A Variable Neighbourhood Descent Algorithm for the Redundancy Allocation Problem

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1 EMS Vol. 4, No., pp. 94-0, June A Varable Neghbourhood Descent Algorthm for the Redundancy Allocaton Problem Yun-Cha Lang Cha-Chuan u Department of ndustral Engneerng and Management, Yuan Ze Unversty No 35 Yuan-Tung Road, Chung-L, Taoyuan County, Tawan 320, R.O.C. Tel: ext. 252, E-mal: yclang@saturn.yzu.edu.tw Abstract. Ths paper presents the frst known applcaton of a meta-heurstc algorthm, varable neghbourhood descent (VND), to the redundancy allocaton problem (RAP). The RAP, a well-known NP-hard problem, has been the subject of much pror work, generally n a restrcted form where each subsystem must consst of dentcal components. The newer meta-heurstc methods overcome ths lmtaton and offer a practcal way to solve large nstances of the relaxed RAP where dfferent components can be used n parallel. The varable neghbourhood descent method has not yet been used n relablty desgn, yet t s a method that fts perfectly n those combnatoral problems wth potental neghbourhood structures, as n the case of the RAP. A varable neghbourhood descent algorthm for the RAP s developed and tested on a set of well-known benchmark problems from the lterature. Results on 33 test problems rangng from less to severely constraned condtons show that the varable neghbourhood descent method provdes comparable soluton qualty at a very moderate computatonal cost n comparson wth the best-known heurstcs. Results also ndcate that the VND method performs wth lttle varablty over random number seeds. Keywords: varable neghbourhood descent, redundancy allocaton problem, seres-parallel system, combnatoral optmsaton. NTRODUCTON The most studed confguraton of the redundancy allocaton problem (RAP) s a seres system of s ndependent k-out-of-n: G subsystems. Because of the need for relablty and ncreased securty requrements, a seres-parallel system has been wdely used. The RAP s NP-hard (Chern 992) and has been studed n many forms as summarzed n Tllman et al., (977a), and by Kuo and Prasad (2000). As shown n Fgure, a k-out-of-n: G subsystem s functonng properly f at least k of ts n components are operatonal. n the formulaton of a seres-parallel system problem, for each subsystem, multple component choces are used n parallel, and each subsystem may have dfferent component selectons. Thus, the RAP can be formulated to select the optmal combnaton of components and redundancy levels to meet system level constrants, cost of C and weght of whle maxmzng system relablty. t s assumed that system weght and system cost are represented by lnear combnatons of component weght and cost. Max Subject to s R =ΠR ( y k ) () = s C( y ) C, (2) = s ( y ), (3) = n addton, f the maxmum number of components allowed n parallel s pre-determned, the followng constrant s added: k a j j= y 2 n n max = = 2 =, 2,..., s. 2 n 2 = s Fgure. Seres-parallel system confguraton 2 n s (4) : Correspondng Author

2 A Varable Neghbourhood Descent Algorthm for the Redundancy Allocaton Problem 95. Notatons and Assumptons k mnmum number of components requred to functon as a pure parallel system n total number of components used n a pure parallel system k-out-of-n: G a system that functons when at least k of ts n components functon R overall relablty of the seres-parallel system C system cost constrant system weght constrant S number of subsystems ndex for subsystem, =,, s j ndex for components n a subsystem a number of avalable component choces for subsystem r j relablty of component j avalable for subsystem c j cost of component j avalable for subsystem w j weght of component j avalable for subsystem y j quantty of component j used n subsystem ( y,..., y ) y a n = j= j a y, total number of components used n subsystem n max maxmum number of components that can be n parallel (user assgned) k mnmum number of components n parallel requred for subsystem to functon R( y k ) relablty of subsystem, gven k C( y ) total cost of subsystem ( y ) total weght of subsystem R u un-penalzed system relablty of soluton u R up penalzed system relablty of soluton u C u total system cost of soluton u u total system weght of soluton u γ C amplfcaton parameter of cost constrant n the penalty functon γ amplfcaton parameter of weght constrant n the penalty functon l max number of neghbourhood structures l ndex for neghbourhood structures N l the l th neghbourhood structure n the search sequence y current soluton y the best neghbourng soluton of y n a neghbourhood structure e constant that controls the lower bound of number of components used n parallel f constant that controls the upper bound of number of components used n parallel Typcal assumptons are consdered as follows: The state of the components and the system ether functon or fal. Faled components do not damage the system, and are not repared. The falure of a component does not lead to other components falng. Components are actve redundant,.e., the falure rates of components when not n use are the same as when n use. Component relablty, weght and cost are known and determnstc. The supply of components s unlmted..2 Lterature Revew Exact methods of the RAP nclude dynamc programmmng (Bellman and Dreyfus 958, Fyffe et al. 968, Nakagawa and Myazak 98, L 996), nteger programmng (Ghare and Taylor 969, Bulfn and Lu 985, Msra and Sharma 99, Cot and Lu 2000), and mxed-nteger and nonlnear programmng (Tllman et al. 977b). Snce exact methods n practce are lmted to the ncrease n problem sze, meta-heurstcs have become a popular alternatve to exact methods. Meta-heurstc approaches to the RAP vary among Genetc Algorthm (GA) (Cot and Smth 996a&b, Levtn et al. 998), Tabu Search (TS) (Huang et al. 2002, Kulturel-Konak et al. 2003), Ant Colony Optmzaton (ACO) (Lang and Smth 999, Huang et al. 2002, Lang 200, Lang and Smth 2004), hybrd Neural Network (NN) and GA (Cot and Smth 996c), and hybrd ACO wth TS (Huang 2003). n partcular, Levtn et al. (998) generalze a redundancy allocaton problem to mult-state systems, where the system and ts components have a range of performance levels-from perfectly functonng to complete falure. Levtn s paper mplements a unversal moment genera-tng functon to estmate system performance (capacty or operaton tme), and a GA s employed as the optmzaton technque. Furthermore, consderng the problem type for maxmzng system relablty, past studes such as TS proposed by Kulturel-Konak et al. (2003), GA de-veloped by Cot and Smth (996a), and ACO by Lang and Smth (200, 2004) have provded comparable results over a set of 4-subsystem benchmark problems. Huang et al. (2002, 2003) proposed ACO, TS, and hybrd ACO/TS algorthms respectvely to solve the system cost mnmzaton RAP. Our study consders one of the latest meta-heurstcs, Varable Neghbourhood Descent (VND), to solve the system relablty maxmzaton RAP. A varaton of the varable neghbourhood search (VNS) method was ntroduced frst by Mladenovć (995). The VND method explores search space based on the systematc change of

3 96 Yun-Cha Lang Cha-Chuan u neghbourhoods wthn the process and has been successfully appled to dverse combnatoral optmsaton problems, e.g. mult-source problem (Brmberg et al. 2000), sngle machne schedulng (Besten and Stützle 200), arc routng problem (Hertz and Mttaz 200), clusterng (Hansen and Mladenovć 2002), mnmum spannng tree problem (Rbero and Souza 2002), and phylogeny problem (Rbero and Vanna 2005). Because of the proper neghbourhood structure of the RAP and lack of domnant soluton technques, t s a good canddate for other metaheurstc approaches ncludng the focus of ths paper, VND. Ths paper s organzed as follows. Secton 2 ntroduces the varable neghbourhood descent algorthm for RAP, and secton 3 provdes our dscusson on computatonal results. Fnally, secton 4 contans concludng remarks and suggestons for future research. 