Internaonal Journal of Dgtal Content Technology and t Applcaon Volume 4, Number 9, December 00 Ulty Baed Opmal Tak Schedulng Problem n a Mul-agent Sytem Xaowe Zhang, Bn L, Junwu Zhu, Jun Wu School of Informaon Engneerng, Yangzhou Unverty, Yangzhou, Chna, E-mal: xwzhang@yzu.edu.cn, lb@yzu.edu.cn, dkr@63.com, _wu@vp.ohu.net do:0.456/dcta.vol4. ue9.4 Abtract A mul-agent ytem (MAS) ha been bult baed on characterc of emergency ytem, n whch tak chedulable and maxmum tak chedulng problem for the MAS ha been dcued n our prevou work. However, the menoned tak chedulng method both uppoed that all tak were equally mportant and agent have the ame capablty n playng a certan role, and evaluated the qualty of tak chedulng only by the number of accomplhed tak, whch gnored the dfference among tak or Agent that conflct wth our real lfe. In order to make the tak chedulng be more cloe to actual proce of emergence, the ulty baed opmal tak chedulng problem (UOTSP) propoed n th paper. The method agn each tak wth a ulty to repreent the earnng from accomplhng t and add preference on each Agent to dnguh the Agent preference and roleplay able. The UOTSP problem proved to be NP complete by reducng MAX-S to t n polynomal me, and two approxmate algorthm (GTS and RTOM) are propoed to olve the UOTSP problem approxmately. The experment reult how that both GTS and RTOM can get excellent approxmate oluon, and the me performance of GTS wonderful whch make t utable for the emergency applcaon of MAS (emergency ytem).. Introducon Keyword: Mul-agent Sytem, Tak Schedulng, Emergency Sytem, Ulty, NP Complete, Approxmate Algorthm Mul-agent ytem an open ytem, and the change of external envronment may connually generate new tak []. However, ngle Agent ha lmted capablty to proce nformaon and olve problem. At th me, mulple Agent are requred to complete thee tak n cooperaon, whch mean tak chedulng a key problem n MAS. An effecve tak chedulng method neceary for MAS to accomplh the tak wth hgh effcency. Recent year have een a lot of work on tak chedulng method, whch can be broadly clafed a follow: () tak chedulng baed on aucon protocol [-4] ; () Coalon formaon method [5-7] ; (4) Socal network baed method [8] ; (5) Decon theory baed method [9] ; (6) Manterk at al. [0] dcue the poble of achevng effcent allocaon n both cooperave and non-cooperave etng. They propoe a centralzed algorthm to fnd the opmal oluon. The above work have mproved the effcency and enrched the theory of MAS. However, thee work can t be ued n many new applcaon of MAS uch a emergency ytem etc. The reaon are a follow: () In thee applcaon, all agent mut mandatory execung tak allocated by dtrbutor, and the agent local goal mut not conflct wth the dtrbutor global goal. However, above tak chedulng method uch a aucon baed one uppoe that the agent are elfh, and have hgh degree of autonomy, o they would not execute the tak f the dtrbutor doen t offer payment they expected. () Stuaon urgency and tak hould be accomplhed a oon a poble n above applcaon. However, tak allocaon of mot above work decentralzed, and the tak chedulng proce carred out by negoang between agent, and the tak chedulng proce wll be uncertanty and me-conumng, whch cannot afy the requrement of mely tak allocaon n urgency tuaon of above applcaon of MAS. In our prevou work [], we have contructed a MAS model baed on the characterc of the emergency ytem and dcued the tak chedulable and maxmum tak chedulng problem baed on the MAS model, whch ddn t conder the dfference among tak and dfference between Agent. However, n pracce, although the tak arrve to the ytem at the ame me, they have dfferent mportance and urgency. What more, dependency relaonhp ext among ome tak. In general, - 7 -
