A NSGA-II algorithm to solve a bi-objective optimization of the redundancy allocation problem for series-parallel systems

Transcription

1 0 nd Internatonal Conference on Industral Technology and Management (ICITM 0) IPCSIT vol. 49 (0) (0) IACSIT Press, Sngapore DOI: 0.776/IPCSIT.0.V49.8 A NSGA-II algorthm to solve a b-obectve optmzaton of the redundancy allocaton problem for seres-parallel systems Mehd Zaraban +, Seyed Tagh Akhavan Nak and Mohammad Sad Mehrabad Department of Industral Engneerng, Islamc Azad Unversty, Qazvn Branch, Qazvn, Iran Department of Industral Engneerng, Sharf Unversty of Technology, Tehran, Iran Department of Industral Engneerng, Iran Unversty of Scence and Technology, Tehran, Iran Emal: Mehdzaraban_p@yahoo.com, Nak@Sharf.edu, mehrabad@ust.ac.r Abstract. Ths paper nvolves developng a new model for the redundancy allocaton problem of seresparallel systems. The system conssts of subsystem n seres, where components are used n parallel for each subsystem. There are varous types of components as canddates for allocaton. These components that are chosen from a lst that s avalable n the market are characterzed by ther cost, weght, and relablty. Besdes, they have varous levels of performance rangng from functonng perfectly to fal completely. The goal s to fnd the optmal combnaton of the components of each subsystem n order to maxmze the system relablty and to mnmze the cost of purchasng components. A NSGA-II algorthm s developed and proposed to solve the b-obectve problem, where the unversal generatng functon for mult-state components s used to obtan the relablty of a gven system. In addton, a penalty functon that encourages the soluton algorthm to explore wthn nfeasble solutons s proposed. At the end, a numercal example s used to valdate the soluton and to assess the performance of the proposed methodology under dfferent confguratons. Keywords: Avalablty; Redundancy allocaton problem; Seres-parallel systems; Mult-state; Unversal Generatng Functon. Introducton The relablty of a system s drectly related to the relabltes of ts components. One of the best and the most effcent methods to mprove the system relablty s to use redundant components besde the man components []. The redundancy allocaton problem (RAP) wth many applcatons n ndustres s a complex combnatonal optmzaton problem that s shown to be NP-hard []. Electronc systems, power statons, and manufacturng systems are some examples of the RAP applcatons []. A seres-parallel system conssts of some subsystems that work n seres each contanng some components that are arranged n parallel. The RAP of these system nvolves obtanng an optmal number of components n each subsystem such that the total system avalablty s maxmzed whle the constrant(s) (usually the system weght) s (are) satsfed. Component performances of real-world systems are classfed nto more than the usually assumed two states of "completely workng" and "completely faled," resultng n mult-state component systems (MSS), nstead of bnary systems. In a MSS, both the system and ts components are allowed to experence more than two possble states, e.g., completely workng, partally workng, partally faled, and completely faled [4]. Snce the number of MSS states ncreases very rapdly wth ncrease n the number of ts elements, the unversal generatng functon (UGF) was frstly ntroduced n [5] to reduce the computatonal complexty nvolved n evaluatng the MSS relablty [6]. + Correspondng author. Tel.: ; fax: E-mal address: Mehdzaraban_p@yahoo.com 8

2 Many researchers usng dfferent approaches and technques studed RAP for seres-parallel systems wth bnary-state components. The approaches were all proposed to solve sngle-obectve (generally maxmzaton of system relablty) RAPs subect to several constrants such as cost, weght, and volume. Tllman et al. [7] presented a revew of the current redundancy allocaton and relablty lterature. Ouzneb et al. [8] developed an effcent Tabu search to solve the bnary RAP seres parallel systems to maxmze system relablty. However, n many real-world optmzaton of the RAP, smultaneous optmzaton of more than one obectve functon s requred. For nstance, mnmzng total system cost n addton to maxmzng the total system relablty mght be a case that s nvestgated n the current research. In ths paper, a new constraned b-obectve model of a mult-state seres-parallel redundancy allocaton problem s presented and solved. The constrants are the system weght and the number of components used. In the proposed formulaton, UGF s employed to evaluate the relablty of each soluton. The goal s to fnd the optmal combnaton of the components of each subsystem n order to maxmze the system relablty and to mnmze the cost of purchasng components. Furthermore, a NSGA-II algorthm s developed to solve the problem.. Problem Formulaton The problem assumptons, notatons, and the model development come n ths secton as follow... Assumptons The assumptons are: Performance state, weght and cost of all components and number of subsystems are determned and known. There are several types of component as canddate for allocaton n each subsystem. All components of a subsystem are selected from a smlar type (brand). The components are chosen from a lst that s avalable n the market. All allocatons are actve and swtchng between components s performed perfectly. System components are mult-state. Mnmum and maxmum number of allocaton for each subsystem s determned and known... Notatons The notatons that are used n the rest of the paper are as follows: s : The number of subsystems n seres n : Total number of component types avalable for subsystem c : Cost of the -th component type n the -th subsystem x : Number of the -th component type used for the -th subsystem ( = n ) C( X ): R( X ): w : W : L : U : Total system cost Total system relablty Weght of the -th component type n the -th subsystem,,..., Total system weght Mnmum number of components that can be used n parallel for the -th subsystem Maxmum number of components that can be used n parallel for the -th subsystem.. The mathematcal model Based on the assumptons and notatons, the mathematcal formulaton of the seres-parallel system can be expressed as M n C ( X ) c x Max R ( X ) s n = = = 9

