Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 124083, 5 pages http://dx.doi.org/10.1155/2013/124083

Research Article A Competitive Two-Agent Scheduling Problem on Parallel Machines with Release Dates and Preemption Yawei Qi1 and Long Wan2 1

School of Information Technology, Jiangxi Key Laboratory of Data and Knowledge Engineering, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China 2 School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China Correspondence should be addressed to Yawei Qi; [email protected] Received 12 August 2013; Accepted 6 October 2013 Academic Editor: Yunqiang Yin Copyright Β© 2013 Y. Qi and L. Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a competitive two-agent scheduling problem on multiple identical machines with release dates and preemption. In the scheduling model, there are two agents π and π each having their own job sets Jπ = {π½1π , . . . , π½πππ } and Jπ = {π½1π , . . . , π½πππ }, respectively. Each job π½π β Jπ βͺ Jπ has a release date ππ and the π = ππ + ππ jobs need to be preemptively scheduled on π identical machines. For π = 2, we show that the trade-off curve of all the Pareto optimal points can be characterized in polynomial time. When π is input, we show that π|ππ , pmtn|πΏπmax : πΏπmax β€ π can be solved in strongly polynomial time.

1. Introduction and Problem Formulation In recent years, multiagent scheduling problems are under extensive research. A multiagent scheduling problem means that there are multiple agents which must compete to perform their own tasks on the common processing resource. Each agent wants to optimize his/her own objective function. The objective function considered in this paper is to minimize the maximum lateness of the jobs. First, let us briefly recall the history of the classic scheduling problems of minimizing the maximum lateness of the jobs that is, there is only one agent in such a problem. Horn [1] considered the problem of scheduling the jobs preemptively on identical machines with release dates and deadlines and showed that it can be determined in strongly polynomial time if the problem has a feasible schedule by reducing it to a network flow problem which is well known to be solved in strongly polynomial time. Sahni [2] presented a faster algorithm to determine if the problem with identical release dates has a feasible schedule. Furthermore, Sahni and Cho [3] showed that it also can be determined in strongly polynomial time if the problem on related machines has a feasible schedule. Lawler and Labetoulle [4] proved that the feasibility problem on unrelated machines can be settled in weakly polynomial time

by means of linear programming. Labetoulle et al. [5] studied the problems of scheduling the jobs on the parallel machines preemptively to minimize the maximum lateness of the jobs with release dates. They gave a strongly polynomial-time algorithm to solve the problem on identical machines which is based on the same network flow structure introduced by Horn [1] and a weakly polynomial-time algorithm to solve the problem on uniform machines in terms of the polymatroidal network flow model introduced by Lawler and Martel [6]. For the latter, a similar result can be found in Martel [7]. The multiagent scheduling models were initially introduced by Baker and Smith [8] and Agnetis et al. [9]. Their research focused on the problems of nonpreemptively scheduling the jobs which belong to two agents on a single machine. Agnetis et al. [10] investigated the multiagent single machine problem of finding a nonpreemptive schedule in which the cost of each agent does not exceed a given threshold value which is also studied in Cheng et al. [11]. Cheng et al. [12] considered the feasibility model of multiagent scheduling on a single machine for which each agent competes to minimize the total weighted number of his/her own tardy jobs and showed that the problem is strongly NP-hard. For more papers about the multiagent problems of scheduling the jobs nonpreemptively on a single machine, the readers are referred to Yuan et al.

2

Mathematical Problems in Engineering

[13], Ng et al. [14], and Mor and Mosheiov [15]. Leung et al. [16] which initiated the preemptively multiagent scheduling problems investigated the two-agent scheduling problems of scheduling the jobs preemptively on a single machine or identical machines. Yuan et al. [17] studied a competitive two-agent scheduling problem on a single machine with release dates and preemption for which the objective of each agent is to minimize the maximum lateness and showed that all Pareto optimal points can be found in strongly polynomial time. Wan et al. [18] investigated the same twoagent scheduling model for which one agentβs objective is to minimize the maximum lateness and the other agentβs objective is to minimize the total completion time of his/her jobs. They proved that the problem is NP-hard in the ordinary sense by means of reduction from even-odd partition which is well known to be ordinarily NP-hard [19]. The problems in the paper are stated as follows. There are two agents π and π each having their own job sets Jπ = {π½1π , . . . , π½πππ } and Jπ = {π½1π , . . . , π½πππ }, respectively. We make the assumption that Jπ β©Jπ = 0. The jobs in Jπ are called π-jobs and the jobs in Jπ are called π-jobs. Each job π½π β Jπ βͺ Jπ has a release date ππ , a due date ππ , and a processing time ππ . And the π = ππ + ππ jobs need to be preemptively scheduled on π identical machines. Two agents have the same objective of minimizing the maximum lateness. We use the following notation throughout this paper.

The rest of the paper is organized as follows. In Section 2, we show that the tradeoff curve of the Pareto optimal points can be characterized in strongly polynomial time for π2|pmtn|(πΏπmax : πΏπmax ). Section 3 gives a polynomial time algorithm to solve Problem π|ππ , pmtn|πΏπmax : πΏπmax β€ π. We draw some conclusions and present some further research.

2. π2|pmtn|(πΏπmax : πΏπmax )

(i) πππ₯ is the processing time of job π½ππ₯ , π₯ β {π, π}.