Ulty Baed Opmal Tak Schedulng Problem n a Mul-agent Sytem Xaowe Zhang, Bn L, Junwu Zhu, Jun Wu tak whch are urgent, mportant and n front of the dependency relaonhp hould be acheved frt. In order to decrbe the dfferent mportance and urgency among tak and guarantee that the mportant and urgent tak to be completed frt, we add ulty on each tak accordng to the degree of mportance and urgency. Meanwhle, n many cae, the qualty of tak accomplhment relate to agent attrbute and tatu (e.g. Both Ronaldo and L Wefeng can play role of Forward when playng football, but the effect enrely dfferent). Thu, we alo ntroduce ulty to dfferenate agent capablty n playng a certan role. The paper organzed a follow. In Secon we gve the formal defnon the UOTSP problem and contruct flow network correpondng to t, and n Secon 3 we analyze the complexty of the problem and prove t to be NP complete by reducng MAX-S problem to t n polynomal me. In Secon 4, we propoe two approxmate algorthm GTS and RTOM to olve the UOTSP problem. We do experment for above algorthm n Secon 5, and the reult how that GTS utable for olvng the UOTSP epecally when the MAS wa appled to emergency ytem. Latly, we conclude our work and preent the future work.. Problem decrpon Suppoe there a et of agent A = {a,, a m }, each of whch encapulate a part of requred reource for compleng tak. Let R = {r,, r l } denote the collecon of role that the agent can play. Each agent a A can play a number of role, on the contrary, a role r R can alo be played by a number of agent, whch defned by role-agent relaon uch a Re R A. Now uppoe there a et of tak T = {t, t,, t n } arrve at the ytem. Each tak-agent mappng then repreented by a tuple < t, Re >, where t T the tak, Re Re. Then, the exact agnment of tak to agent defned by a tak chedulng. Defnon ( Tak Schedulng ) Gven a et of tak T = {t,, t n }, a et of agent A = {a,, a m }, and a et of role R = {r,, r l } the agent can play, t T ( =,,, m ) a mul-et on R, then tak chedulng a mappng φ: R A. whch mut afy the followng contrant: () If tak t can be accomplhed, every role related to t mut have correpondng agent to play, that for each r n mul-et t, there a funcon φ(r) = a ( r, a ) Re, r R, a A. () The agent be cheduled can at mot play a role at one me pont,.e. a A. { r R : ( r) a)} (3) All role related to every tak have correpondng agent to play, that t T. r R. a A. S( r) a Moreover, we aume that each agent can produce ulty whch mean the agent can obtan a certan amount of revenue from execung tak. Agent ulty n playng a role repreented by U: R A N. On the other hand, we alo uppoe that compleng of each tak can produce ulty whch repreented by U: t N, and the ulty of overall tak chedulng repreented by U(φ) = U( r, a) U( t) a A, n whch (r, a) Re, r R, a A and t T. By ntroducng ulty, we defne t T effcent tak chedulng bellow. Defnon (Effcent Tak Schedulng) Let U: R A N repreent the ulty on role-agent mappng, let U: t N be the ulty of compleng the tak t, and U(φ) = U( r, a) U( t) a A the t T overall ulty of the tak chedulng. We ay a tak chedulng effcent f t can maxmze U(φ). After defnng the effcent tak chedulng, we gve the formal defnon of opmal tak chedulng problem wth ulty a follow. Defnon 3 (Ulty baed Opmal tak chedulng, UOTSP). Suppoe there a et of tak T = {t,, t n }, a et of agent A = {a,, a m }, and a et of role R = {r,, r l } the agent can play, t T ( =,,, m ) a mul-et on R. Let U: R A N repreent the ulty on role-agent mappng, let U: t N be the ulty of compleng the tak t. If there ext an effcent tak chedulng that maxmze the ulty of overall ytem (max U(φ) ). - 8 -
