Internatonal Journal of Performablty Engneerng Vol., No., January 0, pp. -. RAMS Consultants Prnted n Inda A Partcle Swarm Algorthm for Optmzaton of Complex System Relablty SANGEETA PANT *, DHIRAJ ANAND and AMAR KISHOR, SURAJ BHAN SINGH Department of Mathematcs, Unversty of Petroleum and Energy Studes, Dehradun, INDIA: 8007 Department of Electrcal and Electroncs, GEU, Dehradun, INDIA: 800 Department of Electroncs and Communcaton Scences Unt, Indan Statstcal Insttute Kolkata, INDIA: 70008. Department of Mathematcs, Statstcs and Computer Scence, G.B.Pant Unversty of Agrculture and Technology, Pantnagar, INDIA: 6 (Receved on January 8, 0, revsed on May 8, June 07 and July 6, 0) Abstract: In recent years, a broad class of stochastc metaheurstcs, such as Tabu search, smulated annealng, genetc algorthm, partcle swarm optmzaton, ant colony optmzaton etc. has been appled for relablty optmzaton problems. In ths paper a partcle swarm optmzaton algorthm s presented. Then, the performance of the proposed algorthm s tested on some complex engneerng optmzaton problems. They are three well-known complex relablty optmzaton problems. Fnally, the results are compared wth those gven by several well-known methods. Numercal experments demonstrate that the proposed method s promsng and the results obtaned by proposed algorthm are ether superor or comparable to the prevously best known results presented n lterature for relablty optmzaton of complex systems n terms of computaton tme as well as soluton qualty. Keywords: Relablty optmzaton, metaheurstcs, partcle swarm optmzaton, relablty allocaton, redundancy allocaton.. Introducton In the past fve decades the problem of relablty optmzaton and redundancy allocaton has been addressed n many studes. Important contrbutons have been devoted snce 970 [-] n order to cope wth optmzaton problem arsng n system relablty. The problem of relablty optmzaton has been wdely treated by many authors. To lst a few of them, Luss [] optmzed such problems by non lnear nteger programmng procedure, Mohan and Shankar [] appled random search technque to optmze complex system, Msra and Sharma [] developed MIP-Technque to solve nteger programmng problems arsng n system relablty desgn. Tllman et al. [6], Lad et al. [7], Chaturved and Msra [8] provded an excellent survey of earler approaches appled to solve these problems. Relablty optmzaton problems are categorzed nto three typcal problems accordng to the types of ther decson varables: () relablty allocaton, () redundancy allocaton, and () relablty-redundancy allocaton. From the vewpont of mathematcal programmng, relablty allocaton s a contnuous nonlnear programmng problem (NLP), redundancy allocaton s a pure nteger nonlnear programmng problem (INLP), and relablty-redundancy allocaton s a mxed nteger nonlnear programmng problem (MINLP). In general, achevng optmal relablty desgn s qute dffcult because relablty optmzaton problems are NP-hard [9]. Further, solvng such problems usng * Correspondng author s emal: pant.sangeet@gmal.com
Sangeeta Pant, Dhraj Anand, Amar Kshor and Suraj Bhan Sngh heurstcs or exact algorthms s more dffcult. Ths s because these optmzaton problems generate a very large search space, and searchng for optmal solutons usng exact methods or heurstcs wll necessarly be extremely tme consumng. Therefore, metaheurstc algorthms are more sutable for solvng relablty optmzaton problems. Recently, many metaheurstcs [0-9] have been employed to solve relablty optmzaton problems. In ths present paper, we appled a PSO to solve relablty optmzaton and redundancy allocaton problems of complex systems. The man concept of PSO s based on the food searchng behavor of brds flockng or fsh schoolng. When PSO s adopted to solve problems, each partcle has ts own locaton and velocty, whch determne the flyng drecton and dstance respectvely. Comparng wth other evolutonary approaches PSO has the followng advantages [0-]: () less parameters () easy mplementaton () fast convergence. These advantages are good for solvng the relablty optmzaton problems because a populaton of partcles n PSO can operate smultaneously so that the possblty of paralyss n the whole process can be reduced. The results obtaned by usng PSO approach are compared wth the results obtaned from other technques n the lterature. As reported, solutons obtaned by the proposed method are better than or as well as the prevously best-known solutons.. Partcle Swarm Optmzaton Partcle swarm optmzaton (PSO) s a populaton-based search algorthm based on the smulaton of the socal behavor of brds wthn a flock [-]. PSO s ntalzed wth randomly generated populaton of partcles (ntal swarm) and a random velocty s assgned to each partcle that propagates the partcle n search