The Ant Colony Paradigm for Reliable Systems Design

Transcription

1 The Ant Colony Paradgm for Relable Systems Desgn Yun-Cha Lang Department of Industral Engneerng and Management, Yuan Ze Unversty Alce E. Smth Department of Industral and Systems Engneerng, Auburn Unversty 1.1 Introducton Ths chapter ntroduces a relatvely new meta-heurstc for combnatoral optmzaton, the ant colony. The ant colony algorthm s a multple soluton global optmzer that terates to fnd optmal or near optmal solutons. Lke ts sblngs genetc algorthms and smulated annealng, t s nspred by observaton of natural systems, n ths case, the behavor of ants n foragng for food. Snce there are many dffcult combnatoral problems n the desgn of relable systems, applyng new meta-heurstcs to ths feld makes sense. The ant colony approach wth ts flexblty and explotaton of soluton structure s a promsng alternatve to exact methods, rules of thumb and other meta-heurstcs. The most studed desgn confguraton of the relablty systems s a seres system of s ndependent k-out-of-n:g subsystems as llustrated n Fgure 1. A subsystem s functonng properly f at least k of ts n components are operatonal and a seres-parallel system s where k = one for all subsystems. In ths problem, multple component choces are used n parallel n each subsystem. Thus, the problem s to select the optmal combnaton of components and redundancy levels to meet system level constrants whle maxmzng system relablty. Such a redundancy allocaton problem (RAP) s NP-hard [6] and has been well studed (see Tllman, et al. [45] and Kuo & Prasad [25]). Y.-C. Lang and A.E. Smth: The Ant Colony Paradgm for Relable Systems Desgn, Computatonal Intellgence n Relablty Engneerng (SCI) 40, 1 20 (2007) Sprnger-Verlag Berln Hedelberg 2007

2 2 Yun-Cha Lang and Alce E. Smth 1 2 s : : : n1 n2 ns k 1 k 2 k s Fg. 1. Typcal seres-parallel system confguraton. Exact optmzaton approaches to the RAP nclude dynamc programmng [2, 20, 35], nteger programmng [3, 22, 23, 33], and mxed-nteger and nonlnear programmng [31, 46]. Because of the exponental ncrease n search space wth problem sze, heurstcs have become a common alternatve to exact methods. Meta-heurstcs, n partcular, are global optmzers that offer flexblty whle not beng confned to specfc problem types or nstances. Genetc algorthms (GA) have been appled by Panton & Campbell [37], Levtn et al. [26], and Cot & Smth [7, 8]. Rav et al. propose smulated annealng (SA) [39], fuzzy logc [40], and a modfed great deluge algorthm [38] to optmze the complex system relablty. Kulturel-Konak et al. [24] use a Tabu search (TS) algorthm embedded wth an adaptve verson of the penalty functon n [7] to solve RAPs. Three types of benchmark problems whch consder the objectves of system cost mnmzaton and system relablty maxmzaton respectvely were used to evaluate the algorthm performance. Lang and Wu [27] employ a varable neghborhood descent (VND) algorthm for the RAP. Four neghborhood search methods are defned to explore both the feasble and nfeasble soluton space. Ant Colony Optmzaton (ACO) s one of the adaptve meta-heurstc optmzaton methods nspred by nature whch nclude smulated annealng (SA), partcle swarm optmzaton (PSO), GA and TS. ACO s dstnct from other meta-heurstc methods n that t constructs a new soluton set (colony) n each generaton (teraton), whle others focus on mprovng the set of solutons or a sngle soluton from prevous teratons. ACO was nspred by the behavor of physcal ants. Ethologsts have studed how blnd anmals, such as ants, could establsh shortest paths from ther nest to

