainenance acion j. Since on he fir ep of he algorihm we have assumed periodic mainenance, A j * is he average availabiliy of componen j for he whole ime horizon excep he ime ha componen j is acually mainained. The average availabiliy of componen j over he ime inerval j is evaluaed from (Lewis, 1994) j A*= 1 j A j () d j 0 where A j () is he availabiliy funcion of componen j given by Eq. (15). Pr k (A*) j is he probabiliy of he syem being in ae k as a funcion of he average availabiliy of each componen j, A*. j Since independen failures have been assumed, hese probabiliies can be evaluaed from Pr k (A*)= j A* j (1 A*) j iop k jo P k Obviously, an approximaion of he expeced revenue generaed by he process in he ime horizon H is given by muliplying he probabiliy of he syem being in ae k by he maximum amoun of ime he syem can spend in ae k. m.co(a h,j ) is he expeced mainenance co expressed as a funcion of he availabiliy hresholds. Noe ha in problem (Eq. (P6)), he average availabiliy levels, A j *, for each componen j are opimized so as o maximize he expeced profi of he process and balance he radeoffs beween process revenue and mainenance cos. The formulaion in problem (Eq. (P6)) is general and no based on any assumpions regarding he probabiliy diribuion funcions describing he reliabiliy characeriics of he componens. In his work, we assume ha equipmen failure rae is linearly increasing, approximaing he wearou phase of a uni he reliabiliy of which is described by a Weibull probabiliy diribuion funcion.
11 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Linearly increasing failure rae. Suppose ha he syem componens operae in he wearou period. To approximae his siuaion, we assume ha componen failure raes are no conan, bu linearly increasing wih ime h j ()=b j +e j In his case i is shown (see Appendix B) ha problem (Eq. (P6)) can be rewrien as max A h, j s.. H k (A h, j ) Pr k (A*) j ERR k m.co(a h, j ) (P7) ks 0A h, j 1, j m.co(a h, j ) H = ((1 A h, j ) j [ e j +e j2 2 b j ln(a h, j )]/b j C corr, j +A h, j C prev, j ), j H k (A h, j ) H =H+. j, j, k jop [ e j e j2 2 b j ln(a h, j )]/b j j =(1 A h, j ) corr, j +A h, j prev, j, j Pr k (A*)= j A* j (1 A*), j j, k iop jo P q A*=0.5 j q e b j /2 [(z q j + j )/2] 2 e j [ (z q j + j )/2], j i=0 Problem (Eq. (P3)) is a nonlinear mahemaical programming formulaion, he soluion of which provides he availabiliy hresholds for each componen in order o mainain average componen availabiliy o a level ha maximizes he expeced profiabiliy of he process, given ha mainenance is performed periodically. From he values of he availabiliy hresholds we can deermine he number of mainenance acions N j required per uni and also an iniial sequence of mainenance acions. The assumpion of periodic mainenance, (which explains why here is only one availabiliy hreshold value for each componen), will be relaxed in he second ep of he algorihm, where he iniial mainenance sequence will be correced, if necessary, and he exac opimal mainenance ime inans will be obained Sep 2: deermining he opimal mainenance sequence and he opimal mainenance ime inans The soluion of he fir ep is used as an iniial poin o obain he opimal mainenance sequence and he opimal exac mainenance ime inans. For his purpose, a series of NLPs solved ieraively in he following way: 1. wih fixed number and sequence of mainenance acions N and ū, j, as hey are provided by ep1, problem (Eq. (P5)) reduces o a andard nonlinear opimizaion problem, where he opimizaion variables are he mainenance ime inans. max N 1 + =2 N 1 =1 N 1 = ks + ks P,k ()ERR k P,k () ERR k d d ū, j [(1 A 1, j ( )) C corr, j +A 1, j ( ) C prev, j ] (P8) s ,2 (17) The soluion of (Eq. (P8)) provides he mainenance ime sequence, [2,, N 1], for he paricular mainenance sequence ū, j, which is imposed by conrain (Eq. (17)) 2. if some of he inequaliy conrains in (Eq. (P4)) are acive, i.e. + +1,2 we go back o ep 2(a) and check wheher changing he sequence of he mainenance acions involved in his conrain will improve he soluion of (Eq. (P8)). 