First, let us state a feasibility problem of scheduling π jobs preemptively on two identical machines with the deadlines and give a characterization of feasibility. Let J = {π½1 , π½2 , . . . , π½π } be the set of the jobs and ππ and ππ are the deadline and processing time of job π½π , respectively; π = 1, 2, . . . , π. The problem is denoted by π2|pmtn; ππ |β in terms of the threefield notation. Without loss of generality, we assume that the jobs are ordered by π1 β€ π2 β€ β β β β€ ππ . Assume that there are total π different deadlines and π(1) , π(2) , . . . , π(π) with π(1) < π(2) < β β β < π(π) are the π different deadlines. Denote by J(π) the set of the jobs of deadline π(π) , π = 1, 2, . . . , π. Let π0 = 0, π1 , . . . , ππ = π be the π + 1 numbers such that J(π) = {π½ππβ1 +1 , π½ππβ1 +2 , . . . , π½ππ }, π = 1, 2, . . . , π. We assume that π0 = 0 and π½ππβ1 +1 is the largest job of J(π) for convenience, π = 1, 2, . . . , π. Before we present the characterization of feasibility of π2|pmtn|(πΏπmax : πΏπmax ), we first state a well-known result from [2].

(ii) πππ₯ is the release date of job π½ππ₯ , π₯ β {π, π}.

Algorithm Sahni. Consider the following steps.

(iii)

πππ₯

π½ππ₯ ,

is the due date of job

(iv)

πΆππ₯

is the completion time of job π½ππ₯ , π₯ β {π, π}.

π₯ β {π, π}.

(v) πΏπ₯π = πΆππ₯ β πππ₯ is the lateness of job π½ππ₯ , π₯ β {π, π}. (vi) πΏπ₯max = max{πΏπ₯π : 1 β€ π β€ ππ₯ } is the maximum lateness of (the jobs of) agent π₯, π₯ β {π, π}. A schedule π is called Pareto optimal if there is no schedule π such that πΏπmax (π) β€ πΏπmax (π), πΏπmax (π) β€ πΏπmax (π), and at least one inequality strictly holds, that is, a schedule is Pareto optimal for any schedule; if it is better for one agent, then it must be worse for the other agent. We say (πΏπmax (π), πΏπmax (π)) is a Pareto optimal point if schedule π is a Pareto optimal schedule. The first problem we consider is to find all the Pareto optimal points and a corresponding schedule for each Pareto optimal point when π = 2 and all the jobs are released at 0. The second problem we consider is to minimize the maximum lateness of agent π΄ with the maximum lateness of agent π΅ bounded by a given threshold π when π is input. By use of the well-known three-field notation [20], the first and second problems can be formulated as follows. π2|pmtn|(πΏπmax

πΏπmax ),

: which is (i) The first problem: a Pareto optimization problem seeking to minimize πΏπmax and πΏπmax simultaneously. (ii) The second problem: π|ππ , pmtn|πΏπmax : πΏπmax β€ π.

Step 1. π = 1, πΏ 1 = πΏ 2 = 0. Step 2. Find a unscheduled job π½π of J(π) . If ππ β₯ π(π) βπΏ 1 , then return infeasibility. Otherwise, we schedule job π½π . If ππ β€ π(π) β πΏ 2 , then we assign time interval [πΏ 2 , πΏ 2 + ππ ] to π½π on machine 2 and reset πΏ 2 := πΏ 2 +ππ ; else, we assign time interval [πΏ 2 , π(π) ] on machine 2 and time interval [πΏ 1 , πΏ 1 +πΏ 2 +ππ βπ(π) ] on machine 1 to π½π and reset πΏ 2 := π(π) , πΏ 1 := πΏ 1 +πΏ 2 +ππ βπ(π) . Step 3. If all the jobs of J(π) are scheduled, then reset π := π+1; else, go back to Step 2. Step 4. If π = π + 1, then stop; else, go back to Step 2. πΏ 1 and πΏ 2 denote the current loads of machine 1 and machine 2 in algorithm Sahni, respectively. And algorithm Sahni first schedules the jobs of J(1) , then schedule J(2) and so on. The following theorem is from [2]. Theorem 1. If problem π2|πππ‘π; ππ |βis feasible, then algorithm Sahni gives a feasible schedule. Theorem 2. Problem π2|πππ‘π; ππ |βis feasible if and only if it satisfies the following conditions: π

πβ1 ππ‘ + πππβ1 +1 , π = 1, 2, . . . , π, π = (1) π(πβ1) + π(π) β₯ βπ‘=1 1, 2, . . . , π;