2. VARABLE NEGHBOURHOOD DESCENT ALGORTHM Hansen and Mladenovć (2003) provde a detaled summary of the state-of-the-art development of VNS and ts varatons. They descrbe VNS-type algorthms as meta-heurstcs, and employ a set of neghbourhood search methods to fnd the local optmum n each neghbourhood teratvely and hopefully to reach the global optmum at the end. f the change of neghbour-hoods s performed n a determnstc way, such a method s called VND. The VND algorthm for RAP (denoted as VND_RAP) s dscussed below. At the ntalzaton step, a set of neghbourhood structures ( N ; l =,..., l ) and the sequence of ther l max mplementaton are determned. Then a feasble ntal soluton s generated and set as the current soluton (y). n the man search loop, startng from the frst neghbourhood,.e., N, a complete neghbourhood search s performed. f the best neghbourng soluton (y') n ths neghbourhood s better than the current soluton (y), y s replaced by y' and l s reset to,.e., start from the frst neghbourhood agan wth the updated y. Otherwse, start the consecutve neghbourhood search wth y. The process contnues untl all neghbourhoods are vsted and no further mprovement can be obtaned to the current soluton. The VND_RAP procedure can be formalzed n the followng flow chart: Three key factors underle the proposed VND_RAP algorthm - how to generate a proper ntal soluton, how to defne a set of neghbourhood structures, and how to desgn a penalty functon to evaluate nfeasble solutons are dscussed n the followng sectons. 2. ntal Soluton Generaton To generate a feasble ntal soluton, s ntegers between k + e and n max - f (nclusve) are randomly selected to represent the number of components n parallel (n ) for each subsystem. e and f denote the constants that control the range of component selecton. Then, the n components are chosen under a unform dstrbuton to the a dfferent types. here a feasble soluton s calculated, the ntal soluton s found; otherwse, the process s repeated untl a feasble ntal soluton s obtaned. l = l + No St ar t Select the set of neghbourhood structures N l, l =,, l max Generate a feasble ntal soluton (y) l = Fnd the best neghbourng soluton (y') of y, y N (y) s y' better than y? No l = l max? St op Yes l Yes Fgure 2. Flow chart of VND_RAP algorthm 2.2 Neghbourhood Structure y' replaces y Ths secton depcts four knds of neghbourhood structures and llustrates wth an example, respectvely: Type structure: Ths structure consders changes on only one subsystem. Smultaneously replace one type of component wth a dfferent type wthn the same subsystem,.e., y j y j + and y m y m - for j m. All possbltes are enumerated and all subsystems are consdered. For example, f there are three avalable components (,, and ) n a subsystem, and the current

4 A Varable Neghbourhood Descent Algorthm for the Redundancy Allocaton Problem 97 component allocaton s as n Fgure 3(a). Fgures 3(b) to 3(e) show four combnatons of neghbourng solutons generated accordngly and the shaded components are newly added to replace the exstng ones. (a) (b) (c) (d) (e) Fgure 3. An example of neghbourhood type Type 2 structure: They are the same as n a Type structure and changes the number of a partcular component type by ether addng or subtractng one,.e., y j y j + or y j y j -. Each component avalable n a subsystem s ndependently consdered. For example, there are three component types (,, and ) avalable n a partcular subsystem and Fgure 4(a) represents the current allocaton wth one Type component and two Type components. Then, a total of fve possble combnatons of neghbourng solutons are derved as Fgures 4(b) to 4(f), where shaded boxes represent the added components, and the removed components are marked by the dashed box. (a) (b) (c) (d) (e) (f) Fgure 4. An example of neghbourhood type 2 Type 3 structure: Ths structure consders changes on two subsystems at a tme. Hence, smultaneously add one component to a certan subsystem and delete one component n another subsystem,.e., y j y j + and y om y om - for o. All possbltes are enumerated and all