Two-agent scheduling with rejection on a single machine

Applied Mathematical Modelling 39 (2015) 1183–1193

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Two-agent scheduling with rejection on a single machine Qi Feng a,⇑, Baoqiang Fan b, Shisheng Li a, Weiping Shang c a b c

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, PR China Department of Mathematics and Information, Ludong University, Yantai 264025, PR China Department of Mathematics, Zhengzhou University, Zhengzhou 450052, PR China

a r t i c l e

i n f o

Article history: Received 5 February 2012 Received in revised form 11 July 2014 Accepted 23 July 2014 Available online 12 August 2014

a b s t r a c t In this paper, we consider the two-agent scheduling problem with rejection on a single machine. In the problem we have two agents A and B with job families J A and J B , respectively. A job in J A or J B is either accepted and processed on the machine, or rejected with a certain rejection penalty having to be paid. The objective is to minimize the sum of the A

Keywords: Agent scheduling Single machine Rejection Pseudo-polynomial-time algorithm Approximation algorithm

given objective function f of the accepted jobs and total rejection penalty of the rejected jobs of the first agent A, given that the second agent B only accepts schedules such that the B

sum of the given objective function f of the accepted jobs and total rejection penalty of the rejected jobs of the agent B does not exceed a fixed value Q (Q is a positive integer), where A

B

f and f are non-decreasing functions on the completion times of their respective jobs. We n    P  A B C Aj ; C Bmax ; C Amax ; C Bmax ; LAmax ; LBmax ; show that, for all pairs ðf ; f Þ 2 P  P    P A B C Aj ; LBmax ; C Aj ; wBj U Bj g, the problems are NP-hard. When ðf ; f Þ ¼ C Amax ; LBmax ,   A B we provide a pseudo-polynomial-time algorithm. Moreover, when ðf ; f Þ ¼ C Amax ; LBmax and all B-jobs are accepted, we give a 2-approximation algorithm and an fully polynonP  P o P A B C Aj ; LBmax ; C Aj ; wBj U Bj K, mial-time approximation scheme. Finally, for ðf ; f Þ 2 we present pseudo-polynomial-time algorithms, respectively. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In the classical scheduling, it is always assumed that any job must be processed. However, in the real world, due to the limited resources, things may be more flexible and we can make a higher-level decision, i.e., we can break the constraint by rejecting a job. For example, in order to obtain the maximum profits, the manufacturer often rejects some jobs which have the larger processing times but bring the relatively smaller profits. Knapsack is likely a pure instance of the problem where only the ‘‘accept’’ of items is considered and no rejection penalty occurs. However, considering the credit and other factors, to reject a job, of course, the manufacturer should pay a corresponding penalty. The machine scheduling problem with rejection was first introduced by Bartal et al. [1]. They studied the off-line and on-line versions on identical parallel machines. The objective is to minimize the sum of makespan of the accepted jobs and total rejection penalty of the rejected jobs. From then on, machine scheduling problems with rejection have received increasing attention. Seiden [2] presented an improved

⇑ Corresponding author. Tel.: +86 13783626979. E-mail address: [email protected] (Q. Feng). http://dx.doi.org/10.1016/j.apm.2014.07.024 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.

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on-line algorithm for the case where job preemption is allowed. Engels et al. [3] considered the single machine scheduling with rejection to minimize the sum of weighted completion times of the scheduled jobs and total rejection penalty of the rejected jobs. Hoogeveen et al. [4] considered the off-line multi-processor (including identical, uniform, unrelated parallel machines, and open shops) scheduling problems with rejection when preemption is allowed. Zhang et al. [5] studied the single-machine scheduling problems with release dates and rejection. Zhang et al. [6] further considered the single-machine scheduling problems in which the total rejection penalties cannot exceed a given upper bound. Lu et al. [7,8] considered the unbounded and bounded parallel-batch machine scheduling problems with release dates and rejection, respectively. Cheng and Sun [9], Li and Yuan [10] considered the scheduling problems with linear deterioration and rejection on the single machine and identical parallel machines, respectively. In recent years, there has been significant interest in multi-agent scheduling. Scheduling problems involving multiple agents competing for a common processing resource arise naturally in many application environments in which the need for negotiation procedures arises. For example, in industrial management, the multi-agent scheduling problem is formulated as a sequencing game, where the objective is to devise some mechanisms to encourage the agents to cooperate with a view to minimizing the overall cost (Curiel et al. [11] and Hamers et al. [12]). In project scheduling, the problem is concerned with negotiation to resolve conflicts whenever the agents find their own schedules unacceptable (Kim et al. [13]). In telecommunication services, the problem is to do with satisfying the service requirements of individual agents, who compete for the use of a commercial satellite to transfer voice, image and text files for their clients (Schultz et al. [14]). The problem here is to fulfill the requirements of various integrated services. A key issue for the problem is to determinate the nondominated (or Pareto-optimal) schedules, i.e., such that a better schedule for one agent necessarily results in a worse schedule for at least another agent. The agents want to list all possible Pareto-optimal solutions, to negotiate the most acceptable trade-off for all the parties. In order to do this, we have to solve the corresponding Constrained Optimization Problems (for detailed explanation see Agnetis et al. [17]). In this paper, we focus our research on the constrained optimization problems of the multi-agent scheduling. Machine scheduling problems with multi-agent were researched widely. Baker and Smith [15] focused on minimizing the linear combination of the objective functions of the two or more agents on a single machine. They examine the implications of minimizing an aggregate scheduling objective function in which jobs belonging to different agents are evaluated based on their individual criteria. Agnetis et al. [16] considered the scheduling models in which two agents compete for the usage of shared processing resource and each agent has his own criterion to optimize. Agnetis et al. [17] studied single-machine scheduling problems with multiple agents, and the objective functions they considered are the maximum of regular functions, number of tardy jobs, and total weighted completion time. Cheng et al. [18] considered the model of multi-agent scheduling on a single machine, where each agent’s objective function is to minimize the total weighted number of tardy jobs. Moreover, Cheng et al. [19] studied multi-agent scheduling on a single machine and the objective functions of the agents are of the max-form. Ng et al. [20] studied the problem of minimizing the total completion time for one agent subject to a given permitted number of tardy jobs for the other agent. Leung et al. [21] studied the identical parallel-machine scheduling problems with two agents, where the jobs of the two agents may have distinct release dates and preemption is allowed. Although much work have been done in multi-agent scheduling and scheduling with rejection, respectively. It seems that few papers can be found in literature for the above two intersection parts, i.e., multi-agent scheduling with rejection. This situation can satisfy the real-life needs due to the rapid development of manufacturing and other areas. In fact, considering the profits or other reasons, agents maybe decide to reject some jobs to obtain maximum profits or satisfy other needs. It is necessary to consider the multi-agent scheduling with rejection in order to give better service to the reality. In this paper, we consider the two-agent scheduling with rejection on a single machine. In Section 2, we give a formal description of the considered problems and relevant notations. In Section 3, we show that all problems studied in this paper are NP-hard. In Section 4, for problem to minimize the sum of the makespan of the accepted A-jobs and the total rejection penalty of the rejected A-jobs, given that agent B only accepts schedules such that the sum of maximum lateness of the accepted B-jobs and the total rejection penalty of the rejected B-jobs do not exceed a fixed value Q, we present a pseudopolynomial-time algorithm. In Section 5, for a special case that all B jobs are accepted, we give a 2-approximation algorithm and an fully polynomial-time approximation scheme (FPTAS), respectively. In Section 6, for other two different objective functions of agents A and B, we present pseudo-polynomial-time algorithms. 2. Problem formulation and preliminaries There are two competing agents A and B, each of them has a set of non-preemptive jobs to be processed on a single common machine. Agent x has to process the job set J x ¼ fJ x1 ; J x2 ; . . . ; J xnx g; x 2 fA; Bg. For convenience, we simply refer to a job of x agent x as an x-job. For x 2 fA; Bg; J xj is associated with a processing time pxj , a due date dj and a weight wxj . For each agent, J xi is either accepted and processed with a processing time pxi on the machine, or rejected with a penalty exi ; x 2 fA; Bg having to be paid. We assume that all jobs are available for processing from time zero onwards. Let r indicate a feasible schedule of the n ¼ nA þ nB jobs, i.e., a feasible assignment of starting times to the jobs of two P A A PnB B PA A PB B pi þ i¼1 pi ; E1 ¼ ni¼1 ei , and E2 ¼ ni¼1 ei . The completion time of job J xi in r is agents. For convenience, write P ¼ ni¼1 x x x denoted as C i ðrÞ. If there is no ambiguity, we use C i instead of C i ðrÞ. Denote by Ax and Rx the sets of accepted x-jobs A