3 The Ant Colony Paradgm for Relable Systems Desgn 3 food sources and found that the medum used to communcate nformaton among ndvdual ants regardng paths s a chemcal substance called pheromone. A movng ant lays some pheromone on the ground, thus markng the path. The pheromone, whle dsspatng over tme, s renforced f other ants use the same tral. Therefore, superor trals ncrease ther pheromone level over tme whle nferor ones reduce to nl. Inspred by the behavor of ants, Marco Dorgo ntroduced the ant colony optmzaton approach n hs Ph.D. thess n 1992 [13] and expanded t n hs further work ncludng [14, 15, 18, 19]. The prmary characterstcs of ant colony optmzaton are: 1. a method to construct solutons that balances pheromone trals (characterstcs of past solutons) wth a problem-specfc heurstc (normally, a smple greedy rule), 2. a method to both renforce and dsspate pheromone, 3. a method capable of ncludng local (neghborhood) search to mprove solutons. ACO methods have been successfully appled to common combnatoral optmzaton problems ncludng travelng salesman [16, 17], quadratc assgnment [32, 44], vehcle routng [4, 5, 21], telecommuncaton networks [12], graph colorng [10], constrant satsfacton [38], Hamltonan graphs [47] and schedulng [1, 9, 11]. A comprehensve survey of ACO algorthms and applcatons can be found n [19]. The applcaton of ACO algorthms to relablty system problems was frst proposed by Lang and Smth [28, 29], and then enhanced by the same authors n [30]. Lang and Smth employ ACO varatons to solve a system relablty maxmzaton RAP. Secton III uses the ACO algorthm n [30] as a paradgm to demonstrate the applcaton of ACO to RAP. Thus far, the applcatons of ACO to relablty system are stll very lmted. Shelokar et al. [43] propose ant algorthms to solve three types of system relablty models: complex (nether seres nor parallel), N-stage mxed seres-parallel, and a complex brdge network system. In order to solve problems wth dfferent number of objectves and dfferent types of decson varables, the authors develop three ant algorthms for sngle objectve combnatoral problem, sngle objectve contnuous problem, and b-objectve contnuous problem, respectvely. The ant algorthm of sngle objectve combnatoral verson use the pheromone nformaton only to construct the solutons, and no onlne pheromone updatng rule s appled. Two local search methods, swap and random exchange, are performed to the best ant. For contnuous problems, the authors dvded the colony nto two groups global ants and local ants. The global ant concept can be consdered as a pure GA mechansm snce these ants apply crossover and mutaton and no pheromone s deposted. Local ants are mproved by a

4 4 Yun-Cha Lang and Alce E. Smth stochastc hll-clmbng technque, and an mprovng ant can depost the mprovement magntude of the objectve on the trals. Lastly, a clusterng technque and Pareto concept are combned wth the contnuous verson of ant algorthms to solve b-objectve problems. The authors compared ther algorthms wth methods n the lterature such as SA, a generalzed Lagrange functon approach, and a random search method. The results on four sets of test problems show the superorty of ACO algorthms. Ouddr et al. [36] develop an ACO algorthm for mult-state electrcal power system problems. In ths system redesgn problem, the objectve s to mnmze the nvestment over the study perod whle satsfyng avalablty or performance crtera. The proposed ant algorthm s based on the Ant Colony System (ACS) of [17] and [30]. A unversal moment generatng functon s used to calculate the avalablty of the reparable multstate system. The algorthm s tested on a small problem wth fve subsystems, each wth four to sx component optons. Samrout et al. [41] apply ACO to determne the component replacement condtons n seres-parallel systems mnmzng the preventve mantenance cost. Three algorthms are proposed two based on Ant System (AS) [18] and one based on ACS [17]. Dfferent transton rules and pheromone updatng rules are employed n each algorthm. Local search s not used. Gven dfferent msson tmes and avalablty constrants, the performance of the ACO algorthms s compared wth a GA from the lterature. In ths paper, results are mxed: one of the AS based methods and the ACS based method outperform the GA whle the other AS algorthm s domnated by the GA. Nahas and Nourelfath [34] use an AS algorthm to optmze the relablty of a seres system wth multple choces and budget constrants. Onlne pheromone updatng and local search are not used. The authors apply a penalty functon to determne the magntude of pheromone deposton. Four examples wth up to 25 component optons are tested to verfy the performance of the proposed algorthm. The computatonal results show that the AS algorthm s effectve wth respect to soluton qualty and computatonal expense. The remanng chapter s organzed as follows. Secton II offers the notaton lst and defnes the system relablty maxmzaton RAP. A detaled ntroducton of an ant colony paradgm on solvng RAP s provded n Secton III usng the work of Lang and Smth as a bass. Computatonal results on a set of benchmark problems are dscussed n Secton IV. Fnally, concludng remarks are summarzed n Secton V.

