Decentralized subcontractor scheduling with divisible jobs

Transcription

1 DOI 0.007/s Decentralzed subcontractor schedulng wth dvsble jobs Behzad Hezarkhan, Wesław Kubak The Authors 05. Ths artcle s publshed wth open access at Sprngerlnk.com Abstract Subcontractng allows manufacturer agents to reduce completon tmes of ther jobs and thus obtan savngs. Ths paper addresses the coordnaton of decentralzed schedulng systems wth a sngle subcontractor and several agents havng dvsble jobs. Assumng complete nformaton, we desgn parametrc prcng schemes that strongly coordnate ths decentralzed system,.e., the agents choces of subcontractng ntervals always result n effcent schedules. The subcontractor s revenue under the prcng schemes depends on a sngle parameter whch can be chosen to make the revenue as close to the total savngs as requred. Also, we gve a lower bound on the subcontractor s revenue for any coordnatng prcng scheme. Allowng prvate nformaton about processng tmes, we prove that the pvotal mechansm s coordnatng,.e., agents are better off by reportng ther true processng tmes, and by partcpatng n the subcontractng. We show that the subcontractor s maxmum revenue wth any coordnatng mechansm under prvate nformaton equals the lower bound of that wth coordnatng prcng schemes under complete nformaton. Fnally, we address the asymmetrc case where agents obtan savngs at dfferent rates per unt reducton n completon tmes. We show that coordnatng prcng schemes do not always exst n ths case. Keywords Schedulng Dvsble jobs Subcontractng Coordnaton Mechansm desgn B Behzad Hezarkhan b.hezarkhan@tue.nl Wesław Kubak wkubak@mun.ca Department of Industral Engneerng, Endhoven Techncal Unversty, Endhoven, The Netherlands Faculty of Busness Admnstraton, Memoral Unversty, St. John s, NL, Canada Introducton As a common supply chan management practce, manufacturers take advantage of external resources to allevate the burden of nternal operatons through subcontractng and outsourcng. Whle outsourcng externalzes some nternal operatons, subcontractng allows manufacturers to carry out jobs both nternally and externally Van Meghem 999. In ths manner, subcontractng enables a manufacturer to speed up the completon tmes of hs jobs. Instances of subcontractng practces can be found n quck-response ndustres characterzed by volatle demand and nflexble capactes, e.g., metal fabrcaton ndustry Parmgan 003, electroncs assembly Webster et al. 997, hgh-tech manufacturng Aydnlym and Varaktaraks 0, textle producton, and engneerng servces Taymaz and Klçaslan 005. Although a consderable number of papers n the lterature analyzes the subcontractng strateges n producton plannng and schedulng problems of manufacturers Kamen and L 990; Tan and Gershwn 004; Chen and L 008; Lee and Sung 008, the subcontractors schedulng problems have receved less attenton. In realty, a subcontractor by tself faces a lmted capacty whle provdng servce to several manufacturers. Gven the tme-senstve nature of subcontractng operatons, the subcontractor s schedule has crtcal mpact on the performance of manufacturers as well as the supply chan. The lack of due attenton to the subcontractors operatons can cause sgnfcant complcatons n the extended supply chan. A well-documented real-lfe example of ths ssue has been reported n Boeng s Dreamlner supply chan where the overloaded schedules of subcontractors, each workng wth multple supplers, resulted n long delays n the overall producton due dates see Varaktaraks 03 and the references theren. 3

2 JSched An mportant feature of subcontractor s schedulng problem s that the jobs belong to dfferent manufacturers who are concerned wth the processng of ther own jobs only. Unlke centralzed systems where a sngle decson maker controls all the relevant decson varables and possesses all the necessary nformaton, the agents n a decentralzed system have some control over ther ndvdual decsons and/or are prvately nformed about some aspects of the system. Therefore, n order for the system to acheve a partcular objectve a mechansm s needed that motvates the agents to make ther decsons and reveal ther prvate nformaton n a way that the system s ultmate objectve s attaned ndrectly. In ths paper, we focus on the effcency objectve, that s, we are seekng mechansms that result n subcontractng schedules that maxmze the total savngs obtaned by all manufacturers. In other words, we address the problem of coordnaton n decentralzed subcontractng systems. The mechansms consdered n ths paper are prcng and payment schemes that the subcontractor announces before the manufacturers choose ther most desrable subcontractng ntervals. Such mechansms are n fact common n practce. An example s the onlne reservaton system mplemented by the Semconductor Product Analyss and Desgn Enhancement SPADE center of the Hong Kong Unversty of Scence and Technology wheren the semconductor companes choose servce tme ntervals n a frst-come-frstbook manner and n consderaton of an announced prce lst for servces n dfferent tme ntervals. Naturally, the desgn of mechansms n the decentralzed subcontractng systems must consder the utlty of agents,.e., savngs due to subcontractng mnus payments, as well as the subcontractor s revenue to ensure that all partes are suffcently motvated to partcpate and operate n the system. In ths paper, we study a subcontractor schedulng problem wth several manufacturer agents and a sngle subcontractor that carres out the agents jobs on a sngle machne. Of partcular nterest to ths paper s the scenaro where a dvsble job can be processed smultaneously on the manufacturer s prvate machne as well as on the subcontractor s machne. The reducton n completon tme of an agent s job obtaned by such parallel processng provdes monetary savng for that agent. The paper s organzed as follows. Secton contans a revew of related work. In Sects. 3 and 4, we assume complete nformaton and gve prcng schemes that coordnate the decentralzed system. That s the prcng schemes enforce a choce of subcontractng tme ntervals that concdes wth effcent allocatons of the centralzed soluton for every agent. In partcular, Sect. 3 characterzes effcent allocatons of the centralzed soluton. Secton 4 provdes suffcent condtons for the exstence of coordnatng prcng schemes and t ntroduces a famly of prcng schemes that are strongly coordnatng. Wth a strongly coordnatng prcng scheme, the effcent allocatons n the centralzed solutons are unquely optmal for all agents. Moreover, we show a lower bound on the total payments for any coordnatng prcng scheme. Fnally we show that wth the approprate choce of a sngle parameter, our proposed prcng schemes enable the subcontractor to obtan a total revenue anywhere between the lower bound and the maxmum total savngs. In Sect. 5, we allow the true processng tme of each job to be a prvate nformaton of ts agent and we address the ntrcaces resultng from agents possbly lyng by reportng false processng tmes n order to obtan ther preferred subcontractng ntervals. In order to acheve effcency, however, the subcontractor elcts the true processng tmes of jobs usng certan mechansm. We draw upon the class of effcent and ncentve compatble mechansms and on the pvotal mechansm Vckrey 96; Clarke 97; Groves 973 n partcular. The underlyng reason for ths choce s that the pvotal mechansm s the only effcent and ncentve compatble mechansm that always results n nonnegatve payments from agents to the subcontractor. Ths s a desrable property as agents must not be pad f ther jobs are processed by the subcontractor n the subcontractor schedulng problem. We obtan a smple closed-form formula for the payments n the pvotal mechansm and prove that the pvotal mechansm results n truth-tellng beng the unque optmal choce of all agents except the one scheduled last on the subcontractor s machne who can possbly exaggerate hs processng tme wthout affectng anyone s utlty. Ths result s partcularly nterestng as unqueness of equlbrum mght be thought of as the excepton rather than the rule Jackson 000. Snce we also show that the mechansm guarantees that all agents are better off by subcontractng, we actually prove that the pvotal mechansm coordnates the decentralzed subcontractng problem under prvate nformaton. Fnally, we show that under any coordnatng mechansm, the subcontractor s revenue s equvalent to the lower bound of that wth the coordnatng prcng schemes. Therefore, coordnated system under prvate nformaton never generates hgher total revenue for the subcontractor than that under complete nformaton. In Sect. 6, we address the asymmetrc case where the agents obtan savngs at dfferent rates. The effcent centralzed allocatons n ths case are more dffcult to characterze. Moreover, we show that t s mpossble to devse a coordnatng prcng scheme n general for ths case. Secton 7 contans the concludng remarks. 3