and rejected x-jobs, x 2 fA; Bg, respectively. The objective is to minimize the sum of the given objective function f of the

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P accepted A-jobs and total rejection penalty JA 2RA eAj of the rejected A-jobs, given that the second agent B only accepts schedj P B A B ules such that its objective function f þ JB 2RB eBj does not exceed a fixed value Q (Q is a positive integer), where f and f are j

non-decreasing functions on the completion times of their respective jobs. Using the general notation for scheduling problem given by Graham et al. [22], the above scheduling problem is denoted P P A B A B by 1jrejectjf þ JA 2RA eAj : f þ JB 2RB eBj . In this paper, we consider various combinations of regular objective functions ðf ; f Þ, j j P P P P for example, ðC Amax ; C Bmax Þ; ðLAmax ; LBmax Þ; ð C Aj ; C Bmax Þ; ð C Aj ; LBmax Þ, and ð C Aj ; wBj U Bj Þ. 3. Complexity analysis P In this section, we show that even all B-jobs are accepted, i.e., problem 1jrejectjC Amax þ JA 2RA eAj : C Bmax and j P P A B 1jrejectj JA 2AA C j þ JA 2RA eAj : C max are NP-hard. The first problem is done by a reduction from the NP-complete Partition j

j

problem (Garey and Johnson [23]) and the second problem is done from the NP-complete Equal-Size Partition problem (Garey and Johnson [23]). P First, we show that problem 1jrejectjC Amax þ JA 2RA eAj : C Bmax is binary NP-hard. j

3.1. Partition problem P

Given t þ 1 positive integers a1 ; a2 ; . . . ; at ; B such that aj 2S aj ¼ B?

Theorem 3.1. Problem 1jrejectjC Amax þ

P

Pt

i¼1 aj

¼ 2B, is there a subset S  fa1 ; a2 ; . . . ; at g such that

: C Bmax is binary NP-hard. P Proof. The decision version of problem 1jrejectjC Amax þ JA 2RA eAj 6 Y : C Bmax is clearly in NP. For a given instance of the PartiP j tion problem, we construct an instance of the decision version of problem 1jrejectjC Amax þ JA 2RA eAj : C Bmax as follows. A J Aj 2RA ej

j

   

A

fJ A1 ; . . . ; J AnA g with nA fJ B1 g with pB1 ¼ 3B.

t; pAj

eAj

For agent A, we have J ¼ ¼ ¼ aj and ¼ 2aj for each 1 6 j 6 nA . For agent B, we have J B ¼ The upper bound Q of C Bmax is defined by Q ¼ 4B, and the threshold value for agent A is defined by Y ¼ 3B. P The decision asks whether there is a schedule r with C Amax ðrÞ þ JA 2RA eAj 6 Y and C Bmax ðrÞ 6 Q . j