subsystems are consdered. For nstance, the current allocaton s demonstrated n Fgure 5(a). One possble neghbourng soluton s to add one Component n subsystem and delete Component from subsystem ; the result s n Fgure 5(b) where the shaded box s the added component wthout replacng any exstng one, and the dashed box s the elmnated one. 2 (a) 2 Fgure 5. An example of neghbourhood type 3 (b) Type 4 structure: They are the same as n a Type 3 structure. Ths neghbourhood extends the Type neghbourhood above to exchange components n two of many subsystems smultaneously,.e., (y j y j + and y m y m - ) ( y oj y oj + and y om y om - ) for j m and o. An example shown n Fgure 6 assumes three component optons (denoted by,, and ) avalable n each subsystem. The example of Fgure 6(a) shows the current status. n subsystem, a Component replaces one of Component s. Also, n subsystem a Component substtutes one of Component s. Fgure 6(b) shows the result of these two subsystems where the new components are shaded. 2 (a) 2 Fgure 6. An example of neghbourhood type Penalty Functon Cot and Smth (996b) suggest the use of a penalty functon to evaluate the objectve functon under a constraned problem. An approprate-desgned penalty functon wll encourage the algorthm to explore the feasble regon and nfeasble regon near the border of the feasble area, and dscourage, but permt, further search nto the nfeasble regon. After generatng all neghbourng solutons, the algorthm uses a penalty functon for nfeasble solutons (Lang and Smth 2004): R up u γc C = R Cu u γ where C u and u are total system cost and weght of soluton u respectvely. R u s the unpenalzed relablty of soluton u calculated usng equaton (). Then, the penalzed relablty R up for systems that exceed cost constrant C and/or weght constrant s calculated by multplyng the un-penalzed objectve wth two penalty factors C ( C ) u γ C and ( ) u γ (b) (5) where the exponents γ and γ C are preset amplfcaton parameters. 3. COMPUTATONAL RESULTS The VND_RAP s coded n Borland C++ Bulder 5.0 and run wth an ntel Pentum V 2.8GHz PC at 52 MB

5 98 Yun-Cha Lang Cha-Chuan u Table. Component data for test problems (Fyffe et al., 968) Component Alternatves (j) Sub-system () r c w r 2 c 2 w 2 r 3 c 3 w 3 r 4 c 4 w n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a RAM. All computatons use real float pont precson wthout roundng or truncatng values. The system relablty of the fnal soluton s rounded to four dgts behnd the decmal pont n order to compare them wth results n the lterature. The 33 varatons of the Fyffe et al. problem (968) as devsed by Nakagawa & Myazak (98) were used to test the performance of VND_RAP. The component optons for the benchmark problems are lsted n Table. Ths problem sets C = 30 and ncrementally decreased from 9 to 59. n Fyffe et al. (968) and Nakagawa & Myazak (98), the optmzaton approaches requre only dentcal components to be placed n redundancy; however, for the VND_RAP approach, dfferent types are allowed to resde n parallel (assumng that values of n max = 8 and k = for all subsystems). The parameters of the penalty functon are set as γ = c and γ = 2. Ten runs were made usng dfferent random number seeds for each problem nstance. The followng two settngs play key roles n the VND_RAP algorthm: the number of ntal components n each subsystem and the sequence of the neghbourhood structures. Therefore, detals of these two factors are nvestgated n sectons 3. and 3.2. n addton, consderng pror RAP work where component mxng was allowed on the heurstc benchmarks, the study chose GA of Cot and Smth (996a), TS of Kulturel-Konak et al. (2003), and ACO of Lang and Smth (2004) for comparson and wll be dscussed n secton Number of ntal Components The generaton of an ntal soluton s controlled by a range between k + e and n max - f (nclusve) determnng the number of components n parallel n each subsystem. n order to obtan a feasble ntal soluton close to the border between the feasble regon and the nfeasble