Internaonal Journal of Dgtal Content Technology and t Applcaon Volume 4, Number 9, December 00 Algorthm. Contrucng Cot Flow Network. Create drected graph G = ( V, E) n accordance wth an ntance of tak chedulng problem, where the vertex et V = T R A and the edge et E = Tr Re, n whch Tr T R. and Re R A.. Create the cot flow network model N a below.. add ource node and nk node t to graph G... for each tak node t ( =,,, n) n T, create an edge (, t ) from to th node wth capacty equal to out-degree of t ( e.g. c(, t ) = d out (t ) ), and cot value -U(t )..3. for each agent node a ( =,,, m) n A, create an edge from a to nk node t wth capacty value, and cot value 0..4. for each edge (r, a) n Re (r R, a A), et the capacty to be and cot value equal to -U(r, a). The decon problem of UOTSP can be decrbed a follow. Whether there ext a tak chedulng φ whoe overall ulty larger than a contant value k, that U(φ) k. Tak chedulng problem and ulty baed opmal tak chedulng problem can be graphcally repreented by cot flow network. The contrucng algorthm hown n algorthm. For example, uppoe there an ntance I of UOTSP. The ntance I nclude a tak et T = {t, t, t 3 }, role et R = {r, r, r 3, r 4 }, and Agent et A = {a, a, a 3, a 4 }. Tak t a mul-et on {r, r } and the ulty value on t, t a mul-et on {r, r 3 } and the ulty value on t, t 3 a mul-et on {r 3, r 4 } and the ulty value on t 3 3, (r, a ), (r, a ), (r 3, a 3 ), (r, a 3 ) and (r 4, a 4 ) are the role-agent mappng, U(r, a ) =, U(r, a ) =, U(r 3, a 3 ) = 3, U(r, a 3 ) = 4 and U(r 4, a 4 ) = 5. Then, the correpondng cot flow network hown n fgure. c =,u=0 c=, u=- c=,u=- c=, u=-3 t t t 3 c =,u=0 c =,u=0 c =,u=0 c =,u=0 c =,u=0 r r r 3 r 4 c=,u=- c=,u=- c=,u=-3 a a a 3 a 4 c=,u=- 3. Complexty analyzng for UOTSP c=,u=- c=,u=- c=,u=-4 c=,u=- c=,u=-5 Fgure. Correpondng cot flow network We have proved that MAX-S n NP complete n []. Generally, UOTSP can be reduced to MAX-S, whch mean t more complex than MAX-S. For example, f we let ulty of each tak and ulty of each role-agent mappng to be zero, then the UOTSP mplfed to MAX-S. Actually, MAX-S pecal cae of UOTSP from a certan pont of vew. Thu, the UOTSP problem mut alo be NP complete. Theorem. The UOTSP problem NP complete. Proof. We frt how that UOTSP NP. Suppoe we are gven an ntance I of UOTSP and an nteger k. The cerfcate we chooe the tak chedulng φ. The verfcaon algorthm - 9 -
Ulty Baed Opmal Tak Schedulng Problem n a Mul-agent Sytem Xaowe Zhang, Bn L, Junwu Zhu, Jun Wu affrm that U(φ) k, and then t check whether φ afe the condon (), () and (3). Th verfcaon can be performed traghtforwardly n polynomal me. Now, we prove that the UOTSP problem NP-hard by howng that MAX-S P UOTSP. Suppoe that I an ntance of MAX-S wth T = {t,, t n }, a et of agent A = {a,, a m }, a et of role R = {r,, r l }, tak-role mappng Tr = T R and role-agent mappng Re = R A. We hall contruct an ntance I of UOTSP uch that the tak chedulng φ max f and only f U(φ) alo max. The contrucng algorthm hould only let the ulty of each tak t T equal to and the ulty of role-agent mappng to be zero. For example, uppoe an ntance I of MAX-S (the correpondng flow network N hown n fgure ) ha tak et T = {t, t, t 3 }, role et R = {r, r, r 3, r 4 }, and Agent et A = {a, a, a 3, a 4 }. Tak t a mul-et on {r, r }, t a mul-et on {r, r 3 }, t 3 a mul-et on {r 3, r 4 }, (r, a ), (r, a ), (r, a ), (r, a 3 ), (r 3, a 3 ), (r 3, a 4 ) and (r 4, a 4 ) are the role-agent mappng. Then, let ulty of each tak and ulty of role-agent mappng are zero, the correpondng cot flow network hown n fgure 3. c=,u=- f=, u=- c=, u=- t t t3 t t t3 c=,u=0 c=,u=0 c=,u=0 c=,u=0 c=,u=0 c=,u=0 r r r3 r4 r r r3 r4 5 c=, u=0 c=, u=0 c=, u=0 c=,u=0 c=,u=0 c=,u=0 c=, u=0 a a a3 a4 a a a3 a4 c=,u=0 c=,u=0 c=,u=0 c=,u=0 Fgure. Flow Network Correpond to I Fgure 3. Cot Flow Network Correpond to I We mut how that th tranformaon of I nto I a reducon. Frtly, we uppoe that φ a maxmum tak chedulng of I. Then we can know from [] that the flow value of flow network N acheve maxmum and have f d out ( t ), whch mean U(φ) = ( ) T d t out t. Note T that we et ule of role-agent mappng are zero. Thu, U(φ) = d out ( t ) maxmum. T Converely, uppoe that U(φ) = d out ( t ) maxmum. Obvouly, the flow of the flow T network hould alo be maxmzed and ha f d ( t ). A know from theorem n [], φ mut be the maxmum tak chedulng, whch complete our proof. By further analyzng the problem, we can ee alo that even though each tak exactly relate to only three role, the problem cannot be olved n polynomal me. Hence, we can derve the concluon hown a corollary. Corollary. Let 3-UOTSP to be the ub-problem of the UOTSP of whch each ntance t exactly relate to three role ( t T. R 3), then 3-UOTSP NP complete. 4. Approxmate algorthm for UOTSP A know from the lat econ that the UOTSP problem NP complete. Even though t may be dffcult to fnd an opmal tak chedulng n an ntance of UOTSP, t not too hard to fnd a tak chedulng that near-opmal. Therefore, we degn two approxmate algorthm to olve the problem n th econ. 4.. Greedy Tak Schedulng (GTS) T out - 30 -
Internaonal Journal of Dgtal Content Technology and t Applcaon Volume 4, Number 9, December 00 The greedy algorthm take a nput an flow network N, tak et T, role et R, Agent et A and return a accomplhed tak et T whoe overall ulty guaranteed near opmal. The greedy method work by pckng, at each tage, the tak t T wth greatet ulty value and the agent wth greatet ulty value on t role-agent mappng. Greedy-Tak-Schedulng ( N, T, R, A) T Whle T and 3 { elect the tak t wth larget ulty from T 4 for each role r R relate to t 5 elect the Agent a A wth larget ulty from A accordng to the role-agent mappng. 6 f there ext an Agent et to play all role relate to tak t, then 7 { T T t 8 T T t 9 0 Delete all tak-role mappng that relate from tak t, and delete all role-agent mappng that relate to each Agent n } ele T T t and delete all tak-role mappng that relate from tak t } Algorthm. Greedy Tak Schedulng Algorthm The algorthm work a follow. The et T contan, at each tage, the et of remanng unelected tak. The et T contan the elected tak. Lne 3 and lne 5 the greedy deconmakng tep. A tak t and a et of Agent are choen that maxmze the ulty of the tak chedulng. After t and are elected, the tak t and element of are removed, and tak t placed n T. When the algorthm termnate, the et T contan the accomplhed tak that approxmately maxmze the ulty of tak chedulng. The greedy algorthm can ealy be mplemented to run n me polynomal n T, R and A. Snce the number of teraon of the loop on lne - bounded from above by T, and the loop body can be mplemented to run n me O( R A ), there an mplementaon that run n me O( T R A ). 