space towards optma over a number of teratons. Each partcle has a memory rememberng best poston attaned by t n the past, whch s called personal best poston (P best ). Each partcle has ts P best and the partcle wth the best value (maxmum or mnmum accordng to the problem) of ftness s called global best partcle (G best ). Suppose that the search space s D dmensonal, the th partcle of the populaton can be represented by a D-dmensonal vector D T X = ( x, x,..., x ). The velocty of ths partcle can be represented by another D-dmensonal vector D T V = ( v, v,..., v ).The prevously best vsted poston of th partcle s denoted by D T P = ( p, p,..., p ) and the best partcle n the swarm s denoted by D T P g = ( pg, pg,..., pg ). Partcle changes ts poston and velocty accordng to the followng equatons: k + k k k k Vd = wvd + cr [ pd ( t) xd ( t)] + cr [ pg ( t) xd ( t)] () k + k k + xd ( t + ) = xd ( t) + vd ( t + ) () where k = teraton number, d=,,, D; =,,.N; N= swarm sze, w = nerta weght, whch controls the momentum of partcle by weghng the contrbuton of prevous velocty, c and c are postve constants called acceleraton coeffcents; r and r random numbers unformly dstrbuted between [0,]. The partcle swarm optmzaton algorthm whch s appled presently to solve relablty optmzaton problem s smlar to the one used by [] for optmzaton of nonlnear taper but here we adopted dynamc changng acceleraton coeffcents as n [6]. These modfcatons can be mathematcally represented by equatons () and () respectvely.
Partcle Swarm Algorthm for Optmzaton of Complex System Relablty c ( t) = c + ( c f c )* ITER / ITERMAX () c ( t) = c + ( c f c )* ITER / ITERMAX () where, c and c are ntal values and c f and c f are fnal values of c and c respectvely, ITERMAX s maxmum number of teratons and ITER s the current teraton. The large value of c and small value of c at begnnng the partcles are allowed to explore new regons n search space nstead of chasng the global best partcle. On the contrary wth small value of c and a large value c allow partcles to move towards the populaton best. Also we set the constant value of nerta weght nstead of lnearly decreasng t and vary c and c accordng to equatons () and (). Maxmum velocty n each dmenson s restrcted tovmax = γ * X, where max γ s a constant and can take any value between 0 and (problem dependent). Also here we have restrcted velocty and poston of the partcle accordng to equatons () and (6) respectvely,. e. Vmax, f Vd > Vmax V d = Vmax, f Vd < V max () Vd, otherwse X max, f xd > X max x d = X mn, f xd < X mn (6) xd, otherwse. Problem Descrpton Relablty engneers often need to work not only on seres or parallel systems but also on the systems whch are nether purely connected n seres nor purely n parallel but may have mxed confguraton. Such systems are called complex systems. To evaluate the performance of the proposed approach two relablty optmzaton problems, () relablty allocaton and () redundancy allocaton, are consdered.. Relablty Allocaton Problems Under ths prsm two case studes are consdered. They are a lfe support system n space capsule and a complex brdge system... Lfe Support System n Space Capsule Fgure shows the frst complex system of the present study [6]. The system relablty R s and system cost C s of lfe support system n a space capsule are gven by: [( )( )] ( )[ { ( )( )}] RS = r r r r r r r (7) α α α α CS = Kr + K r + K r + K r (8) where, K =00, K =00, K =00, K =0and α = 0.6, =,,,. Ths problem s contnuous nonlnear optmzaton problem, where objectve s to determne the mnmum cost of system subject to the constrants on relablty of the system. The problem formulaton s as follows: Mnmze C S
6 Sangeeta Pant, Dhraj Anand, Amar Kshor and Suraj Bhan Sngh subject to 0. =,,, r 0.9 R S where, R s th component s relablty... Complex Brdge System Ths problem consders the relablty optmzaton of complex brdge network [,] as shown n the Fgure wth nonlnear constrant. The system relablty (R s ) and system cost (C s ) are gven by R S = r r + r r + r r r + r r r + r r r r r r r r r (9) r r r r r r r r r r r r r r r r b C S = a exp = ( r ) (0) The mathematcal expresson of the problem s Mnmze C S subject to 0 r, =,,,, a 0.99 R s, = and b =0. 000, for =,,,,. IN OUT IN OUT Fgure : Lfe-Support System n a Space Capsule Fgure : Complex Brdge System. Redundancy Allocaton Problems Here a mxed seres parallel system [] s consdered. The block dagram of a n-stage seres parallel system s presented n Fgure. IN OUT x x x n Fgure : Block Dagram of n-stage Seres Parallel System Ths problem s an nteger non lnear programmng problem where objectve s to fnd the optmal number of redundances n a multstage mxed system n order to acheve the maxmum relablty under the restrcton on cost, weght and volume.