3 Related work The decentralzed schedulng problems have been consdered n cooperatve and non-cooperatve settngs. In the former, agents jobs are able to communcate and coordnate strateges. The early work of Curel et al. 989 studes the cost allocaton problem n a cooperatve game based on the perennal schedulng problem of Smth 956. A survey of related research s gven by Curel et al In the non-cooperatve settngs agents choose ther strateges ndvdually and n competton wth other agents. Ths paper falls n ths category. Heydenrech et al. 007 provde an ntroducton to and a lterature revew of problems arsng n the non-cooperatve decentralzed schedulng. One mportant problem pertans to determnng the equlbra of ndvdual decsons and the qualty of the correspondng soluton compared wth that of the centralzed optmal soluton. The poneerng work of Koutsoupas and Papadmtrou 999,whch gave rse to the lterature on prce of anarchy, analyzes the worst-case performance of a decentralzed schedulng system wth parallel machnes n comparson wth that n the centralzed system. As the outcomes of a decentralzed system depend on the polces employed to handle the agents, another mportant problem addresses a better desgn of such polces. The coordnaton mechansms dscussed by Chrstodoulou et al. 004 seek polces whch mprove the performance of a decentralzed parallel sequencng system Immorlca et al. 005 revew and extend ths lne of research. Ideally, such polces could make the decentralzed system as effcent as the centralzed system. Wellman et al. 00 study prcng schemes for a schedulng problem that could acheve ths goal and show that such prcng schemes mght not exst n general. Kutanoglu and Wu 999 report smlar non-exstence results for another schedulng problem. Even f the exstence of such prcng schemes could be proven, the problem of fndng the prcng scheme mght be NP-hard Chen et al Prvate nformaton further complcates the coordnaton problem. The well-known ncentve compatble mechansms draw upon payment schemes to truthfully elct the prvate nformaton of the agents. The Vckery Clarke Groves VCG mechansms Vckrey 96; Clarke 97; Groves 973 characterze all effcent mechansms that make truthtellng an undomnated strategy of all agents. For sngle machne sequencng, Sujs 996 shows that there exsts no ncentve compatble and ndvdually ratonal mechansm that results n the total payment of zero. Nsan and Ronen 007 propose payment schemes that guarantee a certan performance for a decentralzed schedulng problem for whch the payment schemes of VCG mechansms are NP-hard to compute. The decentralzed subcontractng and outsourcng problems have also been addressed n a number of papers. Aydnlym and Varaktaraks 00 study a cooperatve game where coaltons of agents reschedule ther reserved tme ntervals to obtan savngs. They prove non-emptness of the core and the convexty of the correspondng game. Ca and Varaktaraks 0 nvestgate a problem wth overtme and tardness costs where the subcontractor announces the prces of tme ntervals before agents book ther most preferred tme ntervals n a frst-come-frst-book manner. They show the balancedness of the correspondng cooperatve game and mplement the VCG mechansms to elct the prvate nformaton of agents. Bukchn and Hanany 007 compare the costs of a schedulng system comprsed of multple capactated agents and an uncapactated subcontractor n decentralzed and centralzed systems. Q 0 nvestgates a subcontractor s prcng problem wth a sngle agent havng multple jobs whch can be subcontracted though not partally to reduce the tardness costs. The subcontractng problem related to the one addressed n ths paper was frst studed by Varaktaraks and Aydnlym 007 where they compare the performance n decentralzed and centralzed settngs. Buldng upon the same model, Varaktaraks 03 analyzes the outcomes of a decentralzed subcontractng system under dfferent protocols announced by the subcontractor. However, both papers assume complete nformaton and nether provdes coordnatng prcng schemes for the problem. The model consdered n ths paper generalzes the model of Varaktaraks and Aydnlym 007 by allowng an agent to use more than one nterval on the subcontractor s machne. Ths generalzaton s crtcal for a coordnatng prcng schemes whch need to gve the agents freedom n choosng how many ntervals on the subcontractor s machne to buy the Varaktaraks and Aydnlym 007 model would a pror lmt ths choce to at most a sngle nterval whch does not make ther model adequate for the study of prcng schemes. It s worth notcng that the models n Varaktaraks and Aydnlym 007 and Varaktaraks 03 borrow from the concept of dvsble jobs ntroduced n the context of job shops by Anderson 98, and n the context of dstrbuted computer systems schedulng by Bharadwaj et al. 996, see also Drozdowsk 009 for a more recent revew. The dstrbuted computer systems provde another mportant applcaton area for the results obtaned n ths paper, where agents carry out ther computatons on ther prvate machnes as well as buy computatonal tme on shared CPUs wth avalable capacty. 3 The problem Consder a set of agents N ={,...,n}.attmet = 0, agent N has a dvsble job wth processng tme p > 0. Let We refer to as a job or an agent dependng on the context. 3

