Minimization of the Expected Total Net Loss in a Stationary Multistate Flow Network System

Transcription

1 Appled Mathematcs, 6, 7, Publshed Onlne May 6 n ScRes. Mnmzaton of the Expected Total Net Loss n a Statonary Multstate Flow Networ System Krstna Sutlaberg, Bent Natvg Department of Mathematcs, Unversty of Oslo, Oslo, Norway Receved 5 March 6; accepted May 6; publshed 4 May 6 Copyrght 6 by authors and Scentfc Research Publshng Inc. Ths wor s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract In the present paper, a three-component, statonary, multstate flow networ system s studed. Detaled costs and ncomes are specfed. The am s to mnmze the expected total net loss wth respect to the expected tmes the components spend n each state. Ths represents a novelty n that we connect the expected component tmes spent n each state to the mnmal total net loss of the system, wthout frst fndng the component mportance. Ths s of nterest n the desgn phase where one may tune the components to mnmze the expected total net loss. Due to the complex nature of the problem, we frst study a smplfed verson. There the expected tmes spent n each state are assumed equal for each component. Then a modfed verson of the full model s presented. The optmzaton n ths model s completed n two steps. Frst the optmzaton s carred out for a set of pre-chosen fxed expected lfe cycle lengths. Then the overall mnmum s dentfed by varyng these expectatons. Both the smplfed and the modfed optmzaton problems are nonlnear. The setup used n ths artcle s such that t can easly be modfed to represent other flow networ systems and cost functons. The challenge les n the optmzaton of real lfe systems. Keywords Relablty, Nonlnear Optmzaton, Multstate Flow Networ. Introducton A seres of challenges concernng relablty engneerng s presented n []. Some of these challenges are connected to the representaton and modelng of complex systems, such as multstate systems, and ther How to cte ths paper: Sutlaberg, K. and Natvg, B. (6) Mnmzaton of the Expected Total Net Loss n a Statonary Multstate Flow Networ System. Appled Mathematcs, 7,

2 K. Sutlaberg, B. Natvg operatonal tass, for nstance mantenance optmzaton. Over the past decades varous measures of component mportance have been studed. The use of such measures permts the relablty analyst to prortze the system components n order to allocate resources effcently. In [] a new theory for measures of mportance of system components s presented. Generalzatons of the Brnbaum, Barlow-Proschan and Natvg measures (see [3]-[5] respectvely) from the bnary to the multstate case, both for unreparable and reparable systems are covered. A numercal study of the above mentoned multstate measures of component mportance s also covered n []. Loss of utlty due to the system leavng the dfferent sets of better states are ntroduced n that study. However, no detaled costs or ncomes are specfed. Recently, wor has been done to also nclude costs n the determnaton of component mportance for bnary systems. In [6] and [7] the Brnbaum measure s extended to also nclude both falure nduced and mantenance costs, whle [8] and [9] ntroduce other cost-effectve mportance measures. In mantenance optmzaton studes one s often nterested n choosng a mantenance plan whch mnmzes lfe cycle costs, maxmzes net present value or maxmzes system relablty for a gven system. See for nstance []-[4] for some recent wor on these subjects. In ths artcle we wll loo at one partcular type of mantenance acton, the complete repar. As the components reach the complete falure state, they are repared to what we wll denote the perfect functonng state. The am s to nclude both costs and ncomes n the study of a reparable multstate flow networ system. To acheve ths, we wll defne ncomes and cost functons for the purpose of mnmzng the expected total net loss over a tme perod wth respect to the expected component tmes n the dfferent states. Ths represents a novelty n that we connect the expected component tmes spent n each state to the mnmal total net loss of the system, wthout frst fndng the component mportances. It would of course have been nce to optmze wth respect to probablty dstrbutons nstead of expectatons, but ths s not trval even for a smple three-component system. However, the optmzaton problem consdered n ths artcle s partcularly nterestng n a desgn or re-desgn phase, where one may tune the components n such a way that the expected total net loss s mnmzed. Wth the optmzaton problem consdered n ths artcle we are facng complex dependences. We therefore study both a smplfed verson and a modfed verson of the optmzaton problem. In the smplfed verson we see that the optmal expected tme spent n each state ncreases wth ncreasng operatonal tme for all three cost functon types consdered. However, the extent of the ncrease dffers wth the dfferent basc cost functon types. Due to basc nvestment costs ths s not a trval result. In the modfed verson of the optmzaton problem we only fnd approxmate solutons. We observe that the dfferent types of cost functons nfluence the end results sgnfcantly. For nstance one of the functonng states s redundant for two of the three cost functon types when the cost functon parameter s ncreasng. For both problems we see that the mnmum expected total net loss s ncreasng wth ncreasng component cost per repar. The rest of the artcle s organzed as follows: Secton ntroduces the basc model, the three dfferent types of cost functons and the three-component system of nterest. The smplfed verson of the optmzaton problem wth results s presented n Secton 4. Secton 5 presents the modfed optmzaton problem wth results, and concludng remars are found n Secton 6.. Basc Model Let S be the set of possble system states, and S, =,, n, the set of possble component states. Throughout ths artcle we wll assume that S = S = {,, M}. Snce we are regardng the system as a flow networ, the system state s the amount of flow that can be transported through the networ. In the same way, the component state s the amount of flow that can be transported through each component. Let X ( t) = ( X ( t),, Xn ( t) ) be the vector of component states at tme t. That s X ( t) = f component, =,,, n, s n state S at tme t. A bnary mnmal cut set s a mnmal set of components whch upon falure wll brea the connecton between the endponts of the networ. Let K l, l =,, m, be the bnary mnmal cut sets of the networ. Then, by applyng the max-flow-mn-cut theorem (see [5]), we get that the system state s gven by φ ( X ( t) ) = mn X ( ). t () l m K l 794

