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European Journal of Operational Research 188 (2008) 603–609

www.elsevier.com/locate/ejor

Short Communication

Multi-agent scheduling on a single machine withmax-form criteria

T.C.E. Cheng a,*, C.T. Ng a, J.J. Yuan b

a Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of Chinab Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People’s Republic of China

Received 21 May 2006; accepted 24 April 2007Available online 1 May 2007

Abstract

We consider multi-agent scheduling on a single machine, where the objective functions of the agents are of the max-form. For the feasibility model, we show that the problem can be solved in polynomial time even when the jobs are subjectto precedence restrictions. For the minimality model, we show that the problem is strongly NP-hard in general, but can besolved in pseudo-polynomial-time when the number of agents is a given constant. We then identify some special cases ofthe minimality model that can be solved in polynomial-time.� 2007 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Multi-agent; Fixed jobs

1. Introduction and problem formulation

The following single-machine multi-agent sched-uling problem was introduced by Agnetis et al.(2004) and Baker and Smith (2003). There are sev-eral agents, each with a set of jobs. The agents haveto schedule their jobs on a common processingresource, i.e., a single machine, and each agentwishes to minimize an objective function thatdepends on the completion times of his own set ofjobs. The problem is either to find a schedule thatminimizes a combination of the agents’ objectivefunctions or to find a schedule that satisfies eachagent’s requirement for his own objective function.

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved

doi:10.1016/j.ejor.2007.04.040

* Corresponding author. Tel.: +852 27665216; fax: +85223645245.

E-mail address: lgtcheng@polyu.edu.hk (T.C.E. Cheng).

Scheduling is in fact concerned with the allocationof limited resources over time. Scheduling problemsinvolving multiple customers (agents) competing fora common processing resource arise naturally inmany settings. For example, in industrial manage-ment, the multi-agent scheduling problem is formu-lated as a sequencing game, where the objective is todevise some mechanisms to encourage the agents tocooperate with a view to minimizing the overall cost(see, for example, Curiel et al. (1989), and Hamerset al. (1995)). In project scheduling, the problem isconcerned with negotiation to resolve conflictswhenever the agents find their own schedules unac-ceptable Kim et al. (1999). In telecommunicationservices, the problem is to do with satisfying the ser-vice requirements of individual agents, who competefor the use of a commercial satellite to transfer voice,image and text files for their clients Schultz et al.

.

604 T.C.E. Cheng et al. / European Journal of Operational Research 188 (2008) 603–609

(2002). Following Agnetis et al. (2004) and Bakerand Smith (2003), and intending to generalize theirresults on two-agent scheduling, we focus this studyon deriving feasible or optimal solutions for differentscenarios of competition among the agents, andexamining the computational complexity issues ofsome intractable cases of the single-machine multi-agent scheduling problem.

Now we define the multi-agent scheduling interms of common scheduling terminology. We havem families of jobs Jð1Þ;Jð2Þ; . . . ;JðmÞ, where, foreach i with 1 6 i 6 m, JðiÞ ¼ fJ ðiÞ1 ; J

ðiÞ2 ; . . . ; J ðiÞni

g.The jobs in JðiÞ are called the ith agent’s jobs. Eachjob J ðiÞj has a positive integral processing time pðiÞj , apositive integral due date dðiÞj , and a positive weightwðiÞj . All the jobs have zero release dates. The jobswill be processed on a single machine starting attime zero without overlapping and idle timebetween them. A schedule is a sequence of the jobsthat specifies the processing order of the jobs on themachine. Under a schedule r, the completion timeof job J ðiÞj is denoted by CðiÞj ðrÞ; job J ðiÞj is called

tardy if CðiÞj ðrÞ > dðiÞj ; U ðiÞj ðrÞ ¼ 1 if J ðiÞj is tardy

and zero otherwise; the lateness of J ðiÞj is definedas LðiÞj ðrÞ ¼ CðiÞj ðrÞ � dðiÞj ; the tardiness of J ðiÞj isdefined as T ðiÞj ðrÞ ¼ maxf0; LðiÞj ðrÞg; the maximumlateness of the ith agent is defined asLðiÞmaxðrÞ ¼ max16j6ni L

ðiÞj ðrÞ; and the maximum tardi-

ness of the ith agent is defined as T ðiÞmaxðrÞ ¼max16j6ni T

ðiÞj ðrÞ. For each job J ðiÞj , let f ðiÞj ð�Þ be a

nondecreasing function of the completion time ofjob J ðiÞj (such an objective function is called regularin the scheduling literature). We assume in thispaper that f ðiÞj ð�Þ can be evaluated in constant time.The ith agent’s objective function F ðiÞðrÞ takes eitherone of the following two forms:

max�form F ðiÞðrÞ ¼ max16j6ni

f ðiÞj ðCðiÞj ðrÞÞ;

sum� form F ðiÞðrÞ ¼X

16j6ni

f ðiÞj ðCðiÞj ðrÞÞ:

Throughout this paper, each agent’s objective func-tion is assumed to be of the max-form.