regon, an nvestgaton s conducted. The algorthm consders both (e =, f = 4) and (e =, f = 5); n other words, 2 to 4 components and 2 to 3 components are selected for each subsystem. Fgure 7 ndcates the maxmum relablty over ten runs n 33 nstances. Under the least ( = 9~8) and moderate ( = 80~70) constraned problems, the opton of 2~4 components performs equal or better than that of 2~3 components n 4 out of 22 nstances. hen the weght constrant becomes strcter ( = 69~59), the opton of 2~3 components outperforms the 2~4 components one n 7 out of nstances. Then, wth the decrease of the value, Table 2 shows the computatonal expense of the 2~4 components opton grows tremendously. For the most constraned nstances, the average CPU tme ncreases to approxmately 96 seconds. However, the 2~3 components opton performs wth much less computatonally effort. Our results also show whether the constrants are loose or strct, the average CPU tme always appears lower than second. To balance soluton qualty and computatonal expense, the 2~4 components opton s adopted for the least and moderate constraned nstances,.e., = 9~70, and remanng nstances use the 2~3 components opton. Max Relablty over Ten Runs nstances (denoted by eght Constrant) e=,f=4 (2~4 components) e=,f=5 (2~3 components) Fgure 7. The effect of number of ntal components on system relablty

6 A Varable Neghbourhood Descent Algorthm for the Redundancy Allocaton Problem 99 Table 2. The effect of number of ntal components on average CPU tme (n seconds) 9~8 80~70 69~59 2~ ~ The Sequence of Neghbourhood Structures Hansen and Mladenovć (2003) ndcate that the key to successfully employng the VND conssts of the proper defnton of neghbourhood structures, and the sequence of neghbourhood structures, etc. Four types of neghbourhood structures are proposed n secton 2.2, so the number of possble search sequences s 4! = 24. Fgure 8 shows the average system relablty over 0 runs n each possble sequence of neghbourhood structures. The nstances are dvded nto three groups 9~8, 80~70, and 69~59 dependng on the severty of the weght constrant. The sequence of outperforms other sequences n all three groups of nstances. n addton, when startng wth a Type 2 neghbourhood, the average performance s better than startng wth other structures. Table 3 summarzes the average performance (denoted by R avg ), the best performance (denoted by R max ), and theaverage standard devaton (denoted by Stdev) over 0 runs n all 33 nstances for each possble se-quence. t agan confrms that the sequence of Type 2-Type 3-Type -Type 4 performs the best n all measures. That s ths sequence not only provdes the best soluton qualty but also s the most robust to the random number seed. Therefore, n next secton, ths sequence s used to compare other best-known heurstcs n the lterature. System Relablty Sequence of Neghbourhood Structures Fgure 8. The effect of sequence of neghbourhood structures on average system relablty 3.3 Comparson wth Other Metaheurstcs Consderng pror RAP works where component mxng was allowed on the heurstc benchmarks, the three best-known metaheurstcs - GA of Cot and Smth (996a), TS of Kulturel-Konak et al. (2003), and ACO of Lang and Smth (2004) were chosen for comparson. Each algorthm was run 0 tmes usng dfferent random number seeds for each nstance. Percentages of the devaton between VND_RAP and other meta-heurstcs, GA, ACO, and TS are dsplayed n Fgure 9. The performance measure s defned as ((other methods VND_RAP) / VND_RAP) 00%. The best soluton over 0 runs was used for comparson. The devatons for the less and moderate constraned nstances (the frst 22) scatter between 0.5% and 0.35%, and when the problems become more constraned (the last ), the devatons fluctuate between 0.45% and 0.0%. The Table 3. The effect of sequence of neghbourhood structures on R, R max, and Stdev avg Sequence R avg R max Stdev Sequence R avg R max Stdev