3 K. Sutlaberg, B. Natvg Thus, the system state equals the smallest total flow through the mnmal cut sets of the system. Assume now that no components are n seres wth the rest of the system. Then there must be at least two components n every mnmal cut set. If all components are n the perfect functonng state, M, the system state wll be at least M, and therefore we must have that S S. Thus, the assumpton of equalty between the set of system states and the set of component states, S = S, =,, n, mples that at least one component s n seres wth the rest of the system. For ths reason, we wll n Sectons 4 and 5 focus on the three component system gven n Fgure. Assume that the components deterorate by gong through all states n S, from the perfect functonng state M to the complete falure state, before beng repared bac to M. Let be the expected tme component =,, n spends n state =,, M, and let the vector of postve expected component tmes n each state be denoted (,,, M M =,,, n ). Assume for =,, n that the basc nvestment costs of component spendng the expected amount of tme n state,, M g for,, M h for =. = are gven by the cost functons ( ) These basc costs appear once n the tme nterval [ ] = and ( ),T for each combnaton of =,, n and =,, M. For any gven functonng state =,, M, t seems natural that these basc expenses grow when the expected tmes become large. If however, no tme s spent n a functonng state, there wll not be any basc costs of eepng the component n ths state. Smlarly, the shorter the expected tme spent n the complete falure state, the more expensve t should be. In other words, the faster a complete repar s executed, the more g s ncreasng and the cost functon expensve t should be. Therefore, we assume that the cost functon ( ) ( ) g( ) = h( ) =. Throughout ths artcle we assume the cost functons, g ( ) and h ( ) h s assumed to be decreasng; moreover, both functons are assumed to be twce dfferentable wth the followng types: g c Type : ( ) = and h( ) = c. Type : ( ) ln ( g = c + ) and h( ) = ( c + ) Type 3: ( ) e c g c = and ( ) e h =, ln., to be of one of where c >, =,, n, =,, M are constants. These cost functons are constructed by the authors accordng to the above mentoned crtera to represent a varaton n the potental basc cost development. In ths artcle we only consder perfect repars. Let C > denote the cost per repar from the complete falure state to the perfect functonng state of component =,,, 3. The total number of repars of component =,,, 3 n the nterval [,T ] s denoted N ( T ) for =,, n. Let R be the fxed ncome per unt of tme when the system s n state S, and assume that R < R R M. Ths means that the ncome decreases, startng from the perfect functonng state, from one state to the next untl the system reaches the complete falure state, where the ncome s non-postve. Thus, there s a loss per tme unt that the system spends n the complete falure state. Such negatve ncome mght correspond to nterest rate expenses connected to system buldng nvestments. The presence of such costs wll ncrease the ncentve for reparng the faled components. Fgure. A system wth three components. 795

