A TABU SEARCH FOR MULTIPLE MULTI-LEVEL REDUNDANCY ALLOCATION PROBLEM IN SERIES-PARALLEL SYSTEMS

Transcription

1 Internatonal Journal of Industral Engneerng, 18(3), , A TABU SEACH FO MULTIPLE MULTI-LEVEL EDUNDANCY ALLOCATION POBLEM IN SEIES-PAALLEL SYSTEMS Kl-Woong Jang and Jae-Hwan Km Department of Data Informaton, Korea Martme Unversty, Busan, , South Korea Correspondng author: Jae-Hwan Km, jhkm@hhu.ac.kr The tradtonal AP (edundancy Allocaton Problem) s to consder only the component redundancy at the lowest-level. A system can be functonally decomposed nto system, module, and component levels. Modular redundancy can be more effectve than component redundancy at the lowest-level. We consder a MMAP (Multple Mult-level edundancy Allocaton Problem) n whch all avalable tems for redundancy (system, module, and component) can be smultaneously chosen. A tabu search (TS) of memory-based mechansms that balances ntensfcaton wth dversfcaton va the short-term and long-term memory s proposed for ts soluton. To the best of our knowledge, ths s the frst attempt to use a TS for MMAP. Our algorthm s compared wth the exstng genetc algorthm(ga) for MMAP on the new composed test problems as well as the benchmark problems n the lterature. Computatonal results show that the TS outstandngly outperforms the GA for all test problems. Sgnfcance: To evaluate the performance of metaheurstc approaches for the MMAP, the exstng GA was coded n C/ C++ programmng language and a TS algorthm was developed. From computatonal results, we notced that the proposed TS substantally outperformed the exstng GA. Keywords: tabu search, module, component, redundancy allocaton, relablty optmzaton. 1. INTODUCTION (eceved 1 May 2010; Accepted n revsed form 27 Feb 2011) The relablty of a system can be ncreased by properly allocatng redundant unts to subsystems under varous resource and technologcal constrants. A well-known edundancy Allocaton Problem (AP) s to determne the optmal number of redundant component n order to maxmze the system relablty constraned to resource restrctons or system-level constrants for cost and weght, etc. The AP has become an mportant ssue for system desgns such as semconductor ntegrated crcuts, nanotechnology, and most electronc systems. The AP s typcally classfed nto the fve categores such as seres, parallel, seres-parallel, parallel-seres, and complex systems. In ths study, we only focus on the seres-parallel systems. Solutons for the seres-parallel AP have been suggested by many authors. Fyffe et al. (1968) orgnally set up the problem and suggested a soluton algorthm utlzng a dynamc programmng approach. Nakagawa and Myazak (1981) developed 33 varatons of Fyffe s problem, where the weght constrant vared ts value from 159 to 191. Cot and Lu (2000) proposed zero-one nteger programmng for small sze of problems. They constraned the soluton space so that only the dentcal component type can be allowed for each subsystem. On the contrary, Cot and Smth (1996) extended Fyffe s problem n such a way that the parallel system could be more flexble. They allowed a mxng of component types wthn a subsystem and employed a genetc algorthm to obtan optmal solutons. Better solutons have been presented by tabu search (Kulturel-Konak et al., 2004), ant colony optmzaton (Lang & Smth, 2004), varable neghborhood search (Lang & Chern, 2007), and hybrd metaheurstcs (Nahas et al., 2007; Ouzneb et al., 2010). The above all methods proposed for only best solutons for 14-subsystem problems (Cot & Smth, 1996) wthout referrng to ther global optmal solutons. However, Km and Km (2006) suggested the global optmal solutons for these problems by the transformaton of bnary nteger programmng, and also showed that all solutons suggested by Kulturel-Konak et al. (2004) for the 14-subsystem problems were the global optmum. For lager problems wth up to 56-subsystem, Bae et al. (2007) compared metaheurstcs wth global optmal solutons. In the meanwhle, Yun and Km (2004) proposed a new knd of AP, that s, MAP (Mult-level AP). The tradtonal AP s to consder only the component redundancy at the lowest-level. A system can be functonally decomposed nto system, module, and component levels. Modular redundancy can be more effectve than component redundancy at the lowest-level. MAP s to consder all cases of the redundancy for system, module, and component levels. MAP, however, has a strong restrcton that only one level among component, module and system can be an alternatve for redundancy n AP. Yun et al. (2007) extended MAP to the multple MAP (MMAP) n whch all avalable tems for redundancy ISSN X INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