The scheduling problem studied in this paperincludes the following two models:

• Feasibility model: 1jprecjF ðiÞ 6 Qi; 1 6 i 6 m. Inthis model the goal is to find a feasible scheduler that satisfies F ðiÞðrÞ 6 Qi, 1 6 i 6 m.

• Minimality model: 1jprecjP

16i6mF ðiÞ. In thismodel the goal is to find a schedule r that mini-mizes

P16i6mF ðiÞðrÞ.

In the above scheduling models, prec denotesthat the jobs are subject to precedence restrictions.When the jobs are independent, i.e., there are noprecedence relations among the jobs, we leave thesecond field of the model description blank.

The feasibility model of multi-agent schedulingwas first studied by Agnetis et al. (2004), and theminimality model of multi-agent scheduling wasfirst studied by Baker and Smith (2003). This paperseeks to present general methods to treat the single-machine multi-agent scheduling problem. Theresults presented in this paper embrace some ofthe results on single-machine two-agent schedulingreported in the literature Agnetis et al. (2004), Bakerand Smith (2003), and Yuan et al. (2005).

The work of this paper is related to schedulingwith fixed jobs. This scheduling problem was firstintroduced by Scharbrodt et al. (1999). In thismodel we have two types of jobs: free jobs and fixedjobs. The fixed jobs are already fixed in the schedule.The intervals occupied by the fixed jobs can bereferred to as unavailability intervals of themachine. The remaining free jobs, which consist ofthe jobs of all the agents, are to be assigned to theremaining time slots of the machine in such a waythat they do not overlap with one another nor withthe fixed jobs. In our discussion we allow preemp-tion of the free jobs.

The adding of fixed jobs in the scheduling modelis motivated by the need to perform maintenance onthe machine in real-life settings. Due to the need formaintenance, the machine is unavailable during cer-tain periods of time over the scheduling horizon.Usually machine maintenance operations areplanned in advance and so the unavailable periodson the machine are pre-specified (i.e., fixed). Kov-alyov et al. (2007) provided a detailed survey onfixed interval scheduling. In deriving the results inthis paper, we use a technique to transform a freejob into a fixed job. Hence, we use the terminology‘‘fixed jobs’’ as in Scharbrodt et al. (1999).

The feasibility model of single-machine multi-agent preemptive scheduling with fixed jobs isdenoted by 1jprec; fix; pmtnjF ðiÞ 6 Qi; 1 6 i 6 m,and the minimality version is denoted by 1jprec;fix; pmtn j

P16i6mF ðiÞ. Here fix denotes the constraint

due to the presence of the fixed jobs, and pmtn meansthat the free jobs are preemptively scheduled.Throughout this paper, n* and n are used to denotethe number of fixed jobs and the number of freejobs, respectively. Then we have n ¼ n1 þ n2 þ � � � þnm.

T.C.E. Cheng et al. / European Journal of Operational Research 188 (2008) 603–609 605

Since the objective functions are assumed to beregular, we need only to consider regular schedules,where a schedule is called regular if under the sche-dule there is no idle time on the machine before thecompletion time of the last free job.

The idea in this paper comes partially fromAgnetis et al. (2004), Baker and Smith (2003), andYuan et al. (2005). This paper is organized as fol-lows. In Section 2 we show that the feasibility modelcan be solved in polynomial-time. In Section 3 weshow that the minimality model is strongly NP-hard. In Section 4 we show that, when the numberof agents is a constant, the minimality model canbe solved in pseudo-polynomial-time. In Section 5we give some polynomial-time solvable sub-casesof the minimality model.