4 K. Sutlaberg, B. Natvg The contrbuton from the -th component to the total cost connected to the operaton of the system n the tme CN T, n addton to the basc nvestment costs nterval [,T ], s the total repar cost over the nterval, ( ) related to component spendng the expected amount of tme n state,, M. M =, ( ) ( ) g + h = To get the total costs connected to the operaton of the system we sum over all components. Let I{ A } denote the ndcator functon of the event A. Then the ncome at tme t connected to the operaton of the system s gven by M = { φ ( ( t) ) = } I X R. To fnd the total ncome we ntegrate the ncome at tme t over the tme perod [,T ]. Hence, the total net loss connected to the operaton of the system n the tme nterval [,T ] s n M M T ( ) = ( ) + ( ) + ( ) { φ( X ( )) = } M CN T g h I t R d. t = = = Note that a negatve net loss equals a postve net gan. By tang the expectaton we fnd that the correspondng objectve functon s n M M T ( ) = ( ) = ( ) + ( ) + ( ) ( φ( X ( )) = ) O E M C E P d. N T g h t R t () = = = In the remanng parts of ths artcle, we wll focus on statonary multstate systems. Component avalabltes are now gven by a = lm P( X( t) = ) =, t M l l= for =,, n and S. The statonary system avalabltes are gven by Let (,, M M = a a, a,, a n ) ( φ ( ( )) ) a = lmp X t =. (4) φ t a denote the vector of component avalabltes. When the components operate ndependently the statonary system avalabltes, a φ, equals ( ) a φ a for =,, M. The expected number of repars of component s now gven by ( ) = objectve functon, gven n (), therefore becomes ( ) ( ) ( ) M (3) l EN T = T, =,, n. The n M M M l O = CT + g + h aφ RT, (5) = l= = = whch s determned explctly by, =,, n, =,, M. Thus, the optmzaton problem that we wll consder s to mnmze (5) wth respect to, =,, n and =,, M wth dfferent cost functons g ( ) and h ( ). 3. The Three-Component System For smplcty, the system we wll focus on, s the multstate flow networ system consstng of three components where component s n seres wth the parallel structure of components and 3 (see Fgure ). We wll S = S =,,. assume that all components and the system are n one of three states, that s we assume { } The structure functon of the module consstng of components and 3 n parallel s ( x, x ) mn (, x x ) whereas the structure functon of the system s l ψ = +,

5 K. Sutlaberg, B. Natvg ( ) = ( x ψ ( x x3) ) = ( x x + x3) φ x mn,, mn,, (6) snce x. For the system to be n the perfect functonng state, state, both modules, that s both component and the parallel structure of components and 3, must be n the perfect functonng state. For the system to be n state both modules must be functonng, and at least one of the modules must be functonng at level =. For the system to be n the complete falure state, at least one of the modules must be n the complete falure state. The system avalabltes are hence gven by a = a aa + aa + aa + aa + aa + aa = a aa aa aa 4. The Smplfed Problem φ ( ) ( ) ( ) ( ) ( )( 3 ) a = a aa + a aa + aa φ a = a aa = a + aa aaa. φ Because of the complex nature of the problem presented n Secton we frst study a smplfed verson of the problem. Assume the expected tmes spent n each state to be equal for each component. That s, we assume =, for =,, and =,, 3. It s now natural to also assume c = c for =,, and =,, 3. As a consequence, the component avalabltes are equal to a = for all and. Thus, the system 3 avalabltes are equal to a φ = a φ = a φ = As a consequence, the total ncome, gven by the last term n the objectve functon (5), s constant. Let =,, be the vector of expected component tmes. The smplfed objectve functon s gven by ( ) 3 and the correspondng optmzaton problem s 3 φ = = ( ) ( ) ( ) O = CT 3 + g + h a RT, (8) mnmze O ( ) subject to, =,, 3. Ths s a box constraned nonlnear optmzaton problem. Note that the sum ndependent of and wll therefore not affect the mnmum. 4.. Analyss of Convexty = φ Let H O ( ) denote the Hessan matrx related to the objectve functon (8). Ths s a 3 3 matrx. H O ( ) O O O ( ) 3 D O O O = = D ( ) 3 D 3 O O O ( ), (7) (9) a RT, n (8), s 797