It can be observed that the above construction can be done in polynomial time. We show in the following that the instance of Partition problem has a solution if and only if there exists a feasible schedule r of the scheduling instance with P C Amax ðrÞ þ JA 2RA eAj 6 Y and C Bmax ðrÞ 6 Q . j P First, we assume that the instance of Partition problem has a solution S such that aj 2S aj ¼ B. We construct a schedule r A B by the following way: assign the jobs in fJ j : aj 2 Sg followed by J 1 to be processed on the machine and all other jobs of agent P A are rejected. Since aj 2S aj ¼ B and pB1 ¼ 3B, we have C Bmax ðrÞ ¼ 4B ¼ Q . Then, r is a feasible schedule. Furthermore, we P have that C Amax ðrÞ þ JA 2RA eAj ¼ 3B ¼ Y. P j Conversely, suppose that there is a schedule r with C Amax ðrÞ þ JA 2RA eAj 6 Y and C Bmax ðrÞ 6 Q . We are ready to show that j the instance of Partition problem has a solution. Note that Y ¼ 3B. From the construction of the scheduling instance, All P B accepted A-jobs are processed before the B-job J 1 in schedule r. It follows that C Amax ðrÞ ¼ JA 2AA pAj . P j A B A A B B If C max > B, then C max ðrÞ ¼ JA 2AA pj þ p1 ¼ C max ðrÞ þ p1 > 4B ¼ Q , a contradiction. j P P P P P If C Amax ðrÞ < B, then JA 2AA aj < B. We have JA 2R aj > B. Therefore, C Amax ðrÞ þ JA 2RA eAj ¼ JA 2AA aj þ JA 2RA 2aj > 3B ¼ Y, a j

j

j

j

j

contradiction again. Hence, C Amax ðrÞ ¼ B, and therefore S ¼ faj : J Aj 2 AA g is a solution of the instance of Partition problem. h P P Next, we show that problem 1jrejectj JA 2AA C Aj þ JA 2RA eAj : C Bmax is also binary NP-hard. j j 3.2. Equal-size partition problem

Given 2t þ 1 positive integers a1 ; a2 ; . . . ; a2t ; B such that P P with jS1 j ¼ jS2 j such that aj 2S1 aj ¼ aj 2S2 aj ¼ B? Theorem 3.2. Problem 1jrejectj

P

A J Aj 2AA C j

þ

P

A JAj 2RA ej

P2t

i¼1 aj

¼ 2B, is there a partition of the ai ’s into two sets S1 and S2

: C Bmax is binary NP-hard.

P P Proof. The decision version of problem 1jrejectj JA 2AA C Aj þ JA 2RA eAj 6 Y : C Bmax is clearly in NP. For a given instance of the P j j Equal-Size Partition problem, we construct an instance of the decision version of problem 1jrejectj JA 2AA C Aj P j B A þ JA 2RA ej : C max as follows. j

 For agent A, we have J A ¼ fJ A1 ; . . . ; J AnA g with nA ¼ 2t; pAj ¼ B þ aj and eAj ¼ DðB þ aj Þ for each 1 6 j 6 nA , where D ¼ tðt þ 1ÞB þ 1.

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 For agent B, we have J B ¼ fJ B1 g with pB1 ¼ Y, where Y is defined below.  The upper bound Q of C Bmax is defined by Q ¼ ðt þ 1ÞB þ Y, and the threshold value for agent A is defined by Y ¼ tðt þ 1ÞB þ Dðt þ 1ÞB. P P  The decision asks whether there is a schedule r with JA 2AA C Aj þ JA 2RA eAj 6 Y and C Bmax ðrÞ 6 Q . j

j

It can be observed that the above construction can be done in polynomial time. We show in the following that the instance of Equal-Size Partition problem has solutions if and only if there exists a feasible schedule r of the scheduling P P instance with JA 2AA C Aj þ JA 2RA eAj 6 Y and C Bmax ðrÞ 6 Q . j j First, we assume that the instance of Equal-Size Partition problem has a solution S1 and S2 with jS1 j ¼ jS2 j such that P P A B aj 2S1 aj ¼ aj 2S2 aj ¼ B. We construct a schedule r by the following way: assign the jobs in fJ j : aj 2 S1 g followed by J 1 to be processed on the machine and all other jobs of agent A are rejected, and furthermore, the accepted A-jobs are scheduled in an P P P arbitrary order. Since and pB1 ¼ Y, we have C Bmax ðrÞ ¼ JA 2AA pAj þ pB1 ¼ JA 2AA ðB þ aj Þ þ Y ¼ tBþ aj 2S1 aj ¼ B j j P J Aj 2AA aj þ Y ¼ ðt þ 1ÞB þ Y ¼ Q . Then r is a feasible schedule. Note that the every accepted A-job has a completion time P P A Since we have at most ðt þ 1ÞB (a very rough estimation). Then aj 2S2 aj ¼ B, J Aj 2AA C j 6 tðt þ 1ÞB. P P P P P A A A Then J A 2RA ej ¼ JA 2RA DðB þ aj Þ ¼ DtB þ D J A 2RA aj ¼ DtB þ DB ¼ Dðt þ 1ÞB. J A 2AA C j þ J A 2RA ej 6 tðt þ 1ÞB þ Dðt þ 1Þ j

j

j

j

j

B ¼ Y, as required. P P Conversely, suppose that there is a schedule r with JA 2AA C Aj þ JA 2RA eAj 6 Y and C Bmax ðrÞ 6 Q . We are ready to show that j j the instance of Equal-Size Partition problem has solutions. Note that pB1 ¼ Y. From the construction of the scheduling P instance, all accepted A-jobs are processed before the B-job J B1 in schedule r. It follows that C Amax ðrÞ ¼ JA 2AA pAj . h j

Claim 1. The number of accepted A-jobs is less than or equal to t. Otherwise, the number of accepted A-jobs is larger than t. Then P P C Bmax ¼ JA 2AA pAj þ pB1 ¼ JA 2AA ðB þ aj Þ þ Y > ðt þ 1ÞB þ Y ¼ Q , a contradiction.Claim 2 follows. j

j

Claim 2. The number of rejected A-jobs is less than or equal to t. Otherwise, the number of rejected A-jobs is larger than t. Then P P A J A 2RA ej ¼ JA 2RA DðB þ aj Þ P Dðt þ 1ÞB þ D ¼ Dðt þ 1ÞB þ tðt þ 1ÞB þ 1 > Y, a contradiction. Claim 2 follows. j

j

From Claim 1 and Claim 2, we know that there are just t accepted A-jobs and t rejected A-jobs in r. P P P P P B A B aj þ Y > ðt þ 1ÞBþ Claim 3. A J Aj 2AA aj ¼ B. If JAj 2AA aj > B, then C max ðrÞ ¼ JAj 2AA pj þ p1 ¼ JAj 2AA ðB þ aj Þ þ Y ¼ tB þ P P P PJj 2AA A a < B, then a > B. We have e ¼ Y ¼ Q , a contradiction. If A A A A j j Jj 2AA Jj 2RA Jj 2RA j Jj 2RA DðB þ aj Þ ¼ DBtþ P D JA 2RA aj P DBt þ DðB þ 1Þ ¼ DBðt þ 1Þ þ D ¼ Dðt þ 1ÞB þ tðt þ 1ÞB þ 1 > Y, a contradiction again. Claim 3 j