4 JSched p = p,...,p n be the vector of processng tmes. We assume, except n Sect. 5, that p s gven. An agent has ts own prvate machne to do ts job n tme p, besdes prvate machnes a subcontractor s avalable that can process any porton of job on ts machne that can be shared by all agents. Thus, by subcontractng, agent can reduce the completon tme of ts job to less than p.lett = m be the total T k tme allocated to agent consstng of m non-overlappng subntervals T k =[t k, tk + t k where tk 0 and t k > 0are the start tme and the duraton of the kth subnterval, respectvely. The subntervals of agent are ndexed by the order of ther start tmes. An allocaton T ={T N} s a set of allocatons T on subcontractor s machne. Allocaton T s feasble f and only f for any, j, k, and k the subntervals [t k, tk + t k and [tk j, tk j + t k j do not overlap. Let T denote the set of all feasble allocatons T for the agents n N. Our model allows preemptons on subcontractor s machne, that s the porton of a job allocated to the subcontractor s machne or the subcontracted part of a job s allowed to be executed n more than one dsjont tme nterval on that machne. Ths renders our model to be more general and arguably closer to real-lfe than the one n Varaktaraks and Aydnlym 007 where only at most one tme nterval on subcontractor s machne s allowed for any subcontracted part. Consequently two man decsons need to be made for each job n the model: one s the sze of the subcontracted part of a job ths part can be executed on the subcontractor s machne smultaneously wth the remanng part of the job executed on prvate machne thus the term dvsble jobs, the other as to how to execute the subcontracted part ths part can possbly be executed n several dsjont tme ntervals thus the term preemptons on the subcontractor s machne. The savng obtaned by agent from an allocaton T s calculated recursvely as follows. Assume that ntally, agent uses ts prvate machne only n the nterval [0, p ], N. Take the earlest nterval allocated to, T. If the start tme t < p, then a porton of the remander of job done after t,.e., p t,on s prvate machne can be transferred to the subcontractor. The most effcent way to do ths s for agent to splt the remander equally between ts prvate and subcontractor s machnes, unless the duraton of the nterval s too short see Fg.. If the duraton of the nterval s too short,.e., t <p t /, then T s fully utlzed by on the subcontractor s machne. Therefore, by utlzng the allocated nterval T, agent can reduce the fnsh tme of ts job by an amount equal to { υ T p {mn = max t } }, t, 0. Clearly, the allocaton T cannot be utlzed at all by agent f t p. The savng obtaned by the next nterval can be calculated n the same manner consderng that the new Fg. Valuaton n subcontractng processng tme of job s p υ T on s prvate machne. Thus the savng due to the kth subnterval T k to agent s obtaned recursvely by { υ T k p {mn = max k l= υ } } T l t k, t k, 0 total savng due to the allocaton T s calculated by summng up the savngs obtaned by all of ts subntervals: m υ T = υ T k. If T ={[t, t } s a sngle nterval, the Eq. smplfes to υ T = max { mn { p t /, t }, 0 }. The total savng of a feasble allocaton T s the sum of savngs of all agents, that s υt = N υ T. 3. Characterzaton of effcent centralzed allocatons The objectve of the centralzed problem s to fnd effcent allocatons on the subcontractor machne for N.An effcent allocaton T s a feasble allocaton that maxmzes the total savng, that s T arg max υt. 3 T T Let T ={T N} be an effcent allocaton. We have the followng two smple observatons that hold for any effcent allocaton. Observaton For N, job does not fnsh on the subcontractor s machne later than on agent s prvate machne. 3

5 Observaton The subcontractor s machne s never dle when some agent s prvate machne s busy. We now prove that all effcent allocatons are nonpreemptve. Lemma No effcent allocaton on subcontractor s machne s preemptve. Proof By contradcton. Let T be an effcent and preemptve allocaton and let n n be the number of agents wth allocatons on the subcontractor s machne. We show that then there exsts another feasble allocaton wth hgher total savng. Suppose that m > forsome N. Wthout loss of generalty we take the largest such. LetT m j j = [t m j j, t m j j + t m j j be the allocaton mmedately precedng T m.wehave = j, and t m j j + t m j j = t m by Observaton. Consder the followng modfcaton: [Step ] delay T m by t m t m + t m ; speed up all the allocatons between T m and T m by t m. Thus job would be preempted one less tme. By Observaton ths modfcaton does not change the completon tme of any job. Therefore, total savng remans unchanged. [Step ] Increase the duraton of the last allocaton of j by t m /, so that by Observaton the prvate machne of agent j fnshes earler by t m /. For k =,...,n,.e., jobs ncludng and after, start the last allocaton of k by t m } / k + later and fnsh t mn { t m / k +, t k later on the subcontractor s machne observe that f t m / k + t k then k wll no longer be executed} on the subcontractor s machne and mn { t m / k +, t k later on agent k prvate machne. The resultng allocaton s feasble. Furthermore, jobs n N \{j,,...,n } complete as before Step, and for the jobs n { j,,...,n } the total savng ncreases by t m t m t m 4 t m 4 mn { t m } n +, t n t m n + > 0. Hence, the alternatve allocaton has a hgher total savng whch contradcts the effcency of T. The lemma excludes allocatons wth preemptons on subcontractor s machne from the set of effcent allocatons. Ths result s key for our coordnatng prcng scheme n Sect. 4 snce the effcent allocatons wth preemptons on subcontractor s machne could make the exstence of coordnatng prcng scheme questonable or possbly more dffcult to prove. By Lemma, m = for N n any effcent allocaton T that s T = [t, t + t for N. Varaktaraks and Aydnlym 007 observe that, for any effcent allocaton T wth m =, N, each job fnshes smultaneously on ts own prvate and subcontractor s machnes, that s p t = t + t for N. 4 They also show that for m =, N, the agents effcent allocatons T are ordered n non-decreasng order of p. Thus, we assume hereafter n ths secton and n Sect. 4 that N s arranged n the non-decreasng order of processng tmes. In ths manner, job would be sequenced n th poston on subcontractor s machne. Fnally, by Observaton, there s no dle tme on the subcontractor machne n effcent allocatons, thus we have t = 0 and t = t + t for >. 5 The unque soluton of the recursve equatons 4 and 5 s gven n the followng theorem. Theorem Effcent allocaton Varaktaraks and Aydnlym 007 For effcent allocatons wth m = for N, we have t = p By ths theorem t k = p p k. 6 + k υ T = υ T = t 7 and thus, υt = N t. Equatons 6 and 5 defne vectors of effcent duratons t = t,..., t n and start tmes t = t,...,t n on subcontractor s machne. Fnally, we let t n+ = t n + t n denote the completon tme of the effcent allocatons,.e., the makespan of effcent allocatons. Though multple effcent allocatons exst as long as there are jobs wth equal processng tmes, n each one of them the jobs n the same poston on subcontractor s machne start at the same tme and have the same effcent duratons. Ths observaton s key to the coordnatng prcng scheme developed n the next secton. 4 Strongly coordnatng prcng schemes In contrast to a centralzed system where allocatons are chosen for the agents so as to maxmze the total savng, n a decentralzed system the decsons are made by the agents n a dstrbuted fashon. Though the agents are self-nterested they need, by defnton of coordnatng mechansm, to collectvely converge to an effcent allocaton. The convergence 3