6 K. Sutlaberg, B. Natvg where ( ) g( ) ( ) h( ) ( ) CT D = The objectve functon s convex f and only f the Hessan matrx s postve semdefnte (see for nstance [6]). In our case, the Hessan matrx s a dagonal matrx wth D=,,, 3 on the dagonal. Hence, f all the dagonal elements are non-negatve,.e. D, =,, 3, then the objectve functon s convex and a local mnmum wll also be the global mnmum. 4.. The Objectve Functons 4... Type Cost Functons Let the cost functons be gven by g( ) = c and h( ) now gven by () = c respectvely. The objectve functon (8) s 3 c O( ) = CT 3 + c + aφ RT, = () = and the dagonal elements of the Hessan matrx are, for =,, 3, CT c D = + > Thus, n ths case, the objectve functon s convex. Dfferentatng O ( ) gves ( ( )) For =,, 3 the optmal s gven by ( ) ( ) O CT c = + c. 3 C T = +. () c 6 We see from () that the optmal, =,, 3 s dependng on TC, and c. T s the operatonal tme perod, C s the cost per repar of component and c s a constant connected to the basc nvestment costs of component spendng the expected amount of tme,, n state. Furthermore, the optmal s ndependent of j, j. It s ncreasng n T, as seen by the sold lne n Fgure, ncreasng n C, as seen n Table for = and as the basc nvestment cost parameter, c, ncreases, the optmal decreases. Ths s also seen n Table for =. These latter results are reasonable Type Cost Functons Let for,, 3 = the cost functons be logarthmc and gven by g( ) = ln ( c + ) and h( ) ln ( c ) The objectve functon (8) becomes Table. Cost functons of type. C = C3 = c = c3 =., T =, R =., R =., R =. = 5.. Theoretcal s gven by the expresson (). = +., startng values C c C c Theoretcal Computatonal Mn. expected total net loss

7 K. Sutlaberg, B. Natvg Fgure. Optmal as functon of the operatonal tme T for cost functons of type,,3. C = c =. for all. 3 CT O( ) = + ln ( c + ) + ln ( c + ) aφ RT, = 3 (3) = The dagonal elements, (), n the Hessan matrx are n ths case gven by for =,, 3. The numerator, A, s gven by ( c + ) CT c c A D = + = ( c + ) c + c + 3 ( c ) ( ) ( ) + 3 ( ) ( ) ( ) ( )( ) A= CT c + c + 6c c + + 3c c + c +. Ths s a 5-th order polynomal n. As grows large A s domnated by the 5 term whch has negatve sgn. Hence, the numerator, and D, are negatve for large. On the other hand, when approaches, D s postve. The objectve functon (3) s thus nether convex nor concave. We see that O approaches nfnty when, =,, 3, approaches ether or. Therefore the objectve functon (3) ( ) has mnmum values. Dfferentatng (3) wth respect to, =,, 3 gves the followng: The solutons to O ( ) = ( ) O CT c c = c c are the zeroes of the thrd degree polynomal n 799

8 K. Sutlaberg, B. Natvg ( ) ( ( )) c + c c CT c + CT + c CTc (4) , whch can be solved numercally. Every thrd degree polynomal has at least one real root, and snce expected tme component spends n each state, we are only nterested n postve solutons of Type 3 Cost Functons c The cost functons are n ths secton gven by ( ) e g,,, 3 h c ( ) e,,, 3 O ( ) s the =. = = for the two functonng states, and = = for the complete falure state. The objectve functon (8) now becomes 3 CT c c O( ) = + e + e 9 aφ RT = 3 (5) = The dagonal elements of the Hessan matrx are, for all =,, 3, gven by CT c c D = + ( c ) e + + e >. 3 c c 3 3 ( ) ( ) Hence, (5) s convex and therefore t has a global mnmum value Results In ths secton the ncomes per tme unt are chosen to be R =., R =. and R =.. For =,, 3 the startng values for the numercal computatons are chosen to be = 5.. The assumpton, =, =,, 3, =,,, mples that the total ncome term n the objectve functon, (8), s constant, as has already been stated. Thus, the optmal 's only depend on the parameter values, and not on the structural placements of the components. Therefore, only results for component are gven n the followng Effect of T Fgure shows the development of the optmal expected tmes spent n each state (the optmal, =,, 3 ) as functon of the operatonal tme T. Note that n ths case, because of the chosen parameter values C = c =., =,,3, from () vald for cost functon of type, the optmal = = 3 =, and the optmal expected lfe cycle length s 3 for all components. In Fgure we see that, and hence the optmal lfe cycle length, ncreases when the operatonal tme ncreases. Due to the basc nvestment costs ths s not a trval result. However, the extent of the ncrease dffers wth the dfferent cost functon types. For cost functons of type we see some ncrease n the optmal expected. Ths s n complance wth (). We also see that s by far the largest for cost functons of type for all T. For cost functons of type 3, on the other hand, we only observe a slght ncrease n the optmal as T ncreases. From Fgure 3 we see that the mnmum expected net loss as functon of the operatonal tme T behaves dfferently wth dfferent types of cost functons. For cost functons of type and, the mnmum expected net loss s decreasng wth ncreasng operatonal tme T. For type 3 the mnmum expected net loss s ncreasng at frst before t starts to decrease. The mnmum expected net loss s postve for T > when we use cost functons of type Effect of C For cost functons of type, we see from Table that the theoretcal results gven by () are equal to the computatonal results. For constant C c the theoretcal and computatonal results are also constant, whch s n accordance wth (). Even though C c s held constant (see Table for C c =. ), we see an ncrease n the mnmum expected net loss. The mnmum expected net loss s dependent on the values of C and c respectvely. We see that when these values ncrease, the net loss ncreases, as s obvous from (). For cost functons of type we found n Secton.. that the optmal s are the zeroes of the cubc 8