3 Subect to: s n = = wx W L x U, for = k =,,..., s; =,,..., n x = 0, for k =,,..., s; =,,..., n x {0,,,..., U } The obectve functons are mnmzaton of system cost and maxmzaton of system relablty, where the frst constrant mples that the total weght of the desgned seres-parallel system should be less than W. The second constrant s related to maxmum and mnmum number of component that s allocated to a subsystem. The thrd constrant shows ths assumpton that all components of a subsystem are selected from a smlar type. It means when x = 0, then there s no allocaton for the -th subsystem of the -th component.. UGF and NSGA-II algorthm Smlar to other RAP problems, the b-obectve optmzaton problem that has been formulated above s NP-hard. In ths secton a NSGA-II algorthm n whch the UGF approach s used to determne the relablty of a system based on ts components relablty s developed... UGF In UGF, the probablty dstrbuton functon of each element s frst evaluated usng the ( ) uzfuncton gven n equaton (). Then, to evaluate the probablty dstrbuton functon of MSS, equaton () s utlzed for a par of components connected n parallel and equaton () s employed for a par of components connected n seres. ( ) u z l = p z () = g. k k k k b a a+ b π( u( z), u( z) ) = π pz, qz = pq z = = = = () k k k k b a mn{ a, b } π( u( z), u( z) ) = π pz, qz = pq z () = = = = where the parameters I, k and k are numbers of possble performance levels for components; g, a and b are physcally nterpreted as the performance levels of these components; whle p and q are steady-state probabltes of the dfferent performance levels for the components. At the end, the relablty of the entre system s evaluated based on the mnmum measure of demand... NSGA-II The steps nvolved n the developed NSGA-II algorthm are. Intalzaton of parameters and the frst populaton generaton of n chromosomes. Ftness evaluaton of chromosomes. Fast non-domnated sortng and crowdng dstance 4. Selectng parents based on the crowded tournament selecton operator 5. Applyng contnuous unform crossover and swap mutaton 6. Evaluaton of offsprng and combnng parent and offsprng populaton 7. Evaluaton of fast non-domnated sort and crowdng dstance agan for solutons 8. Sortng populaton based on crowdng dstance and selectng N ndvdual for the new populaton 9. When a maxmum number of generaton s reached, stop or return to step 4 In order to acheve better convergence of NSGA-II algorthm to a good soluton, a local search heurstc procedure s proposed n ths paper. In ths procedure, for each type of component, we frst generate a nteger number between L and U, randomly. Then, based on the generated numbers, the relabltes of the 40

4 correspondng subsystems are calculated. Next, the -th component type wth the maxmum relablty s selected. An adaptve penalty functon P( x ) s also used to penalze nfeasble solutons for weght constrant. Ths functon s defned as λ W Px ( ) = Mn, W( x) (4) Then, the ftness functons are penalzed by F(x) = R(x) P(x) (5) F(x) = C(x) P(x) (6) Where F( x ) s the ftness of soluton x, R( x)and C(x ) are the overall relablty and overall cost of the soluton x, respectvely. Also λ s the penalty factor. 4. Numercal example One example s consdered that conssts of three man unts connected n seres. For each unt, there are several components avalable. Each component of the system can have dfferent levels of performance, whch range from maxmum capacty to total falure (Table ). NSGA-II was employed based on a populaton sze of 00 and 0 generatons. Also maxmum total weght of system s 700 and mnmum measure of demand s. The computaton tme was 6 seconds. Fg. shows solutons found n the Pareto front. 5. Concluson In ths paper, a seres-parallel redundancy allocaton problem wth mult-state components was consdered. The obectve functons were the maxmzaton of the total system relablty and the mnmzaton of the total system cost. The components were characterzed to have dfferent performance levels, costs, weghts, and relabltes. The UGF approach was employed to evaluate the relablty of each soluton to reduce computatonal complexty of the MSS. In order to acheve better convergence of NSGA-II algorthm to a good soluton, a local search heurstc procedure was proposed. The soluton to the multobectve problem s a set of solutons, known as the Pareto-front, from whch the analyst may choose one soluton for system mplementaton. 6. References [] C. Ha, W. Kuo, Relablty redundancy allocaton: an mproved realzaton for non-convex nonlnear programmng problems, European Journal of Operatonal Research, vol. 7, pp. 4 8, 006. [] Chern M.S, On the computatonal complexty of relablty redundancy allocaton n a seres system, European Journal of Operatonal Research, vol., pp. 09 5, 99. [] Prasad VR. Raghavachar M. Optmal allocaton of s-dentcal mult-functonal speares n a seres system, IEEE Transactons on Relablty, vol. 48, pp.8-6, 999. [4] Cot DW., Hed A. Taboada, and Jose F. Esprtu, MOMS-GA: A mult-obectve mult-state genetc algorthm for system relablty optmzaton desgn problems, IEEE Transactons on Relablty, vol.57, pp. 8 9, 008. [5] I. Ushakov, Unversal generatng functon, Sovet Journal of Computer and Systems Scences, vol. 4, pp. 8 9, 986. [6] Y Dng, Lsnask A, Fuzzy unversal generatng functon for mult-state system relablty assessment, Fuzzy Sets and Systems, vol. 59, pp.07-4, 008. [7] Tllman FA, Hwang CL, Kuo W. Optmzaton technques for system relablty wth redundancy: a revew, IEEE Transactons on Relablty, vol. 6, pp , 997. [8] Ouzneb M, Nourelfath M, Genderu M, Tabu search for the redundancy allocaton problem of homogenous seres parallel mult-state systems, Relablty Engneerng and System Safety, vol.9, pp. 57-7,

5 Sub System Component Type Table : Input data of the numercal example Avalablty (p ) Feedng Capacty (%) Cost Weght Fg. : The NSGA-II Pareto front 4