6 JSched depends on a mechansm used n the collectve decson makng. In ths secton we desgn such a mechansm based on a prcng scheme. A prcng scheme s a functon q, defned on t 0, whch determnes a prce qt 0 for acqurng the tme t on the subcontractor s machne by any agent. The mechansm based on a prcng scheme q worksasfollows. The subcontractor announces ts prcng scheme, and subsequently agent buys T n a frst-come-frst-serve manner. The agent makes ts decson so as to maxmze ts utlty, whch for a gven q, and assumng quaslnear utltes, equals u q T = υ T π q T, 8 where π q T = qtdt t T 9 s the agent s payment for T made to the subcontractor. We requre q to be a locally ntegrable functon so that 9 always exsts. Though each agent maxmzes ts utlty wth respect to q, the choce of q must ensure coordnaton n the decentralzed system. That s q must guarantee that the agents choose ther subcontractng ntervals n the exact same manner as n some effcent allocaton T. We call such prcng schemes coordnatng. Let E be the set of all effcent allocatons. We formally defne. Defnton Coordnatng prcng scheme The prcng scheme q s coordnatng f for any T = T,...,T n, T E, and any T = T,...,T n, T T \ E, π q T π q T υ T υ T 0 for each N. The prcng scheme q s strongly coordnatng f 0 holds strctly for each N. A coordnatng prcng scheme results n the stuaton where no agent would be worse off by choosng an effcent allocaton. However, the mplementaton of a coordnatng prcng scheme may not necessarly result n the effcency of the system. Ths s due to the possblty that an agent chooses an nterval whch s not a part of any effcent allocaton. Such a choce could hnder forthcomng agents to select ther effcent allocatons. Therefore, the strong coordnaton requres all agents to exclusvely choose effcent allocatons. A natural class of prcng schemes to consder for schedulng conssts of those wth non-ncreasng q where agents have to pay a hgher prce for earler ntervals. When the choce of agent conssts of sngle nterval only,.e., T =[t, t + t, we let π q T = π q t, t + t. We now show that the class of non-ncreasng prcng schemes contan coordnatng prcng schemes. Theorem Suffcent condtons A non-ncreasng prcng scheme q s coordnatng f q0 < and the followng two condtons are met: C. For any =,...,n and every ɛ, 0 <ɛ t /: π q t ɛ, t π q t + ɛ, t + ɛ; C. For any =,...,n and every ɛ, 0 <ɛ t /, and for = n and every ɛ, 0 <ɛ t n : π q t, t + ɛ π q t +, t + + ɛ ɛ. Proof All T E defne the same start tmes t = t,...,t n and duratons t = t,..., t n as shown n the prevous secton. The ndvdual ratonalty requres that for 0 t < t n+, qt must be less than, otherwse jobs would be ether better off by not selectng the tme wth subcontractng prce exceedng one or ndfferent f the prce equals one. Snce q s non-ncreasng, the ndvdual ratonalty holds f q0 <. For q to be a coordnatng prcng scheme, t must be that no agent can devate from an effcent allocaton and mprove ts utlty. The frst-come-frst-booked order breaks tes between jobs wth equal processng tmes, f any, and pcks a unque effcent allocaton T from E.InT agent has the utlty υ T qt. The fact that t + t = p and q s non-ncreasng, mples that agent cannot choose another nterval to ncrease ts valuaton and, at the same tme, reduce ts payment. In order to mprove ts utlty, the agent may be able to choose another nterval to: a ncrease ts valuaton as well as payment such that the added valuaton s greater than the addtonal payment, or b decrease ts valuaton as well as payment such that the savng n payment s greater than the reducton n valuaton. We enforce condtons on q such that nether a nor b could possbly happen for any job. a Suppose that the agent chooses T n a way that υ T υ T = ɛ, forsomeɛ > 0. Note that ths s not an opton for =. As q s non-ncreasng, T has the cheapest payment f t starts as late as possble. Therefore, T =[t ɛ, t + ɛ s the cheapest alternatve nterval for agent whch results n ɛ mprovement n ts valuaton. Note that cannot have negatve start tme, thus ɛ t /. From Defnton t follows that for the prcng scheme q to be coordnatng, t must hold for any =,...,n and for every 0 <ɛ t / that π q t ɛ, t + ɛ π q t, t + ɛ. By the defnton of π q n 9, the last nequalty can be rewrtten as see Fg.. We use nducton and show that the latter would be the case f for any =,...,n, 3

7 Thus, 3 would be the case f for every ɛ such that 0 <ɛ t j /wehave, π q t j ɛ, t j π q t j+ ɛ, t j+ ɛ. Condton C for the job j ensures that the latter holds. Therefore, f for any =,...,n, holds strctly for every 0 <ɛ t /, then holds strctly for every 0 <ɛ t /. b It s straghtforward to see that the nterval whch reduces the valuaton of the agent by ɛ>0, whle havng the largest decrease n ts payment, s T = t + ɛ, t + + ɛ, as long as ɛ t. Therefore, for q to be coordnatng t must hold for every =,...,n and for every ɛ such that 0 <ɛ t that, π q t, t + π q t + ɛ, t + + ɛ ɛ. Fg. Non-ncreasng prcng schemes holds for every 0 <ɛ t /,.e., the condton stated n C. Ths holds trvally for = ast = t.fx = 3,...,n and suppose that for j, j <, holds for every ɛ such that 0 <ɛ t k /. We show that holds as well for every ɛ such that, j+ 0 <ɛ t k /. Observe that ths would be the case f for every ɛ such that 0 <ɛ t j /wehave, π q t j ɛ, t j π q t + t k / ɛ, t + t k / ɛ. π q As q s non-ncreasng, we have t + t k / ɛ, t + π q t j+ ɛ, t j+ t k / 3 By the defnton of π q n 9, the last nequalty can be rewrtten as. We use nducton to show that for =,...,n the latter would be the case f holds for every 0 <ɛ t /,.e., the condton stated n C. Fx =,...,n and suppose that for j, j <, holds for every ɛ such that, 0 <ɛ mn t +k /, t. We show that holds as well for every ɛ such that, j+ 0 <ɛ mn t +k /, t. In case j t +k / < t, observe that the latter would hold f for every 0 <ɛ mn { t + j /, t } we have, π q t + j, t + j + ɛ π q t + + t +k /, t + + π q t +k / + ɛ ɛ. 4 As q s non-ncreasng, we have t + + t +k /, t + + π q t + j+, t + j+ + ɛ t +k / + ɛ 3