follows. From Claim 3, we conclude that S1 ¼ faj : J Aj 2 AA g and S2 ¼ faj : J Aj 2 RA g form a solution of the instance of Equal-Size Partition problem. P P B Note that the NP-hardness of problem 1jj C Aj : C j was provided in [16]. Thus, for the SUM-form objective functions of P P A B A B f and f studied in this paper, problems 1jrejectjf þ JA 2RA eAj : f þ JB 2RB eBj are NP-hard. j

j

4. Problem 1jrejectjC Amax þ +J A ‰RA eAj : LBmax þ +J B ‰RB eBj j

j

In this section, we consider problem 1jrejectjC Amax þ Lemma 4.1. For problem 1jrejectjC Amax þ properties.

P

A J Aj 2RA ej

P

: LBmax þ

A J Aj 2RA ej

P

: LBmax þ

B J Bj 2RB ej ,

P

B J Bj 2RB ej .

there exists an optimal schedule satisfying the following two

(i) The accepted A-jobs are scheduled consecutively in an arbitrary order. (ii) The accepted B-jobs are scheduled in earliest due date (EDD) order. Furthermore, the accepted B-jobs are partitioned into two subsets (if exist) and the jobs of each subset are scheduled in EDD order, i.e., some accepted B-jobs (if exist) are scheduled consecutively in EDD order at the beginning of the schedule, and the remaining accepted B-jobs are scheduled consecutively in EDD order at the end of the schedule. Proof. The above two properties both can be easily proved by the standard job-shifting argument. h We call a schedule effective if it satisfies the two properties in Lemma 4.1. For convenience, we re-label the jobs of agent A B B B and B such that pA1 6 pA2 6    6 pAnA and d1 6 d2 6    6 dnB . Let f ði; j; t; EA ; EB Þ be the optimal objective function value of the following scheduling problem: (1) The jobs in consideration are J A1 ; J A2 ; . . . ; J Ai and J B1 ; J B2 ; . . . ; J Bj . (2) The makespan of the accepted jobs among J A1 ; J A2 ; . . . ; J Ai and J B1 ; J B2 ; . . . ; J Bj is t.

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(3) The total rejection penalty among J A1 ; J A2 ; . . . ; J Ai is EA . (4) The total rejection penalty among J B1 ; J B2 ; . . . ; J Bj is EB . In an optimal schedule for the first i A-jobs and the first j B-jobs, either J Ai is rejected, or J Bj is rejected, or the last scheduled job is one of J Ai and J Bj . We distinguish the following four cases. Case 1. J Ai is rejected. Since J Ai is rejected, in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai1 , and J B1 ; J B2 ; . . . ; J Bj , the makespan of the accepted jobs among J A1 ; J A2 ; . . . ; J Ai and J B1 ; J B2 ; . . . ; J Bj is t. The total rejection penalty of A-jobs and B-jobs are EA  eAi and EB , respectively. The contribution of J Ai to objective function is eAi . Then we have f ði; j; t; EA ; EB Þ ¼ f ði  1; j; t; E  eAi ; EB Þ þ eAi . Case 2. The last scheduled job is J Ai . Since the last scheduled job is J Ai , in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai1 , and J B1 ; J B2 ; . . . ; J Bj , the makespan of the accepted jobs is t  pAi . The total rejection penalty of A-jobs and B-jobs are EA and EB , respectively. Then we have f ði; j; t; EA ; EB Þ ¼ f ði  1; j; t  pAi ; EA ; EB Þ þ pAi . Case 3. J Bj is rejected. Since J Bj is rejected, in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai , and J B1 ; J B2 ; . . . ; J Bj1 , the makespan of the accepted jobs is t. The total rejection penalty of A-jobs and B-jobs are EA and EB  eBj , respectively. The contribution of J Bj to objective function is eBj . Then we have f ði; j; t; EA ; EB Þ ¼ f ði; j  1; t; EA ; EB  eBj Þ. Case 4. The last scheduled job is J Bj . Since the last scheduled job is J Bj , in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai , and J B1 ; J B2 ; . . . ; J Bj1 , the makespan of the accepted jobs is t  pBj . The total rejection penalty of the A-jobs and B-jobs B

are EA and EB , respectively. Then we have f ði; j; t; EA ; EB Þ ¼ f ði; j  1; t  pBj ; EA ; EB Þ. In this case, t  dj þ EB 6 Q . Combining the above four cases, we have the following dynamic programming algorithm. Algorithm DP1 The boundary conditions:

f ð0; 0; t; EA ; EB Þ ¼



0;

if t ¼ EA ¼ EB ¼ 0;

þ1; otherwise:

The recursive function:

f ði; j; t; EA ; EB Þ ¼ min

8 f ði  1; j; t; EA  eAi ; EB Þ þ eAi ; > > > > < f ði  1; j; t  pAi ; EA ; EB Þ þ pAi ; f ði; j  1; t; EA ; EB  eBj Þ; > > > > : f ði; j  1; t  pBj ; EA ; EB Þ;

B

if t  dj þ EB 6 Q:

The optimal value is given by n o P P min f ðnA ; nB ; t; EA ; EB Þ : 0 6 t 6 P; 0 6 EA 6 16j6nA eAj ; 0 6 EB 6 16j6nB eBj . Theorem 4.2. Algorithm DP1 solves 1jrejectjC Amax þ

P

A J Aj 2RA ej

: C Bmax þ

P

B JBj 2RB ej

in OðnA nB PE1 E2 Þ time.