7 00 Yun-Cha Lang Cha-Chuan u average percentage of the devaton VND_RAP over all 33 nstances was less than 0.3% compared wth other three methods. n addton, the average standard devaton of VND_RAP over 330 runs (33 nstances wth 0 runs each) was Thus, VND_RAP was able to fnd comparable solutons to the best-known methods n the lterature and was robust to random number seeds. Percentage Devaton of VND_RAP versus Other Metaheurstcs nstances (from the least to the most constraned) GA ACO TS Fgure 9. Comparsons of VND_RAP versus GA, ACO, and TS t s dffcult to make a precse comparson on computatonal expense. CPU seconds vary accordng to hardware, software and programmer s codng skll. Therefore, the number of solutons generated may provde a better dea on how effcent an algorthm performs. The number of solutons generated n GA (Cot and Smth 996b) was 48,040 (a populaton sze of 40 and stopped after 200 teratons), the ACO algorthm n Lang and Smth (2004) needed about 00,000 evaluatons (a colony sze of 00 wth up to 000 teratons), and TS n Kulturel-Konak et al. (2003) evaluated an average of 350,000 solutons. The number of solutons searched n VND_RAP was approxmately 49,000 on average. That s, VND_RAP requred only a smlar number of evaluatons to GA, less than half of the ones n ACO, and /7 of TS to get the comparable performance. n addton, the average CPU tme of VND_RAP was 0.73 seconds that was consdered as a reasonable tme n solvng such a large system. 4. CONCLUSONS Ths paper uses a varable neghbourhood descent meta-heurstc method to solve the seres-parallel redundancy allocaton problem. The RAP s a well-known NPhard problem generally n a restrcted form where each subsystem must consst of dentcal components n parallel to make computatons tractable. The newer metaheurstc methods overcome ths lmtaton and offer a practcal way to solve large nstances of the relaxed RAP where dfferent components can be placed n parallel. Gven the well-structured neghbourhood of the RAP, varable neghbourhood descent method that has not been used n relablty desgn yet s lkely to be more effectve and more effcent than one that does not. A VND_RAP algorthm s proposed and tested on the most well known benchmark problems from the lterature. t has been shown that VND_RAP performs comparably well to other meta-heurstcs n soluton qualty and very effcently n computatonal expense consderaton. Therefore, VND and ts varatons seem very promsng for other NP-hard relablty desgn problems such as those found n networks and complex structures. ACKNOLEDGEMENT The authors would lke thank Dr. Shu-Ka S. Fan for hs precous comments on ths research. REFERENCES Bellman, R. and Dreyfus, S. (958), Dynamc Programmng and the Relablty of Multcomponent Devces, Operatons Research, 6, Besten, M. D. and Stützle, T. (200), Neghborhoods Revsted: an Expermental nvestgaton nto the Effectveness of Varable Neghborhood Descent for Schedulng, Proceedngs of the 4 th Metaheurstcs nternatonal Conference, Proto, Portugal, Brmberg, J., P. Hansen, Mladenovć, N. and Tallard, É. (2000), mprovements and Comparson of Heurstcs for Solvng the Multsource eber Problem, Operatons Research, 48(3), Bulfn, R. L. and Lu, C. Y. (985), Optmal Allocaton of Redundant Components for Large Systems, EEE Transactons on Relablty, R-34(3), Chern, M. S. (992), On the Computatonal Complexty of Relablty Redundancy Allocaton n a Seres System, Operatons Research Letters,, Cot, D.. and Lu, J. (2000), System Relablty Optmzaton wth k-out-of-n Subsystems, nternatonal Journal of Relablty, Qualty and Safety Engneerng, 7(2), Cot, D.. and Smth, A. E. (996a), Relablty Optmzaton of Seres-Parallel Systems Usng a Genetc Algorthm, EEE Transactons on Relablty, 45(2), Cot, D.. and Smth, A. E. (996b), Penalty Guded Genetc Search for Relablty Desgn Optmzaton, Computers & ndustral Engneerng, 30(4), Cot, D.. and Smth, A. E. (996c), Solvng the Redundancy Allocaton Problem Usng a Combned Neural Network/Genetc Algorthm Approach, Computers & Operatons Research, 23(6),

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