5 The Ant Colony Paradgm for Relable Systems Desgn Problem Defnton Notaton Redundancy Allocaton Problem (RAP) k mnmum number of components requred to functon 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 cost constrant W weght constrant s number of subsystems a number of avalable component choces for subsystem r relablty of component j avalable for subsystem c cost of component j avalable for subsystem w weght of component j avalable for subsystem y quantty of component j used n subsystem y ( y 1,..., y a ) a n = y, total number of components used n subsystem j= 1 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 W ( y ) total weght of subsystem R u unpenalzed system relablty of soluton u R up penalzed system relablty of soluton u R mp penalzed system relablty of the rank m th soluton total system cost of soluton u C u

6 6 Yun-Cha Lang and Alce E. Smth W u AC total system weght of soluton u set of avalable component choces Ant Colony Optmzaton (ACO) ndex for subsystem, = 1,..., s j ndex for components n a subsystem τ pheromone tral ntensty of combnaton (, j ) old τ pheromone tral ntensty of combnaton (, j ) before update τ pheromone tral ntensty of combnaton (, j ) after up- new τ 0 date =1/a, ntal pheromone tral ntensty of subsystem P transton probablty of combnaton (, j ) η problem-specfc heurstc of combnaton (, j ) α β l ρ q q 0 E m γ relatve mportance of the pheromone tral ntensty relatve mportance of the problem-specfc heurstc ndex for component choces from set AC [0,1], tral persstence [0,1], a unformly generated random number [0,1], a parameter whch determnes the relatve mportance of explotaton versus exploraton number of best solutons chosen for offlne pheromone update ndex (rank, best to worst) for solutons n a gven teraton amplfcaton parameter n the penalty functon Redundancy Allocaton Problem The RAP can be formulated to maxmze system relablty gven restrctons on system cost of C and system weght of W. It s assumed that system weght and system cost are lnear combnatons of component weght and cost, although ths s a restrcton that can be relaxed usng heurstcs. s max R = Π R ( y k ) (1) = 1 Subject to the constrants

7 The Ant Colony Paradgm for Relable Systems Desgn 7 s C (y ) C, (2) = 1 s W ( y ) W, (3) = 1 If there s a known maxmum number of components allowed n parallel, the followng constrant s added: k a j= 1 y n =1,2,...,s max (4) Typcal assumptons are: The states of components and the system are ether operatng or faled. Faled components do not damage the system and are not repared. The falure rates of components when not n use are the same as when n use (.e., actve redundancy s assumed). Component attrbutes (relablty, weght and cost) are known and determnstc. The supply of any component s unconstraned. 1.3 Ant Colony Optmzaton Approach Ths secton s taken from the authors earler work n usng the ant colony approach for relable systems optmzaton [28, 29, 30]. The generc components of ant colony are each defned and the overall flow of the method s defned. These should be applcable, wth mnor changes, to many problems n relable systems combnatoral desgn Soluton Encodng As wth other meta-heurstcs, t s mportant to devse a soluton encodng that provdes (deally) a one to one relatonshp wth the solutons to be consdered durng search. For combnatoral problems ths generally takes the form of a bnary or k-nery strng although occasonally other representatons such as real numbers can be used. For the RAP, each ant represents a desgn of an entre system, a collecton of n components n parallel ( k n nmax ) for s dfferent subsystems. The n components are chosen from a avalable types of components. The a types are sorted n descendng order of relablty;.e., 1 represents the most relable component