Proof. The correctness of the algorithm DP1 is guaranteed by the above discussion. The recursive function has at most P P OðnA nB P 16j6nA eAj 16j6nB eBj Þ states, each iteration costs a constant time. Hence, the total running time of the algorithm P P DP1 is bounded by OðnA nB PE1 E2 Þ, where E1 ¼ 16j6nA eAj and E2 ¼ 16j6nB eBj Þ. h 5. Problem 1jrejectjC Amax þ +J A ‰RA eAj : LBmax j

P P In this section, we consider a special case of problem 1jrejectjC Amax þ JA 2RA eAj : LBmax þ JB 2RB eBj in which all B-jobs are j j P accepted, then the problem can be written as 1jrejectj C Amax þ JA 2RA eAj : LBmax . For the problem, it can be observed that if j

the rejection penalty of an A-job (say job J Ak ) does not exceed its processing time, then there exists an optimal schedule such that job J Ak is always rejected. Hence, in the following, we assume that pAj < eAj for all j ¼ 1; . . . ; nA . We will give a dynamic programming algorithm, a 2-approximation algorithm and an fully polynomial-time approximation scheme (FPTAS), respectively. 5.1. Dynamic programming algorithm It is obvious that there exists an optimal schedule satisfying the properties in Lemma 4.1. We consider an arbitrary optimal feasible schedule r. If there is an A-job being processed after all the jobs of agent B in r, then from Lemma 4.1, all

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accepted A-jobs should be processed at the end in r. Thus, r can be easily constructed in OðnB log nB þ nA Þ time. In the following discussion, we assume that there exists at least one A-job scheduled before the last B-job J BnB . Then, all the accepted A-jobs are scheduled before J BnB . Suppose that there are totally k (0 6 k < nB ) B-jobs J B1 ; . . . ; J Bk scheduled consecutively at the beginning in r. Denote by P Bk the total processing time of these k B-jobs. Then the remaining ðnB  kÞ B-jobs J Bkþ1 ; . . . ; J BnB are scheduled at the end of the schedule. And all accepted A-jobs are scheduled between the above two job sets of agent B. We call r a k-schedule. Since LBmax ðrÞ 6 Q , the starting time of job J Bkþ1 in any k-schedule is less than or equal to

(

S

ðkÞ

¼ max sk : sk þ

j X

pBi i¼kþ1

)



B dj

6 Q ; j ¼ k þ 1; . . . ; nB ;

for k < nB :

ð1Þ

Write

Y k ¼ SðkÞ  P Bk : PAnA

ð2Þ

PnA

P Bk .

If ¼ i¼1 pAi 6 Y k , we can obtain an optimal k-schedule by processing all A-jobs with starting time Hence, we assume that P AnA > Y k . P Now, we give a dynamic programming algorithm for problem 1jrejectjC Amax þ JA 2R eAj : LBmax to construct an optimal j

k-schedule. The dynamic programming algorithm goes through nA phases. At every phase l (l ¼ 1; 2; . . . ; nA ), a state set Cl is generated. Any state in Cl is a vector ½f ; t which corresponds to a feasible k-schedule for the first l A-jobs: The coordinate f measures the optimal objective value of such k-schedule, and the coordinate t measures the total processing time of the accepted A-jobs (note that these accepted A-jobs start at time PBk ). The state sets C1 ; C2 ; . . . ; CnA are constructed iteratively. For any state ½f ; t 2 Cl , there are two possible cases: either J Al is rejected or accepted. Case 1. J Al is rejected. The contribution of J Al to objective function value is eAl . Then ½f ; t ¼ ½f  eAl ; t. Case 2. J Al is accepted. If t 6 Y k , The contribution of J Al to objective function value is pAl . Thus, we have ½f ; t ¼ ½f  pAl ; t  pAl . If t > Y k , Case 2 does not occur. Combining the above two cases, we have the following dynamic programming algorithm. Algorithm DP2 Initialization: Set C0 ¼ f½PBk ; 0g and C1 ¼ f½P Bk þ pA1 ; pA1 ; ½P Bk þ eA1 ; 0g, where 0 6 k < nB . Phase l. For 2 6 l 6 nA , the state set Cl is computed inductively as follows: For every ½f ; t 2 Cl1 , put ½f þ eAl ; t into Cl always; If t þ pAl 6 Y k , then put ½f þ pAl ; t þ pAl  into Cl . Output. Output the vector ½f ; t 2 CnA that minimizes the f value. PA A By taking an upper bound E1 on f and PAnA ¼ ni¼1 pi on t, the complexity of algorithm DP2 is OðE1 P AnA Þ.  Now, let f k be the output of the algorithm DP2 and f be the optimal objective value of our problem. Note that k have at most nB distinct choices. Thus, we can enumerate all possibilities and select the feasible schedule with the minimum objec tive value, i.e. f ¼ minff k : 0 6 k < nB g. Hence, we conclude the following theorem. Theorem 5.1. Problem 1jrejectjC Amax þ

P

B JAj 2R ej

: LBmax can be solved in OðnB E1 P AnA Þ time.

5.2. 2-approximation algorithm and fully polynomial-time approximation scheme P In this section, for problem 1jrejectj C Amax þ JA 2RA eAj : LBmax , we will give a 2-approximation algorithm and an fully polynoj mial-time approximation scheme (FPTAS). eAn eA A We assume that all A-jobs in J are re-indexed such that p1A P    P pAA . Let J 0A # fJ A1 ; . . . ; J AnA g. Also, all jobs in J 0A are ordered in nonincreasing order of

eAi

pAi

1

nA

, where J Ai 2 J 0A . Write jJ 0A j ¼ n0A . For convenience, we give the following terms.

 PAr ðJ 0A Þ the total processing time of the first r jobs of J 0A .  Erþ1 ðJ 0A Þ the total rejection penalty of the last n0A  r jobs of J 0A . Let P A0 ðJ 0A Þ ¼ 0 and En0A þ1 ðJ 0A Þ ¼ 0. If J 0A ¼ J A , then we denote P Ar ðJ 0A Þ and Erþ1 ðJ 0A Þ by PAr and Erþ1 , respectively. First, we consider the relaxation of the problem where job splitting of agent A is allowed. ‘‘splitting’’ means that a piece (say a; 0 6 a 6 1) of an A-job (say J Ak ) is accepted (i.e., processed with processing time apAk ) and the remaining piece is rejected with rejection penalty ð1  aÞeAk . We denote the splitting version of the problem as 1jsplt; rejectjC Amax þ P B A J A 2RA ej : Lmax . j

Let J Ar be the job satisfying PAr1 < Y k and P Ar > Y k , where Y k is defined by (2). If the jobs J A1 ; . . . ; J Ar1 are accepted completely. Then, the job J Ar cannot be accepted completely and only a piece can be accepted. By above observations, we have the following lemma.