Partcle Swarm Algorthm for Optmzaton of Complex System Relablty 7 Two dfferent cases for two dfferent values of n (n= and ) are consdered. As the value of n (no. of stages) ncreased the complexty of the system wll ncrease as well.... Case - Mxed Seres Parallel System wth Fve Parallel Unts The mathematcal formulaton of the problem s: Fnd the optmal x, =,,,, to maxmze R subject to V S S = = x ( r ) = = P x 0 x CS = C x + exp 7 = W exp 00 S = W x = x The relevant data for ths problem s gven n Table... Case - Mxed Seres Parallel System wth Ffteen Parallel Unts The problem formulaton s as follow: Fnd the optmal x, =,,,..., whch maxmzes R subject to C x S = [ ( r = = = 00 W x ) x ] Table : Data Used for Problem.. Table : Data Used for Problem.. r 0.80 7 7 o.8 7 8 0.90 8 0.6 9 6 0.7 9 r C W R C W 0.90 8 9 0.78 7 0.7 9 0 0.9 8 0.6 9 6 0.79 6 9 0.80 7 7 0.77 7 7 0.8 7 8 0.67 9 6 6 0.9 8 0.79 8 7 0.78 6 9 0.67 6 7 8 0.66 9 6. Smulaton Results and Dscusson The PSO s coded n Bloodshed Dev C++. All the programs were run on an Intel (R) Pentum (R), D CPU,.86GHz processors wth 0.99 GB of Random Access Memory (RAM). Table shows the setup of parameters c c, c, c, γ and w used for, f f relablty optmzaton problems, whch affect convergence rate and robustness of the
8 Sangeeta Pant, Dhraj Anand, Amar Kshor and Suraj Bhan Sngh search procedure. Dfferent parameter settngs have been checked and tuned for each problem, the parameter settng correspondng to best results are reported n Table. Tables to 7 presents a comparson between the best results obtaned n ths paper wth other results n lterature. The comparson was made on the bass of the number of functon evaluatons as prevously have been done by [], [] and []. Table : Parameters for PSO Examples Pop sze Lfe support system 0 0.0 0. n space capsule Complex brdge system 0.... 0.70 0.0 Mxed seres parallel system consstng fve parallel unts Mxed seres parallel system consstng ffteen parallel unts 0 0.0 0. 0. 0. 0.. 0.70 0.7 For lfe support system n space capsule, as mentoned n Table, total 00 cost functon evaluatons are made and PSO provded 6.86 as the optmal system cost and wth 0.900000 system relablty. Results obtaned due to present study have been lsted along wth the results obtaned for the same n the past as gven n Table. ACO [] also obtaned same results but wth hgher (0,00) functon evaluatons. Whereas MGDA [] and C-SOMGA [] obtaned 6.8608 and 6.8000 optmal cost wth 6,60 and,00,000 functon evaluatons respectvely. Thus, for ths problem where objectve s to mnmze system cost subject to the constrants on system relablty, PSO obtaned the dentcal soluton wth ACO but consumed less functon evaluatons. Further, comparson ndcates that soluton obtaned by PSO scores over C-SOMGA [], MGDA [], RMMM-CES [7], INESA [] and SA [] n terms of both accuracy as well as convergence speed. Table : Result Comparson for Lfe Support System n Space Capsule PSO C-SOMGA MGDA ACO CEA (best) INESA SA r 0.00000 0.000 0.0000 0.00000 0.00000 0.00060 0.0090 r 0.8890 0.88900 0.8899 0.8890 0.8890 0.88870 0.8770 r 0.00000 0.6000 0.00000 0.00000 0.00000 0.0000 0.000 r 0.00000 0.00000 0.00000 0.00000 0.00000 0.0000 0.000 R S 0.900000 0.90000 0.900000 0.900000 0.900000 0.90000 0.90000 C 6.86 6.8000 6.8608 6.86 6.877 6.800 6.90000 FE 00 00000 6,60 0,00 - - - For the complex brdge system, Table presents the results obtaned by PSO along wth all prevously reported results. Table shows that PSO provdes most relevant mprovement to the prevous sx best known solutons. It obtaned mnmum system cost.0998 at,0,000 functon evaluatons. Although, t consumed more functon evaluatons than C-SOMGA [], MGDA [] and ACO [] but stll t s able to gve mnmum system cost.