3. if here are no acive conrains or no beer soluion is obained, op. I should be noed ha he woep raegy does no heoreically guaranee o idenify he opimal soluion of problem (Eq. (P1)) in a ric mahemaical sense bu only a lower bound soluion (valid if each NLP is solved o global opimaliy). If he approximaion, however, used in he fir ep is good, hen he soluion is ofen very close o he opimal soluion as will be discussed in he following secion. Nex, he opimal inspecion policy will be idenified in he case of equipmen unis wih linearly increasing failure raes; subcases are examined o illurae he impac of process uncerainy, mainenance cos and mainenance acion duraion. 5. Numerical example The chemical complex shown in Fig. 2a convers species A o C eiher hrough he producion of he inermediae produc B or by direc conversion of A o C. The supply of raw maerial A and he demand of produc C are considered o be coninuous uncerain parameers, described by normal probabiliy diribuion funcions. The process model, he mahemaical
12 228 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Table 3a Case 1, failure rae characeriics and resuls of eps 1 and 2 Table 3c Uni b e Iem ainenance sequence Profi ,3,4,1,3,2,4,3,4,1,3,2,4,3, ,3,4,1,3,2,4,1,3,4,3,2,4,3, ,3,4,1,3,2,4,1,3,4,2,3,4,3, ,3,4,1,2,3,4,1,3,4,2,3,4,3, descripion of he uncerainy, he nine rucurally operable aes and heir expeced revenue raes are presened in Tables 2a and 2b. The conversions facors j for each uni j are 0.92, 0.9, 0.85, 0.75 and he equipmen volumes d j are 5, 5, 7, 9, respecively. The revenue rae for each syem ae is evaluaed by inegraing he probabiliy diribuion funcion of he revenue rae funcion over he feasible operaing region of each ae (problem Eq. (P2)). The soluion o his problem is obained using a Gaussian inegraion scheme wih en quadraure poins for boh coninuous uncerainies, similar o he one presened in Acevedo and Piikopoulos (1998). The resuls are summarized in Table 2b. All he NLP formulaions were solved using he GAS modeling package (Brooke, Kendrick & eeraus, 1988). In his case, here are nine aes wih a nonzero revenue rae Increasing failure rae Case 1: finding he opimal mainenance policy The daa required o describe he componen failure raes is given in Table 3a. The co and he duraion of prevenive and correcive mainenance acions are he same as in he previous case. In he fir ep of he algorihm, he opimal availabiliy hreshold and he opimal average availabiliy for each componen are obained by solving problem (Eq. (P7)). These values yield a number of mainenance acions per componen and an iniial mainenance sequence. The resuls of he fir ep are shown in Table 3b. In he second ep of he algorihm, he mainenance sequence and he number of mainenance acions per uni obained from he fir ep are used o solve problem (Eq. (P8)). By reversing he order of he acive conrains (see Vassilaidis, 2000) he opimal soluion is obained in four ieraions (Table 3c). Finally, he opimal mainenance schedule obained is shown in Fig. 4a Case 2: analyzing he impac of uncerainy Suppose here is no uncerainy in he process. The values of supply and demand are considered conan and equal o he nominal values (S N =12, D N =7) of normal probabiliy diribuion funcions ha described hem when hey were uncerain. The new operable process aes and heir corresponding maximum revenue raes are summarized in Table 4a. Implemening he fir ep of he mainenance opimizaion algorihm, he opimal availabiliy hreshold values and average availabiliies for each componen are obained (see Table 4b). Then, he number of mainenance acions and an iniial mainenance sequence can be deermined. Noe ha due o he fac ha here are only hree operable aes, he average availabiliy levels required per uni are higher. As a resul, in order o achieve hese levels, he number of mainenance acions is bigger. Noe, also, ha he hreshold values for unis 3 and 4 are idenical. This is expeced since neiher in he process (boh componens paricipae in all he operable aes), nor in he