8 8 Yun-Cha Lang and Alce E. Smth type, etc. An ndex of a + 1 s assgned to a poston where an addtonal component was not used (that s, was left blank) wth attrbutes of zero. Each of the s subsystems s represented by n postons wth each component lsted accordng to ts relablty ndex, as n [7], therefore a complete system desgn (that s, an ant) s an nteger vector of length nmax s. max Soluton Constructon Also, as wth other meta-heurstcs, an ntal soluton set must be generated. For global optmzers the soluton qualty n ths set s not usually mportant and that s true for the ant approach as well. In the ACO-RAP algorthm, ants use problem-specfc heurstc nformaton, denoted by η, along wth pheromone tral ntensty, denoted by τ, to construct a soluton. n components ( k + 1 n nmax 4 ) are selected for each subsystem usng the probabltes calculated by equatons 5 and 6, below. Ths range of components encourages the constructon of a soluton that s lkely to be feasble, that s, be relable enough (satsfyng the k + 1 lower bound) but not volate the weght and cost constrants (satsfyng the n max 4 upper bound). Solutons whch contan more or less components per subsystem than these bounds are examned durng the local search phase of the algorthm (descrbed n Secton III D). r The ACO problem specfc heurstc chosen s η = where r, c + w c, and w represent the assocated relablty, cost and weght of component j for subsystem. Ths favors components wth hgher relablty and smaller cost and weght. Adherng to the ACO meta-heurstc concept, ths s a smple and obvous rule. Unform pheromone tral ntenstes for the ntal teraton (colony of ants) are set over the component choces, that s, τ =1/a. The pheromone tral ntenstes are subsequently changed as de- 0 scrbed n Secton III E. A soluton s constructed by selectng component j for subsystem accordng to: β arg max[( τ l ) α ( ηl ) ] q q0 l AC j = (5) J q > q0

9 The Ant Colony Paradgm for Relable Systems Desgn 9 and J s chosen accordng to the transton probablty mass functon gven by ( τ ) α ( η ) β a ( τ ) α l ( η ) β l j AC l = 1 P = (6) 0 Otherwse where α and β control the relatve weght of the pheromone and the local heurstc, respectvely, AC s the set of avalable component choces for subsystem, s a unform random number, and q determnes the q 0 relatve mportance of the explotaton of superor solutons versus the dversfcaton of search spaces. When q explotaton of known good solutons occurs. The component selected s the best for that partcular subsystem, that s, has the hghest product of pheromone ntensty and rato of relablty to cost and weght. When q > q 0, the search favors more exploraton as all components are consdered for selecton wth some probablty. q Objectve Functon Ftness (the common term for the analogy to objectve functon value for nature nspred heurstcs) plays an mportant role n the ant colony approach as t determnes the constructon probabltes for the subsequent generaton. After soluton u s constructed, the unpenalzed relablty R u s calculated usng equaton (1). For solutons wth cost that exceeds C and / or weght that exceeds W, the penalzed relablty s calculated: γ γ W C R up = Ru (7) Wu Cu where the exponent γ s an amplfcaton parameter and W and C R up are the system weght and cost of soluton u, respectvely. Ths penalty functon encourages the ACO-RAP algorthm to explore the feasble reu u