π

π ππ , π = 1, 2, . . . , π. (2) 2π(π) β₯ βπ=1

Mathematical Problems in Engineering Proof. βonly if β part. Problem π2|pmtn; ππ |βis feasible and denoted by π a feasible schedule for π2|pmtn; ππ |. For any π and any π with π β€ π, only π½ππβ1 +1 out of {π½1 , π½2 , . . . , π½ππβ1 } βͺ {π½ππβ1 +1 } can be processed in [π(πβ1) , π(π) ] in π. Note that π½ππβ1 +1 must be completed by π(π) and all the jobs of {π½1 , π½2 , . . . , π½ππβ1 } must be completed by π(πβ1) . Then we have 2π(πβ1) + (π(π) β ππβ1 π(πβ1) ) β₯ ββ=1 πβ + πππβ1 +1 , which means that (1) holds. Note that all the jobs of all the jobs of {π½1 , π½2 , . . . , π½ππ } must be ππ completed by π(π) . Then we can get that 2π(π) β₯ βπ=1 ππ , which means that (2) holds. βIf β Part. When π = 1, which means that all the jobs have the same deadline π. By the conditions, we have πmax = π1 β€ π and βππ=1 ππ /2 β€ π. According to the result of [21], we know that there exists a feasible schedule of the π jobs meeting with the same deadline. Now assume that the conclusion holds for π = π‘. We consider the case of π = π‘ + 1. Let J = {π½1 , π½2 , . . . , π½ππ‘ } = βͺπ‘π=1 J(π) . By the assumption of π = π‘, we know that there exists a feasible schedule π for J. Furthermore, by Theorem 1, we can assume that π is generated by algorithm Sahni. Denote by π₯ and π¦ the loads of machine 1 and machine 2, respectively. By the assumption of π = π‘ + 1 and by the algorithm Sahni, we have that π₯ + πππ‘ +1 β€ ππ‘+1 π(π‘+1) and π₯+π¦+βπ=π ππ β€ 2π(π‘+1) . Similar to McNaughtonβs π‘ +1 algorithm of [21], we can schedule the jobs of J(π‘+1) starting from π to meet with the deadlines of the jobs of J(π‘+1) . Then (π) we can get a feasible schedule for the job set βͺπ‘+1 π=1 J .

Remark 3. Theorem 2 still holds for π(1) β€ π(2) β€ β β β β€ π(π) . By Theorem 2, we can easily get the following lemma. Lemma 4. Problem π2|πππ‘π|πΏ max can be solved in π(π2 ) time. In the following, let us consider problem π2|pmtn|(πΏπmax : Let (π₯1 , π¦1 ) and (π₯2 , π¦2 ) be two Pareto optimal points of π2|pmtn|(πΏπmax : πΏπmax ); then π₯1 > π₯2 means that π¦1 < π¦2 . So we can assume π¦ = π(π₯) such that (π₯, π¦) is a Pareto optimal point of π2|pmtn|(πΏπmax : πΏπmax ), and we know that π¦ = π(π₯) is a strictly decreasing function on π₯. In order to calculate the tradeoff curve of π2|pmtn|(πΏπmax : πΏπmax ), we must determine the domain π· of π₯ and present an efficient calculation of π(π₯). Let π‘(π₯) be the optimal value of π2|pmtn|πΏπmax β€ π₯ : πΏπmax then {(π₯, π‘(π₯)) : 0 β€ π₯ β€ π₯0 } includes all the Pareto optimal points of π2|pmtn|(πΏπmax : πΏπmax ); that is, {(π₯, π(π₯)) : π₯ β π·} is the remaining set after {(π₯, π‘(π₯)) : 0 β€ π₯ β€ π₯0 } deletes all the non-Pareto optimal points. We call (π₯, π‘(π₯)) a transit point if the due date order of the jobs in J with the due date of π½ππ πΏπmax ).

considered as πππ + π₯ β π and the due date of π½ππ considered as

πππ +π‘(π₯βπ) are different from the due date order of the jobs in J with the due date of π½ππ considered as πππ + π₯ + π and the due

date of π½ππ considered as πππ +π‘(π₯+π) for sufficiently small π. By Lemma 4, we first get the optimal value π₯0 of π2|pmtn|πΏπmax : πΏπmax < +β and the optimal value π0 of π2|pmtn|πΏπmax

π¦0 is determined in such a way that the capacity of this cut is increased to exactly π. The procedure is then repeated in the network induced by π¦1 . This process yields a sequence of increasing trial values π¦π . It terminates when the minimum cut capacity is exactly π, that is, at an iteration π§ where π¦π§ is the first feasible trial value and therefore the optimum value of πΏπmax . We will show that π§ = π(ππ max{π, π}). Suppose a minimum cut with capacity π0 < π is found in the network for π¦0 . Consider how the capacity of this cut is changed when π¦0 is increased by some positive amount πΏ. The capacity ππ+1 β ππ of an arc (π½π , πΈπ ) (a) stays the same if both of ππ and ππ+1 are induced deadlines or not induced deadlines, (b) increases by πΏ if ππ is not induced deadline and ππ+1 is induced deadline, and (c) decreases by πΏ if ππ is induced deadline and ππ+1 is not induced deadline. A similar situation holds for the capacities of the arcs (πΈπ , π), except that they change by ππΏ or βππΏ rather than by πΏ or βπΏ. All arcs whose capacities are increased are incident with a vertex πΈπ of type (π) of which there are at most π. If (πΈπ , π) is in the cut, then no (π½π , πΈπ ) can be forward arc in the cut, so that the cut capacity increases in all arcs incident with πΈπ is at most ππΏ. It follows that eh capacity of the cut