4.. Retrcted tak orderng method (RTOM) 4... Orderng relaon decrpon For an ntance I of UOTSP, we ort the tak accordng to ulty value of tak. For example, uppoe there ext a et of tak T = {t, t,, t n }, each tak ha the ulty value,,,n repecvely, then the decendng order of the tak t n >t n- > > t. Now, we retrct that tak n T mut be elected accordng to the order relaon among tak, whch mean tak t mut be accomplhed before t f t t'. We call the opmal tak chedulng problem that retrct the tak execung order retrcted opmal tak chedulng problem (mplfed a ROTSP). Suppoe there an ntance I of UOTSP wth a tak et T = {t, t,, t n } of whch the tak decendng order t >t > > t. Let n n- S { t, t, t } 5 n and S { t, t6, tn } two oluon of UOTSP, then we ay that S better than S becaue t 6 t 5. Generally, let S :... k and S : t t... t be two oluon of a UOTSP ntance I, f there ext an pove h nteger ( mn{ h, k} ) afe that t t, and for any 0 m, there t m t m, then - 3 -
Ulty Baed Opmal Tak Schedulng Problem n a Mul-agent Sytem Xaowe Zhang, Bn L, Junwu Zhu, Jun Wu we ay S better than S whch repreented a S > S. Bede, we can decrbe the addonal opmal relaon a follow: mn{ h, k}. t and k h, then S S () mn{ h, k}. t and k h, then S S () mn{ h, k}. t t and k h, then S S (3) For example, uppoe there ext two oluon S: t t t and 3 S: t t t, f t 3 t or t and t t of t, t t and t 3 t 3, then S>S. Otherwe, S<S or S=S. In parcular, f there a oluon S whch afe that for any other oluon, there ha, then we call S the opmal. Defnon 4(Retrcted Tak Schedulng Problem, ROTSP). Suppoe there a tak chedulng wth a et of tak T = {t,, t n }, a et of agent A = {a,, a m }, and a et of role R = {r,, r l } the agent can play, all tak n T are ordered accordng to tak ulty value n decendng order (t n >t n- > > t ), and we retrct that tak t mut be accomplhed before t f. Then, the problem derve the opmal oluon S max afyng that for any other oluon S, all have S max >S. 4... Approxmate olvng UOTSP by ROTSP Obvouly, ROTSP can be tranferred nto a cot flow network by a contrucng algorthm mlar to algorthm. However, t very crcal to et cot value for each arc. If we et correct cot value, the ROTSP can be olved by a reved mn-cot flow algorthm. Otherwe, the algorthm wll get wrong reult. If we et the cot value of each arc (, t ) of flow network N wth the negave ulty value of t ( =,,, n), and et cot value of all other arc to be zero, then we can ue mn-cot flow algorthm on N to olve ROTSP. The fgure 4 how a cot flow network correpond to an ntance of ROTSP. c=, u=- c=, u=-3 c =,u=0 t c =,u=0 c =,u=0 t3 c =,u=0 r r r3 r4 c =,u=0 c =,u=0 c =,u=0 a a a3 a4 c =,u=0 c =,u=0 t c=,u=- c =,u=0 c =,u=0 c =,u=0 Fgure 4. Cot on Arc of Flow Network N Theorem. Suppoe I an ntance of ROTSP wth a tak et T = {t,, t n } whch afe t n >t n- > > t, and we et cot of arc cot(, t )= -u(t ). Then, for any two tak node t and t n whch t > t, there ext a publc role r R between t and t (note that cot(, t ) < cot(, t )), and let f mnmum cot flow of N, then there mut have f ( t, r) f ( t, r). Proof. To how contradcon, we uppoe there ext two tak t and t wth publc role r R between them, afyng that t > t and f ( t, r) f ( t, r). Then, we can know from the cot flow network contrucng proce that f (, r) 0 and f ( t, r), whch mean f (, t ) capacty (, t ) - 3 -
