A New Mathematical Model for a Redundancy Allocation Problem with Mixing Components Redundant and Choice of Redundancy Strategies

Transcription

1 Appled Mathematcal Sceces, Vol, 2007, o 45, A New Mathematcal Model for a Redudacy Allocato Problem wth Mxg Compoets Redudat ad Choce of Redudacy Strateges R Tavakkol-Moghaddam Departmet of Idustral Egeerg, Faculty of Egeerg Uversty of Tehra, PO Box: 365/4563, Tehra, Ira J Safar Departmet of Idustral Egeerg, Scece ad Research Brach Islamc Azad Uversty, Tehra, Ira Abstract Ths paper presets a ew mathematcal model of a redudacy allocato problem wth mxg compoets redudat subsystems of a seres-parallel system whe the redudacy strategy ca be chose for dvdual subsystems I practce both actve ad coldstadby redudaces are used wth a partcular system desg, ad the choce of the redudacy strategy becomes a addtoal decso varable The proposed model selects the best redudacy strategy, combato of compoets, ad levels of redudacy for each subsystem order to maxmze the system relablty uder systemlevel costrats Keywords: Redudacy Allocato Problem; Relablty; Seres-Parallel Systems; Redudacy Strateges Correspodg author e-mal: tavakol@utacr

2 2222 R Tavakkol-Moghaddam ad J Safar Itroducto The prmary goal of a relablty desg s to mprove the system relablty I the tal desg actvty, a redudacy allocato s a drect way of ehacg the system relablty The redudacy allocato problem (RAP) volves the smultaeous selecto of compoets ad a system-level desg cofgurato, whch ca collectvely meet all desg costrats order to optmze some obectve fuctos such as system cost ad/or relablty [3] I ths problem, there are several dfferet compoet types wth dfferet levels of cost, relablty, weght, ad other characterstcs, ad the compoets redudat wth the subsystem are the same type The RAP s a NP-hard problem [2] solved by usg may dfferece optmzato approaches for dfferet formulatos as summarzed Kuo, et al [0] ad Ge, et al [8] Whle ths problem has bee studed great detals, oe area, whch has ot bee suffcetly aalyzed, s the use of mxg compoets redudat subsystems Ths kd of the redudacy allocato problem s to select the optmal combato of compoets ad levels of redudacy to collectvely meet weght ad cost costrats, whle maxmzg the system relablty [9] Cot ad Smth [4] preseted a ew formulato ad soluto method for the redudacy allocato problem wth mxg compoets redudat subsystems They used a geetc algorthm (GA) to solve ths problem k-out-of-:g system Che ad You [] also preseted the use of a mmue algorthm to solve such a problem a seres system I geeral, ths relablty desg problem has bee formulated by cosderg actve redudacy However, the choce of redudacy strateges for each subsystem s much more realstc ad t provdes a better tool for the desgers Ths becomes a addtoal decso varable a redudacy allocato problem wth mxg compoets redudat subsystems Cot ad Lu [5] preseted a ew formulato ad soluto method for the RAP whe a system desg cludes multple subsystems desged wth ether actve or cold-stadby redudacy Ths soluto method assumes that the redudacy strategy (e, actve or cold-stadby) for each subsystem s predetermed Cot [6] preseted a ew formulato ad soluto method to the RAP whe there are some subsystems usg actve redudacy ad cold-stadby

3 New mathematcal model 2223 redudacy, or selectg the best redudacy strategy Furthermore, Tavakkol- Moghaddam, et al [] used the geetc algorthm to solve the above-metoed problem Ufortuately, the redudacy allocato problem wth mxg compoets subsystems s ot cosdered whe the redudacy strategy ca be chose for dvdual subsystems Thus, the problem s to select the best redudacy strategy, combato of compoets, ad redudacy level for each subsystem order to maxmze the system relablty uder system-level costrats The seres-parallel system depcted Fgure s a commo system structure that s used most system desgs Thus ths paper, the seres-parallel redudacy allocato problem wth mxg compoets, ad the choce of redudacy strateges subsystems s cosdered Fally, a ew mathematcal model s preseted to ths problem Fgure Seres-parallel system The structure of ths paper s orgazed as follows Secto 2 presets a revew o the redudacy strateges I Secto 3, the problem formulato s preseted ad proposed for a redudacy allocato problem wth mxg compoets subsystems whe ether actve or cold-stadby redudacy ca be selected for dvdual subsystems Fally, Secto 4 presets cocluso