8 JSched Thus, 4 would be the case f for every ɛ such that 0 <ɛ mn { t + j /, t } we have π q t + j, t + j + ɛ π q t + j+, t + j+ + ɛ ɛ. Condton C for the job + j ensures that the latter holds. Therefore, f for =,...,n, holds strctly for every 0 < ɛ t /, then holds strctly for every 0 <ɛ t. The condtons n Theorem are desgned to make the devatons from t = t,...,t n undesrable for all agents n the sense that the agent s utltes would be reduced f they choose any ntervals other than those defned by t. Clearly wth quaslnear utltes any agent can ncrease ts utlty ether by choosng an nterval that ether ncreases ts valuaton savng due to reducton n completon tme or reduces ts payment see Fg.. Wth a non-ncreasng prcng scheme attanng both of these at the same tme s mpossble. Therefore, a devaton whch ncreases decreases an agent s valuaton, smultaneously ncreases decreases ts payment. Only strongly coordnatng prcng schemes can guarantee effcent schedules n the decentralzed system. In order to ntroduce a famly of strongly coordnatng prcng schemes, we need the followng techncal lemma whch states that the effcent duraton of any job on subcontractor s machne s not longer than twce the effcent duraton of the job proceedng t. Lemma For =,...,n, t t +. Proof From 6 we get t t + = p + t k + = p p +. t k p + p The agents n N are sequence accordng to non-decreasng order of processng tmes, thus we have p p + whch obtans t t +. Note that n Lemma, the equalty holds for jobs wth equal processng tmes. More precsely, p = p + f and only f t = t +. We are now ready to gve a famly of non-ncreasng prcng schemes that are strongly coordnatng, that s, any devaton from effcent allocaton by any agent results n utlty loss for that agent. Theorem 3 Consder the coeffcent set κ ={κ,...,κ n } such that for =,...,n, 0 <κ <κ + / and 0 < t k κ n n δ/3 where 0 <δ 3/ n+. The followng prcng scheme s strongly coordnatng: δ [ ] q O t= +κ t t / t f t t < t + t, N n δ κ n f t t n + t n 5 Proof The assumptons on κ ensure that: a q O s nonncreasng, b q O 0 <, and c q O t 0fort 0. To verfy a, t suffces to consder the ponts of possble dscontnuty,.e., t = t + for =,...,n. There, t must hold that δ κ δ + κ + or κ + κ + δ. 6 By extendng the assumpton κ <κ + / we get κ <κ n / n 7 for =,...,n. Thus, κ + κ + < 3κ n / n. It follows that for 6 to hold, we need κ n n δ/3. 8 In order for b to hold we must have δ + κ < or κ <δ. Note that 7 and 8 yeld κ < δ/3 for =,...,n whch mples that κ <δ/3. Thus b holds. Gven that q O s non-ncreasng, to verfy c we must check the non-negatvty of q O t at t t n + t n. Note that from 7 we get n δ κ n n δ n δ/3. Hence, for c to hold we need δ 3/ n+. We now check the condtons n Theorem. Observe that q O s pecewse lnear and has a constant slope n between any consecutve ponts of possble dscontnuty. Wth regard to condton C, note that for 0 < ɛ t / both π q O t ɛ, t and π q O t + ɛ, t + can be obtaned by calculatng the areas of the correspondng trapezods. Let t = lm ɛ 0 +t ɛ. For any =,...,n 3

9 we have π q O t ɛ, t π q O t + ɛ, t + = ɛ [ q O t ɛ + q O t ] ɛ [ q O t + ɛ + q O t + ] = [ ɛ δ κ 4ɛ ] + t δ κ [ ɛ δ κ ɛ ] + t δ κ = ɛ [ κ ɛ + κ ɛ t ]. t From Lemma we know that t t and consequently ɛ/ t ɛ/ t. Byassumptonwehaveκ <κ whch leads to the observaton that for every 0 <ɛ< t /, π q O t ɛ, t π q O t + ɛ, t + >ɛ 9 wth regard to ɛ = t / we consder two cases. If t / = t then at ɛ = t /, 9 holds strctly as well whch mples that 9 holds strctly for every 0 <ɛ t.otherwse,f t / = t, then at ɛ = t /, 9 holds as equalty. However, ths requres that p = p whch corresponds to an alternatve effcent allocaton wheren job s allocated wth the nterval allocated to job. Hence, the choce of any earler ntervals for a gven agent whch s not part of an effcent allocaton results n loss of utlty for that agent. Also, for jobs wth equal processng tmes the choce of an earler alternatve effcent ntervals does not alter the utlty to the correspondng agents. Wth regard to condton C, note that for 0 <ɛ t / both π q O t, t + ɛ and π q O t +, t + + ɛ can also be obtaned by calculatng the areas of the correspondng trapezods. for any =,...,n wehave π q O t, t + ɛ π q O t +, t + + ɛ = ɛ [q O t + q O t + ɛ] ɛ [ q O t + + q O t + + ɛ ] = ɛ [ δ + κ + δ + κ 4ɛ t ] [ ɛ δ + κ + + δ + κ + ɛ ] t + = ɛ [ + κ ɛ κ + ɛ ]. t t + From Lemma we know that t t + and consequently ɛ/ t ɛ/ t +. Byassumptonwehaveκ <κ + /. Therefore, for 0 <ɛ< t / we would have π q O t, t + ɛ π q O t +, t + + ɛ <ɛ 0 If t / = t + then the latter also holds for ɛ = t /. In case of t / = t +, 0 would hold as equalty for ɛ = t /. However, ths requres p = p + whch mples the exstence of an alternatve effcent allocaton. Hence, the choce of any later ntervals for a gven agent whch s not part of an effcent allocaton results n the loss of utlty for that agent. Also, for jobs wth equal processng tmes the choce of a later alternatve effcent ntervals does not alter the utlty to the correspondng agents. Consderng the above, we conclude that q O s a coordnatng prcng scheme where for any T E and every T T\E, t holds for all N that π q T π q T <υ T υ T. To see how the prcng scheme ntroduced n Theorem 3 can coordnate the subcontractor schedulng problem frst note that for all agents wth unequal processng tmes, correspondng effcent allocatons are exclusvely the best choces of subcontractng ntervals. For agents wth equal processng tme jobs, however, the best choces are multple. Nevertheless, q O would result n the stuaton where the number of best choces for the jobs wth equal processng tmes are exactly the same as the number of those jobs. Therefore, a frst-come-frst-serve rule would result n strct coordnaton of ndvdual choces. 4. Subcontractor s revenue wth coordnatng prcng schemes Let Π q = N π q T be the total payment from agents to the subcontractor under the prcng schemes q. We call Π q the subcontractor s revenue under q. In ths secton we characterze the range of Π q wth q beng a coordnatng prcng scheme. The coordnatng prcng schemes ntroduced n Theorem 3 can extract almost all of the total savng υt from the agents by selectng δ small enough whch s shown n the followng theorem. Theorem 4 Π q O = υt δ N t. Proof For every N we have, π q O T = t δ + κ + δ + κ κ / = t δ. Therefore Π q O = N t δ N t. Hence, the total payments can be made arbtrary close to the maxmum possble whch s the total savng υt. We now provde a lower bound for the total subcontractor s revenue attanable under any coordnatng prcng 3