2. Polynomial-time algorithm for the feasibility model

Consider the feasibility problem 1jprec; fix; pmtnjF ðiÞ 6 Qi; 1 6 i 6 m, where each F ðiÞ is of the max-form F ðiÞðpÞ ¼ max16j6ni f

ðiÞj ðpÞ. We present an

Oðn� þ n2Þ time algorithm. When the free jobs areindependent, the running time of the algorithm isOðn� þ n log nÞ. The idea of our discussion comesfrom Lawler’s (1973) algorithm for the singlemachine scheduling 1jprecjfmax. The same procedurewas also used in Agnetis et al. (2004).

The first technical issue to deal with is that, if ablock of free jobs of total length l is to be processedpreemptively under a regular schedule on themachine with fixed jobs, how do we determine thecompletion time DðlÞ of the last free job in the block?

Let the n* intervals occupied by the fixed jobs be½s1; t1Þ; ½s2; t2Þ; . . . ; ½sn� ; tn� Þ, where 0 6 s1 < t1 6 s2 <t2 6 � � � 6 sn� < tn� . Note that the sorting of theseintervals needs Oðn� log n�Þ time. We further definet0 ¼ 0 for convenience. We first calculate allFBj ¼

P16i6jðti � siÞ in Oðn�Þ time, where FBj is

the total length of the first j intervals occupied bythe fixed jobs. Clearly, DðlÞ > tj if and only iflþ FBj > tj. Hence, we have

Lemma 2.1. The value DðlÞ can be calculated in

Oðn�Þ time by the formula DðlÞ ¼ lþ FBj� , where j*

is the maximum index j such that lþ FBj > tj.

For each job J ðiÞj , we define its deadline as

DðiÞj ¼ maxfC : C is a positive integerand f ðiÞj ðCÞ 6 Qig:

In the case that DðiÞj > DðP Þ, we re-define DðiÞj ¼ P ,where P is the sum of the length of all the free jobs.

As stated in Agnetis et al. (2004), if the inverse func-tion of f ðiÞj is available, the deadline DðiÞj can be com-puted in constant time; otherwise this requiresOðlog P ðiÞj Þ time. We assume the former case forevery job.

Now we denote the set of all the free jobs byJ ¼ fJ 1; J 2; . . . ; J ng. That is, J is the union of thefree jobs from each agent. The processing timeand deadline of Jj are denoted by pj and Dj, respec-tively. We write J i � J j if job Ji must be processedbefore job Jj.

If Ji and Jj are two jobs such that J i � J j, thenthe completion time Ci of Ji must be less than orequal to Cj � pj under any feasible schedule. Modi-fying the deadline of job Ji by setting Di :¼minfDi;Dj � pjg, we lose nothing, but we obtain abeneficial relation Di < Dj. Hence, we can recur-sively modify the deadlines of the jobs such that,for each pair of jobs Ji and Jj, if J i � J j, thenDi < Dj. This procedure can be performed inOðn2Þ time using the standard ‘‘Algorithm Modifydj’’ given by Brucker (2001).

Clearly, the feasibility problem has a solution ifand only if the regular schedule obtained by theEDD rule on the modified deadline meets everyjob’s deadline. Suppose that the deadlines have beenmodified. Then we re-label the free jobs in the EDDorder, i.e., D1 6 D2 6 � � � 6 Dn, which will takeOðn log nÞ time. Let p be the regular preemptiveschedule obtained by the free job sequenceðJ 1; J 2; . . . ; J nÞ. Then the feasibility problem has asolution if and only if Dðp1 þ p2 þ � � � þ pjÞ 6 Dj

for 1 6 j 6 n.Note that all the values P ðjÞ ¼ p1 þ p2 þ � � � þ pj,

1 6 j 6 n, can be calculated in OðnÞ time recur-sively, and all the values DðP ðjÞÞ, 1 6 j 6 n, canbe calculated in Oðn� þ nÞ time recursively by thefollowing dynamic programming recursion: Letk0 ¼ 0 and, for j from +1 to n, let kj P kj�1 be themaximum value such that PðjÞ þ FBkj > tkj , andthen set DðP ðjÞÞ ¼ P ðjÞ þ FBkj . From this, we con-clude the following result.

Theorem 2.2. The feasibility problem 1jprec; fix;pmtnjF ðiÞ 6 Qi; 1 6 i 6 m can be solved in Oðn�log n� þ n2Þ time.

When the set of fixed jobs is empty, the problemunder study is equivalent to 1j � jF ðiÞ 6 Qi; 1 6 i 6m. Hence, we have

Corollary 2.3. 1jprecjF ðiÞ 6 Qi; 1 6 i 6 m can be

solved in Oðn2Þ time.