Partcle Swarm Algorthm for Optmzaton of Complex System Relablty 9 Table : Result Comparson for Complex Brdge System PSO C SOMGA MGDA ACO I-NESA SA Random Search Technques[] r 0.98 0.99 0.900 0.9869 0.9770 0.9660 0.990 r 0.908 0.90 0.90 0.907 0.990 0.9670 0.90 r 0.7998 0.790 0.78807 0.7986 0.7880 0.79990 0.770 r 0.900 0.90 0.9060 0.980 0.960 0.9870 0.9980 r 0.97 0.97 0.9 0.9 0.90 0.9860 0.980 R S 0.990000 0.990000 0.990000 0.99000 0.990000 0.99000 0.99000 C S.0998.0990.0999.099.0990.09970.0000 FE,0,000 00000 0,9 8060 - - - Same nvestgaton has been done for mxed seres parallel system wth consstng fve parallel unts. The results are presented n Table 6. Agan PSO s able to obtan best known solutons whch were earler obtaned by C-SOMGA [], ACO [], SA [], CEA [7], INESA [] and generalzed Lagrange functon approach [6]. PSO only requres 0 (lowest among all) functon evaluatons to acheve ths value. Thus PSO agan yelded superor soluton n comparson to all other sx methods n terms of consumpton of CPU tme. Table 6: Result Comparson for Mxed Seres Parallel System Consstng Fve Parallel Unts PSO C-SOMGA ACO CEA (best) INESA SA Generalzed Lagrange Functon Approach x x x x x Rs 0.9000 0.9000 0.9000 0.9000 0.9000 0.9000 0.9000 FE 0 00000 800 - - - - Fnally, for mxed seres parallel system wth consstng ffteen parallel unts, Table 7 shows the results obtaned by PSO along wth other past reported solutons. The best result yelded by PSO s (,, 6,,,,,,,,,,,, ). The system relablty and cost have been obtaned as 0.96 and 9.0 respectvely. The results are compared wth the results obtaned by other fve optmzaton methods, whch are C-SOMGA [], MGDA [], ACO [], INESA [], SA [] and nteger programmng technque []. MGDA and ACO obtaned the same system relablty wth 77 and 000 functon evaluatons. Hence t can be concluded easly that the accuracy obtaned by PSO, MGDA,
0 Sangeeta Pant, Dhraj Anand, Amar Kshor and Suraj Bhan Sngh ACO are exactly matchng. whle PSO found to be superor to MGDA, ACO when speed s compared. Whle C-SOMGA obtaned 0.90 system relablty wth 00000 functon evaluaton. Thus PSO acheved hgher relablty than C-SOMGA. Further, present study shows that results obtaned by PSO s agan far better than the earler reported results correspondng to INESA and SA both n terms of accuracy and speed. Thus, Observaton of Table 7 fnds that the soluton provded by PSO gves better relablty at relatvely lower functon evaluatons than C-SOMGA, MGDA, ACO, INESA and SA. Table 7: Result Comparson for Mxed Seres Parallel System Consstng Ffteen Parallel Unts PSO C- SOMGA MGDA ACO INESA SA Integer Programmng x x x 6 6 6 x x x 6 x 7 x 8 x 9 x 0 x x x x x R S 0.96 0.9000 0.96 0.96 0.979 0.99 0.979 FE.6 0 0 77 000 - - -. Concluson In ths paper, a new PSO was proposed for solvng complex network relablty. In general, system relablty optmzaton problems are nonlnear programmng problems and proved to be NP-hard from computatonal pont of vew. That s, they are more dffcult to solve than any general nonlnear programmng problem. Moreover, the comparson wth studes n the lterature nvolvng