10 JSched scheme. We show n Corollary that ths bound s an upper bound on the subcontractor s revenue attanable by any coordnatng mechansm wth prvate nformaton. Theorem 5 Lower bound For any coordnatng prcng scheme q, we have Π q N t / n. Proof Let T E be an effcent allocaton. If q s coordnatng, then for all N there s no T whch results n hgher utlty than T. In partcular, for any agent =,...,n, the followng devaton from effcent allocaton should not be proftable: construct T by removng the last ɛ>0 nterval at the end of T and add nstead another nterval wth duraton ɛ startng at tme t ɛ to have feasble allocatons t must be that ɛ t /. Wth the new allocaton, fnshes ɛ unts of tme earler,.e., υ T υ T = ɛ.tomakeths devaton unproftable, the condton n 0 requres that π q t ɛ, t π q t + ɛ, t + ɛ. We construct a prcng scheme q L whch satsfes the above condton for every =,...,n and every 0 <ɛ t /, and has the lowest possble total revenue. Let q L t = 0fort t n. Observe that n the latter range q L has the lowest possble prces, so under q L the agent n pays 0 for ts effcent allocaton. The condton n for = n and any 0 <ɛ t n / requres that π q L t n ɛ, t n ɛ. To obtan the cheapest prcng scheme we requre q L to yeld π q L t n ɛ, t n = ɛ for every 0 <ɛ t n /. In partcular for ɛ = t n / we get π q L t n, t n = t n /. Ths means that under q L the agent = n pays t n /. Next, the condton for agent = n and every 0 <ɛ t n / requres that π q L t n ɛ, t n π q L t n ɛ, t n ɛ. 3 To obtan the cheapest prcng scheme we can consder q L to be such that 3 holds as equalty for every 0 <ɛ t n /. In ths case, for ɛ = t n / we get π q L t n, t n π q L t n t n /, t n = t n /. 4 By Lemma, we know that t n / t n so t n t n / t n. Hence we have π q L t n t n /, t n = t n /4 and eventually π q L t n, t n = 3 t n /4. 5 The latter mples that agent = n under q L pays t n 3/4 to the subcontractor. By nducton on t s easly verfable that under the prcng scheme that satsfes the condton and has the lowest possble prce, every agent N pays an amount equal to t / n to the subcontractor. Thus for every coordnatng prcng scheme q, t must be the case that Π q N t / n. We close wth the observaton that the lower bound n Theorem 5 s attanable by the followng prcng scheme { / n f t t < t +, N; q L t = 6 0 ft t n. Clearly q L s non-ncreasng, and coordnatng. To see the latter we check the condtons n Theorem : C holds snce we have / n + ɛ / n ɛ = ɛ. Also, we have / n ɛ / n ɛ = ɛ. Thus C holds as 0 ɛ 0 ɛ ɛ. Clearly, q L 0 <. Fnally we get Π q L T = N t / n. 5 Prvate processng tmes Secton 4 shows how to coordnate a set of self-nterested agents through a coordnatng prcng scheme under the assumpton that the vector of true processng tmes p s gven and known to the subcontractor. In ths secton, we relax ths assumpton by allowng true processng tmes to be prvate nformaton of agents. Thus reportng an untruthful processng tme by an agent can deceve the subcontractor who consequently moves the agent s job n the sequence of jobs n effcent allocatons whch can potentally ncrease the agent s tme savng or reduce hs payment. Ths secton presents a mechansm that guarantees that the agents are always better off by reportng ther true processng tmes, and that they are also better off by partcpatng n the subcontractng such mechansm s refereed to as the coordnatng mechansm for the subcontractor schedulng problem under prvate nformaton. Fnally, the secton calculates the amount the subcontractor forfets to the agents to extract the true processng tmes from them. We draw upon mechansm desgn theory Nsan 007; Jackson 000 and ntroduce a payment scheme that motvates agents to report true processng tmes knowng the subcontractor s ntenton to maxmze the total savngs. A prcng scheme n Sect. 4 determnes payments made by agents to the subcontractor for tme ntervals that they choose to utlze on the subcontractor s machne. In ths secton the payments are made for reportng partcular processng tmes by the agents to the subcontractor. Formally, each agent reports a processng tme r from ts ndvdual processng tme space P R +.Weusep to denote the true processng tme of agent.letp = P P n be the processng tme space of all agents. 3