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When the free jobs are independent, the proce-dure for modifying the deadlines can be omitted.Hence, we have

Theorem 2.4. 1jfix; pmtnjF ðiÞ 6 Qi; 1 6 i 6 m can be

solved in Oðn� log n� þ n log nÞ time.

Corollary 2.5. 1kF ðiÞ 6 Qi; 1 6 i 6 m can be solved

in Oðn log nÞ time.

3. NP-hardness for the minimality model

P P

Theorem 3.1. Both 1k 16i6mLðiÞmax and 1k 16i6mT ðiÞmax are binary NP-hard.

Proof. Recall that, by Du and Leung (1990), thefeasibility scheduling problem 1k

PT j 6 Y is binary

NP-complete.Let us be given an instance I of 1k

PT j 6 Y :

ðp1; p2; . . . ; pn; d1; d2; . . . ; dn; Y Þ, where pi and di arethe processing time and due date of the ith job Ji,1 6 i 6 n, and Y P 0 is the threshold value of thetotal tardiness. The decision asks whether there is aschedule r for the n jobs such that

P16i6nT iðrÞ 6 Y .

Write P ¼P

16i6npi. By the proof of the NP-completeness of 1k

PT j 6 Y in Du and Leung

(1990), we can further assume that 0 6 di 6 P foreach i, and Y 6 nP . We construct an instance I* ofthe feasibility scheduling problem 1k

P16i6mLðiÞmax 6

Q as follows:

• n agents with each agent having exactly two jobs,i.e., JðiÞ ¼ fJ ðiÞ1 ; J ðiÞ2 g; 1 6 i 6 n.

• Processing times of the jobs are defined bypðiÞ1 ¼ pi and pðiÞ2 ¼ X , 1 6 i 6 n, where X ¼ nPþY þ 1. The jobs J ðiÞ2 ; 1 6 i 6 n, are called large

jobs, and the jobs J ðiÞ1 ; 1 6 i 6 n, are called smalljobs.

• Due dates of the jobs are defined by dðiÞ1 ¼di and dðiÞ2 ¼ P þ iX ; 1 6 i 6 n.

• Threshold value is defined by Q = Y, and thedecision asks whether there is a schedule p suchthat

P16i6mLðiÞmaxðpÞ 6 Q.

The above construction can be done in polyno-mial-time. We show in the following that I isfeasible if and only if I* is feasible.

If there is a feasible schedule p for I*, we claimthat: CðiÞ1 ðpÞ 6 P ; and CðiÞ2 ðpÞ ¼ dðiÞ2 ¼ P þ iX ; 1 6i 6 n.

We notice that LðiÞmaxðpÞP �P for each i, sincedi 6 P . If CðxÞ1 ðpÞ > P for a certain x, then there

must be at least one large job completed before J ðxÞ1 ,and thus CðxÞ1 ðpÞP X . This means that LðxÞmaxðpÞPX � P . Consequently,

P16i6mLðiÞmaxðpÞP X � nP ¼

Y þ 1 > Q, contradicting the assumption that p isfeasible. So, under p, any small job is processedbefore any large job. This also means that thecompletion times of the large jobs are P þ iX ,1 6 i 6 n. If possible, let y, 1 6 y 6 n, be theminimum such that CðyÞ2 ðpÞ > dðyÞ2 ¼ P þ yX . Then,LðyÞmaxðpÞP X . Again, this implies

P16i6mLðiÞmaxðpÞ >

Q, contradicting the assumption that p is feasible.We conclude that the claim holds.

The above claim means that under the feasibleschedule p each large job completes at its due dates,and LðiÞmaxðpÞ ¼ T ðiÞmaxðpÞ ¼ T ðiÞ1 ; 1 6 i 6 n: Since thesmall jobs in I* correspond one-to-one to the jobs inI with the same processing times and due dates, wededuce that I is feasible if and only if I* is feasible.We conclude that 1k

P16i6mLðiÞmax 6 Q is NP-

complete.The above proof is also valid for 1k

P16i6m

T ðiÞmax 6 Q. Hence, the result follows. h

The classical scheduling problem 1kP

wjT j isstrongly NP-hard. One of the well-known proofsfor the strongly NP-hardness of this problem wasgiven by Lawler (1977) by using a reduction fromthe strongly NP-complete 3-Partition problem(Garey and Johnson, 1979) to the decision versionof the scheduling problem. In the constructedinstance of 1k