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Lemma 5.2. For problem 1jsplt; rejectjC Amax þ

P

A JAj 2RA ej

: LBmax , there exists an optimal k-schedule

p such that:

(a) The accepted A-jobs are fJ A1 ; . . . ; J Ar1 ; J Ar g. (b) There is only one splitted A-job J Ar and the processing time of accepted piece of J Ar is Y k  PAr1 . eA

eA

i

j

Proof. If there exist two jobs J Ai and J Aj with piA P pjA in schedule p, such that a piece of J Ai with processing time pAi1 ð0 < pAi1 < pAi Þ

is rejected and a piece of J Aj with processing time pAj1 ð0 < pAj1 6 pAj Þ is accepted. Let p0 ¼ minfpAi1 ; pAj1 g. A new schedule obtained from C Amax ð

pÞ¼ 0

p by the following transformation. We replace job JAj by JAi with the processing time p0 . Clearly, we have that pÞ. The rejection penalty of p0 is increased by

C Amax ð

eAj eA eA p A  p iA ¼ p0 A  iA pj pi pj pi 0

eAj

p0 is

0

! 6 0:

Then, p0 is still an optimal k-schedule. Note that P Ar1 < Y k ; PAr > Y k and eAr > pAr . Therefore, by repeating this exchanging process, we can eventually get an optimal k-schedule with the above two properties. h 

Let f k ðspltÞ denote the objective value of the optimal k-schedule of the splitting problem. By Lemma 5.2, we have 

f k ðspltÞ ¼ PBk þ Y k þ ðPAr  Y k Þ

eAr þ Erþ1 : pAr



Let f k denote the objective value of the optimal k-schedule of problem 1jrejectjC Amax þ 

P

A JAj 2RA ej

: LBmax . Then



f k P f k ðspltÞ P PBk þ PAr1 þ Erþ1 : R

ð3Þ

 Assume Ak ¼ fJ Ai : eAi > f k ; i ¼ 1; . . . ; nA g. Then, A A A ¼ fJ i : pi > Y k  PjAk j ðAk Þ; J Ai 2 J A n Ak g and J 0A ¼

jobs in Ak should be accepted in any optimal k-schedule. Let S J A n ðAk RÞ. Then we have



f k P PBk þ PAjAk j ðAk Þ þ P Ar0 1 ðJ 0A Þ þ Er0 þ1 ðJ 0A Þ þ E1 ðRÞ:

ð4Þ

where r 0 satisfies conditions PAr0 1 ðJ 0A Þ < Y k  PAjAk j ðAk Þ and P Ar0 ðJ 0A Þ > Y k  PAjAk j ðAk Þ. P We now give an approximation algorithm for 1jrejectjC Amax þ JA 2RA eAj : LBmax . j

Algorithm Hk Step 0: Computer SðkÞ and Y k by (1) and (2). Set R ¼ ;; A ¼ ; and i ¼ 0. Step 1: Set J A ¼ fJ A1 ; . . . ; J AnA g and computer ri such that P Ari 1 6 Y k and PAri > Y k . Step 2: Obtain a schedule pi with accepted A-jobs fJ A1 ; . . . ; J Ari 1 g. S S S Step 3: Set Y k :¼ Y k  pAri ; R :¼ R fJ Aj 2 J A : pAj > Y k ; j – r i g; A :¼ A fJ Ari g, and J A :¼ J A n ðAk RÞ. If J A ¼ ;, go to Step 4. Otherwise, set i :¼ i þ 1, go to Step 1. Step 4: Select the schedule with the minimum objective value among all schedules obtained in Step 2. Let f k be the objective value of the schedule obtained by algorithm Hk . 

Lemma 5.3. f k 6 2f k . Proof. All jobs in A are re-indexed in appear sequence in algorithm Hk , such that A ¼ fJ Ar1 ; . . . ; J Arh g, where 1 6 h < nA . Let P be the set of schedules obtained by algorithm Hk . Then jPj ¼ h þ 1.   If there exists j with 1 6 j 6 h such that eri > f k for i ¼ 1; . . . ; j  1, and erj 6 f k . Then there is a schedule pj 2 P in which A A A A A jobs fJ r1 ; . . . ; J rj1 g are accepted and J rj is rejected. Let Rj ¼ fJ i : pi > Y k  P Aj1 ðAÞ; J Ai 2 J A n fJ Ar1 ; . . . ; J Arj1 gg and J 0A ¼ J A n fJ Ar1 ; . . . ; J Arj1 g n Rj . By (4), the objective value of pj is 

f ðpj Þ 6 PBk þ PAj1 ðAÞ þ P~Ar1 ðJ 0A Þ þ E~rþ1 ðJ 0A Þ þ E1 ðRj Þ þ erj 6 2f k : where ~r satisfies conditions P~Ar1 ðJ 0A Þ 6 Y k  PAj1 ðAÞ and P~Ar ðJ 0A Þ > Y k  P Aj1 ðAÞ.  If eri > f k for i ¼ 1; . . . ; h, then there exists a schedule phþ1 2 P in which the jobs of A are accepted. Let  Rh ¼ fJ Ai : pAi > Y k  PAh ðAÞ; J Ai 2 J A n Ag. By algorithm Hk , we have J 0A ¼ J A n A n Rh ¼ ;. It implies that f ðphþ1 Þ ¼ f k . The conclusion follows. h Note that k have at most nB distinct choices. Thus, we can enumerate all possibilities and select the feasible schedule with the minimum objective value f, i.e. f ¼ minff k : 0 6 k < nB g. Since f P B A JA 2RA ej : Lmax , by Lemma 5.3, we conclude the following result.



is the optimal objective value of 1jrejectjC Amax þ

j

Theorem 5.4. For problem 1jsplt; rejectjC Amax þ

P

A J Aj 2RA ej



: LBmax , a schedule can be obtained with f 6 2f in OðnB nA log nA Þ time.

The standard scaling techniques can help obtain an FPTAS from the DP2 in Section 5.1 and the above 2-approximation algorithm. We omit it here. We demonstrate the results of the Theorem 5.4 in the following example.