2 Multple Mult-level edundancy Allocaton (system, module, and component) can be smultaneously chosen, by relaxng the strong restrcton of MAP, and also proposed a smple GA n order to obtan the optmal soluton for MMAP. In ths paper, we propose a TS of memory-based mechansms, balancng ntensfcaton wth dversfcaton, for MMAP. To the best of our knowledge, ths s the frst attempt to use a TS for MMAP. Our TS s also compared wth the exstng GA (Yun et al., 2007) for MMAP on the new composed test problems as well as the benchmark problems n the lterature. Computatonal results show that the TS outstandngly outperforms the GA (Yun et al., 2007) for all test problems. The rest of ths paper s organzed as follows: n secton 2 we descrbe the MMAP. In secton 3, we propose a TS for MMAP. In secton 4 and 5, numercal examples and computatonal results for the performance of TS are provded wth the benchmark problems. Fnally, conclusons are dscussed n secton MULTIPLE MULTI-LEVEL EDUNDANCY ALLOCATION POBLEM 2.1 Acronym AP MAP MMAP GA TS TSMMAP redundancy allocaton problem Mult-level AP multple MAP genetc algorthm tabu search tabu search for MMAP 2.2 Notatons n the number of basc tems m the number of constrants x the number of redundancy allocated to the th basc tem (x) the system relablty p (x) the penalty functon of the system relablty g j (x) the jth constrant functon r the relablty of the th basc tem c the cost of the th basc tem b j the amount of allowable resource l the lower bound of x u the upper bound of x 2.3 Assumptons In MMAP, the number of avalable redundancy structure ncreases exponentally as the sze of problem becomes large. In ths paper, two assumptons are set up to exclude mpossble redundancy structures (Yun et al., 2007). (1) The combnaton of the basc tems for redundancy should satsfy the functon at the system level. If a basc tem s used, all ts sblng tems should be used or ts functon should be satsfed by correspondng chld tems. (2) The basc tems for redundancy should be used n parallel at one combnaton. 2.4 Problem Formulaton The MMAP optmzaton model can be generally formulated as the followng nonlnear nteger programmng problem: Maxmze (x) subject to g j (x) b j for j=1, 2,, m... (1) l x u x s a postve nteger number for =1,2,, n Ths problem was proven to be a NP-hard problem (Chern, 1992). We refer nterested readers to Yun et al. (2007) for the detaled formulaton of MMAP. 121