10 10 Yun-Cha Lang and Alce E. Smth gon and nfeasble regon that s near the border of the feasble area, and dscourages, but allows, search further nto the nfeasble regon Improvng Constructed Solutons Through Local Search After an ant colony s generated, each ant s mproved usng local search. Local search s an optonal, but usually benefcal, aspect of the ant colony approach that allows a systematc enhancement of the constructed ants. For the RAP, startng wth the frst subsystem, a chosen component type s deleted and a dfferent component type s added. All possbltes are enumerated. For example, f a subsystem has one of component 1, two of component 2 and one of component 3, then one alternatve s to delete a component 1 and to add a component 2. Another possblty s to delete a component 3 and to add a component 1. Whenever an mprovement of the objectve functon s acheved, the new soluton replaces the old one and the process contnues untl all subsystems have been searched. Ths local search does not requre recalculatng the system relablty each tme, only the relablty of the subsystem under consderaton needs to be recalculated Pheromone Tral Intensty Update The pheromone tral s a unque concept to the ant approach. Naturally, ths dea s taken drectly from studyng physcal ants and ther deposts of the pheromone chemcal. For the RAP, the pheromone tral update conssts of two phases onlne (ant-by-ant) updatng and offlne (colony) updatng. Onlne updatng s done after each soluton s constructed and ts purpose s to lessen the pheromone ntensty of the components of the soluton just constructed to encourage exploraton of other component choces n the later solutons to be constructed. Onlne updatng s by τ new old = ρ τ + ( 1 ρ) τ o (8) where ρ [0,1] controls the pheromone persstence;.e., 1 ρ represents the proporton of the pheromone evaporated. After all solutons n a colony have been constructed and subject to local search, pheromone trals are updated offlne. Offlne updatng s to reflect the dscoveres of ths teraton. The offlne ntensty update s: E τ new = old ρ τ + ( 1 ρ) ( E m + 1) Rmp (9) m= 1

11 The Ant Colony Paradgm for Relable Systems Desgn 11 where m = 1 s the best feasble soluton yet found (whch may or may not be n the current colony) and the remanng E-1 solutons are the best ones n the current colony. In other words, only the best E ants are allowed to contrbute pheromone to the tral ntensty and the magntudes of contrbutons are weghted by ther ranks n the colony Overall Ant Colony Algorthm Generally, ant colony algorthms are smlar to other meta-heurstcs n that they terate over generatons (termed colones for ACO) untl some termnaton crtera are met. If an algorthm s eltst (as most genetc algorthms and ant colones are) the best soluton found s also contaned n the fnal teraton (colony). The termnaton crtera are usually a combnaton of total solutons consdered (or total computatonal tme) and lack of best soluton mprovement over some number teratons. These are expermentally determned. Of course, there s no downsde to runnng the ACO overly long except waste of computer tme. The flow of the ACO-RAP s as follows: Set all parameter values and ntalze the pheromone tral ntenstes Loop Sub-Loop Construct an ant usng the pheromone tral ntensty and the problem-specfc heurstc (eqs. 5, 6) Apply the onlne pheromone ntensty update rule (eq. 8) Contnue untl all ants n the colony have been generated Apply local search to each ant n the colony Evaluate all ants n the colony (eqs. 1, 7), rank them and record the best feasble one Apply the offlne pheromone ntensty update rule (eq. 9) Contnue untl a stoppng crteron s reached 1.4 Computatonal Experence To show the effectveness of the ant colony approach for relable systems desgn results from [30] are gven here. The ACO s coded n Borland C++ and run usng an Intel Pentum III 800 MHz PC wth 256 MB 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 wth results n the lterature.