PwjT j in Lawler (1977), the jobs

are of two families Jð1Þ and Jð2Þ: each jobJ j 2 Jð1Þ has a due date dj ¼ 0 and so T j ¼ Cj inany schedule; and each job J j 2 Jð2Þ has a positivedue date dj and a sufficiently large weight wj sothat T j ¼ 0 in any feasible schedule. This just

implies that the feasibility problem 1kP

wð1Þj Cð1Þj 6

Y : Lð2Þmax 6 0 is strongly NP-complete. Note thatAgnetis et al. (2004) have proved the ordinary NP-completeness of the same problem.

Theorem 3.2. 1kP

16i6mmax16j6ni wðiÞj CðiÞj is strongly

NP-hard.

Proof. Let us be given an instance I of 1kPwð1Þj Cð1Þj 6 Y : Lð2Þmax 6 0:

ðp1; . . . ; pn; w1; . . . ;wk; dkþ1; . . . ; dn; Y Þ;

where J 1; . . . ; J k are the first agent’s jobs andJ kþ1; . . . ; J n are the second agent’s jobs; pj P 1 isthe integral processing time of job Jj, 1 6 j 6 n;wj P 1 is the weight of job Jj, 1 6 j 6 k; dj P 1 is

T.C.E. Cheng et al. / European Journal of Operational Research 188 (2008) 603–609 607

the integral due date of job Jj, k þ 1 6 j 6 n; andY > 0 is the threshold value of the total weightedcompletion time of the first agent’s jobs. Thedecision asks whether there is a schedule r on themachine such that Lð2ÞmaxðrÞ 6 0 and

P16i6kwi

CiðrÞ 6 Y .Since the completion time of each job cannot

exceed P ¼P

16j6npj, we can assume that Y 6 WP ,where W ¼

P16j6kwj. It is also reasonable to

suppose that dj 6 P for every Jj with j > k. Weconstruct an instance I* of the feasibility schedul-ing problem 1k

P16i6mmax16j6ni w

ðiÞj CðiÞj 6 Q as

follows:

• k + 1 agents and n + 1 jobs J 01; J02; . . . ; J 0nþ1 with

J 0j, 1 6 j 6 n, corresponding to job Jj in I. For1 6 i 6 k, the ith agent has exactly one job J 0i.The job set of the ðk þ 1Þth agent is defined byfJ 0kþ1; J

0kþ2; . . . ; J 0nþ1g.

• Processing times of the jobs are defined by p0j ¼ pj

for 1 6 j 6 n and p0nþ1 ¼ M � P , where M ¼dmaxðY þ W Þ þ Y þ P with dmax ¼ maxkþ16j6ndj.

• Weights of the jobs are defined by w0i ¼ 2wi for1 6 i 6 k, w0j ¼ M=dj for k þ 1 6 j 6 n, andw0nþ1 ¼ 1.

• Threshold value is defined by Q ¼ 2Y þM , andthe decision asks whether there is a schedule psuch that

P16i6mmax16j6ni w

ðiÞj CðiÞj 6 Q.

The above construction can be done in polyno-mial-time. We show in the following that I isfeasible if and only if I* is feasible.

Suppose I has a feasible schedule p. Then weobtain a corresponding sequence p* of the jobs in I

apart from J 0nþ1. We insert job J 0nþ1 in thelast position of p* to obtain a schedule r. It canbe seen that C0jðrÞ 6 CjðpÞ for 1 6 j 6 n, andmaxkþ16j6nþ1w0jC

0jðrÞ ¼ M . It follows that the objec-

tive value of the instance I* is no more than2Y þM ¼ Q.

Conversely, suppose that p is a feasible scheduleof I*. From the construction of I*, we see thatP

16j6nþ1p0j ¼ P þM � P ¼ M , 2M > Q andw0jðdj þ 1Þ > 2Y þM ¼ Q for k þ 1 6 j 6 n. If, forsome j with 1 6 j 6 k, J 0j is the last job in p, thenQ P w0jC

0jðpÞP 2M > Q; a contradiction. If, for

some j with k þ 1 6 j 6 n, C0jðpÞP dj þ 1, thenQ P w0jC

0jðpÞP w0jðdj þ 1Þ > Q; a contradiction

again. It follows that the last job in p must beJ 0nþ1, and C0jðpÞ 6 dj for each j with k þ 1 6 j 6 n.Hence, we must have maxfw0jC0jðpÞ : 1 6 j 6nþ 1g ¼ M . Since the cost of p is no more

than Q ¼ 2Y þM , we deduce thatP

16j6kw0jC0jðpÞ 6 2Y .

