Maintenance Scheduling by using the Bi-Criterion Algorithm of Preferential Anti-Pheromone

Transcription

1 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS Department of Informatcs, Unversty of Praeus, 80, Karaol & Dmtrou Str., 8534 Praeus, Greece E-mals: Abstract Ths paper presents the soluton for the Thermal Generator Mantenance Schedulng Problem, usng a b-crteron Ant Colony Optmzaton Algorthm called Preferental Ant-pheromone (PAP). Ths method allows the agents of an ant colony to depost a small amount of pheromone tral to every path that has been used, to construct the potental solutons, but also to gve extra emphass to the best soluton found at the end of an teraton of the algorthm. In the same tme that good solutons are beng nvestgated from the agents, bad solutons are examned too, wth the am to avod short-term poor solutons and lead to long-term good and, respectvely, global best solutons. In ths way, through the teratons of the algorthm, we end up to the fnal solutons. The algorthm s appled to a real-scale problem, and further nvestgaton s beng made so as to fnd the best possble soluton. Keywords Thermal Generator; Mantenance Schedulng; Ant Colony Optmzaton; Ant Colony System; Preferental Ant-Pheromone. Introducton The Thermal Generator Mantenance Schedulng Problem s a complex problem that s 43

2 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS necessary for the relablty and rght operaton of a generator system, gven that the whole producton cost s dependent on the mantenance and operaton cost. Thus, the mantenance procedure has to be scheduled and compled wth the best possble way, mnmzng these two costs and at the same tme, coverng the energy demands, so as every constrant of the problem s satsfed. The problem has been studed n the past wth a varety of methods and algorthms []. The ntal formulaton was made by Gruhl [2, 3] n 973. He presented an umbrella of schedulng problems, one of whch was the mantenance schedulng problem, wth a lnear approach. Durng 975, Dopazo and Merll [4] developed a model whch was clamed to have the ablty of fndng the best soluton. But ths approach was lacng n real-scale problems applcaton, somethng that Zurn and Quntana [5] later acheved to do. In 983, Yamayee and Sdenblad [6] mproved the cost functon that was used tll then, wth great mprovements n executon tme. Eght years later, Satoh and Nara [7] appled for the frst tme a stochastc method, called Smulated Annealng wth very good results n large-scale systems as well, that were mpossble to be solved wth lnear methods n the past. They also nvestgated the problem wth genetc algorthms [8] and tabu-lst methods [9] wth smlar results, but wth the ablty to solve real-scale problems, too. In 993 Charest and Ferland [0] tred to modfy the lnear method wth successful results n executon tme, whle Dahal and Donald [] appled a genetc algorthm n Boolean representaton [2] whch had also some good results. In 997, Bure and Smth [3] tred to create a hybrd model of the smulated annealng and the tabu-lst method wthout success, followng another attempt to mae another hybrd model wth memetc and tabu-lst methods three years later, whch resulted n better results, but wth a small ncrease n executon tme. Ths paper studes the results of the Ant-pheromone method, an ACS-le algorthm based on the behavor of true ants. It s organzed as follows: In secton 2, we revew the prncpal framewor of the ACO, Ant Colony System and Prefferental Ant-Pheromone methods. In secton 3, we represent the formulaton of the Mantenance Schedulng problem. In secton 4, we represent the mplementaton of PAP for the problem and the algorthm used. The paper ends wth case studes on a real system n secton 5 and conclusons n secton 6. 44

3 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p Ant Colony Optmzaton (ACO) Overvew Ant Colony Optmzaton s a pac of Artfcal Intellgence algorthms that rely on the mtaton of the socal nsects behavor, and especally ants. These algorthms use agents, that we call ants, for the nvestgaton of the best soluton of a problem, for example the shortest path between some places that mght be food for the colony, just le happenng wth the true ant colones. Fgure. Real ants after a whle tend to choose the shortest path between nest and food These agents, are constructng through teraton the solutons of the problem. The probablty for an ant to vst a town s affected from the amount of pheromone that every agent detects durng ts exploraton. Pheromone s a substance that ants produce and depost along the paths that have traversed, mang them more attractve for the next ones that mght pass from the same pont, whle the already exstent pheromone s vaporzng as tme passes. Durng the progress of the algorthm, the artfcal pheromone s placed after the constructon of a complete tour-soluton on each and every town that was chosen and vsted for the constructon of t. In ths way, the amount of pheromone s the heurstc nformaton at a gven tmng pont, reflectng the experence of the colony about the feasble solutons of the problem. If we consder the possble solutons as a graph wth the ants movng from town sr to 45