3 Jang and Km 3. TS ALGOITHM The TS algorthm, frst proposed by F. Glover (1986), s a metaheurstc method to expand ts search beyond local optmalty usng adaptve memory. The adaptve memory s a mechansm based on the tabu lst of prohbted moves. The tabu lst s one of the mechansm to prevent cyclng and gude the search towards unexplored regon of the soluton space. The TS generally adopts the penalty functon to allow to explore the search towards the attractve nfeasble regon. The TS has been successfully appled to many combnatoral optmzaton problems such as vehcle routng problems, travellng salesman problems, tme tablng problems, and resource allocaton problems, etc. In AP, Kultruel-Konak et al. ( 2004) developed the TS algorthm for the AP of seres-parallel system, and t s htherto known that ther solutons are the best among the other metaheurstcs such as genetc algorthm (Cot & Smth, 1996), ant colony optmzaton (Lang & Smth, 2004), varable neghborhood search (Lang & Chern, 2007), and hybrd metaheurstcs (Nahas et al., 2007). ecent hybrd metaheurstc (Ouzneb et al., 2010) obtaned the same results of Kultruel-Konak et al. ( 2004). In ths paper, we develop a knd of TS called TSMMAP whch s based on the TS of Kultruel-Konak et al. (2004). The TSMMAP conssts of 4 parts whch are constructon of ntal solutons, the tabu lst, the penalty functon, and the structure of generatng the neghborhood solutons. Among them, the structure of generatng the neghborhood solutons s usually dfferent accordng to the characterstcs of the decson varables for the problem. The TSMMAP can then be summarzed as follows: Step 0. Generate randomly an ntal feasble soluton. Step 1. Explore all the neghborhood solutons for the defned moves. Step 2. If the best soluton of the step 1 s n the tabu lst and s not better than the current best, then repeat step 1 to fnd the next best. Otherwse, select the soluton as the next best move. Step 3. If the stoppng crteron s satsfed, then stop. Otherwse, go to step 1. The ntal solutons are randomly generated, and the scheme of randomly constructng the ntal solutons s nearly dentcal to the general scheme. In our experments, we try to fnd the optmal soluton 10 tmes wth dfferent ntal solutons for each problem. The stoppng crteron of TSMMAP was defned as 500 teratons wthout fndng an mprovement n the best feasble soluton. For the sze of a tabu lst, Kultruel-Konak et al. (2004) showed that the dynamc sze of the tabu lst plays a crtcal role n fndng the better solutons for AP. In our experment, the sze of the tabu lst s reset every 20 teratons to the value of between [n, 3n] unformly dstrbuted for the long-term memory. Once the lst s full, the oldest element of the tabu lst should be removed as a new one s added. The TS generally adopts the penalty functon to allow to explore the search towards the promsng nfeasble regon. The the same type of the penalty functon of Kultruel-Konak et al. (2004) s adopted n our TSMMAP. The penalty functon for only one cost constrant s as follows: p ( x) = ( x) ( ) all feas Δc NFT c k... (2) where s the unpenalzed (feasble or nfeasble) system relablty of the best soluton found so far, all feas s the system relablty of the best feasble soluton found so far, and Δc represent the magntude for the volaton of the cost constrant. The value of NFT s used to control the range of explored nfeasble regon. Namely, If the current move s feasble, c NFT has a property of encouragng to search the neghborhood solutons towards the nfeasble regon by decreasng the c penalzed value of (2). eversely, f the current move s nfeasble, NFT s controlled to ncrease the penalzed value for c explorng the search towards the feasble regon. The ntal value of NFT s set to 1% of the constrant lmt for the cost c and k s set to 1, though computatonal results are nsenstve to these values. In TSMMAP, the structure of the neghborhood solutons s generated by smple two moves. The frst move s to add one for all the allocated redundant number of the module or component. The second move s to subtract one for all the allocated number of the module or component. For example, let us consder the followng system structure of mult-level n x Table 1. The cost functon s g( x) = c x + λ for =1, 2,, n. The constrant lmt for the cost s set to 95. The system structure s shown n Fg. 1. Let us assume the current soluton durng the teratons of TSMMAP to obtan the optmal soluton for our problem to be gven n Table 2, that s, the encodng status of (0, 2, 1, 1, 1, 1, 1). From the current soluton, all possble neghborhood solutons by the frst and second move are 7 and 5 cases shown n Table 2, respectvely. Specfcally, when the value of any component or module equals to 1, the second move s somewhat complcated. Namely, f the value of any component s 1, then the values of all components n the same subsystem s set to zero and the value of the module n the same subsystem s compensated to the approprate values to mantan the feasble soluton, and vce versa. For example, n Table 2, when the value of component (112) s 1, t s replaced wth the approprate values (1 or 2) of module (11). That s, the redundant value of module (11) can be assgned up to the maxmum redundant value of all 122

4 Multple Mult-level edundancy Allocaton components ((111), (112), and (113)). eversely, when the value of module (12) equals to 1, t s replaced wth the value of all components added to each component by 1, (121, 122) = (2, 2). Namely, t s allowed to add the redundant value of all components up to the redundant value of module (12). Table 1. The nput data for an llustraton Basc Item Parent unt r c λ 1(system) S Fgure 1. The system structure of Table 1 In Table 2, the system relablty for each neghborhood soluton s evaluated by the penalty functon of the equaton (2). The neghborhood soluton (1, 2, 1, 1, 1, 1, 1), whch has the maxmum value of among the 12 cases, s selected for the next best move n our TSMMAP. Table 2. All possble neghborhood solutons by two moves x=(11, 111, 112, 113, 12, 121, 122) (x) g ( x) current soluton ( 0, 2, 1, 1, 1, 1, 1) ( 1, 2, 1, 1, 1, 1, 1) ( 0, 3, 1, 1, 1, 1, 1) ( 0, 2, 2, 1, 1, 1, 1) frst move ( 0, 2, 1, 2, 1, 1, 1) ( 0, 2, 1, 1, 2, 1, 1) ( 0, 2, 1, 1, 1, 2, 1) ( 0, 2, 1, 1, 1, 1, 2) second move ( 0, 1, 1, 1, 1, 1, 1) ( 1, 0, 0, 0, 1, 1, 1) ( 2, 0, 0, 0, 1, 1, 1) ( 0, 2, 1, 1, 0, 2, 2) ( 0, 2, 1, 1, 2, 0, 0) p