Mathematical Problems in Engineering is increased by π0 πΏ, where π0 is an integer multiplier with π0 β€ ππ max{π, π}. We assert that π0 β₯ 1, and let π¦0 be the next critical value after π¦0 . Note that The vertex-arc structure of the network remains unchanged for π¦0 and π¦0 . Since π¦0 is a feasible critical value, by the max-flow min-cut theorem, we have that π0 + π0 (π¦0 β π¦0 ) β₯ π, which means that π0 β₯ 1. Accordingly, we set πΏ = (π β π0 )/π0 , π¦1 = π¦0 + πΏ and repeat. Each cut in the network can be characterized by a pair (π, πσΈ ), where π is its multiplier and πσΈ its capacity. When π¦π is increased to π¦π+1 , the multipliers of cuts do not change, although their capacities indeed do. Suppose that the minimum cut found at iteration π has multiplier ππ β₯ 1 and capacity ππ and consider the replacement of π¦π by π¦π+1 . Each cut with multiplier π β₯ ππ will have its capacity increased to at lease π. Hence, ππ+1 < ππ . Note that ππ β₯ 1 for all π and π0 β€ ππ max{π, π}. It follows that there can be at most π0 β€ ππ max{π, π} iterations.

5

[13]

[14]

[15]

[16]

[17]

Conflict of Interests

[18]

The authors declare that there is no conflict of interests regarding the publication of this paper.

[19]

References [1] W. A. Horn, βSome simple scheduling algorithms,β Naval Research Logistics Quarterly, vol. 21, pp. 177β185, 1974. [2] S. Sahni, βPreemptive scheduling with due dates,β Operations Research, vol. 27, no. 5, pp. 925β934, 1979. [3] S. Sahni and Y. Cho, βScheduling independent tasks with due times on a uniform processor system,β Journal of the Association for Computing Machinery, vol. 27, no. 3, pp. 550β563, 1980. [4] E. L. Lawler and J. Labetoulle, βOn preemptive scheduling of unrelated parallel processors by linear programming,β Journal of the Association for Computing Machinery, vol. 25, no. 4, pp. 612β619, 1978. [5] J. Labetoulle, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, βPreemptive scheduling of uniform machines subject to release dates,β in Progress in Combinatorial Optimization, pp. 245β261, Academic Press, Toronto, Canada, 1984. [6] E. L. Lawler and C. U. Martel, βComputing maximal βpolymatroidalβ network flows,β Mathematics of Operations Research, vol. 7, no. 3, pp. 334β347, 1982. [7] C. Martel, βPreemptive scheduling with release times, deadlines and due times,β Journal of the Association for Computing Machinery, vol. 29, no. 3, pp. 812β829, 1982. [8] K. R. Baker and J. C. Smith, βA multiple-criterion model for machine scheduling,β Journal of Scheduling, vol. 6, no. 1, pp. 7β 16, 2003. [9] A. Agnetis, P. B. Mirchandani, D. Pacciarelli, and A. Pacifici, βScheduling problems with two competing agents,β Operations Research, vol. 52, no. 2, pp. 229β242, 2004. [10] A. Agnetis, D. Pacciarelli, and A. Pacifici, βMulti-agent single machine scheduling,β Annals of Operations Research, vol. 150, pp. 3β15, 2007. [11] T. C. E. Cheng, C. T. Ng, and J. J. Yuan, βMulti-agent scheduling on a single machine with max-form criteria,β European Journal of Operational Research, vol. 188, no. 2, pp. 603β609, 2008. [12] T. C. E. Cheng, C. T. Ng, and J. J. Yuan, βMulti-agent scheduling on a single machine to minimize total weighted number of tardy

[20]

[21] [22]

[23]

jobs,β Theoretical Computer Science, vol. 362, no. 1β3, pp. 273β 281, 2006. J. J. Yuan, W. P. Shang, and Q. Feng, βA note on the scheduling with two families of jobs,β Journal of Scheduling, vol. 8, no. 6, pp. 537β542, 2005. C. T. Ng, T. C. E. Cheng, and J. J. Yuan, βA note on the complexity of the problem of two-agent scheduling on a single machine,β Journal of Combinatorial Optimization, vol. 12, no. 4, pp. 387β 394, 2006. B. Mor and G. Mosheiov, βScheduling problems with two competing agents to minimize minmax and minsum earliness measures,β European Journal of Operational Research, vol. 206, no. 3, pp. 540β546, 2010. J. Y.-T. Leung, M. Pinedo, and G. Wan, βCompetitive two-agent scheduling and its applications,β Operations Research, vol. 58, no. 2, pp. 458β469, 2010. J. J. Yuan, C. T. Ng, and T. C. E. Cheng, βA note on twoagent scheduling on a single machine with release dates and preemption,β Unpublished Manuscript, 2011. L. Wan, J. J. Yuan, and Z. C. Gen, βA note on the preemptive scheduling to minimize total completion time with release time and deadline constraints,β In Submission, 2012. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, A Series of Books in the Mathematical Sciences, W. H. Freeman, San Francisco, Calif, USA, 1979. R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, βOptimization and approximation in deterministic sequencing and scheduling: a survey,β Annals of Discrete Mathematics, vol. 5, pp. 287β326, 1979. R. McNaughton, βScheduling with deadlines and loss functions,β Management Science, vol. 6, pp. 1β12, 1959. A. V. Karzanov, βDetermining the maximal flow in a network by the method of preflows,β Soviet Mathematics Doklady, vol. 15, pp. 434β437, 1974. R. E. Tarjan, βA simple version of Karzanovβs blocking flow algorithm,β Operations Research Letters, vol. 2, no. 6, pp. 265β 268, 1984.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Research Article A Competitive Two-Agent Scheduling Problem on Parallel Machines with Release Dates and Preemption Yawei Qi1 and Long Wan2 1