12 12 Yun-Cha Lang and Alce E. Smth The parameters of the ACO algorthm are set to the followng values: α = 1, β = 0. 5, q0 = 0.9, ρ = 0. 9 and E = 5. Ths gves relatvely more weght to the pheromone tral ntensty than the problem-specfc heurstc and greater emphass on explotaton rather than exploraton. The ACO s not very senstve these values and tested well for qute a range of them. For the penalty functon, γ = 0. 1 except when the prevous teraton has 90% or more nfeasble solutons, then γ = Ths ncreases the penalty temporarly to move the search back nto the feasble regon when all or nearly all solutons n the current colony are nfeasble. Ths b-level penalty mproved performance on the most constraned nstances of the test problems. Because of varyng magntudes of R, C and W, all η and τ are normalzed between (0,1) before soluton constructon. 100 ants are used n each colony. The stoppng crteron s when the number of colones reaches 1000 or the best feasble soluton has not changed for 500 consecutve colones. Ths results n a maxmum of 100,000 ants. The 33 varatons of the Fyffe et al. problem [20] as devsed by Nakagawa and Myazak [35] were used to test the performance of ACO. In ths problem set C =130 and W s decreased ncrementally from 191 to 159. In [20] and [35], the optmzaton approaches requred that dentcal components be placed n redundancy, however for the ACO approach, as n Cot and Smth [7], dfferent component types are allowed to resde n parallel (assumng that a value of = 8 for all subsystems). Ths n max makes the search space sze larger than Snce the heurstc benchmark for the RAP wth component mxng s the GA of [7], t s chosen for comparson. Ten runs of each algorthm (GA and ACO) were made usng dfferent random number seeds for each problem nstance. The results are summarzed n Table 1 where the comparsons between the GA and ACO results over 10 runs are dvded nto three categores: maxmum, mean and mnmum system relablty (denoted by Max R, Mean R and Mn R, respectvely). The shaded box shows the maxmum relablty soluton to an nstance whle consderng all GA and ACO results. The ACO solutons are equvalent to or superor to the GA over all categores and all problem nstances. When the problem nstances are less constraned (the frst 18), the ACO performs much better than the GA. When the problems become more constraned (the last 15), ACO s equal to GA for 12 nstances and better than GA for three nstances n terms of the Max R measure (best over ten runs). However, for Mn R (worst over 10 runs) and Mean R (of 10 runs), ACO domnates GA.

13 The Ant Colony Paradgm for Relable Systems Desgn 13 Table 1. Comparson of the GA [7] and the ACO over 10 random number seeds each for the test problems from [35]. These results are from [30]. C&S [7] GA - 10 runs ACO-RAP - 10 runs No C W Max R Mean R Mn R Max R Mean R Mn R Thus, the ACO tends to fnd better solutons than the GA, s sgnfcantly less senstve to random number seed, and for the 12 most constraned nstances, fnds the best soluton each and every run. Whle these dfferences n system relablty are not large, t s benefcal to use a

14 14 Yun-Cha Lang and Alce E. Smth search method that performs well over dfferent problem szes and parameters. Moreover, any system relablty mprovement whle adherng to the desgn constrants s of some value, even f the relablty mprovement realzed s relatvely small. The best desgn and ts system relablty, cost and weght for each of the 33 nstances are shown n Table 2. For nstances 6 and 11, two desgns wth dfferent system costs but wth the same relablty and weght are found. All but nstance 33 nvolve mxng of components wthn a subsystem whch s an ndcaton that superor desgns can be dentfed by not restrctng the search space to a sngle component type per subsystem. It s dffcult to make a precse computatonal comparson. CPU seconds vary accordng to hardware, software and codng. Both the ACO and the GA generate multple solutons durng each teraton, therefore the computatonal effort changes n drect proporton to number of solutons consdered. The number of solutons generated n [7] (a populaton sze of 40 wth 1200 teratons) s about half of the ACO (a colony sze of 100 wth up to 1000 teratons). However, gven the mproved performance per seed of the ACO, a drect comparson per run s not meanngful. If the average soluton of the ACO over ten seeds s compared to the best performance of GA over ten seeds, n 13 nstances ACO s better, n 9 nstances GA s better and n the remanng nstances (11) they are equal, as shown n Fgure 2. Snce ths s a comparson of average performance (ACO) versus best performance (GA), the addtonal computatonal effort of the ACO s more than compensated for. In summary, an average run of ACO s lkely to be as good or better than the best of ten runs of GA. The dfference n varablty over all 33 test problems between ACO and the GA s clearly shown n Fgure 3. Gven the well-structured neghborhood of the RAP, a meta-heurstc that explots t s lkely to be more effectve and more effcent than one that does not. Whle the GA certanly performs well relatve to prevous approaches, the largely random mechansms of crossover and mutaton result n greater run to run varablty than the ACO. Snce the ACO shares the GA s attrbutes of flexblty, robustness and mplementaton ease and mproves on ts random behavor, t seems a very promsng general method for other NP-hard relablty desgn problems such as those found n networks and complex structures.