Soluon me(m) Internaonal Journal of Dgtal Content Technology and t Applcaon Volume 4, Number 9, December 00 and f (, t ) 0. Th how that there mut ext a cycle (,, r, t, ) wth negave cot n redual network N f. From theorem. n [3], we can conclude that f cannot be the mnmum cot flow whch conflct wth our hypothe. Thu, f ( t, r) f ( t, r). The theorem how that the ROTSP problem can be olved by Mn-cot flow algorthm. We olve the ROTSP problem by terated Mn-cot flow algorthm (IMCA) [] and let the opmal oluon of ROTSP to be approxmate oluon of UOTSP. 5. Experment We mplemented the algorthm (GTS) and the terave mn-cot flow algorthm for RTOM method n C++ whch were teted on a Wndow PC. The purpoe of thee experment to tudy the performance of the algorthm n dfferent problem etng ung dfferent ntance of the problem. The performance meaurement are the oluon qualty and compung me, where the oluon qualty (SQ) computed a follow. When the number of tak mall, we compare the output of the above algorthm wth the opmal oluon whch wa calculated by brute force algorthm, but f t not feable to compute the opmal oluon, we ue the tak compleon rate to compare the oluon qualty. In the followng, we decrbe the etup of all experment, and preent the reult. 5.. Experment etng Setng. In order to be able to compare the output of the algorthm wth the opmal oluon, the cale of problem relavely mall. The number of tak 0, the number of role 0, and the number of agent ncreaed gradually from 5 to 40. Durng the expermenng, the average reource requrement reman unchanged, the role-play ablty of agent mantaned around 3, and the ulty value of all tak and role-agent mappng are generated randomly. The reource requrement can be meaured by reource rao derved from dvdng the number of Agent by the number of tak. Setng. The etng ued to evaluate the computaon me of the algorthm mplemenng on large cale problem. The number of the vertex vare from 70 to 90, and the number of tak vare from 00 to 800, and the average out-degree of the tak vertex and agent vertex n correpondng flow network reman around 4. The rao between the number of tak and the number of agent 0.5. 5.. Experment reult The frt experment mplement on expermental etng. We would lke to ee how the algorthm behave when the reource rao vared gradually. 0 The opmalty of GTS and RTOM 9 x 04 The me performance of GTS and RTOM 9 8 8 7 runnng me of GTS *00 runnng me of RTOM 7 6 5 4 3 opmal oluon oluon of RTOM oluon of GTS 0 0. 0.4 0.6 0.8..4 The rao of reource Fgure 5. Soluon of GTS and RTOM 6 5 4 3 0 0 00 400 600 800 000 00 400 the number of vertex Fgure 6. Tme Performance of GTS and RTOM In Fgure 5 we ee the oluon quale of GTS algorthm and the RTOM algorthm dependng on the reource rao. Remarkably, for hgher reource rao, the quale of the algorthm are better. - 33 -
Ulty Baed Opmal Tak Schedulng Problem n a Mul-agent Sytem Xaowe Zhang, Bn L, Junwu Zhu, Jun Wu When the reource rao grow above., the oluon of the algorthm almot opmal oluon, and the oluon qualty of the RTOM better than GTS all the me. Another experment to tet the me performance of above algorthm. We would lke to ee the run me of two algorthm when the cale of the UOTSP ntance ncreaed. The fgure 6 how the me performance of two algorthm, n whch the run me of GTS are mulpled by 00 to how t more clearly. We can obvouly ee from the fgure 6 that the run me of GTS much le than RTOM. Therefore, the GTS algorthm more utable for emergency ytem whch requre hgh real-me repone, epecally when the cale of the ytem very large. 