4 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS sr+, then the pheromone level that exsts after the passng of an agent from a feasble soluton s gven from equaton: τ (s r,s r+ ) = ( ρ) τ (s r,s r+ ) + τ (s r,s r+ ) () where: ρ s the pheromone tral evaporaton factor n τ ( sr, sr+ ) = τ ( sr, sr+ ) = n the number of ants the number of each ant. The probablty of the th ant to follow the path from town sr to sr+, s: T where ( s, a, s r r r+ ) = τ ( s, s ) (2) N r r r+ τ( s, s ) r N r s a feasble soluton when ant s on node r, and a r the acton requred for the ant to move from node s r+ tos r. Ant Colony System (ACS) The Ant Colony System (ACS) [4] meta-heurstc algorthm belongs to the umbrella of ACO algorthms. On each step of the algorthm (specfcally for the Travelng Salesman Problem - TSP, but respectvely for other problems), the transton of an ant from town to town j, depends on:if town has already been vsted. For each ant, a tabu lst s beng saved, whch grows each tme t reaches a new town, end emptes whenever a full soluton for the problem s accomplshed. In ths way, towns are never vsted more than once. The nverse proporton of the dstance between two towns η = d that s called vsblty, and descrbes the heurstc preference of an ant between two towns. The amount of pheromone τ that exsts between two towns and j, on whch the experence of the colony s dependant, and descrbes the preference for town j, accordng to the experence from prevous teratons. When the th ant s on town, the probablty to move to town j, s gven from equaton: 46

5 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p β argmax { τ [η ] }, f q q (3) u N u u 0 j =, otherwse where: q s a random varable unformly dstrbuted between [0,] q 0 s a determnng parameter between [0,] N the number of nodes that are not n tabu lst of ant yet. s a town randomly chosen under equaton: p where: τ [ η ] = τ η 0, j N β, j N β u [ u ] u Nl β the vsblty varable for the transton rule. N the number of nodes that are not n tabu lst of ant yet. In the end of every teraton the pheromone tral s updated under equaton: τ ( t + ) ( ρ) τ ( t) + ρ τ where: t the number of teraton tang place ρ the evaporaton tral factor wth 0< ρ < τ 0 the ntal pheromone amount that s placed on every town. 0 After completng an teraton (when all ants have created ther won feasble solutons) the global update rule s beng appled: τ ( t + ) ( γ ) τ ( t) + γ τ ( t) whereγ the global pheromone evaporaton factor Q t, τ t = L ( ) 0, where: Town Best Soluton otherwse Q the amount of pheromone added L t the best soluton found tll teraton t In ths way, the equaton above s appled only to towns that belong to the best soluton durng the executon (all teratons) and ths enables them wth a bgger amount of pheromone. (4) (5) (6) 47

6 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS Preferental Ant-pheromone Durng the ant exploraton n the search space, they gan nowledge concernng the most desrable nodes. Ths often leads to short-term solutons that are not the best ones that can be found. Randal and Montgomery [5] are tang nto account ths fact wth the Accumulated Experence Ant Colony - AEAC usng a method that can fnd whch solutons can lead to long-term good solutons. Ant-pheromone s a substance that can have smlar effects and s placed n nodes that belong to the worst solutons, nformaton that s lost wth other methods. Irad, Merle and Mddendorf [6] suggest a method called Preferental Antpheromone (PAP) that uses pheromone, as much as ant-pheromone for the exploraton of the search space. The use of ths b-crteron exploraton s mproved durng the teraton process of the algorthm. Ths mprovement s feasble due to the dfference of ants concernng ther preference between these two nds of pheromone. For ths reason, we use a parameter λ, whch s gven for the th ant by equaton (-)/(m-), where =[,m]. In ths way, we gve some ants the opportunty to explore the best, but also to some others the worst solutons. In ths way, the transton rule of an ant from town I to town j that we dscussed on classcal ACO s: β argmax {[ λτ + (-λ)τ' ] [η ] }, f q q (7) u N u u u 0 j =, otherwse where s a town chosen by equaton: β [ λτ + (- λ)τ' ] [η ] (8) l l l, j N β p = + [ λτ u (- λ)τ' u ] [ηu ] u N 0, j N Pheromone and ant-pheromone are updated n the same way when ants traverse the problem s nodes to form the solutons, as local updatng gnores the cost of solutons produced. So, n addton wth the local updatng rule of classcal ACO for pheromone, antpheromone s updated accordng to the followng rule: τ ' ( t + ) ( ρ) τ ' ( t) + ρ τ 0 (9) 48