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Example 5.1. There are four agent A’s jobs J A ¼ fJ A1 ; J A2 ; J A3 ; J A4 g with pA1 ¼ 1; pA2 ¼ 3; pA3 ¼ 1; pA4 ¼ 1 and eA1 ¼ 6; eA2 ¼ 17; eA3 ¼ 2; B

eA4 ¼ 2. Agent B has one job J B1 with pB1 ¼ 15 and d1 ¼ 18. Let the upper bound on LBmax be Q ¼ 0. B Firstly, we computer Sð0Þ ¼ Q  d1  pB1 ¼ 3; Y 0 ¼ 3, which mean that all accepted jobs of agent A are processed in time slot ½0; 3Þ, and Y 1 ¼ 1, which means that all accepted jobs are processed in ½15; 1Þ. According to algorithm Hk , we can get P schedules r0 ¼ ðJ A1 ; J A3 ; J A4 ; J B1 Þ with objective value C Amax þ JA 2RA eAj ¼ 3 þ 17 ¼ 20 and r1 ¼ ðJ B1 ; J A1 ; J A2 ; J A3 ; J A4 Þ with C Amax þ j P A e ¼ 15 þ 6 ¼ 21. Finally, we select the schedule r with the minimum objective value 20. In fact, the optimal A 0 J 2RA j j



objective value is f ¼ 3 þ 6 þ 2 þ 2 ¼ 13 and the optimal schedule is

r ¼ ðJA2 ; JB1 Þ. Thus, we have that f ðfr0 Þ ¼ 20 13 < 2.

6. Pseudo-polynomial-time algorithm for other two objective functions In this section, we present dynamic programming algorithms for two agents scheduling with rejection when the two agents have other objective functions. 6.1. Problem 1jrejectj

P

A JAj 2AA C j

þ

P

A JAj 2RA ej

: LBmax þ

P

B JBj 2RB ej

In this subsection, we consider the problem in which agent A wants to minimize the sum of the total completion time of the accepted A-jobs and the total rejection penalty of the rejected A-jobs, given that the sum of the maximum lateness of the accepted B-jobs and the total rejection penalty of the rejected B-jobs does not exceed a fixed value Q. Lemma 6.1. There is an optimal schedule for problem 1jrejectj

P

A J Aj 2AA C j

þ

P

A J Aj 2RA ej

: LBmax þ

P

B J Bj 2RB ej

in which the accepted

A-jobs are sequenced in SPT order and the accepted B-jobs are sequenced in EDD order. Proof. It is easy to show using job-interchanging and job-shifting arguments. h Based on Lemma 6.1, we only need to consider these schedule with the property of Lemma 6.1. Re-label the jobs of agents B B B A and B such that pA1 6 pA2 6    6 pAnA and d1 6 d2 6    6 dnB . Let f ði; j; t; EB Þ be the optimal objective function value of the following scheduling problem: (1) The jobs in consideration are J A1 ; J A2 ; . . . ; J Ai and J B1 ; J B2 ; . . . ; J Bj . (2) The makespan of the accepted A-jobs among J A1 ; J A2 ; . . . ; J Ai and the accepted B-jobs among J B1 ; J B2 ; . . . ; J Bj is exactly t. (3) The total rejection penalty among J B1 ; J B2 ; . . . ; J Bi is EB . In an optimal schedule for the first i A-jobs and the first j B-jobs, either J Ai is rejected, or J Bi is rejected, or the last scheduled job is one of J Ai and J Bj . we distinguish the following four cases. Case 1. J Ai is rejected. Since J Ai is rejected, in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai1 , and J B1 ; J B2 ; . . . ; J Bj , the makespan is t. The total rejection penalty of the B-jobs is EB . The contribution of J Ai to objective function is eAi . Then we have f ði; j; t; EB Þ ¼ f ði  1; j; t; EB Þ þ eAi . Case 2. The last scheduled job is J Ai . Since the last scheduled job is J Ai , the makespan is t  pAi in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai1 and J B1 ; J B2 ; . . . ; J Bj . The total rejection penalty of the rejected B-jobs is EB . The contribution of J Ai to objective function is t. Then we have f ði; j; t; EB Þ ¼ f ði  1; j; t  pAi ; EB Þ þ t. Case 3. J Bj is rejected. Since J Bj is rejected, in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai , and J B1 ; J B2 ; . . . ; J Bj1 , the makespan is t. The total rejection penalty of the rejected B-jobs is EB  eBj . Then we have f ði; j; t; EB Þ ¼ f ði; j  1; t; EB  eBj Þ. Case 4. The last scheduled job is J Bj . Since the last scheduled job is J Bj , in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai , and J B1 ; J B2 ; . . . ; J Bj1 , the makespan is t  pBj . The total rejection penalty of the rejected B-jobs is EB . In this case, we have B t  dj þ EB 6 Q . Then we have f ði; j; t; EB Þ ¼ f ði; j  1; t  pBj ; EB Þ. Combining the above four cases, we have the following dynamic programming algorithm. Algorithm DP3 The boundary conditions:

f ð0; 0; t; EB Þ ¼



0;

if t ¼ EB ¼ 0;

þ1; otherwise:

The recursive function:

f ði; j; t; EB Þ ¼ min

8 f ði  1; j; t; EB Þ þ eAi ; > > > > < f ði  1; j; t  pAi ; EB Þ þ t; f ði; j  1; t; EB  eBj Þ; > > > > : f ði; j  1; t  pBj ; EB Þ;

B

if t  dj þ EB 6 Q:

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The optimal value is given by n o P min f ðnA ; nB ; t; EB Þ : 0 6 t 6 P; 0 6 EB 6 16j6nB eBj . Theorem 6.2. Algorithm DP3 solves 1jrejectj

P

A JAj 2AA C j

þ

P

A J Aj 2RA ej

: LBmax þ

P

B J Bj 2RB ej

in OðnA nB PE2 Þ time.