5 Jang and Km 4. NUMEICAL EXAMPLES We consder two examples to evaluate the performance of the exstng GA (Yun et al., 2007) and the proposed TSMMAP. Two algorthms are coded n C/C++ programmng language, and the numercal experments are executed on an IBM-PC Table 3. The nput data for the frst example Basc Item Parent unt r c λ 1(system) compatble wth a Pentum IV 3.0 GHz. Two benchmark examples are taken from Yun et al. (2007). The data for the frst and the second example are shown n Table 3 and 4, respectvely. The frst example conssts of 11 basc tems and 20 test problems varyng the cost lmt from 150 to 340. The second example conssts of 16 basc tems and 10 test problems varyng the cost lmt from 260 to 350. For the two examples, we compared the TSMMAP wth the exstng GA (Yun et al., 2007). Computatonal results for the frst and second example are shown n Table 5 and 6, respectvely. As the results n Table 5 ndcate, GA and TSMMAP obtan the same results for 20 test problems for the frst example. For larger sze of the second example, however, TSMMAP obtans the better optmal solutons for 7 cases of 10 test problems than the GA (Yun et al., 2007). In addton, to show the dfference between MMAP and MAP, the system structure of the optmal soluton for MMAP and MAP for the cost lmt of 160 of the frst example are shown n Fg. 2(a) and Fg. 2(b), respectvely. In ths case, the system relablty for MMAP and MAP are gven by and , respectvely. Ths result ndcates that MMAP consderng the redundancy for the module and component smultaneously, provdes the possblty of havng the better system relablty than MAP. Table 4. The nput data for the second example Basc Item Parent unt r c λ 1(system)

6 Multple Mult-level edundancy Allocaton Table 5. Computatonal results for the frst example Metaheurstc methods No. Cost lmt GA TSMMAP (a) MMAP (b) MAP Fgure 2. The system structure for MMAP and MAP 125

7 Jang and Km Table 6. Computatonal results for the second example Metaheurstc methods No. Cost lmt GA TSMMAP * * * * * * * * * * * * * *: the case of obtanng the best soluton 5. COMPUTATIONAL ESULTS To addtonally evaluate the performance of the exstng GA (Yun et al., 2007) and the TSMMAP, the new test problems for lager sze of system structure are desgned. They are composed of combnng the frst example wth the second example. The resultng data for the composed problem s shown n Table 7. The system structure s shown n Fg. 3. Ths problem conssts of 26 basc tems and the system cost constrants ranges from 700 to 880 n step sze of 20. They are composed of 10 sets of 10 test problems. Totally, 100 test benchmark problems are desgned. Two algorthms are coded n C/C++ programmng language, and experments are performed on a Pentum IV 3.0 GHz PC. Performances of GA and TSMMAP are assessed n terms of average relatve error (A), maxmum relatve error (M), optmalty rate (O) and average executon tme (sec.) of 100 problems (T) defned as follows. A = * ( j j ) * j= 1 j * ( ) j j M = max *, for j=1, 2, 10 j O = the number of tmes (out of 10 problems) that each method yelds the best soluton. j = the system relablty obtaned by each method for each test problem j. * j = the best system relablty obtaned by both of GA and TSMMAP. In our experments, the stoppng crteron of TSMMAP was defned as 500 teratons wthout fndng an mprovement n the best feasble soluton, and TSMMAP was appled 10 tmes wth dfferent startng ntal solutons. The computatonal results for evaluatng the performance between the GA (Yun et al., 2007) and TSMMAP are summarzed n Table 8. As the results n Table 8 ndcate, TSMMAP outstandngly outperformed the GA. TSMMAP obtaned the hgher system relablty for all test problems than the GA. The GA faled to obtan the best soluton for all cases of 100 test problems. Even for the best case of cost lmt (880), the average relatve error (A) of GA s very poor, scorng the value of In partcular, for the case of cost lmt (700), GA has the worst value of maxmum relatve error (M=0.042). In terms of computng tme, however, we notced that TSMMAP requres almost 2 tmes as much as the computng tme of GA. 6. CONCLUSIONS In ths study, a new verson of TS called TSMMAP was proposed for MMAP (Multple Mult-level edundancy Allocaton Problem) n whch all avalable tems for redundancy (system, module, and component) can be smultaneously chosen. To the best of our knowledge, ths s the frst attempt to use a TS for MMAP. The TSMMAP was also compared wth the exstng GA (Yun et al., 2007) on the new composed test problems as well as benchmark problems n the lterature. 126