7 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p After completng a tour and each ant has chosen a feasble soluton, pheromone s placed on towns that belong to the best soluton accordng the global updatng rule of classc ACS, and pheromone to the worst ones, accordng to the followng rule: τ ' ( t + ) ( γ ) τ ' ( t) + γ τ ' ( t) (0) where: Q t, Town WorstSoluton τ' t = L w ( ) 0, otherwse t th and L w the worst length (cost) soluton found on t teraton. Formulaton of the Problem The objectve of the Thermal Generator Mantenance Schedulng Problem s the mantenance of the energy producton unts of a system n a gven horzon, usually n wees, wth the lowest possble cost. The lst of symbols that descrbe the problem s as follows [, 7]: : Generator number Ι : Number of generators j : Number of wee : Horzon n number of wees x : Mantenance start perod; x {, 2,, } X : Set of preferred mantenance start perods; X : {, 2,, } M : : Mantenance length n wees Y : : State varable;, f unt s n ma nt enance Υ = at perod j 0, otherwse p : generator output of unt- at perod-j f : fuel cost coeffcent (lnear cost functon) c (x ): mantenance cost of unt- when the mantenance s commtted at perod x P : capacty of unt D j : antcpated demand at perod-j R j : requred reserve at perod-j 49

8 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS The generator mantenance schedulng problem s formulated as shown below: Objectve functon The objectve s to mnmze the sum of the followng two terms: I I () Mn f p + c ( x ) = j= = where the frst term s the producton cost and the second s the mantenance cost. Constrants ) The nomnal startng perod of mantenance s pre-specfed for each generatng unt: {,2 } x X,..., (2) 2) Once the mantenance of unt- starts, the unt must be n the mantenance state for just M perods: 0, j =,2,..., x (3) Υ =, j = x,..., x + M 0, j = x + M,..., 3) If unt- and unt- 2 cannot be mantaned n a gven wee because of the crew constrant, the followng constrant (combnatonal constrant) s mposed: Y( ) j + Y( 2) j, j =,2,..., (4) 4) If the mantenance of unt- must be fnshed pror to the startng of that of unt- 2, the followng constrant (order constrant) s added: x + M x 2 5) The generator output must be less than ts upper lmt; and the output of the generator n mantenance must be equal to zero. Such an operaton constrant s expressed by: 0 p P ( y), =,...,, I j=,..., (6) 6) The demand constrant must be met: I = (5) (7) p = D j, j =,2,..., 7) In order to ensure that the total avalable power s greater than the demand D j even when a unt random outage occurs, the reserve constrants are mposed. That s, the total avalable power from unts whch are not commtted must be greater than the demand plus reserve: 50

9 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p I = P ( y ) D j + R j, j =,2,..., (8) Penalty Functon In the mantenance schedulng problem, the constrants are classfed nto two groups; easy constrants and dffcult constrants. The easy constrants are eq (2), (3), (5), (6), the dffcult constrants are eq (4), (7), (8). Snce the set X s gven, the value of x can be selected as a member of X so that eq (2), (5) are satsfed. Then the value of y s drectly defned by eq (3), and eq (6) becomes a smple bound on p. On the other hand, t s very dffcult to fnd a feasble soluton whch satsfes eq (4), (7) and (8). So, the artfcal varables z, u, and v are ntroduced correspondng to eq. (4), (7) and (8), wth assocated postve penalty parameters α, β, and γ. Then the problem s re-formulated as follows: I f p + = j= Mn + α + z β j= x X {,2,..., } = I c ( x ) u + γ j= j= v (9) (20) 0, j =,2,..., x (2) Υ =, j = x,..., x + M 0, j = x + M,..., Y( ) j + Y( 2) j z j, j =,2,..., (22) x + M x (23) 2 0 p P ( y), =,...,, I j=,..., (24) I (25) p + u j = D j, j =,2,..., = I (26) P ( y) + vj Dj + Rj, j =,2,..., = z n {0,} (27) uj, vj 0 j, j =,2,..., (28) By usng the above formulaton, once the value of x s determned, the value of y s drectly defned, and the value of p s calculated through the equal ncremental method (equal method) for the economc dspatch problem []. Therefore, the value of the objectve functon can be effcently evaluated f the value of x s specfed. 5