9 K. Sutlaberg, B. Natvg Fgure 3. Mnmum expected net loss as functon of the operatonal tme T for cost functons of type,,3. C = c =. for all. polynomal gven n (4). For the parameter values n Table ths polynomal has one postve root, whch equals the results obtaned from the optmzaton routne. We see an ncrease n the optmal as C ncreases. Fgure 4 shows the development of the mnmum expected net loss as the repar costs, C, of component ncreases. We see that the mnmum expected net loss s negatve for cost functons of type. That s, we have a postve maxmum expected net gan for ths cost functon. For the other two types of cost functons we have a postve mnmum expected net loss. For all C, the loss s greater for cost functons of type 3 than t s for cost functons of type. The correspondng optmal, and 3 are shown n Fgure 5. From Fgure 5 we see that for cost functons of type t s optmal to spend longer tme n each state than t s for the other two cost functons. The optmal expected tme spent n each state for component s ncreasng wth ncreasng repar costs C. Ths seems natural. The results for components and 3 are equal because the parameter values concernng these two components are equal. From the rght plot n Fgure 6 we see that the ncrease n the repar costs of component has no nfluence on the optmal expected tme spent n each state for components and 3, thus we see that the optmal, =,3 are constant Effect of c Fgure 6 shows the mnmum expected net loss as a functon of the cost functon parameter c. The mnmum expected net loss s ncreasng wth an ncreasng c. We see that an ncrease n c has much larger effect on the mnmum expected net loss when we use cost functons of type 3 than when we use the other two types of cost functons. Ths s natural snce the type 3 cost functons are exponental. The correspondng optmal, =,, 3 are shown n Fgure 7. We see that s constant., on the other hand, seems to be decreasng wth ncreasng c when we have cost functons of type and 3. Ths behavor seems reasonable. For cost functons of type we see that eventually starts to ncrease when c ncreases, whch seems unnatural. 8

10 K. Sutlaberg, B. Natvg Fgure 4. Mnmum expected net loss as functon of C for cost functons of type,,3. C = C3 = c =. for all and T =. Fgure 5. Optmal as functon of C for cost functons of type,,3. C = C3 = c =. for all and T =. 8

11 K. Sutlaberg, B. Natvg Fgure 6. Mnmum expected net loss as functon of c for cost functons of type,,3. c = c3 = C =. for all and T =. Fgure 7. Optmal as functon of c for cost functons of type,,3. c = c3 = C =. for all and T =. 83