reliabiliy and mainenance daa, any diinguishing elemens beween hese wo unis can be found. On he oher hand, he availabiliy hreshold value for uni 1 is larger han ha for uni 2, A h,1 A h,2, since uni 1 paricipaes in he mo profiable aes of he process. In he second ep of he algorihm an opimal soluion is obained, by reversing he order of he acive conrains, in nine ieraions. The resuls are summarized in Table 4c. The opimal mainenance schedule is shown in Fig. 4b. Table 3b Uni A h A* ainenance inans ainenance acions , , , 684, 1026, 1368, , 630, 945, 1260, 1575, Iniial mainenance sequence 4,3,4,1,3,2,4,3,4,1,3,2,4,3,4
13 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Fig. 4. Cases 1 4, opimal mainenance schedules. Comparing cases 1 and 2, we observe ha he opimal prevenive mainenance schedule for he same syem is differen in he presence of uncerainy. This should no be surprising despie he fac ha uncerainy is process relaed and no reliabiliy or mainenance relaed. If here is uncerainy in he process, hen process revenue changes. Therefore since mainenance opimizaion defined as idenifying he balance beween process revenue and mainenance cos ha maximizes profiabiliy, he opimal mainenance policy has o change in he presence of uncerainy. The error induced if uncerainy is no aken ino accoun when rying o find he opimal prevenive mainenance policy is eimaed in Table 4d. If uncerainy had been ignored, he opimal mainenance policy along wih he values of expeced profi and mainenance cos would have been as shown in he fir column ( imaginary case) of Table 4d. However, he expeced profi would have been much smaller if he policy depiced as opimal in he imaginary case was implemened for he same syem in he presence of uncerainy (second column), because i does no accoun for i. On he oher hand, if uncerainy is aken ino accoun, he opimal mainenance policy is differen leading o a larger expeced profi and smaller mainenance cos (hird column) Case 3: effec of mainenance ask duraions In his case, he process characeriics remain he same while he duraion of prevenive and correcive mainenance asks for he fir uni is increased o prev,1 =100 and corr,1 =20, respecively. The impac of his increase o he mainenance schedule is capured quaniaively in he fir ep of he algorihm where he opimal availabiliy hreshold values and average availabiliies are deermined. Noe ha he availabiliy hreshold value of he fir uni is dramaically decreased compared wih case 1 while here is a big increase in he availabiliy hreshold value of he second uni, which operaes in parallel wih he fir. These changes sugge ha uni one should be mainained less frequenly since mainenance aciviies are economically unaracive due o he poenial loss of producion from he large repair imes of his uni. On he oher hand, uni 2 should be mainained more frequenly o compensae for he reducion in he availabiliy of he fir uni. As a resul he number of mainenance acions for he fir uni is reduced o one and he number of mainenance acions for he second uni is increased o hree. The resuls are depiced in Table 5a. In he Table 4a Case 2, revenue rae characeriics and resuls of eps 1 and 2 Sae ERR k S=12; co rae of A=40; D=7; revenue rae from C=200
14 230 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Table 4b Uni A h A* ainenance inerval ainenance acions Iniial mainenance sequence 4,3,4,1,3,2,4,3,1,4,3,2,4,3,1,4,3 second ep of he algorihm, he informaion derived from he fir ep is used in order o obain he opimal mainenance sequence and he opimal exac mainenance ime inans in hree ieraions. The resuls of he second ep are depiced in Table 5b. The opimal mainenance schedule is shown in Fig. 4c Case 4: effec of mainenance cos In his case, he cos of prevenive and correcive mainenance of unis 3 and 4 are increased o c prev,3 = c prev,4 =5000 and c corr,3 =c corr,4 =25 000, respecively. This increase of cos resuls in a significan decrease of he opimal availabiliy hresholds and average levels of availabiliy of unis 3 and 4. Subsequenly, he opimal number