10 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS Implementaton of PAP for the Mantenance Schedulng Problem Expresson approach For the mplementaton of the problem, we used a sort of graph. Every node of the graph represents a feasble soluton, and more specfcally a feasble wee that the mantenance of every generator can be started. In ths way, every ant traverses one by one the generator unts, choosng one of the feasble mantenance startng perods and, n the end, constructng a complete soluton. When all ants complete ther tours, the teraton s completed and a new one taes place. t I t I 2 I m t t 2 t 2 t 2 t n t n t n Fgure 2. Representaton of the problem usng graph So, on every step, all unts are selected, and the total cost of the soluton s calculated, summng up the total mantenance cost, plus the total generator operaton cost needed for every wee (plus the penalty of erroneous solutons, f any). The probablty for an ant- that s on town- to move to town-j, s gven by the randomproportonal rule of PAP: β argmax {[ λτ + (-λ)τ' ] [η ] }, f q q (29) u N u u u 0 j =, otherwse where s a town chosen by equaton: 52

11 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p β [ λτ + (- λ)τ' ] [η ] (30) l l l, j N β p = + [ λτ u (- λ)τ' u ] [ηu ] u N 0, j N th where N are the towns that are not yet ncluded on the agent s tabu lst. As vsblty η between towns, we wll use the equaton η ( t) = + PCV ( t) where PCV (t) s a method countng the total number of the Problem Constrant Volatons. We wll bas each constrant volaton usng weghts whch correspond to the relatve mportance of each constrant. Thus, the nodes that cause volatons wll be less desrable by ants. equaton: The pheromone tral s updated every tme an ant traverses a node, accordng to the τ ( t + ) ( ρ) τ ( t) + ρ τ where: t the number of teraton tang place ρ the pheromone evaporaton factor, 0< ρ < τ 0 the ntal amount of pheromone added on each node whle the ant-pheromone tral s updated accordng to the rule: τ ' ( t + ) ( ρ) τ ' ( t) + ρ τ 0 0 Fnally, the nodes pheromone trals that belong to the best soluton found are updated accordng to equaton: where τ ( t + ) ( γ ) τ ( t) + γ τ ( t) Q t, f Town BestSolut on τ t = Cost ( ) best 0, otherwse and the nodes pheromone trals that belong to the worst soluton, accordng to equaton: τ ' where ( t + ) ( γ ) τ ' ( t) + γ τ ' ( t) (3) (32) (33) Q t ( t) = Cost 0 τ' worst, f Town Worst Soluton, otherwse 53

12 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS where: γ the global pheromone evaporaton factor Q the amount of pheromone added Cost t best the best (lowest) cost found tll teraton t Cost t worst the worst (hghest) cost found on teraton t The algorthm. Defne problem parameters for each agent and generator. 2. For every ant and for every generator select a power level randomly. 3. Evaluaton of every soluton constructed by the ants Cost best = mn { Cost best, Cost teraton-best } Cost worst = mn { Cost worst, Cost teraton-worst } 4. Update the amount of pheromone and ant-pheromone trals for every traversed soluton (local) 5. Update the amount of pheromone for the best soluton path. (global) 6. Update the amount of ant-pheromone for the worst soluton path. (global) 7. For every ant and for every generator select a power level based on the randomproportonal rule. 8. Repeat algorthm from Step 3. Case study on a real-scale system of generators The algorthm just descrbed, was mplemented on a real scale system of generator unts [7] wth 22 power generator unts that have to be mantaned wthn a 52-wee horzon. The table shows the parameters that descrbe every generator unt s operaton and mantenance. Ι R E L M Table. System Parameters a b c f Crew constrant for every mant. wee