6. Concluon Wherever Tme New Roman pecfed, Tme Roman, or Tme may be ued. If nether avalable on your word proceor, pleae ue the font cloet n appearance to Tme New Roman that you have acce to. Pleae avod ung bt-mapped font f poble. True-Type font are preferred. In th paper, we tuded the opmal tak chedulng problem wth ulty n a pecal MAS baed on our prevou work. Frtly, we defned the tak chedulng and ulty baed opmal tak chedulng problem (UOTSP). Then we analyzed the complexty of UOTSP and proved t to be NP complete by reducng MAX-S to t n polynomal me. In order to get the approxmate oluon of UOTSP, we propoe greedy algorthm and retrcted tak orderng method for t. The experment reult how that the me performance of GA very well and the oluon qualty of retrcted tak orderng method wonderful, both of whch are utable for our applcaon of MAS. There are many ntereng extenon to our current work. In th paper, we only tet the performance of approxmaon algorthm by experment. In our future work, we would alo lke to analyze the algorthm opmalty theorecally. Bede, we wll dcu tak chedulng problem n open and dynamc envronment n whch both Agent and tak can enter or ext the ytem at any me. Another ntereng topc for further work tudyng the relaon among role whch wll further complcate the tak chedulng. 7. Acknowledgement Th paper upported by the Naonal Scence Foundaon of Chna under Grant No. 607033, the Naonal Scence Foundaon of Chna under Grant No. 6070047, and the Natural Scence Foundaon of the Jangu Provnce of Chna under Grant No. BK 003. 8. Reference [] Woolrdge M. An ntroducon to mul-agent ytem. New York: John Wley & Son, 00. [] D. Sarne, M. Hadad and S. Krau, Aucon equlbrum tratege for tak allocaon n uncertan envronment, Proceedng of Eureopean Conference on Arfcal Intellgence, Sprnger, pp. 7-85, 004. [3] K La, L Ramuon, E Adar, L Zhang et al. Tycoon: An mplementaon of a dtrbuted, marketbaed reource allocaon ytem, Journal of Mulagent and Grd Sytem, vol, no. 3, pp. 69-8, 005. [4] D. Sarne and S. Krau, Solvng the aucon-baed tak allocaon problem n an open envronment, Proc of Tweneth Naonal Conference on Arfcal Intellgence and the Seventeenth Innovave Applcaon of Arfcal Intellgence Conference, MIT Pre, pp. 64-69,005. [5] O. Shehory and S. Krau, Method for tak allocaon va agent coalon formaon, Arfcal Intellgence, vol. 0, pp. 65-00, 998. [6] P. Toc and G. Agha, Maxmal clque baed dtrbuted coalon formaon for tak allocaon n large-cale mul-agent ytem, Proceedng of Mavely Mul-Agent Sytem. Sprnger, pp. 04-0, 005. - 34 -
Internaonal Journal of Dgtal Content Technology and t Applcaon Volume 4, Number 9, December 00 [7] S. Aknne and O. Shehory, A feable and praccal coalon formaon mechanm leveragng comprome and tak relaonhp, Proceedng of IEEE/WIC/ACM Internaonal Conference on Intellgent Agent Technology, IEEE Computer Socety, pp. 436-439, 006. [8] M. Weerdt, Y. Q. Zhang and T. B. Klo, Dtrbuted tak allocaon n ocal network, Proceedng of Internaonal Conference on Autonomou Agent and Mulagent Sytem, ACM pre, pp. 488-495, 007. [9] S. Abdallah and V. Leer, Modelng tak allocaon ung a decon theorec model, Proceedng of Internaonal Conference on Autonomou Agent and Mulagent Sytem, ACM pre, pp. 79-76, 005. [0] E. Manterk, E. Davd, S. Krau and N. R. Jennng, Formng effcent agent group for compleng complex tak, Proceedng of Internaonal Conference on Autonomou Agent and Mulagent Sytem, ACM pre, pp. 57-64, 006. [] Z. Xaowe. Reearch on Tak Schedulng problem for MAS baed Emergency Sytem, Mater degree the, 009. [] X. W. Zhang, J. Wu, and B. L, Tak chedulable problem n mul-agent ytem, JOURNAL OF SOUTHEAST UNIVERSITY (Natural Scence Edon), Vol. 38, up (), pp. 50-53, 008. [3] A. V. Goldberg, R. E. Taran, An effcent mplementaon of a calng mnmum-cot flow algorthm, Journal of Algorthm, (): -9,997. - 35 -