4 2224 R Tavakkol-Moghaddam ad J Safar 2 Redudacy Strateges There are two types of redudacy strateges, amely actve ad stadby If all redudat compoets operate smultaeously from tme zero, eve though the system eeds oly oe at ay gve tme, such a arragemet s called actve redudacy There are three varats of the stadby redudacy, referred to as cold, warm, ad hot I the cold stadby redudacy, the compoet does ot fal before t operates I the warm stadby redudacy, the compoet s more proe to falure before operato tha the cold stadby compoets I the hot stadby redudacy, the falure patter of compoet does ot deped o whether the compoet s dle or operato The mathematcal models for hot stadby ad actve redudacy arragemets are the same I the stadby redudacy arragemet, the redudat compoets are sequetally used the system at compoet falure tmes Each redudat compoet the stadby arragemet ca operate oly whe t s swtched o Whe the compoet operato fals, oe of the redudat uts s swtched o to cotue the system operato [7] I the stadby redudacy, there are two scearos frst detectg falure ad the swtchg to good compoets These are classfed as Case ad Case 2 For the Case, the falure detecto ad swtchg hardware or software cotually motors the system performace Whe t detects a falure, t actvates a redudat compoet For the Case 2, a falure s oly possble whe a swtch s requred At ay tme the swtch s requred, there s a costat probablty (ρ ) that the swtchg wll be successful [6] Ths paper cosders redudacy strateges cosstg of oly actve (e, hot) ad cold-stadby redudacy The approach used categorzes all subsystems to four sets accordg to the followg deftos: N: Set of all subsystems wth o redudacy A: Set of all subsystems wth actve redudacy S: Set of all subsystems wth cold-stadby redudacy A&S: Set of all subsystems wth actve or cold-stadby redudacy

5 New mathematcal model Problem Formulato I each subsystem, all assgable compoet types are sorted descedg order by the compoet relablty ad put a set of compoet types From ow, compoet types are show by ther rak orders ths set Therefore, compoets of type are more relable tha compoets of type 2 ad compoets of type 2 are more relable tha compoets of type 3, ad the lke For crease the total relablty of a system, compoets of the frst type that are more relable all compoet types are used After all compoets of type are dow ad falure, compoets of type 2 that are more relable backup compoets are used, ad so oe Sequetal usage of backup compoets cotues utl falure of all compoets The ew mathematcal model of the redudacy allocato problem wth mxg compoets redudat subsystems for the seres-parallel system s preseted as the followg teger olear programmg problem whe the redudacy strategy ca be chose for dvdual subsystems ad two separable lear costrats 3 Assumptos Two redudacy strateges (e, actve redudacy, cold stadby) are cosdered The states of compoets ad the system have oly two optos: good or bad The compoet attrbutes (e, relablty, cost, ad weght) are kow ad determstc There s o compoet repar or prevetve mateace Falures of compoets are depedet evets Faled compoets do ot damage the system 32 Idces Idex for subsystems ( =,2,, s) Idex for compoet type =,2,, m ) (

6 2226 R Tavakkol-Moghaddam ad J Safar k Idex for umber of falure of type compoets each subsystem ( k =,2,, ) l Idex for umber of the allocated compoet types z Idex of compoet choces used for subsystem, z {,2,, } l z Set of z l, (z, z 2, ), for example (, 3, 4) l m 33 Decso Varables ARS Allocated redudacy strategy subsystem ARS = ARS, ARS,, ARS ) Vector of allocated redudacy strategy ( 2 s subsystems Number of compoets of type used subsystem, {,2,, } max, = s Type of compoets s2 2 s m m 2m 2 m sm s Subsystem 34 Parameters s Number of subsystems m Number of avalable compoet choces for subsystem, ( =, 2,, s) m Upper boud for m, (m m ) max, Upper boud for, ( max, )

7 New mathematcal model 2227 t Msso tme r (t) Relablty at tme t for the th avalable compoet for subsystem c, w Cost ad weght for the th avalable compoet for the subsystem C, W System-level costrat lmts for cost ad weght R(t; ; AR S) System relablty at tme t for desgg matrx ad vector ARS ρ (t) Falure-detecto/swtchg relablty at tme t (Case ) ρ Falure-detecto/swtchg success probablty (Case 2) ( k ) f, ( t) pdf of the k th falure tmes of the type compoets for subsystem at tme t 35 Mathematcal model Max R(t; ; ARS) () st s m = = c C (2) s m = = w W (3) Wth respect to Equato (), the obectve s to determe the redudacy strategy, combato of compoets, ad the quatty of compoets each subsystem to acheve the maxmum system relablty Costrats gve Equatos (2) ad (3) cosder the avalable cost ad weght, respectvely To calculate R ( t;, ARS), Equatos (4) ad (5) are preseted for the system relablty two cases as follows: Case : Cotuous detector/swtch operato: m t ( k ) R(; t, ARS) = r () t ( r () t ) rz () t + ( ) ρ 0 u f N A = S z k= t () ( u) r ( t u) du ρ ( ) ( ) ( ) 0 u fz u r z t u du (4)