School of Information Technology, Jiangxi Key Laboratory of Data and Knowledge Engineering, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China 2 School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China Correspondence should be addressed to Yawei Qi; [email protected] Received 12 August 2013; Accepted 6 October 2013 Academic Editor: Yunqiang Yin Copyright Β© 2013 Y. Qi and L. Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a competitive two-agent scheduling problem on multiple identical machines with release dates and preemption. In the scheduling model, there are two agents π and π each having their own job sets Jπ = {π½1π , . . . , π½πππ } and Jπ = {π½1π , . . . , π½πππ }, respectively. Each job π½π β Jπ βͺ Jπ has a release date ππ and the π = ππ + ππ jobs need to be preemptively scheduled on π identical machines. For π = 2, we show that the trade-off curve of all the Pareto optimal points can be characterized in polynomial time. When π is input, we show that π|ππ , pmtn|πΏπmax : πΏπmax β€ π can be solved in strongly polynomial time.

1. Introduction and Problem Formulation In recent years, multiagent scheduling problems are under extensive research. A multiagent scheduling problem means that there are multiple agents which must compete to perform their own tasks on the common processing resource. Each agent wants to optimize his/her own objective function. The objective function considered in this paper is to minimize the maximum lateness of the jobs. First, let us briefly recall the history of the classic scheduling problems of minimizing the maximum lateness of the jobs that is, there is only one agent in such a problem. Horn [1] considered the problem of scheduling the jobs preemptively on identical machines with release dates and deadlines and showed that it can be determined in strongly polynomial time if the problem has a feasible schedule by reducing it to a network flow problem which is well known to be solved in strongly polynomial time. Sahni [2] presented a faster algorithm to determine if the problem with identical release dates has a feasible schedule. Furthermore, Sahni and Cho [3] showed that it also can be determined in strongly polynomial time if the problem on related machines has a feasible schedule. Lawler and Labetoulle [4] proved that the feasibility problem on unrelated machines can be settled in weakly polynomial time

by means of linear programming. Labetoulle et al. [5] studied the problems of scheduling the jobs on the parallel machines preemptively to minimize the maximum lateness of the jobs with release dates. They gave a strongly polynomial-time algorithm to solve the problem on identical machines which is based on the same network flow structure introduced by Horn [1] and a weakly polynomial-time algorithm to solve the problem on uniform machines in terms of the polymatroidal network flow model introduced by Lawler and Martel [6]. For the latter, a similar result can be found in Martel [7]. The multiagent scheduling models were initially introduced by Baker and Smith [8] and Agnetis et al. [9]. Their research focused on the problems of nonpreemptively scheduling the jobs which belong to two agents on a single machine. Agnetis et al. [10] investigated the multiagent single machine problem of finding a nonpreemptive schedule in which the cost of each agent does not exceed a given threshold value which is also studied in Cheng et al. [11]. Cheng et al. [12] considered the feasibility model of multiagent scheduling on a single machine for which each agent competes to minimize the total weighted number of his/her own tardy jobs and showed that the problem is strongly NP-hard. For more papers about the multiagent problems of scheduling the jobs nonpreemptively on a single machine, the readers are referred to Yuan et al.

2

Mathematical Problems in Engineering

[13], Ng et al. [14], and Mor and Mosheiov [15]. Leung et al. [16] which initiated the preemptively multiagent scheduling problems investigated the two-agent scheduling problems of scheduling the jobs preemptively on a single machine or identical machines. Yuan et al. [17] studied a competitive two-agent scheduling problem on a single machine with release dates and preemption for which the objective of each agent is to minimize the maximum lateness and showed that all Pareto optimal points can be found in strongly polynomial time. Wan et al. [18] investigated the same twoagent scheduling model for which one agentβs objective is to minimize the maximum lateness and the other agentβs objective is to minimize the total completion time of his/her jobs. They proved that the problem is NP-hard in the ordinary sense by means of reduction from even-odd partition which is well known to be ordinarily NP-hard [19]. The problems in the paper are stated as follows. There are two agents π and π each having their own job sets Jπ = {π½1π , . . . , π½πππ } and Jπ = {π½1π , . . . , π½πππ }, respectively. We make the assumption that Jπ β©Jπ = 0. The jobs in Jπ are called π-jobs and the jobs in Jπ are called π-jobs. Each job π½π β Jπ βͺ Jπ has a release date ππ , a due date ππ , and a processing time ππ . And the π = ππ + ππ jobs need to be preemptively scheduled on π identical machines. Two agents have the same objective of minimizing the maximum lateness. We use the following notation throughout this paper.

The rest of the paper is organized as follows. In Section 2, we show that the tradeoff curve of the Pareto optimal points can be characterized in strongly polynomial time for π2|pmtn|(πΏπmax : πΏπmax ). Section 3 gives a polynomial time algorithm to solve Problem π|ππ , pmtn|πΏπmax : πΏπmax β€ π. We draw some conclusions and present some further research.

2. π2|pmtn|(πΏπmax : πΏπmax )

(i) πππ₯ is the processing time of job π½ππ₯ , π₯ β {π, π}.