8 Multple Mult-level edundancy Allocaton Table 7. The nput data for larger system Basc Item Parent unt r c λ 1(system) S Fgure 3. System structure From the computatonal results, TSMMAP substantally outperformed the GA (Yun et al., 2007). The TSMMAP obtaned the hgher system relablty for all test problems than the GA, even though TSMMAP requred almost 2 tmes as much as the computng tme of GA. In terms of soluton qualty, our TSMMAP s recommended as a promsng metaheurstc n ths feld. To acheve the better soluton qualty, the development of more powerful metaheurstcs for MMAP would be performed n the future research. 127

9 Jang and Km Table 8. Computatonal results No. Cost lmt GA TSMMAP A M O A M O / / / / / / / / / / / / / / / / / / / /10 T 5.24(sec.) T 9.67(sec.) 7. EFEENCES 1. Bae, C.O., Km, H.G., Km, J.H., Son, J.Y., and Yun, W.Y.(2007). Solvng the redundancy allocaton problem wth multple component choces usng metaheurstcs. Internatonal Journal of Industral Engneerng 14(4): Chern, M.S. (1992). On the computatonal complexty of relablty redundancy allocaton n a seres system. Operatons esearch Letters, 11: Cot, D.E. and Smth, A.E. (1996). elablty optmzaton of seres-parallel systems usng a genetc algorthm. IEEE Transactons on elablty, 45(2): (1996). 4. Cot, D.E. and Lu, J. (2000). System relablty optmzaton wth k-out-of-n subsystems. Internatonal Journal of elablty, Qualty and Safety Engneerng, 7(2): Fyffe, D.E., Hnes, W.W., and Lee, N.K. (1968). System relablty allocaton and a computatonal algorthm. IEEE Transactons on elablty, 17(2): Glover, F. (1986). Future Paths for Integer Programmng and Lnks to Artfcal Intellgence. Computers and Operatons esearch, 13: Km, J.H. and Km, J.S.(2006). Globally solvng the redundancy allocaton problem for the case of seres-parallel systems. Proceedngs of the 2 nd Asan Internatonal Workshop(AIWAM 2006), Kulturel-Konak, S., Norman, B.A., Cot, D.W., and Smth, A.E. (2004). Explotng tabu search memory n constraned problems. INFOMS Journal on Computng, 16(3): Lang, Y.C. and Smth, A.E. (2004). An ant colony optmzaton algorthm for the relablty allocaton problem (AP). IEEE Transactons on elablty, 53(3): Lang, Y.C. and Chen, Y.C. (2007). edundancy allocaton of seres-parallel systems usng a varable neghborhood search algorthm. elablty Engneerng and System Safety, 92: Nahas, N., Nourelfath, M., and At-Kad D.(2007). Couplng ant colony and the degraded celng algorthm for the redundancy allocaton problem of seres-parallel systems. elablty Engneerng and System Safety, 92: Nakagawa, Y. and Myazak, S. (1981). Surrogate constrants algorthm for relablty optmzaton problems wth two constrants. IEEE Transactons on elablty, 30(2): Ouzneb, M., Nourelfath, M., and Gendreau, M.(2010). An effcent heurstc for relablty desgn optmzaton problems. Computers & Operatons esearch, 37: Yun, W.Y. and Km, J.W. (2004). Mult-level redundancy optmzaton n seres systems. Computers & Industral Engneerng, 46: Yun, W.Y., Song, Y.M. and Km, H.G. (2007). Multple mult-level redundancy allocaton n seres systems. elablty Engneerng and System Safety, 92:

10 Multple Mult-level edundancy Allocaton BIOGAPHICAL SKETCH Kl-Woong Jang receved the B.S., M.S. and Ph.D. n Computer Engneerng from Kyungpook Natonal Unversty, Korea n 1997, 1999 and 2002, respectvely. Snce 2003, he has been Assocate Professor of Data Informaton at Korea Martme Unversty. Hs research nterests nclude the development of metaheurstc algorthms and wreless network protocols. Jae-Hwan Km s Professor and Char of Data Informaton at Korea Martme Unversty. He receved hs B.S. n Industral Engneerng from Korea Unversty and hs M.S. & Ph.D. n Industral Engneerng from Korea Advanced Insttute of Scence and Technology (KAIST). He wll be a vstng scholar of Industral and Systems Engneerng at Auburn Unversty from August 2011 to August Hs research centers on relablty optmzaton, applcaton and development of metaheurstc algorthms, and combnatoral optmzaton. 129