of mainenance acions for unis 3 and 4 is smaller, since he big increase in mainenance cos renders mainenance aciviies economically unaracive. The resuls of he fir ep are shown in Table 6a. In he second ep, solving he NLP formulaion in Eq. (P8) for he resuling iniial mainenance sequence (4, 3, 1, 2, 4, 3, 4, 3), he soluion depiced in Table 6b is obained which corresponds o he mainenance schedule depiced in Fig. 4d. Because of he small number of mainenance acions, unis are more likely o fail and he syem is more likely o operae in less profiable aes. Furhermore, expeced mainenance cos are high. As a resul, he expeced profi of he syem is significanly reduced compared wih case Concluding remarks In his work, a rigorous opimizaion framework for he opimizaion of prevenive mainenance in process syems is presened. The proposed mehodology differs from radiional approaches in ha: he objecive is o idenify he mainenance policy ha opimized process profiabiliy inead of an availabiliyrelaed crierion; in addiion o he mainenance model, he ineracions of mainenance, process characeriics and he process uncerainy are aken ino consideraion by including he full process model. Therefore, he opimal mainenance policy and he opimal processoperaing paern are idenified via he soluion of he same opimizaion problem, having accouned for heir ineracions. This approach faciliaes he quanificaion of he impac of process characeriics, including he exiing uncerainy, upon he opimal mainenance policy. I is shown ha alhough uncerainy is relaed o process and no reliabiliymainenance parameers, i rongly influences he de Table 4c Iem ainenance sequence Profi 1 4,3,4,1,3,2,4,3,1,4,3,2,4,3,1,4, ,3,4,1,3,2,4,3,1,4,2,3,4,3,1,4, ,3,4,1,3,2,4,3,1,2,4,3,4,3,1,4, ,3,4,1,3,2,4,3,2,1,4,3,4,3,1,4, ,3,4,1,2,3,4,3,2,1,4,3,4,3,1,4, ,3,4,2,1,3,4,3,2,1,4,3,4,3,1,4, ,3,4,2,1,3,4,2,3,1,4,3,4,3,1,4, ,3,2,4,1,3,4,2,3,1,4,3,4,3,1,4, ,3,2,4,1,3,4,2,3,4,1,3,4,3,1,4, Table 4d No uncerainy ( Imaginary case) Uncerainy Would ve Should ve Expeced profi ainenance 17 (3,2,6,6) 17 (3,2,6,6) 15 (2,2,5,6) acions (N) ainenance co Table 5a Case 3, resuls of eps 1 and 2 Uni A h A* ainenance acions Iniial mainenance 4,3,2,4,3,1,4,3,2,4,3,4,2,3,4 sequence
15 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Table 5b Iem ainenance sequence Profi 1 4,3,2,4,3,1,4,3,2,4,3,4,2,3, ,3,4,2,3,1,4,3,2,4,3,4,2,3, ,3,4,2,3,1,4,3,2,4,3,4,2,3, Table 6a Case 4, resuls of eps 1 and2 poraion of arkov s mainenance models, which have he flexibiliy o describe a wide variey of mainenance policies (Vassiliadis & Piikopoulos, 1999b). Finally, in erms of he required compuaional effor, an advanageous feaure of he proposed soluion raegy is he approximaion of he original INLP formulaion by a series of NLPs. This grealy increases he number of he mainenance decision (in paricular he 01) variables ha can be handled o address bigger problems. Uni A h A* ainenance acions Iniial mainenance sequence 4,3,1,2,4,3,4,3 Table 6b Iem ainenance sequence Profi 1 4,3,1,2,4,3,4, erminaion of he opimal mainenance policy. This is aribued o he fac ha, in he presence of uncerainy, process profiabiliy changes and, as a resul, he balance beween mainenance benefis and cos, which deermines he soluion of he mainenance opimizaion problem, also changes. The proposed mehodology allows for he udy of he ineracions beween various reliabiliymainenance characeriics and process operaion in he deerminaion of he opimal mainenance policy. This is illuraed in cases 3 and 4 of he numerical example for increasing failure rae, where changes in mainenance cos and mainenance ask duraions are aken ino accoun. This significanly affecs expeced process profiabiliy and he mainenance policy is modified accordingly. Anoher imporan issue concerns he ineracions of design and mainainabiliy, since he selecion of process design criically deermines fuure reliabiliy characeriics as well as prevenive mainenance policies and mainenance cos. Exensions of his work oward incorporaing mainainabiliy issues a he design