the same problems demonstrated that PSO has better effcency n solvng the complex network relablty optmzaton problem as t can provde a soluton whch s ether comparable to or superor to the best avalable results. The smulaton results ndcates that our approach s a vable alternatve snce PSO s able to obtan solutons not only n terms of accuracy and but also n terms of speed than those obtaned from prevously publshed n lterature. References [] Msra, K. B., and M. D. Ljubojevc. Optmal Relablty Desgn of A System: A New Look. IEEE Transactons on Relablty, 97; R-(): -8. [] Msra, K. B. Optmal Relablty Desgn of A System Contanng Mxed Redundances. IEEE Transactons on Power Apparatus & Systems, 97; 9(): 98-99. [] Luus, R. Optmzaton of System Relablty by A New Nonlnear Integer Programmng Procedure. IEEE Transactons on Relablty, 97; R-(): -6.
Partcle Swarm Algorthm for Optmzaton of Complex System Relablty [] Mohan, C. and K. Shanker. Relablty Optmzaton of Complex Systems usng Random Search Technque. Mcroelectroncs Relablty, 988; 8 (): -8. [] Msra, K. B. and U. Sharma. An Effcent Algorthm to Solve Integer-Programmng Problems Arsng n System-Relablty Desgn. IEEE Transactons on Relablty, 99; 0(): 8-9. [6] Tllman, F. A., C. L. Hwang, and W. Kuo. Optmzaton of Systems Relablty. Marcel Dekker Inc., New York, 980. [7] Lad, B. K., M. S. Kulkarn, and K. B. Msra. Optmal Relablty Desgn of a System. In: Handbook of Performablty Engneerng. Msra, K. B. (Ed.) Sprnger, Verlag, London, 008, 99 9. [8] Chaturved, S. K., and K. B. Msra. MIP: A Versatle Tool for Relablty Desgn of A System. In: Handbook of Performablty Engneerng. Msra, K.B. (Ed.) Sprnger Verlag, London, 008, -. [9] Chern, M. S. On the Computatonal Complexty of Relablty Redundancy Allocaton n a Seres System. Operatons Research Letters, 99; (): 09-. [0] Kuo, W., and R. Wan. Recent Advances n Optmal Relablty Allocaton. IEEE Transactons on System, Man and Cybernetcs, Part A: System Humans, 007; 7():- 6. [] Rav, V., B. S. N. Murty and J. Reddy. Nonequlbrum Smulated-Annealng Algorthm Appled to Relablty Optmzaton of Complex Systems. IEEE Transactons on Relablty, 997; 6(): -9. [] Shelokar, P. S., V. K. Jayaraman, and B.D. Kulkarn. Ant Algorthm for Sngle and Multobjectve Relablty Optmzaton Problems. Qualty and Relablty Engneerng Internatonal, 00; 8(6): 97-. [] Rav, V. Optmzaton of Complex System Relablty by A Modfed Great Deluge Algorthm. Asa-Pacfc Journal of Operatonal Research, 00; (): 87 97. [] Deep, K., and Deept. Relablty Optmzaton of Complex Systems through C-SOMGA. Journal of Informaton and Computng Scence, 009; (): 6-7. [] Yeh, W. C. A Two-Stage Dscrete Partcle Swarm Optmzaton for the Problem of Multple Mult-Level Redundancy Allocaton n Seres Systems. Expert Systems wth Applcatons, 009; 6(): 99 900. [6] Pant, S., and S. B. Sngh. Partcle Swarm Optmzaton to Relablty Optmzaton n Complex System. IEEE Int. Conf. on Qualty and Relablty, Bangkok, Thaland, Sept - 7, 0; -. [7] Ran, M., S. P. Sharma and H. Garg. A Novel Approach for Analyzng the Behavour of Industral Systems under Uncertanty. Internatonal