11 Defnton A mechansm M s defned by a n + -tuple f M,π M,...,πM n consstng of an allocaton functon f M : P T, and a payment scheme π M : P R n. Gven a vector of reported processng tmes r and the sequence n whch the agents report them to break tes n the allocaton, the payment scheme of a mechansm determnes the monetary amount π M r that N pays n return for recevng the allocaton prescrbed for by f M. Wth quaslnear utltes, a mechansm M would result n the utlty u M f M r = υ f M r π M r 7 for N. The subcontractor s seekng to allocate the subcontractng ntervals effcently, thus we focus on mechansms whose allocaton functons yeld effcent allocatons,.e., f M T. We call such mechansms effcent. We concentrate on ncentve compatble mechansm snce they have the potental to motvate the agents to report ther true processng tmes. Defnton 3 A mechansm M sncentve compatble ffor every reported processng tme vector r = r,...,r n and for every agent N u M f M p, r u M f M r, 8 where r s the vector r wthout ts th coordnate. Although ncentve compatblty mples that agents are not better off by lyng about ther true processng tmes, t does not generally mply that the agents are better off by reportng ther true processng tmes there could generally exst untruthful agent reports that result n the same utlty as the truthful ones. Therefore, unfortunately, agents may generally choose to be untruthful after all even wth the ncentve compatble mechansms. Ths s generally the case for VCG mechansms, n partcular the pvotal mechansm, whch characterze the class of effcent and ncentve compatble mechansms for quaslnear utltes Green and Laffont 977. However, we show later n Theorem 6 that for the subcontractor schedulng problem studed n ths paper the pvotal mechansm makes the agents always better off by reportng ther true processng tmes. 5. Payment scheme We draw upon the VCG mechansms to obtan a desrable payment scheme for the subcontractor schedulng problem. The payment scheme for VCG mechansms s defned as follows π VCG r = h r j N\{} υ j T r, where h s a functon ndependent of agent s reported processng tme r. Among the class of VCG mechansms, the pvotal mechansm wth h r = j N\{} υ j T r 9 guarantees that all payments from the agents to the mechansm are non-negatve. Here T r s an effcent allocaton for the set of jobs excludng job and for processng tme vector r. Thus, the payment by pvotal mechansm for N equals π PM r = j N\{} [ t j r ] t j r, 30 where t j r s the duraton of subcontractng tme allocated to j by the effcent optmal allocaton for the reported r and t j r s the duraton of subcontractng tme allocated to j by the effcent allocaton for the reported r that excludes. By 30 the payment π PM r charged by the pvotal mechansm to agent s the total subcontractng tme that could have been allocated to all other agents had not asked for subcontractng and reported r. It follows from the characterzaton of effcent solutons that the excluson of an agent reportng r would only affect the agents that follow n T r. Letσ be the permutaton of jobs n T r and let [], for N, be the poston of n T r. Lemma 3 For all N, we have π PM r = t r / n []. 3 Proof We start by showng that for any, j N, wehave t j r t j r = 0f[ j] < [] and t j r t j r = t r/ [ j] [] f [ j] > []. By 6 r does not appear n t j r for [ j] < []. Therefore, t j r t j r = 0for[ j] < []. If [ j] > [], then the excluson of affects the duraton of allocated tme to jobs j wthout alterng the relatve poston of other jobs. By 4 and 5, we have t j r = r j t r + t r + and t j r = r j t r + []<[k]<[ j] []<[k]<[ j] t k t k r r 3

12 JSched Thus t j r t j r = t r []<[k]<[ j] [ t k r t k r] By nducton on [ j] [] we obtan t j r t j r = t r/ [ j] [] f [ j] > []. Based on the above observaton, the total payment n 30 becomes π PM r = t r/ [k] [] [k]>[] = t r / [k] [] = t r [k]>[] / n []. 5. Coordnatng meachansm Lemma 3 provdes a closed-form formula for the pvotal mechansm payments. Wth these payments, each agent pays an amount drectly proportonal to the duraton of ts allocaton on subcontractor s machne and nversely proportonal to ts order n the sequence of jobs on subcontractor s machne. Therefore f an agent announces a shorter than true processng tme, whch could possbly precptate ts poston n the sequence and thus result n the agent s hgher tme savng, then the agent pays more. We show that as a result of ths msrepresentaton, the agent s utlty decreases. At the same tme, we show that reportng longer than true processng tme reduces agent s utlty as well, unless the agent comes n the last poston n the sequence. Ths poston s specal snce t nether ncurs any payment by 3 nor t affects any other agent. Therefore, whether the agent n ths poston les or not s rrelevant. Alternatvely, one can assume that the longest job always belongs to the subcontractor ths dummy job could represent the contnuaton of subcontractor s operatons. Consequently, the mechansm enforces agents to report ther actual processng tme whch we now formally prove. Theorem 6 Let r = r,...,r n be the reported processng tmes and let p be true processng tme of agent. Then u PM T p, r > u PM T r, provded that p = r and s not n the last poston of T r. Proof Assume p = r P for some agent. Letp = p, r. Suppose job s n poston h and l n effcent solutons T p and T r, respectvely, and assume l = n. By 7, 3, 7, and 4, for p, the utlty of agent s u PM T p = p t p p t p = p t p n h. 3 n h There are two cases for r :ar > p, and b r < p. a Wth r > p we have { u PM p t r T r = max r t r }, 0 n l, 33 where n > l h and t r s the start tme of job n T r. Suppose l = h. Inthscasewehavet r = t p. Snce h < n, t drectly follows from 3 and 33 that u PM T r < u PM T p. Next, suppose l > h. Letσk be the job n poston k n T p. By4 and 5 and nducton on l h we have, t r = t p + l k=h+ r σk t p l+ k = r σl + + r σh+ l h + t p l h Usng the fact that r σk p for k h we get t r l k=h+ p l+ k + t p l h = p p l h + t p l h 34 For a contradcton assume u PM T r u PM T p,.e., by 3 and 33 assume that { } p t r max, 0 r t r p t p n h. n l If p t r 0, then the contradcton readly follows. Otherwse, suppose p t r > 0. Then, we have p t r r t r n l The last nequalty smplfes to p t p l h [r t r] [p t p] n h [r p ]. n h. Usng 34, note that the last nequalty remans vald f t r s replaced wth p p / l h + t p/ l h.the nequalty then smplfes to l h [r p ] n h [r p ]. 3