Proof. The correctness of the algorithm is guaranteed by the above discussion. The recursive function has at most P OðnA nB P 16j6nB eBj Þ states, each iteration costs a constant time. Hence, the total running time is bounded by OðnA nB PE2 Þ, where P E2 ¼ 16j6nB eBj . h 6.2. Problem 1jrejectj

P

A JAj 2AA C j

þ

P

A JAj 2RA ej

:

P

wBj U Bj

When the objective function of agent A is to minimize the sum of the total completion time of the accepted A-jobs and total rejection penalty of the rejected A-jobs such that the weighted number of late B jobs does not exceed a fixed value Q. The following Lemma is easy to show and we omit the proof. Lemma 6.3. There is an optimal schedule for the problem 1jrejectj

P

A J Aj 2AA C j

þ

P

A JAj 2RA ej

:

P

wBj U Bj in which the accepted A-jobs

are sequenced in SPT order and the accepted early B-jobs are sequenced in EDD order. Moreover, the accepted A-jobs and the accepted early B-jobs are scheduled first, followed by late B-jobs. Based on Lemma 6.3, we only need to consider the schedules with the property of Lemma 6.3. Re-label the jobs of agent A and B B B B such that pA1 6 pA2 6    6 pAnA and d1 6 d2 6    6 dnB . Let f ði; j; t; WÞ be the optimal objective function value of the following scheduling problem: (1) The jobs in consideration are J A1 ; J A2 ; . . . ; J Ai and J B1 ; J B2 ; . . . ; J Bj . (2) The makespan of the accepted jobs of J A1 ; J A2 ; . . . ; J Ai and the accepted early jobs of J B1 ; J B2 ; . . . ; J Bj is exactly t. That is, the late B-jobs are also be rejected. (3) The weighted number of late jobs among J B1 ; J B2 ; . . . ; J Bj is exactly W ðW 6 Q Þ. In an optimal schedule for the first i A-jobs and the first j B-jobs, either J Ai is rejected, or J Bj is rejected, or the last scheduled job is one of J Ai and J Bj . We distinguish the following four cases. Case 1. J Ai is rejected. Since J Ai is rejected, in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai1 , and J B1 ; J B2 ; . . . ; J Bj , the makespan is t. The weighted number of late jobs among J B1 ; J B2 ; . . . ; J Bj is W. The contribution of J Ai to objective function is eAi . Then we have f ði; j; t; WÞ ¼ f ði  1; j; t; WÞ þ eAi . Case 2. The last scheduled job is J Ai . Since the last scheduled job is J Ai , the makespan is t  pAi in optimal schedule for the accepted jobs among J A1 ; J A2 ; . . . ; J Ai and the accepted early jobs among J B1 ; J B2 ; . . . ; J Bj . The weighted number of late jobs among J B1 ; J B2 ; . . . ; J Bj is still W. Then we have f ði; j; t; WÞ ¼ f ði  1; j; t  pAi ; WÞ þ t. B Case 3. The last scheduled job is J Bj . In this case, we have t 6 dj . Since the last scheduled job is J Bj , in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai , and J B1 ; J B2 ; . . . ; J Bj1 , the makespan is t  pBj . The weighted number of late jobs among J B1 ; J B2 ; . . . ; J Bj is still W. Then we have f ði; j; t; WÞ ¼ f ði; j  1; t  pBj ; WÞ. Case 4. J Bj is late. Since J Bj is late, in the optimal schedule for J A1 ; J A2 ; . . . ; J Ai , and J B1 ; J B2 ; . . . ; J Bj1 , the makespan is t. The weighted number of late jobs among J B1 ; J B2 ; . . . ; J Bj1 is W  wBj . Then we have f ði; j; t; WÞ ¼ f ði; j  1; t; W  wBj Þ.

Combining the above four cases, we have the following dynamic programming algorithm. Algorithm DP4 The boundary conditions:

 f ð0; 0; t; WÞ ¼

0;

if t ¼ W ¼ 0;

þ1; otherwise;

and

f ði; j; ;t; WÞ ¼ þ1;

if W > Q:

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The recursive function:

8 f ði  1; j; t; WÞ þ eAi ; > > > > < f ði  1; j; t  pA ; WÞ þ t; i f ði; j; t; WÞ ¼ min f ði; j  1; t  pBj ; WÞ; > > > > : f ði; j  1; t; W  wB Þ; if W 6 Q: j The optimal value is given by minff ðnA ; nB ; t; WÞ : 0 6 t 6 Pg. Theorem 6.4. Algorithm DP4 solves 1jrejectj

P

A JAj 2AA C j

þ

P

A J Aj 2RA ej

:

P

wBj U Bj in OðnA nB PÞ time.

Proof. The correctness of the algorithm is guaranteed by the above discussion. The recursive function has at most OðnA nB PÞ states, each iteration costs a constant time. Hence, the total running time is bounded by OðnA nB PÞ. h 7. Concluding remarks In this paper, we have studied the two-agent scheduling on a single machine with rejection. The objective is to minimize A

the sum of the given objective function f of the accepted jobs and total rejection penalty of the rejected jobs of the first B

agent A, given that the second agent B only accepts schedules such that the sum of the given objective function f of the A

accepted jobs and total rejection penalty of the rejected jobs of the agent B does not exceed a fixed value Q, where f and P P B f are regular scheduling criteria, e.g., C max ; Lmax ; C j , and wj U j . We show that the problems are NP-hard for various comP A B A B and f , and provide pseudo-polynomial-time algorithms for ðf ; f Þ 2 fðC Amax ; LBmax Þ; ð C Aj ; LBmax Þ; binations of f P A P B B A B ð C j ; wj U j Þg. Moreover, when ðf ; f Þ ¼ ðC Amax ; LBmax Þ and all B-jobs are accepted, we give a 2-approximation algorithm and an fully polynomial-time approximation scheme. For the further research, it is interesting to analyze the scheduling problems with more than two agents. Another direction for future research is to provide more efficient approximate algorithms for the intractable problems. Acknowledgments We are grateful to the editor and the anonymous referees for their helpful comments on an earlier version of this paper. This paper was partially supported by the National Natural Science Foundation of China under Grant Nos. 11271338 and 11326191. In addition, the first author was supported by the Henan Province Natural Science Foundation under Grant No. 132300410392 and the second author was supported by the Shandong Province Natural Science Foundation under Grant No. ZR2011AL016. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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