Since Jj and J 0j have the same processing time for1 6 j 6 n, p, restricted on the jobs J 0j, 1 6 j 6 n, canalso be seen as a schedule for the instance I. Now,w0j ¼ 2wj for each j and

P16j6kw0jC

0jðpÞ 6 2Y

implies thatP

16j6kwjCjðpÞ 6 Y . Consequently, pis a feasible schedule for I. The result follows. h

It is still open whether the problem 1kP

16i6mLðiÞmax

is strongly NP-hard.

4. Pseudo-polynomial-time algorithms for given m

Suppose that F(i) is an integral regular functionfor each agent i. When m, the number of agents, isa given constant, even the general problem 1jprec;fix; pmtnj

P16i6mF ðiÞ can be theoretically solved in

pseudo-polynomial-time provided that, for eachagent i, the upper bound UBðiÞ and the lower boundLBðiÞ of F ðiÞ, with LBðiÞ 6 F ðiÞðpÞ 6 UBðiÞ for anyregular schedule p, can be determined in pseudo-polynomial-time, say PðiÞ.

By the discussion of Section 2, for every setfQ1;Q2; . . . ;Qmg with LBðiÞ 6 Qi 6 UBðiÞ, 1 6i 6 m, the feasibility problem 1jprec; fix; pmtnjF ðiÞ 6 Qi; 1 6 i 6 m can be solved in Oðn� log n�þn2Þ time. By enumerating all the possibilities ofQ1;Q2; . . . ;Qm and choosing the best one, we even-tually solve the problem. Let BðiÞ ¼ UBðiÞ�LBðiÞ þ 1, 1 6 i 6 m. Then each Qi has at mostB(i) choices. Hence, we have

Theorem 4.1. 1jprec; fix; pmtnjP

16i6mF ðiÞ can be

solved in OðP

16i6mPðiÞ þ ðn� log n� þ n2Þ P16i6m

BðiÞÞ time.

Similarly, we have

Theorem 4.2. 1jprecjP

16i6mF ðiÞ can be solved in

OðP

16i6mPðiÞ þ n2P16i6mBðiÞÞ time.

Theorem 4.3. 1jfix; pmtnjP

16i6mF ðiÞ can be solved in

OðP

16i6mPðiÞ þ ðn� log n� þ n log nÞP16i6mBðiÞÞtime.

Theorem 4.4. 1kP

16i6mF ðiÞ can be solved inOðP

16i6mPðiÞ þ ðn log nÞP16i6mBðiÞÞ time.

When each F(i) is in a form familiar to us, thecomplexity can be reduced to a normal size. Forexample, if F ðiÞ ¼ maxjw

ðiÞj T ðiÞj , then we can choose

UBðiÞ ¼ max16j6ni wðiÞj � P and LBðiÞ ¼ 0; in this

case, PðiÞ can be omitted from the complexity.

608 T.C.E. Cheng et al. / European Journal of Operational Research 188 (2008) 603–609

5. Polynomial-time algorithms for special models

Let K be a subset of the free jobs, and let p be aschedule of 1jfix; pmtnj

P16i6mF ðiÞ. We use SK and

CK to denote the minimum starting time, and themaximum completion time of the jobs in K underp, respectively. We say the jobs in K are consecu-tively processed in p if, under the schedule p, themachine has no idle time in ½SK ;CKÞ, and no freejobs other than that in K are processed in ½SK ;CKÞ.

The following observation, which holds for two-agent scheduling (see Baker and Smith (2003), andYuan et al. (2005)), still holds for multi-agentscheduling.

Observation 5.1. For the problem 1jfix; pmtnjP16i6mF ðiÞ, there is an optimal schedule r such that,

(1) if there is a certain i such that F ðiÞ ¼max16j6ni w

ðiÞCðiÞj , then the jobs in JðiÞ are con-secutively processed;

(2) if there is a certain i such that F ðiÞ ¼max16j6ni w

ðiÞj CðiÞj , then the jobs in JðiÞ are pro-

cessed in the maximum weight (MAX-W)order; and furthermore, the jobs with the sameweights are consecutively processed; and

(3) if there is a certain i such that F ðiÞ ¼ wðiÞLðiÞmax,then the jobs in JðiÞ are processed in the EDDorder; and furthermore, the jobs with thesame due dates are consecutively processed.