12 K. Sutlaberg, B. Natvg Table. Cost functons of type. C = C3 = c = c = c3 =., T =, R =., R =., R =., startng values = 5.. Theoretcal s the root of the polynomal (4). C Theoretcal Computatonal Mn. expected total net loss Modfcatons of the Full Model Optmzaton Problem The orgnal problem, represented by the objectve functon (5), turned out to be qute complex even though we only consdered a smple three-component system wth three possble system and component states. Thus, the optmzaton of ths problem was not straghtforward. In order to overcome dffcultes wth startng value senstve optmzaton results, we reformulated the orgnal optmzaton problem n order to fnd an approxmate soluton. Let α = + +, =,, 3, be the fxed expected length of the lfe cycle of component. Then, the objectve functon s gven by where (,,, 3 ) 3 φ = = ( ) α ( ) ( ) ( ) O = CT + g + g + h a RT, (6) = 9 s a vector of the expected tmes spent n each state for each component. We are nterested n mnmzng the expected total net loss (6) subject to the fxed expected length of the lfe cycles α, =,, 3. The optmzaton of the modfed problem goes as follows: Step : Choose values for α, =,, 3. For every combnaton of these α, =,, 3, the nonlnear optmzaton problem wth both equalty and nequalty constrants, (7), s solved. mnmze μ O ( ) subject to α =, =,, 3. =, =,,3, =,, Step : Identfy the overall mnmum from the optmzaton results from step. The correspondng wll approxmately mnmze the expected total net loss over the tme perod [,T ]. Optmzaton problems as the ones n step may be solved usng the augmented Lagrange multpler method, for nstance usng the SOLNP algorthm as descrbed n [7]. Ths algorthm s mplemented n the Rsolnp pacage, see [8], n R. Note that mnmzng the expected total net loss s equvalent to maxmzng the expected total net gan, and that a negatve expected total net loss s a postve expected net gan. Snce we are usng mnmzaton algorthms nstead of maxmzaton algorthms, the focus has been on mnmzng the total net loss rather than maxmzng the total net gan. 5.. Results In ths secton lower bounds on, =,, 3 and =,, are chosen to be -5 for cost functons of type and, and. for cost functons of type 3. The upper bound s chosen to be equal to the operatonal tme T. As n the prevous secton, the ncomes per tme unt when the system s n state =,,, are R =., R =. and R =. respectvely. The possble lfe cycle lengths are chosen to be α = ( 4, 6,8,,,4,6 ), =,,3, and the startng values are =. for =,, 3 and =,,. Component and component 3 are n parallel. Ther roles n the system are therefore nterchangeable. Snce (7) 84

13 K. Sutlaberg, B. Natvg we assume that the components cost functons are of the same type and that we are varyng one parameter at a tme, we are n the followng only varyng the parameters connected to component and component. When the parameters of component are vared the results for components and 3 are dentcal. Hence, results for component 3 are then omtted Effect of T Fgure 8 shows the mnmum expected net loss as a functon of T. We see that wth cost functons of type and, the mnmum expected net loss s negatve and decreasng for the chosen values of T. Ths means that for these cost functons we have an ncreasng maxmum expected net gan. The loss s smaller for cost functons of type than t s for the other two types of cost functons. For cost functons of type 3 the mnmum expected net loss s postve for small T. The correspondng optmal s, =, and =,,, are gven n Fgure 9. It seems le the optmal s stablzes as T becomes large Effect of an Increasng Cost Per Repar C, =, For ths, and the followng sectons, the operatonal tme s set to T = 3. Fgure and Fgure shows the mnmum expected net loss as a functon of the repar cost of component and respectvely. We see that the mnmum expected net loss s ncreasng wth ncreasng repar costs C, =,, for all three cost functons. Ths seems natural. The correspondng optmal, =,, =,, as functons of C are shown n Fgure and for C n Fgure 3 for =,, 3, =,,. As the repar costs of component, C, ncreases, the optmal expected lfe cycle length ( α = ) = for component ncreases for cost functon types and 3. For cost functons of type the optmal,,, = are constant. For cost functons of type and type, t s optmal to eep Fgure 8. Mnmum expected net loss as functon of T for cost functons of type,,3. C = C = C3 = c =. for =,,3, =,,. 85

14 K. Sutlaberg, B. Natvg Fgure 9. Optmal =,,3, =,,., =,, =,,, as functon of T for cost functons of type,,3. C = C = C3 = c =. for Fgure. Mnmum expected net loss as functon of C for cost functons of type,,3. C = C3 = c =. for =,,3, =,,. 86

15 K. Sutlaberg, B. Natvg Fgure. Mnmum expected net loss as functon of C for cost functons of type,,3. C = C3 = c =. for =,,3, =,,. Fgure. Optmal, =,, =,,, as functon of C for cost functons of type,,3. C = C3 = c =. for =,,3, =,,. 87