phase of a process are discussed in Piikopoulos and Vassiliadis (1998) and Vassiliadis and Piikopoulos (1999a). Depending on he complexiy of he mainenance procedures and mainenance policy assumpions, i is no always possible o express he mainenance model in an analyical form. The rucure of he proposed opimizaion framework, however, allows for he incor Nomenclaure h k g k z k x k r k FOR k ERR k j ( ) j d j Pr k () N u, j A, j () A, j () Pr k, () vecor of process equaliy conrains a ae k vecor of process inequaliy conrains a ae k vecor of degrees of freedom a ae k vecor of process variables a ae k vecor of uncerain process parameers revenue rae funcion a ae k feasible operaing region a ae k expeced revenue rae a ae k binary parameer denoing wheher uni j is up or down rucure funcion of he syem Conversion facor for uni j equipmen volume for uni j probabiliy of being a ae k as a funcion of ime ineger variable denoing he number of mainenance acions for all unis of he syem 01 variable denoing wheher he ( 1)h mainenance acion is going o be performed on uni j or no coninuous variable denoing he ( 1)h mainenance ime inan availabiliy of uni j afer he ( 1)h and before he h mainenance acion as a funcion of he mainenance opimizaion variables (N, u, j, ) and ime availabiliy of uni j during he ( 1)h mainenance acion as a funcion of he mainenance opimizaion variables (N, u, j, ) and ime probabiliy of being a ae k afer he ( 1)h and before he h mainenance acion as a funcion of he mainenance opimizaion variables (N, u, j, ) and ime
16 232 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) P r k, () R j () A h,j A j * * j b j,e j c prev,j c corr,j prev,j corr,j H S OP k (O P k ) probabiliy of being a ae k during he ( 1)h mainenance acion as a funcion of he mainenance opimizaion variables (N, u, j, ) and ime expeced duraion of he ( 1)h mainenance acion as a funcion of he mainenance opimizaion variables (N, u, j, ) reliabiliy funcion of uni j availabiliy hreshold value for uni j average availabiliy for uni in he ime horizon j mean o ime o failure of uni j in he case of exponenially diribued reliabiliy funcions linear failure rae parameers for uni j in he case of unis ha are in he wearou phase co of a prevenive mainenance ask for uni j co of a correcive mainenance ask for uni j duraion of a prevenive mainenance ask for uni j duraion of a correcive mainenance ask for uni j ime horizon of operaion se of operable syem aes se of(in)operable componens in ae k componen j reains he same availabiliy characeriics as in he previous mainenance inerval A, j ()=A 1, j (), +1 u,j =1: hen mainenance acion (correcive or prevenive) a is performed on uni j which, according o our assumpions, becomes AGAN A, j ()=A 1, j ( ), +1 Funcion A 1,j ( ) is simply funcion A 1,j () shifed in ime by unis. Wih he use of Eq. (18) we can, recursively, eablish analyical expressions for he availabiliy funcions of all unis in he ime inervals afer each mainenance acion and before he nex one. Therefore, he availabiliy of equipmen uni j afer he fir mainenance acion and before he second will be A 2, j ()=(1 u 2, j ) A 1, j ()+u 2, j A 1, j ( 2 ), 2 3 where 2 and 3 are mainenance opimizaion corresponding o he fir and he second mainenance ime inans, respecively. Similarly, he availabiliy of uni j afer he second and before he hird mainenance acion would be Appendix A. Analyical expressions for problem (Eq. (P3)) A.1. Componen aailabiliy Suppose ha he mainenance acion ( 1) akes place a ime inan and is performed on one of he equipmen componens (boh he ime and he uni of mainenance are unknown o be deermined). The availabiliy of each componen j, 1j of he syem afer his mainenance acion ( 1) and before he nex one () can be expressed as follows: A, j ()=(1 u, j ) A 1, j ()+u, j A 1, j ( ), +1 (18) where = +, being he expeced duraion of mainenance acion ( 1) (see A.3). Eq. (18) describes he following wo cases: u,j =0: hen mainenance a has been planned o ake place on a componen oher han j. Hence, A 3, j ()=(1 u 3, j ) [(1 u 2, j ) A 1, j () +u 2, j A 1, j ( 2 )]+u 3, j A 1, j ( 3 ) =(1 u 3, j ) (1 u 2, j ) A 1, j () +(1 u 3, j ) u 2, j A 1, j ( 2 ) +u 3, j A 1, j ( 3 ) 3 4 In his manner, we can recursively eablish analyical expressions for he availabiliy of uni j in he ime inerval afer he ( 1)h and before he h mainenance acion: n A, j ()= (1 u i, j ) u k, j A 1, j ( k ),2 k=1 i=k+1 (19) where, by definiion, u 1,j =1. During he mainenance acion ( 1), on he oher hand, he availabiliy of each componen j of he syem can be expressed as follows: A, j ()=(1 u, j ) A 1, j (), (20)