15 The Ant Colony Paradgm for Relable Systems Desgn 15 Table 2. Confguraton, relablty, cost and weght of the best soluton to each problem. These results are from [30]. No. W R Cost Weght Soluton ,11,111,2222,333,22,333,3333,23,122,333,4444,12, ,11,111,2222,333,22,333,3333,22,112,333,4444,11, ,11,111,2222,333,22,333,3333,22,122,11,4444,11, ,11,111,2222,333,22,333,3333,23,112,13,4444,12, ,11,111,2222,333,22,333,3333,23,122,13,4444,11, ,11,111,2222,333,22,333,3333,22,122,11,4444,11, ,11,111,2222,333,24,333,3333,33,122,13,4444,12, ,11,111,2222,333,22,333,3333,13,122,13,4444,11, ,11,111,222,333,22,333,3333,33,112,11,4444,11, ,11,111,222,333,22,333,3333,33,112,13,4444,11, ,11,111,222,333,22,333,3333,33,122,13,4444,11, ,11,111,222,333,22,333,3333,13,122,13,4444,11, ,11,111,222,333,22,333,3333,23,122,11,4444,11, ,11,111,222,333,22,333,3333,33,122,11,4444,11, ,11,111,222,333,22,333,3333,33,122,13,4444,11, ,11,111,222,333,22,333,3333,33,222,13,4444,11, ,11,111,222,333,22,333,133,33,122,13,4444,11, ,11,111,222,333,22,333,133,33,222,13,4444,11, ,11,111,222,333,22,13,3333,33,122,11,4444,11, ,11,111,222,333,22,13,3333,33,122,13,4444,11, ,11,111,222,333,22,13,3333,33,222,13,4444,11, ,11,111,222,333,22,13,133,33,122,13,4444,11, ,11,111,222,333,22,13,133,33,222,13,4444,11, ,11,111,222,333,22,13,133,33,222,33,4444,11, ,11,111,222,333,22,33,133,33,222,13,4444,11, ,11,111,222,333,22,33,133,33,222,33,4444,11, ,11,111,222,33,22,13,133,33,222,33,4444,11, ,11,11,222,333,22,13,133,33,222,33,4444,11, ,11,111,222,33,22,33,133,33,222,33,4444,11, ,11,11,222,333,22,33,133,33,222,33,4444,11, ,11,11,222,33,22,13,133,33,222,33,4444,11, ,11,11,222,33,22,33,133,33,222,13,4444,11, ,11,11,222,33,22,33,133,33,222,33,4444,11, ,11,11,222,33,22,33,333,33,222,13,4444,11, ,11,11,222,33,22,33,333,33,222,33,4444,11,22

16 16 Yun-Cha Lang and Alce E. Smth Mean ACO System Relablty - Max GA System Relablty Problem Instance Fg. 2. Comparson of mean ACO wth best GA performance over 10 seeds. These results are from [30]. 1.5 Conclusons Ths chapter ctes the latest developments of ACO algorthms to relablty system problems. The man part of the chapter gves detals of a general ant colony meta-heurstc to solve the redundancy allocaton problem (RAP) whch was devsed over the past several years by the authors and publshed n [28, 29, 30]. The RAP s a well known NP-hard problem that has been the subject of much pror work, generally n a restrcted form where each subsystem must consst of dentcal components n parallel to make computatons tractable. Heurstc methods can overcome ths lmtaton and offer a practcal way to solve large nstances of a relaxed RAP where dfferent components can be placed n parallel. The ant colony algorthm for the RAP s shown to perform well wth lttle varablty over problem nstance or random number seed. It s compettve wth the bestknown heurstcs for redundancy allocaton. Undoubtedly there wll be much more work forthcomng n the lterature that uses the ant colony paradgm to solve the many dffcult combnatoral problems n the feld of relable system desgn.

17 The Ant Colony Paradgm for Relable Systems Desgn System Relablty Problem Instance (a) ACO System Relablty Problem Instance (b) GA Fg. 3. Range of performance over 10 seeds wth mean shown as horzontal dash. These results are from [30].

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