First, let us state a feasibility problem of scheduling π jobs preemptively on two identical machines with the deadlines and give a characterization of feasibility. Let J = {π½1 , π½2 , . . . , π½π } be the set of the jobs and ππ and ππ are the deadline and processing time of job π½π , respectively; π = 1, 2, . . . , π. The problem is denoted by π2|pmtn; ππ |β in terms of the threefield notation. Without loss of generality, we assume that the jobs are ordered by π1 β€ π2 β€ β β β β€ ππ . Assume that there are total π different deadlines and π(1) , π(2) , . . . , π(π) with π(1) < π(2) < β β β < π(π) are the π different deadlines. Denote by J(π) the set of the jobs of deadline π(π) , π = 1, 2, . . . , π. Let π0 = 0, π1 , . . . , ππ = π be the π + 1 numbers such that J(π) = {π½ππβ1 +1 , π½ππβ1 +2 , . . . , π½ππ }, π = 1, 2, . . . , π. We assume that π0 = 0 and π½ππβ1 +1 is the largest job of J(π) for convenience, π = 1, 2, . . . , π. Before we present the characterization of feasibility of π2|pmtn|(πΏπmax : πΏπmax ), we first state a well-known result from [2].

(ii) πππ₯ is the release date of job π½ππ₯ , π₯ β {π, π}.

Algorithm Sahni. Consider the following steps.

(iii)

πππ₯

π½ππ₯ ,

is the due date of job

(iv)

πΆππ₯

is the completion time of job π½ππ₯ , π₯ β {π, π}.

π₯ β {π, π}.

(v) πΏπ₯π = πΆππ₯ β πππ₯ is the lateness of job π½ππ₯ , π₯ β {π, π}. (vi) πΏπ₯max = max{πΏπ₯π : 1 β€ π β€ ππ₯ } is the maximum lateness of (the jobs of) agent π₯, π₯ β {π, π}. A schedule π is called Pareto optimal if there is no schedule π such that πΏπmax (π) β€ πΏπmax (π), πΏπmax (π) β€ πΏπmax (π), and at least one inequality strictly holds, that is, a schedule is Pareto optimal for any schedule; if it is better for one agent, then it must be worse for the other agent. We say (πΏπmax (π), πΏπmax (π)) is a Pareto optimal point if schedule π is a Pareto optimal schedule. The first problem we consider is to find all the Pareto optimal points and a corresponding schedule for each Pareto optimal point when π = 2 and all the jobs are released at 0. The second problem we consider is to minimize the maximum lateness of agent π΄ with the maximum lateness of agent π΅ bounded by a given threshold π when π is input. By use of the well-known three-field notation [20], the first and second problems can be formulated as follows. π2|pmtn|(πΏπmax

πΏπmax ),

: which is (i) The first problem: a Pareto optimization problem seeking to minimize πΏπmax and πΏπmax simultaneously. (ii) The second problem: π|ππ , pmtn|πΏπmax : πΏπmax β€ π.

Step 1. π = 1, πΏ 1 = πΏ 2 = 0. Step 2. Find a unscheduled job π½π of J(π) . If ππ β₯ π(π) βπΏ 1 , then return infeasibility. Otherwise, we schedule job π½π . If ππ β€ π(π) β πΏ 2 , then we assign time interval [πΏ 2 , πΏ 2 + ππ ] to π½π on machine 2 and reset πΏ 2 := πΏ 2 +ππ ; else, we assign time interval [πΏ 2 , π(π) ] on machine 2 and time interval [πΏ 1 , πΏ 1 +πΏ 2 +ππ βπ(π) ] on machine 1 to π½π and reset πΏ 2 := π(π) , πΏ 1 := πΏ 1 +πΏ 2 +ππ βπ(π) . Step 3. If all the jobs of J(π) are scheduled, then reset π := π+1; else, go back to Step 2. Step 4. If π = π + 1, then stop; else, go back to Step 2. πΏ 1 and πΏ 2 denote the current loads of machine 1 and machine 2 in algorithm Sahni, respectively. And algorithm Sahni first schedules the jobs of J(1) , then schedule J(2) and so on. The following theorem is from [2]. Theorem 1. If problem π2|πππ‘π; ππ |βis feasible, then algorithm Sahni gives a feasible schedule. Theorem 2. Problem π2|πππ‘π; ππ |βis feasible if and only if it satisfies the following conditions: π

πβ1 ππ‘ + πππβ1 +1 , π = 1, 2, . . . , π, π = (1) π(πβ1) + π(π) β₯ βπ‘=1 1, 2, . . . , π;