17 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) Eq. (20) describes he following wo cases: If mainenance is performed (u,j =1) hen he componen is unavailable A, j ()=0, if no mainenance acion akes place (u,j =0), hen componen j has he same availabiliy funcion as in inerval ( 1) A, j ()=A 1 (), This means ha he componen is up wih a probabiliy of A 1,j () or down wih a probabiliy of 1 A 1,j (), By combining Eqs. (19) and (20) he availabiliy of equipmen uni j during he ( 1)h mainenance acion is given by 1 1 A, j ()=(1 u, j ) (1 u i, j ) k=0 i=k+1 u k, j A 1, j ( k ) n,2 (21) Eqs. (19) and (21) describe equipmen availabiliy wihin he ime horizon of operaion as a funcion of he iniial reliabiliy characeriics of he equipmen (A 1,j ()) and he mainenance opimizaion variables (, u,j ) A.2. Probabiliy of being a each ae k of he syem Each operable syem ae k has a probabiliy of occurrence which depends on he reliabiliy characeriics of he equipmen, heir degradaion wih ime and he mainenance acion aken o increase he availabiliy of he componens. In our model, in paricular, since we have assumed independen failures and repairs, he probabiliy of he syem being in ae k afer he ( 1)h and before he h mainenance acion is given by Pr k, ()= A, j () (1 A, j ()), +1 jop jo P (22) where OP(O P) is he se of operable (inoperable) componens in syem ae k and A j () is he availabiliy of componen j afer he ( 1)h and before he h mainenance acion given by Eq. (19). During he ( 1)h mainenance acion, on he oher hand, he probabiliy of occurrence of ae k will be given by P r k, ()= A, j () (1 A, j ()), jop jo P (23) where A,j () is he availabiliy of componen j during he ( 1)h mainenance acion. Noe ha Eqs. (22) and (23) describe he probabiliy of occurrence of each syem ae k as a funcion of ime, equipmen reliabiliy and mainenance opimizaion variables (, u,j ). A.3. Duraion of mainenance asks Suppose ha he ( 1)h mainenance acion akes place a ime and is performed on componen j. According o he mainenance policy assumpions aed in Secion 3.2, a mainenance acion can be eiher correcive or prevenive depending on wheher he unis is up or down a he ime of inspecion. The probabiliy of componen j being up a ime is equal o A 1 ( ) while he probabiliy of j being down is (1 A 1 ( )), where A 1 () is he availabiliy of componen j in he ime inerval afer he ( 2)h and before he ( 1)h mainenance acion. Therefore, he expeced duraion of mainenance ask ( 1), given ha his is performed on uni j, would be, j =(1 A 1, j ( )) corr, j +A 1, j ( ) prev, j where corr,j and prev,j are he fixed duraions of correcive and prevenive mainenance asks for uni j, respecively. Since, however, we do no know wheher he ( 1)h mainenance acion is going o be performed on uni j or on a differen uni bu his remains o be deermined by he soluion of he opimizaion problem, he expeced duraion of mainenance acion ( 1) is given by = u, j [(1 A 1, j ( )) corr, j +A 1, j ( ) prev, j ] or, by making use of Eq. (19) = u, j 1 k=1 1 1 i=k+1 u k, j A 1, j ( k ) n corr, j + k=1 1 1 i=k+1 prev, j (1 u i, j ) (1 u i, j ) u k, j A 1, j ( k ) n (24) A.4. Process reenue Having derived expressions for he probabiliy of he syem being in each ae k during and afer mainenance acions as well as expressions for he expeced duraion of each mainenance ask, he revenue of he process hroughou he ime horizon, as given by Eq. (4), can be analyically expressed as a funcion of he mainenance opimizaion variables.