16 K. Sutlaberg, B. Natvg Fgure 3. Optmal, =,,3, =,, as functon of C for cost functons of type,,3. C = C3 = c =. for =,,3, =,,. component n the perfect functonng state for as long as possble. Hence, we see a large for these two cost functon types. For cost functons of type 3, on the other hand, the ncrease n expected lfe cycle length s placed n state =, and we see an ncrease n for ths cost functon. The extra costs connected to an ncrease n the expected tmes spent n ether of the two functonng states are much larger for cost functons of type 3 than for the other two types of cost functons. As the repar costs of component, C, ncreases, we see from Fgure 3 that the results for component are constant. Hence, for component the optmal expected tme spent n each state s ndependent of the repar cost of component. We also see that an ncreasng repar cost results n an ncreasng optmal expected lfe cycle length for component for all cost functons. For cost functons of type and the ncrease s on the expected tme spent n state,, whle the ncrease s on the expected repar tme,, for cost functons of type 3. Snce component s crtcal to the functonng of the system we see from Fgure an ncrease n for cost functons of type 3 as C ncreases. Component s n parallel wth component 3. It s therefore possble to extend the expected repar tme of ths component when C ncreases, whle at the same tme the expected repar tmes of component 3,, are ept low. Ths s seen n Fgure 3 for cost functons of type Effect of an Increasng c, =,, As the cost functon parameter c, =, for component ncreases, the mnmum expected net loss also ncreases for cost functons of type and 3. The mnmum expected net loss remans unchanged when cost functons of type are used. See Fgure 4 and Fgure 5. An ncrease n c,, = has largest mpact on the mnmum expected net loss when the cost functons are of type 3, that s when we have exponental cost 88

17 K. Sutlaberg, B. Natvg Fgure 4. Mnmum expected net loss as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =,3, =,, and =, =,. Fgure 5. Mnmum expected net loss as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =,3, =,, and =, =,. 89

18 K. Sutlaberg, B. Natvg functons. In state = there was no ncrease n the mnmum expected net loss wth ncreasng Fgure 6. Fgures 7-9 show the development n the optmal ncreases. The effect of an ncreasng c s evdent n the ncreasng optmal seen n Fgure 7, where there s also a slght ncrease n and c, as seen n, =,, =,, as c,,, = for cost functons of type 3. Ths s for cost functons of type. The behavour of dffers for the three cost functon types as the cost functon parameter c ncreases. In state =, s close to for both cost functons of type and, whle postve for c > for cost functons of type 3. For = the optmal s hgh and (close to) constant for cost functons of type (). For cost functons of type 3 the optmal s lower. An ncreasng c has no effect on the optmal, =,, =,,. Ths s seen n Fgure 8. As c ncreases, we see from Fgure 9 that the optmal expected tme spent n state for component,, s decreasng for both cost functons of type and 3. As the costs of eepng at a fxed level s ncreasng, t becomes less desrng to mantan ths level, and we see a decrease. The decrease s faster n for cost functons of type 3. For ths cost functon we see an ncrease n wth ncreasng c that s n contrast to the results for cost functons of type and whch are ndependent of the ncrease n to. c and equal Fgure 6. Mnmum expected net loss as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =,3, =,, and =, =,. 8

19 K. Sutlaberg, B. Natvg Fgure 7. Optmal, =,, =,, as functon of c =. for =,3, =,, and =, =,. c for cost functons of type,,3. C =. for =,,3 and Fgure 8. Optmal, =,, =,, as functon of c for cost functons of type,,3. C =. for all and c =. for =,3, =,, and =, =,. 8

20 K. Sutlaberg, B. Natvg Fgure 9. Optmal, =,, =,, as functon of c =. for =,3, =,, and =, =,. c for cost functons of type,,3. C =. for =,,3 and Effect of an Increasng c, =,, An ncreasng c, =,, has lttle effect on the mnmal expected net loss. Ths s seen n Fgures - where the mnmal expected net loss wth cost functons of type seems to be constant and the result for cost functons of type and 3 s ncreasng slghtly for small c,,, = before t seems to be constant. Fgures 3-5 show the optmal expected tmes spent n each state for each component as the cost functon parameter, c,,, =, ncreases. The optmal expected tmes spent n each state for component,,,, = reman unchanged for all types of cost functons. For every cost functon type we see from Fgure 3, for component, an ncrease n and a decrease n as c ncreases. For component 3 we have the opposte behavour. As the costs of eepng the expected repar tmes of component low ncreases, t s optmal to spend more expected tme reparng ths component. At the same tme, t wll be more mportant to eep the expected repar tmes of component 3 low. We see from Fgure 4 that the optmal s slghtly decreasng wth ncreasng c for both cost functons of type and 3. 3 s also slghtly decreasng for these two cost functons, but the optmal s ncreasng for these cost functons. For cost functons of type we see from Fgure 4 that the optmal s hgh (approxmately 5) for c >, whle s below 5 for all values of c for cost functons of type 3. For component 3 we see that the results for cost functons of type and are lower than the correspondng results for cost functons of type 3 n state =, and above n state =. The optmal, =,, 3 and =,, for ncreasng c are shown n Fgure 5. We see that the results for cost functons of type are constant. Furthermore, for cost functons of type and 3, component 8