Proof. By the job-shifting argument. h

Theorem 5.2. 1kP

16i6mwðiÞCðiÞmax can be solved in

Oðnþ m log mÞ time.

Proof. By Observation 5.1(1), we first combine thejobs in each JðiÞ into a large job Ji with processingtime pi ¼

P16j6ni

pðiÞj and weight w(i). As a result, theproblem is transformed into the standard single-machine scheduling problem 1k

PwðiÞCi, which

can be solved by the SWPT rule in Oðm log mÞ time.The result follows. h

In the following, we consider that, for each agenti, either F ðiÞ ¼ max16j6ni w

ðiÞj CðiÞj or F ðiÞ ¼ wðiÞLðiÞmax. By

Observation 5.1, the ith agent’s jobs with the sameweights (F ðiÞ ¼ max16j6ni w

ðiÞj CðiÞj ) or the same due

dates (F ðiÞ ¼ wðiÞLðiÞmax) can be combined into a largejob. Hence, in the following, we suppose that theith agent’s jobs have distinct weights or due dates,according to F ðiÞ ¼ max16j6ni w

ðiÞj CðiÞj or F ðiÞ ¼

wðiÞLðiÞmax.

Furthermore, we suppose that, when F ðiÞ ¼max16j6ni w

ðiÞj CðiÞj , the jobs in JðiÞ ¼ fJ 1ðiÞ; J 2ðiÞ;

. . . ; J niðiÞg are indexed in the MAX-W order

wðiÞ1 < wðiÞ2 < � � � < wðiÞni, and when F ðiÞ ¼ wðiÞLðiÞmax,

the jobs in JðiÞ ¼ fJ ðiÞ1 ; JðiÞ2 ; . . . ; J ðiÞni

g are indexed in

the EDD order dðiÞ1 < dðiÞ2 < � � � < dðiÞni. By Observa-

tion 5.1 again, there is an optimal schedule suchthat, for each agent i, its jobs are processed in theorder ðJ ðiÞ1 ; J

ðiÞ2 ; . . . ; J ðiÞni

Þ. It follows that in an opti-mal schedule, the completion time of any job is ofthe form DðP ðx1; x2; . . . ; xmÞÞ for some x1; x2; . . . ; xm

with 0 6 xi 6 ni, 1 6 i 6 m, where Pðx1; x2; . . . ;xmÞ ¼

P16i6m

P16j6xi

pðiÞj and Dð�Þ is the same as thatin Section 2. Consequently, in an optimal schedule,the possible value of F ðiÞ belongs to the setRðiÞ ¼ ff ðiÞj ðDðP ðx1; x2; . . . ; xmÞÞ : 0 6 xi 6 nig. Notethat RðiÞ ¼ Oðn1n2 � � � nmÞ for each i.

By the discussion of Section 2, for every sequenceQ1;Q2; . . . ;Qm with Qi 2 RðiÞ, 1 6 i 6 m, the feasi-bility problem 1jFB; pmtnjF ðiÞ 6 Qi; 1 6 i 6 m canbe solved in Oðn� log n� þ n log nÞ time. By enumer-ating all possibilities of Q1;Q2; . . . ;Qm and choosingthe best one, we eventually solve the problem. Sincethe sequence Q1;Q2; . . . ;Qm has at most Oððn1n2

� � � nmÞmÞ choices and all the values DðPðx1; x2; . . . ;xmÞÞ can be calculated in Oðn�n1n2 � � � nmÞ time, thetotal complexity is Oððn� log n� þ n log nÞðn1n2

� � � nmÞmÞ, which is polynomial when m is fixed.

Theorem 5.3. The problem 1jfix; pmtnjP

16i6mF ðiÞ

can be solved in Oððn� log n� þ n log nÞ ðn1n2 � � �nmÞmÞ time.

Similarly, we have

Theorem 5.4. 1kP

16i6mF ðiÞ can be solved inOððlog nÞðn1n2 � � � nmÞmÞ time.

Acknowledgements

We are grateful for the helpful comments of twoanonymous referees on an earlier version of this pa-per. This research was supported in part by TheHong Kong Polytechnic University under grantnumber S818 and The National Natural ScienceFoundation of China under grant number10671183.

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