18 234 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) This revenue is for a paricular realizaion of he process variables and uncerain parameers in each ae (z k, x k, k ). Consider he ime inerval afer he ( 1)h and before he h mainenance acion. The revenue generaed from all he aes of he process, if he operaing program of he process corresponds o a vecor (z k, x k, k ), in his inerval is given by +1 + ks Pr,k () r(x k, z k, k )d, +1 During a mainenance acion, he process may go on operaing in one of he degraded aes. Therefore, here is an expeced revenue generaed during mainenance acion. The revenue generaed during he ( 1)h is given by + ks P r,k () r(x k, z k, k )d, Then, he process revenue over he whole ime horizon H, if he operaing program of he process corresponds o a vecor (z k, x k, k ), can be expressed as ER H = +1 =1 + ks + + =2 P,k () ERR k d ks P,k () ERR k d (25) Eq. (25) comprises wo erms. The fir erm is equivalen o he revenue generaed from he process, operaing a a cerain program (z k, x k, k ), beween he N 2 mainenance acions, while he second corresponds o he revenue generaed during he N 2 mainenance acions, for a paricular realizaion of he process variables and uncerain parameers in each ae (z k, x k, k ). Boh erms are clearly associaed wih he reliabiliy characeriics of he equipmen and he mainenance opimizaion variables N, u,j, highlighing he impac of he implemened mainenance policy o he profiabiliy of he process. A.5. ainenance cos ainenance cos form a key erm in he objecive funcion. Clearly, his erm is influenced direcly by he mainenance policy o be implemened since i is a funcion of he number, he sequence and exac ime inans of mainenance acions o be performed. Similar o he duraion of mainenance asks, he co of a mainenance ask depends on wheher his ask corresponds o a prevenive or a correcive mainenance acion. Following an analysis similar o he one presened in (A.3), he expeced of he ( 1)h mainenance acion, if his performed o uni j is shown o be m.co, j =(1 A 1, j ( )) C corr, j +A 1, j ( ) C prev, j where C corr,j and C prev,j are he fixed cos of correcive and prevenive mainenance, respecively, for uni j. Therefore, he expeced duraion if he ( 1)h mainenance acion is m.co, j = u, j [(1 A 1, j ( )) C corr, j +A 1, j ( ) C prev, j ] The oal expeced correcive mainenance co in he ime horizon is given by m.co corr = =2 u, j (1 A 1, j ( )) C corr, j and ha of prevenive mainenance is m.co prev = =2 u, j A 1, j ( ) C prev, j The oal expeced mainenance co in he ime horizon H is, hen, expressed as he sum of correcive and prevenive mainenance co. m.co= =2 u, j 1 k=1 1 1 i=k+1 n (1 u i, j ) u k, j A 1, j ( k ) C corr, j 1 n 1 i=k+1 (1 u i, j ) u k, j A 1, j ( k ) + k=1 C prev, j Appendix B (26) Availabiliy hresholds values problem (Eq. (P6)) If he failure rae of componen j is linearly increasing wih ime h j ()=b j +e j hen he reliabiliy funcion of componen j is given by R j ()=e (b j +e j )d =e b j /2 2 e j 0 Eq. (27) in conjuncion wih Eq. (15) sugges ha in order o mainain an availabiliy hreshold A h,j for componen j, a mainenance acion, which reores he componen o an AGAN condiion, should be performed every j = e j+e j2 2 b j ln(a h, j ), j b j The average number of mainenance acions for componen j hroughou he ime horizon can, hen, be expressed as