13 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p ,

14 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Table 2. Weely demand j Demand D j j Demand D j Trantafyllos MYTAKIDIS and Arstds VLACHOS where: R the hghest level of energy can be produced. E and L the earlest and latest perod that the mantenance can start. M the mantenance perod length (n wees). a, b, c, the cost parameters for the operaton of the generators. f the fuel cost coeffcent (lnear functon). and, fnally, the mantenance crew needed for every mantenance wee. The hghest level of energy that can be produced from all generators s zero. The rght table, also, represents the antcpated demand of the system for every wee wthn the horzon. 56

15 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p The requred reserve for each wee of the horzon can be defned usng one of the followng approaches:. As a constant percentage of the demand D j... As equal to the sze of the largest generatng unt. In dependence of other necessary crtera. Here, we appled the frst approach, wth a 20% percentage on demand D j. That s: R j = 20%, D j, j =,2 It s mportant to defne some determnant parameters for the soluton of the problem. The followng executons of the problem are loong nto the followng matters: As we are worng on a real system, t s easy to apprecate the fact that we need a mantenance crew constrant that wll be: o Flexble, concernng the solutons that can be produced, wthout confnng them. o Bg enough to produce solutons wthout volatons-penaltes and, respectvely, non-feasble. o Small enough so as to mnmze the exstence of not needed crew. For all these reasons, after close study of the problem constrant table and the solutons produced, the crew number was set to 30. As we can see, the objectve functon descrbes the constrant volatons as extra cost added to the producton cost. These solutons are not feasble, thus we have to defne the postve penalty weght parameters α, β and γ to be analogous wth the soluton cost of the problem, so as to be added an extra cost feasble to reject them. After expermental executons, we found that a soluton wthout volatons s n the order of hundreds mllons cost unts (0 8 ). So, forasmuch as each volaton can occasonally occur, the parameters were defned as follows: o α = 00 o β = 00 o γ = 20 The followng varables were defned emprcally: o τ 0 =0,0000 o Q= 57

16 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS o Maxmum number of teratons=2000, so as executon s not stopped early, and consequently wthout a good soluton. o Maxmum number of teratons wthout a better soluton=50, number bg enough for a decent result. The defnton of the heurstc nformaton weght parameter β, the local and global pheromone tral evaporaton factor ρ and γ, as well as the q 0 parameter, s also essental. The followng tables represent the results gven by expermental executons. All soluton costs are ndcatve and expressed n 0 8 cost unts. Table 3. β, ρ, q 0 and γ versus Best soluton β Best Cost ρ Best Cost q 0 Best Cost γ Best Cost

17 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p Fgure 3. β versus Best soluton Fgure 4 - ρ versus Best soluton Fgure 5. q 0 versus Best soluton 59

18 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS Fgure 6. γ versus Best soluton We have to emphasze that the above results are not unque, as randomness s taen nto account, although effort was made to mnmze ths factor. The followng parameters were chosen for the resoluton of the problem: β=0,5 ρ= q 0 =0,6 γ=0,3 The program of the proposed method s wrtten n Matlab 6.5 and t s run on an AMD Athlon GHz processor gvng the followng results: The best soluton found s [35,5,36,34,30,9,29,4,46,6,40,7,0,6,25,24,2,26,3,8,48,4] That s the perod that mantenance for every unt of the system can start (For unt, mantenance starts at wee 35, for unt 2 on wee 5, and so on). The mantenance perods are represented n detal on the followng dagram: 60

19 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p Fgure 7. Mantenance perods of Best soluton where the feasble perods are represented wth x and the selected perods wth * The best soluton cost found s The best soluton was found on teraton number 203. The best soluton progress versus teratons s represented to the followng fgure: Fgure 8. Best soluton cost versus Iteratons The executon tme s 0:3: As we can see n Fgure 8, the PAP algorthm has produced a very good soluton wth only a small number of teratons. From the frst steps, t contnuously fnds better solutons 6