π

π ππ , π = 1, 2, . . . , π. (2) 2π(π) β₯ βπ=1

Mathematical Problems in Engineering Proof. βonly if β part. Problem π2|pmtn; ππ |βis feasible and denoted by π a feasible schedule for π2|pmtn; ππ |. For any π and any π with π β€ π, only π½ππβ1 +1 out of {π½1 , π½2 , . . . , π½ππβ1 } βͺ {π½ππβ1 +1 } can be processed in [π(πβ1) , π(π) ] in π. Note that π½ππβ1 +1 must be completed by π(π) and all the jobs of {π½1 , π½2 , . . . , π½ππβ1 } must be completed by π(πβ1) . Then we have 2π(πβ1) + (π(π) β ππβ1 π(πβ1) ) β₯ ββ=1 πβ + πππβ1 +1 , which means that (1) holds. Note that all the jobs of all the jobs of {π½1 , π½2 , . . . , π½ππ } must be ππ completed by π(π) . Then we can get that 2π(π) β₯ βπ=1 ππ , which means that (2) holds. βIf β Part. When π = 1, which means that all the jobs have the same deadline π. By the conditions, we have πmax = π1 β€ π and βππ=1 ππ /2 β€ π. According to the result of [21], we know that there exists a feasible schedule of the π jobs meeting with the same deadline. Now assume that the conclusion holds for π = π‘. We consider the case of π = π‘ + 1. Let J = {π½1 , π½2 , . . . , π½ππ‘ } = βͺπ‘π=1 J(π) . By the assumption of π = π‘, we know that there exists a feasible schedule π for J. Furthermore, by Theorem 1, we can assume that π is generated by algorithm Sahni. Denote by π₯ and π¦ the loads of machine 1 and machine 2, respectively. By the assumption of π = π‘ + 1 and by the algorithm Sahni, we have that π₯ + πππ‘ +1 β€ ππ‘+1 π(π‘+1) and π₯+π¦+βπ=π ππ β€ 2π(π‘+1) . Similar to McNaughtonβs π‘ +1 algorithm of [21], we can schedule the jobs of J(π‘+1) starting from π to meet with the deadlines of the jobs of J(π‘+1) . Then (π) we can get a feasible schedule for the job set βͺπ‘+1 π=1 J .

Remark 3. Theorem 2 still holds for π(1) β€ π(2) β€ β β β β€ π(π) . By Theorem 2, we can easily get the following lemma. Lemma 4. Problem π2|πππ‘π|πΏ max can be solved in π(π2 ) time. In the following, let us consider problem π2|pmtn|(πΏπmax : Let (π₯1 , π¦1 ) and (π₯2 , π¦2 ) be two Pareto optimal points of π2|pmtn|(πΏπmax : πΏπmax ); then π₯1 > π₯2 means that π¦1 < π¦2 . So we can assume π¦ = π(π₯) such that (π₯, π¦) is a Pareto optimal point of π2|pmtn|(πΏπmax : πΏπmax ), and we know that π¦ = π(π₯) is a strictly decreasing function on π₯. In order to calculate the tradeoff curve of π2|pmtn|(πΏπmax : πΏπmax ), we must determine the domain π· of π₯ and present an efficient calculation of π(π₯). Let π‘(π₯) be the optimal value of π2|pmtn|πΏπmax β€ π₯ : πΏπmax then {(π₯, π‘(π₯)) : 0 β€ π₯ β€ π₯0 } includes all the Pareto optimal points of π2|pmtn|(πΏπmax : πΏπmax ); that is, {(π₯, π(π₯)) : π₯ β π·} is the remaining set after {(π₯, π‘(π₯)) : 0 β€ π₯ β€ π₯0 } deletes all the non-Pareto optimal points. We call (π₯, π‘(π₯)) a transit point if the due date order of the jobs in J with the due date of π½ππ πΏπmax ).

considered as πππ + π₯ β π and the due date of π½ππ considered as

πππ +π‘(π₯βπ) are different from the due date order of the jobs in J with the due date of π½ππ considered as πππ + π₯ + π and the due

date of π½ππ considered as πππ +π‘(π₯+π) for sufficiently small π. By Lemma 4, we first get the optimal value π₯0 of π2|pmtn|πΏπmax : πΏπmax < +β and the optimal value π0 of π2|pmtn|πΏπmax

π¦0 is determined in such a way that the capacity of this cut is increased to exactly π. The procedure is then repeated in the network induced by π¦1 . This process yields a sequence of increasing trial values π¦π . It terminates when the minimum cut capacity is exactly π, that is, at an iteration π§ where π¦π§ is the first feasible trial value and therefore the optimum value of πΏπmax . We will show that π§ = π(ππ max{π, π}). Suppose a minimum cut with capacity π0 < π is found in the network for π¦0 . Consider how the capacity of this cut is changed when π¦0 is increased by some positive amount πΏ. The capacity ππ+1 β ππ of an arc (π½π , πΈπ ) (a) stays the same if both of ππ and ππ+1 are induced deadlines or not induced deadlines, (b) increases by πΏ if ππ is not induced deadline and ππ+1 is induced deadline, and (c) decreases by πΏ if ππ is induced deadline and ππ+1 is not induced deadline. A similar situation holds for the capacities of the arcs (πΈπ , π), except that they change by ππΏ or βππΏ rather than by πΏ or βπΏ. All arcs whose capacities are increased are incident with a vertex πΈπ of type (π) of which there are at most π. If (πΈπ , π) is in the cut, then no (π½π , πΈπ ) can be forward arc in the cut, so that the cut capacity increases in all arcs incident with πΈπ is at most ππΏ. It follows that eh capacity of the cut