21 K. Sutlaberg, B. Natvg Fgure. Mnmum expected net loss as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =,3, =,, and =, =,. Fgure. Mnmum expected net loss as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =,3, =,, and =, =,. 83

22 K. Sutlaberg, B. Natvg Fgure. Mnmum expected net loss as functon of c =. for =,3, =,, and =, =,. c for cost functons of type,,3. C =. for all =,,3 and Fgure 3. Optmal, =,,3, =,, as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =,3, =,, and =, =,. 84

23 K. Sutlaberg, B. Natvg Fgure 4. Optmal, =,,3, =,, as functon of c for cost functons of type,,3. C =. for =,,3 and c =. for =, 3, =,, and =, =,. Fgure 5. Optmal, =,, 3, =,, as functon of c for cost functons of type,, 3. C =. for all =,,3 and c =. for =,3, =,, and =, =,. 85

24 K. Sutlaberg, B. Natvg has ncreasng and decreasng wth ncreasng c, whereas, component 3 has decreasng 3 and ncreasng 3. Ths s the same behavour as observed for ncreasng 6. Concludng Remars c for these cost functons. In the present paper we have been mnmzng the expected total net loss over a tme perod [,T ] as a functon of the expected component tmes n each state for a three-component flow networ system. Frst the basc model was presented. The assumpton of equalty between the set of system states and the set of component states mpled that an approprate flow networ system would have at least one component n seres wth the rest of the system. Hence, the three-component system gven n Fgure was chosen as a case. Wth three possble system and component states and three components, the orgnal box-constraned optmzaton problem had 9 varables. Due to the complexty of ths problem, we frst studed the smplfed problem, where = and c = c for =,, 3 and =,,, and then a modfcaton of the orgnal problem where the optmzaton was done n two steps (see Secton 5). Ths method found an approxmate soluton. The ndcaton of lac of constructve conclusons s manly due to that we are facng complex dependences. The varables TC, and c for =, n the smplfed problem, and the varables TC, and c for =, and =,, n the modfed full problem, were vared one at a tme wth three dfferent types of cost functons. For the smplfed problem we were able to fnd expressons for the optmal, =,, 3 for cost functons of type. For cost functons of type and 3, the objectve functon, (8), turned out to be a convex functon. Wth cost functons of type the objectve functon s nether convex nor concave. The type cost functons are logarthmc, and hence concave whle the other two types are convex. For ths cost functon, we saw n Fgure 7 that the optmal has a mnmum as c ncreased, whch seems unnatural. In both the smplfed problem and the modfed full model, the mnmum expected net loss was ncreasng wth ncreasng C, =, for every cost functon type (as seen n Fgure 4, Fgure and Fgure respectvely). As the operatonal tme T ncreased we saw a decrease n the mnmum expected net loss n the modfed full model for all three cost functons (as seen n Fgure 8). Ths s n contrast to the results wth the smplfed model when the exponental cost functons were used. Then, the mnmum expected net loss ncreased at frst, before t started to decrease (as seen n Fgure 3). For every cost functon parameter, c, =,,, we vared, we saw n Fgures 7-9 that the optmal, =, was constant for cost functons of type. The same observaton of constant, =,, 3 for cost functons of type was done n Fgures 3-5 where c,,, = were vared. The values were also close to zero. Hence, t was, for cost functons of type, optmal to spend as lttle tme as possble n state ndependent of the values of the parameters. Wth cost functons of type we observed the same, except from when c was ncreasng where decreased from around 5 to close to as c ncreased from to. For c > stayed constant and close to. Thus, t seems le the functonng component state s n a way redundant for cost functons of type and. Ths was not the case wth cost functons of type 3. The general objectve functon (5) can qute easly be modfed to represent the expected net loss of other networ flow systems, and to nclude other types of cost functons. However, wth larger systems, wth more components and possbly more component states, the optmzaton problem qucly becomes large. Hence, the real challenge les n the optmzaton of real lfe systems. Acnowledgements The authors than Professor Ger Dahl for the dea on how to modfy the full optmzaton problem and Ph.D Olav Sutlaberg for valuable feedbac throughout the process. References [] Zo, E. (9) Relablty Engneerng: Old Problems and New Challenges. Relablty Engneerng and System Safety, 86

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