Journal of Performablty Engneerng, 0; 9(): 0 0. [8] Zo, E., F. D Mao, and S. Martorell. Fuson of Artfcal Neural Networks and Genetc Algorthms for Mult-Objectve System Relablty Desgn Optmzaton. Journal of Rsk and Relablty, 008; (): -6. [9] Kshore, A., S. P. Yadav, and S. Kumar. A Mult Objectve Genetc Algorthm for Relablty Optmzaton Problem. Internatonal Journal of Performablty Engneerng, 009; (): 7. [0] Hassan, R., B. Cohanm, O. De Weck, and G. Venter. A Comparson of Partcle Swarm Optmzaton and the Genetc algorthm. In 6th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conference, Austn, TX, Aprl 8-, 00. [] Clow, B., and T. Whte. An Evolutonary Race: A Comparson of Genetc Algorthms and Partcle Swarm Optmzaton used for Tranng Neural Networks. Int. Conf. on Artfcal Intellgence, Las Vegas, Nevada, USA, June -, 00; 8-88.
Sangeeta Pant, Dhraj Anand, Amar Kshor and Suraj Bhan Sngh [] Hodgson, R. J. W. Partcle Swarm Optmzaton appled to the Atomc Cluster Optmzaton Problem. The Genetc and Evolutonary Computaton Conference, New York, USA, July 9-, 00; 68-7. [] Engelbrecht, A. P. Fundamentals of Computatonal Swarm Intellgence. Jhon Wley and Sons Ltd., Chchester, England, 00. [] Kennedy, J., and R. Eberhart. Partcle Swarm Optmzaton. IEEE Int. Conf. on Neural Networks, Perth, Australa, 7 Nov- Dec, 99; 9-98. [] Chauhan, N., A. Mttal, D. Wagner, M. V. Kartkeyan, and M. K. Thumm. Desgn and Optmzaton of Nonlnear Tapers usng Partcle Swarm Optmzaton. Internatonal Journal of Infrared and Mllmeter Waves, 008; 9(8): 79-798. [6] Ratnaweera, A., S. K. Halgamuge, and H. C. Watson. Self-Organzng Herarchcal Partcle Swarm Optmzer wth Tme-Varyng Acceleraton Coeffcents. IEEE Transactons on Evolutonary Computaton, 00; 8(): 0-. [7] Rocco, C. M., A. J. Mller, J. A. Moreno and N. A. Carrasquero. Cellular Evolutonary Approach Appled to Relablty Optmzaton of Complex Systems. Annual Relablty and Mantanablty Symposum, Los Angeles, CA, Jan -7, 000; 0-. Sangeeta Pant s currently workng as Assstant Professor n Unversty of Petroleum and Energy Studes, Dehradun, Inda. She holds Ph.D. n Mathematcs wth concentraton on Metaheurstcs and Relablty Optmzaton from G. B. Pant Unversty of Agrculture and Technology, Pantnagar, Inda. Dheeraj Anand holds B.Tech. n Electroncs & Communcaton Engneerng from Graphc Era Unversty, Dehradun, Inda. Hs area of nterest s computer programmng and on self-motvatng projects. Amar Kshor s currently a postdoctoral fellow at Electroncs and Communcaton Scences Unt, Indan statstcal Insttute Kolkata, Inda. Earler, he worked as Assstant professor, Department of Mathematcs, School of Engneerng and Technology, Sharda Unversty, Greater Noda, Inda. He Holds Ph.D. n Mathematcs from I. I. T. Roorkee. Hs current research areas nclude Aggregaton Operators, OWA Operators, Multobjectve Optmzaton, Relablty Optmzaton, Evolutonary Algorthms and Fuzzy Logc. Suraj Bhan Sngh s Professor n G. B. Pant Unversty of Agrculture and Technology, Pantnagar, Inda. He holds Ph.D. n Mathematcs. Hs current research area ncludes, Relablty Analyss and Operatons Research.