13 Ths s a contradcton snce we have l h < n h. Thus u PM T r < u PM T p for r > p. b Wth r < p we have u PM T r = r t r r t r = r t r n l, 35 n l where l h. Suppose h = l. Then we have t r = t p, and t follows from 3 and 35 that u PM T r < u PM T p. Next, suppose l < h.by4 and 5 and nducton on h l h t p = t r + = r σh k=l r σk t r h k + + r σl h l + t r h l. Usng the fact that r σk p for k h we get t p h p h k + t r h l = p p h l + t r h l 36 For a contradcton assume u PM T r u PM T p or, by 3 and 35, r t r n l p t p The last smplfes to r t r h l [p t p]. n h. Usng 36, the last nequalty remans vald f t p s replaced wth p p / h l + t r/ h l. The nequalty then smplfes to r p whch s a contradcton. Thus u PM T r < u PM T p for r < p. Fnally, to ensure agents partcpaton n subcontractng, the effcent mechansm M must satsfy the strct ndvdual ratonalty condton requrng that for all agents, truth-tellng results n postve utlty, rrespectve of the reported processng tmes of other agents. That s, for every r = r,...,r = p,...,r n and all N, u M T r > 0. The effcent mechansm T,π PM satsfes the strct ndvdual ratonalty condton snce, by 7, 3 and 7, we have u PM T r = t r/ n [] > 0, for r = p. Thus we showed the mechansm T,π PM guarantees, by Theorem 6, that the agents are always better off by reportng ther true processng tmes, and that, by strct ndvdual ratonalty, they are better off by partcpatng n the subcontractng. Therefore the mechansm coordnates the subcontractor schedulng problem under prvate nformaton. One possble dsadvantage of coordnaton based on mechansm desgn, the VCG mechansms n partcular, s that ths coordnaton s mplctly centralzed the effcent schedule s computed by the subcontractor and then reported to the agents. Although dstrbuted mplementatons of the pvotal mechansm where the schedule s determned by the self-nterested agents themselves are beyond the scope of ths paper, we refer the reader to Parkes and Shnedman 004 where gudelnes for the dstrbuted mplementatons of the VCG mechansms are proposed. 5.3 Subcontractor s revenue wth coordnatng mechansms The payment scheme π PM provdes the subcontractor wth a postve revenue for any gven r. Let us defne Π PM r = N π PM r as the total payment from agents to the subcontractor n pvotal mechansm. By Theorem 6 all agents except the last n T r report ther true processng tmes whle the last one pays nothng to the subcontractor regardless of hs report. Therefore, we focus on the subcontractor s revenue for the true processng tmes p. Although the subcontractor s servces result n the total savngs of υt p for all agents, t follows from Lemma 3 that the mechansm T,π PM can only acheve revenue Π PM p for the subcontractor whch s always less than vt p. In fact, nstances can be found for whch the rato of subcontractor s revenue to total savngs obtaned va subcontractng,.e., Π PM p/υt p, s arbtrarly close to zero. For example, consder a set of two jobs wth p = and p = 0. Despte havng a total savng of 5, we have π PM = 0.5 and π PM = 0 resultng n the rato υt p/π PM p <0.0. Nevertheless, Mouln 986 shows that f for all agents t holds that mn T υ T = 0 whch s the case n subcontractor s subcontractng problem wth dvsble jobs then among all effcent, ncentve compatble, and ndvdually ratonal mechansms, the pvotal mechansm generates the hghest total payment. Therefore, the subcontractor cannot gan a hgher revenue from any other effcent, ncentve compatble, and ndvdually ratonal mechansms than t does from T,π PM. Thus, by the juxtaposton of Lemma 3 and Theorem 5 we get the followng observaton. Corollary Subcontractor s revenue wth any effcent, ncentve compatble, and ndvdually ratonal mechansms under prvate nformaton never exceeds that wth a coordnatng prcng scheme under complete nformaton. However, the constructon of a coordnatng prcng scheme q requres the knowledge of true processng tmes p, thus the dfference Π q p Π PM p may be consdered 3

14 JSched as the amount the subcontractor forfet to the agents to extract the true processng tmes from them. 6 Asymmetrc valuatons We now relax the assumpton of symmetrc valuatons and assume that the value of a unt tme savng for agent s w. Let w = w N. The savng to agent resultng from the allocaton T thus equals υ w T = w υ T. 37 The total savng for all agents s defned as the sum of ndvdual savngs, that s υ w T = N υw T. The effcent allocaton s T = arg max T υ w T. Unlke the symmetrc case where each job n N s allocated a non-empty nterval on subcontractor s machne, n asymmetrc case effcent allocatons may exclude some jobs from the subcontractor s schedule. Let I ={ N t > 0} be the set of jobs that receve non-empty allocatons, and O ={ N t = 0} the set of jobs that receve no such allocatons. We denote the cardnalty of I wth l. In the remander of ths secton, we focus on coordnatng prcng schemes n stuatons wth asymmetrc valuatons. In these stuatons, any coordnatng prcng scheme q meets the followng condtons: for t t < t + t holds that qt <w, that s, t s requred that subcontractng prce across the allocaton of job do not exceed ts unt savng from subcontractng. We show by a counterexample that t s generally mpossble to construct coordnatng prcng schemes n the asymmetrc case. Wthout loss of generalty we assume I = {,...,l} and that the jobs are sequenced n effcent solutons by ther ndex order. Theorem 7 There exsts no coordnatng prcng scheme n the asymmetrc valuaton case. Proof To construct coordnatng prcng schemes, we must ensure that agents would not beneft by devatng from the effcent allocatons. In partcular, devatons of the smlar type as n Theorem 5 must be unproftable for any agent who receves non-zero allocatons on the subcontractor s machne. That s for every =,...,l and every 0 <ɛ<t t must hold that, q t ɛ, t qt + ɛ, t + w ɛ. 38 Consder the followng example. There are four jobs p = p = 6, p = 8, p 3 = 9, p 4 = 5 wth w = w = 9,w =,w 3 =,w 4 =. The unque effcent allocaton s I ={, 3, 4} wth T =[0, 3, T 3 =[3,, T 4 =[, 8, and O ={}. For any coordnatng prcng scheme, we must have qt >for0 t < 8 so that job would not be able to choose any nterval on subcontractor s machne. The condton n 38forjob = 4 and ɛ = 0.5 requres that, q 0.5, q7.75, Consderng that we need qt >for0 t < 8, t must be that q7.75, 8 >0.5 and consequently q 0.5, > 3.5. The condton n 38 forjob = 3 and ɛ = 0.5 requres that, q, 3 q0.5, We already know that q 0.5, > 3.5 thus t must be that q, 3 > 9.5. Ths mples that n the cheapest coordnatng prcng scheme, agent must pay 9.5 for acqurng the tme nterval [, 3 on subcontractor s machne. However, snce the valuaton of agent s only 9, ths means that he would be better off by not acceptng that nterval. Ths contradcts the fact that q s coordnatng. Therefore, n stuatons wth asymmetrc valuatons, coordnatng prcng schemes may be mpossble to fnd. 7 Fnal remarks and conclusons We studed the decentralzed subcontractor schedulng problem under complete and prvate nformaton. We desgned the coordnatng prcng schemes n case of complete nformaton and studed the range of subcontractor s revenue for these schemes. Our coordnatng prcng schemes are pecewse lnear and non-ncreasng. We also devsed a mechansm that s ncentve compatble, ndvdually ratonal, and effcent n case of prvate nformaton. The mechansm s based on the pvotal mechansm and t s moreover coordnatng. Another desrable property of the mechansm, that mmedately follows from Theorem 6, s Envy-Freeness Foley 967 whch s defned as the condton where no agent prefers another agent s allocaton to ts own, that s u T p u T j p for all j and all. The mechansm produces the hghest subcontractor revenue among effcent, ncentve compatble, and ndvdually ratonal mechansms. For the asymmetrc case, we showed that the coordnaton through prcng schemes s not always possble. An open problem s the complexty of fndng the effcent allocatons for the asymmetrc case. Several other extensons are possble for ths problem: job s release dates and due dates, setup tmes requred for subcontractor to prepare for processng dfferent jobs, and 3