Mathematical Problems in Engineering is increased by π0 πΏ, where π0 is an integer multiplier with π0 β€ ππ max{π, π}. We assert that π0 β₯ 1, and let π¦0 be the next critical value after π¦0 . Note that The vertex-arc structure of the network remains unchanged for π¦0 and π¦0 . Since π¦0 is a feasible critical value, by the max-flow min-cut theorem, we have that π0 + π0 (π¦0 β π¦0 ) β₯ π, which means that π0 β₯ 1. Accordingly, we set πΏ = (π β π0 )/π0 , π¦1 = π¦0 + πΏ and repeat. Each cut in the network can be characterized by a pair (π, πσΈ ), where π is its multiplier and πσΈ its capacity. When π¦π is increased to π¦π+1 , the multipliers of cuts do not change, although their capacities indeed do. Suppose that the minimum cut found at iteration π has multiplier ππ β₯ 1 and capacity ππ and consider the replacement of π¦π by π¦π+1 . Each cut with multiplier π β₯ ππ will have its capacity increased to at lease π. Hence, ππ+1 < ππ . Note that ππ β₯ 1 for all π and π0 β€ ππ max{π, π}. It follows that there can be at most π0 β€ ππ max{π, π} iterations.

5

[13]

[14]

[15]

[16]

[17]

Conflict of Interests

[18]

The authors declare that there is no conflict of interests regarding the publication of this paper.

[19]

References [1] W. A. Horn, βSome simple scheduling algorithms,β Naval Research Logistics Quarterly, vol. 21, pp. 177β185, 1974. [2] S. Sahni, βPreemptive scheduling with due dates,β Operations Research, vol. 27, no. 5, pp. 925β934, 1979. [3] S. Sahni and Y. Cho, βScheduling independent tasks with due times on a uniform processor system,β Journal of the Association for Computing Machinery, vol. 27, no. 3, pp. 550β563, 1980. [4] E. L. Lawler and J. Labetoulle, βOn preemptive scheduling of unrelated parallel processors by linear programming,β Journal of the Association for Computing Machinery, vol. 25, no. 4, pp. 612β619, 1978. [5] J. Labetoulle, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, βPreemptive scheduling of uniform machines subject to release dates,β in Progress in Combinatorial Optimization, pp. 245β261, Academic Press, Toronto, Canada, 1984. [6] E. L. Lawler and C. U. Martel, βComputing maximal βpolymatroidalβ network flows,β Mathematics of Operations Research, vol. 7, no. 3, pp. 334β347, 1982. [7] C. Martel, βPreemptive scheduling with release times, deadlines and due times,β Journal of the Association for Computing Machinery, vol. 29, no. 3, pp. 812β829, 1982. [8] K. R. Baker and J. C. Smith, βA multiple-criterion model for machine scheduling,β Journal of Scheduling, vol. 6, no. 1, pp. 7β 16, 2003. [9] A. Agnetis, P. B. Mirchandani, D. Pacciarelli, and A. Pacifici, βScheduling problems with two competing agents,β Operations Research, vol. 52, no. 2, pp. 229β242, 2004. [10] A. Agnetis, D. Pacciarelli, and A. Pacifici, βMulti-agent single machine scheduling,β Annals of Operations Research, vol. 150, pp. 3β15, 2007. [11] T. C. E. Cheng, C. T. Ng, and J. J. Yuan, βMulti-agent scheduling on a single machine with max-form criteria,β European Journal of Operational Research, vol. 188, no. 2, pp. 603β609, 2008. [12] T. C. E. Cheng, C. T. Ng, and J. J. Yuan, βMulti-agent scheduling on a single machine to minimize total weighted number of tardy

[20]

[21] [22]

[23]

jobs,β Theoretical Computer Science, vol. 362, no. 1β3, pp. 273β 281, 2006. J. J. Yuan, W. P. Shang, and Q. Feng, βA note on the scheduling with two families of jobs,β Journal of Scheduling, vol. 8, no. 6, pp. 537β542, 2005. C. T. Ng, T. C. E. Cheng, and J. J. Yuan, βA note on the complexity of the problem of two-agent scheduling on a single machine,β Journal of Combinatorial Optimization, vol. 12, no. 4, pp. 387β 394, 2006. B. Mor and G. Mosheiov, βScheduling problems with two competing agents to minimize minmax and minsum earliness measures,β European Journal of Operational Research, vol. 206, no. 3, pp. 540β546, 2010. J. Y.-T. Leung, M. Pinedo, and G. Wan, βCompetitive two-agent scheduling and its applications,β Operations Research, vol. 58, no. 2, pp. 458β469, 2010. J. J. Yuan, C. T. Ng, and T. C. E. Cheng, βA note on twoagent scheduling on a single machine with release dates and preemption,β Unpublished Manuscript, 2011. L. Wan, J. J. Yuan, and Z. C. Gen, βA note on the preemptive scheduling to minimize total completion time with release time and deadline constraints,β In Submission, 2012. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, A Series of Books in the Mathematical Sciences, W. H. Freeman, San Francisco, Calif, USA, 1979. R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, βOptimization and approximation in deterministic sequencing and scheduling: a survey,β Annals of Discrete Mathematics, vol. 5, pp. 287β326, 1979. R. McNaughton, βScheduling with deadlines and loss functions,β Management Science, vol. 6, pp. 1β12, 1959. A. V. Karzanov, βDetermining the maximal flow in a network by the method of preflows,β Soviet Mathematics Doklady, vol. 15, pp. 434β437, 1974. R. E. Tarjan, βA simple version of Karzanovβs blocking flow algorithm,β Operations Research Letters, vol. 2, no. 6, pp. 265β 268, 1984.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014