20 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS wth lower costs, untl teraton 203 that no better soluton s found for the next 50 teratons, so the algorthm s termnated. It s remarable that the algorthm produced much better solutons step by step, wthout beng trapped on local best solutons due to pheromone trals (a problem that characterzes most ant-based algorthms). Conclusons Ths paper looed nto the Thermal Generator Mantenance Schedulng Problem of a real-scale system. The problem has been studed wth many mathematcal and heurstc approaches n the past. In ths project, the preferental Ant-pheromone approach was used, whch s a varaton of the ACS algorthm based on the behavor of real ant colones. The results produced prove that the algorthm can be appled successfully to the problem so as the optmum solutons can be found, even n real energy producton systems that the complexty rases sgnfcantly, because of the number of generatng unts, but also due to the number of feasble solutons that have to be produced n a reasonable tme nterval. References. Bure E. K., Smth A.., Hybrd Evolutonary Technques for the Mantenance Schedulng Problem, IEEE Transactons on Power Systems, 2000, 5(), pp Gruhl., Electrc generaton producton schedulng usng a quasoptmal sequental technque, Research Note MIT-EL , MIT Energy Lab, Aprl Gruhl., Electrc power unt commtment schedulng usng a dynamcally evolvng mxed nteger program, Research Note MIT-EL , MIT Energy Lab, anuary Dopazo. F., Merrll H. M., Optmal generator mantenance schedulng usng nteger programmng, IEEE Transactons on Power Apparatus and Systems, 975, PAS-94(5), p Zurn H. H., Quntana V. H., Generator mantenance schedulng va successve approxmatons dynamc programmng, IEEE Transactons on Power Apparatus and 62

21 Leonardo ournal of Scences ISSN Issue 2, anuary-une 2008 p Systems, 975, 94(2), p Yamayee Z. A., Sdenblad K., Yoshmura M., A computatonally effcent optmal mantenance schedulng method, IEEE Transactons on Power Apparatus and Systems, 983, 02(2), p Satoh T., Nara K., Mantenance schedulng by usng the smulated annealng method, IEEE Transactons on Power Systems, 99, 6, p Km H., Nara K., A method for mantenance schedulng usng GA combned wth SA, Selected papers from the 6th annual conference on Computers and ndustral engneerng, 994, p Km H., Hayash Y., Nara K., An algorthm for thermal unt mantenance schedulng through combned use of GA, SA and TS, IEEE Transactons on Power Systems, 997, 2(), p Charest M., Ferland. A., Preventatve mantenance schedulng of power generaton unts, Annals of Operatons Research, 993, 4, p Dahal K. P., McDonald. R., Generatonal and steady state genetc algorthms for generator mantenance schedulng problems, Proceedngs of the Internatonal Conference on Artfcal Neural Networs and Genetc Algorthms, 997, pp Dahal K. P., McDonald. R., Generator mantenance schedulng of electrc power systems usng genetc algorthms wth nteger representaton, Submtted to GALESIA 97, 997, p Bure E. K., Clar. A., Smth A.., Four methods for mantenance schedulng, In Proceedngs of the Internatonal Conference on Artfcal Neural Networs and Genetc Algorthms, 997, pp Dorgo M., Gambardella L.M., Ant Colones for the travellong salesman problem, BoSystems, 997, 43, p Randall M., Montgomery. Q., The Accumulated Experence Ant Colony for the Travellng Salesman Problem, Proceedngs of Inaugural Worshop on Artfcal Lfe, Adelade, Australa, 200, pp Ired S., Merle D., Mddendorf M.Q., B-Crteron Optmzaton wth Mult Colony Ant 63

22 Mantenance Schedulng by usng the B-Crteron Algorthm of Preferental Ant-Pheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS Algorthms. Evolutonary Mult-Crteron Optmzaton, Frst Internatonal Conference (EMO 0), Zurc, 200, pp Ibrahm El-Amn, Salf Duffuaa, Mohammed Abbas, A tabu search algorthm for mantenance schedulng of generatng unts, Kng Fahd Unversty of Petroleum and Mnerals, , Electrc Power Systems Research 54, 2000, pp Wood., Wood B. F., Woolenberg B. F., Power Generaton, Operaton, and Control, ohn Wley,