19 C.G. Vassiliadis, E.N. Piikopoulos / Compuers and Chemical Engineering 25 (2001) H N j =, j [ e j +e j2 2 b j ln(a h, j )]/b j which sugges ha he expeced oal mainenance co as a funcion of he availabiliy hresholds is m.co(a h, j ) H = ((1 A h, j ) j [ e j +e j2 2 b j ln(a h, j )]/b j C corr, j +A h, j C prev, j ) The maximum amoun of ime he syem may spend in ae k is given by H k (A h, j ) =H+ jop H j [ e j +e j2 2 b j ln(a h, j )]/b j where j is he expeced duraion of eing on uni j, given by j =(1 A h, j ) corr, j +A h, j prev, j Finally, he average availabiliy of componen j over he ime horizon is equal o A j *= 1 j j 0 A j ()d= 1 j j 0 e b j /2 2 +e j d For he evaluaion of he above inegral we use a Gaussian quadraure numerical inegraion scheme and we find q A*=0.5 j w q e b j /2 [(z q j + j )/2]2 e j [(z q j + j )/2] i=0 (27) where z q are he roos of he Legendre polynomials and w q are he weigh facors for he q Gauss Legendre quadraures (Carnahan, Luher & Wilkes, 1969). Using he above expressions, (Eq. (P6)) can be rewrien as max A h, j ks m.co(a h, j ) s.. H k (A h, j ) Pr k (A* h, j ) ERR k (P9) 0A h, j 1, j m.co(a h, j ) H = ((1 A h, j ) j [ e j +e j2 2 b j ln(a h, j )]/b j C corr, j +A h, j C prev, j ), j H k (A h, j ) H =H+ j, j, k jop [ e j +e j2 2 b j ln(a h, j )]/b j j =(1 A h, j ) corr, j +A h, j prev, j, j Pr k (A*)= j A* j (1 A*), j j, k iop io P q A*=0.5 j q e b j /2 [(z q j + j )/2]2 e j [(z q j + j )/2], j i=0 References Acevedo, J., & Piikopoulos, E. N. (1998). Sochaic opimisaionbased algorihms for process synhesis and design uncerainy. Compuers and Chemical Engineering, 22(4 5), Alkhamis, T.., & Yellen, J. (1995). Refinery unis mainenance using ineger programming. Applied ahemaical odelling, 19(9), Anderson, R. T., & Neri, L. (1990). Reliabiliy cenered mainenance. New York: Elsevier. Biegler, L., Grossmann, I., & Weerberg, A. (1997). Syemaic mehods of chemical process design. Prenice Hall. Brooke, A., Kendrick, D., & eeraus, A. (1988). Gams: A user guide Redwood Ciy, CA. Brunelle, R. D., & Kapur, K. C. (1997). Cuomercenered reliabiliy mehodology. Proceedings of he annual reliabiliy and mainainabiliy symposium (pp ). BS4778. (2000). Glossary of erms used in qualiy assurance including reliabiliy and mainainabiliy erms. London: Briish Sandards Iniuion. Carnahan, B., Luher, H. A., & Wilkes, J. O. (1969). Applied numerical mehods. Wiley. Cassandras, C. G. (1993). Discree een syems: modeling and performance analysis. Boon, A: Richard D. Irwin Inc, and Aksen Associaes Inc. Dedopoulos, I. T., & Shah, N. (1996). Longerm mainenance policy opimizaion in mulipurpose process plans. Transacions of he Iniue of Chemical Engineers, 74(A3), Dekker, R. (1996). Applicaions of mainenance opimizaion models: a review and analysis. Reliabiliy Engineering and Syem Safey, 51(3), Dekker, R., & Scarf, P. A. (1998). On he impac of opimisaion models in mainenance decision making: he ae of he ar. Reliabiliy Engineering and Syem Safey, 60(2), Gersbakh, I. B. (1977). odels of preenie mainenance. New York: Elsevier. Grievink, J. K., Smi, K., Dekker, R., & VanRijn, C. F. H. (1993). anaging reliabiliy and mainenance in he process indury. Proceedings of he conference on foundaion of compuer aided operaions, FOCAPO (pp ). Creed Bue, CO. Hoyland, A., & Rausand,. (1994). Syem reliabiliy heory. Wiley Inerscience. Kapur, K. C., & Lamberson, L. R. (1977). Reliabiliy in engineering design. New York: Wiley. Lamb, R. G. (1996). Reengineer mainenance for corporae compeiiveness. Hydrocarbon Processing, 75(1), Lewis, E. E. (1994). Inroducion o reliabiliy engineering. Wiley. ichelsen, O. (1998). Use of reliabiliy echnology in he process indury. Reliabiliy Engineering and Syem Safey, 60(2), Nahara, K. (1993). Toal producive managamen in he refinery of he 21 cenury. Proceedings of he conference on foundaion of compuer aided operaions, FOCAPO (pp ) Park Ciy, UT. Nakajima, S. (1988). Inroducion o TP: oal producie mainenance. Cambridge, A: Produciviy Press.
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