8 2228 R Tavakkol-Moghaddam ad J Safar Case 2: Swtch actve oly respose to a falure: m k+ h h= z ρ N A = S z k= ( ) R(; t, ARS) = r () t r () t r () t + t ( k ) t () f ( u) r ( t u) du ρ fz ( u) r ( ) z t u du 0 0 (5) After allocatg a redudacy strategy to each subsystem ad accordg to the defto of ARS, all subsystems ca be classfed by the allocated redudacy strategy as follows: N : Set of all subsystems wth o redudacy A : Set of all subsystems wth actve redudacy S : Set of all subsystems wth cold-stadby redudacy Therefore, Equato (4) s costructed three segmets Segmets, 2, ad 3 calculate the relablty of all subsystems wth o redudacy, actve redudacy, ad cold-stadby redudacy respectvely I segmet 3, the relablty of cold-stadby subsystem s the summato of the relablty of compoets operato as well as backup compoets Accordg to the defto of, at the begg operato of a cold-stadby subsystem, oe compoet of type z s operatg Whe a compoet operato fals, oe of the backup compoets s swtched o to cotue the system operato Falure ( k ) each subsystem s accrued at tme u wth probablty of f, ( u) ad the the swtchg wll be successful wth probablty of (ρ (u)), fally oe backup compoets wth relablty of r ( t u) s operated Equato 5 s the same as Equato (4) The oly dfferece s calculatg the relablty of detectg falure ad swtchg system I ths equato, the power of ρ s the umber of falure subsystem tll tme t

9 New mathematcal model Cocluso Ths paper proposes a ew mathematcal model for a redudacy allocato problem wth mxg compoets redudat subsystems for the seres-parallel system whe ether actve or cold-stadby redudacy s selected for dvdual subsystems Most mathematcal models of the geeral, foregog problems assume that the redudacy strategy for each subsystem s predetermed ad kow I geeral, the actve redudacy has receved more atteto the past The choce of redudacy strateges for each subsystem s much more realstc ad t provdes a better tool for the desgers Ths problem s formulated as a olear mxed-teger programmg model uder a umber of costrats Ths problem s ot easy to solve real cases, especally for large-szed systems Therefore, the use of heurstc or meta-heurstc methods for solvg such a hard problem s proposed for future research Refereces [] T Che ad P You, Immue algorthms-based approach for redudat relablty problems wth multple compoet choces, Computers Idustry, 56 (2005), [2] MS Cher, O the computatoal complexty of relablty redudacy allocato a seres system, Operato Research Letters, (992), [3] DW Cot ad A Smth, Optmzato approaches to the redudacy allocato to the redudacy allocato problem for seres-parallel systems, Proceedgs of the Fourth Idustral Egeerg Research Coferece, Nashvlle, TN, (May 995), [4] DW Cot ad A Smth, Relablty Optmzato of Seres-Parallel Systems Usg a Geetc Algorthm, IEEE Trasactos o Relablty, 45 (996), [5] DW Cot ad J Lu, System relablty optmzato wth k-out-of- subsystems, Iteratoal Joural of Relablty, Qualty & Safety Egeerg, 7 (2000), 29-43

10 2230 R Tavakkol-Moghaddam ad J Safar [6] DW Cot, Maxmzato of system relablty wth a choce of redudacy strateges, IIE Trasactos, 35 (2003), [7] CE Eblg, A troducto to relablty mataablty egeerg, McGraw-Hll, New York (997) [8] M Ge ad YS Yu, Soft computg approach for relablty optmzato: State-of-the-art survey, Relablty Egeerg ad System Safety, 9 (2006), [9] M Ge ad R Cheg, Geetc algorthm ad egeerg desg, Joh Wley & Sos, (997) [0] W Kuo, VR Prasad, FA Tllma ad C Hawag, Optmal relablty desg fudametal ad applcato, Cambrdge Uversty Press, Lodo (200) [] R Tavakkol-Moghaddam, J Safar ad F Sassa, Relablty optmzato of seres-parallel systems wth a choce of redudacy strateges usg a geetc algorthm, Relablty Egeerg ad System Safety, do:006/ress , (2007) Receved: March 24, 2007