Two-agent single-machine scheduling with unrestricted due date assignment

Computers & Industrial Engineering 79 (2015) 148–155

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Two-agent single-machine scheduling with unrestricted due date assignment q Yunqiang Yin a, T.C.E. Cheng b, Xiaoqin Yang a, Chin-Chia Wu c,⇑ a

Faculty of Science, Kunming University of Science and Technology, Kunming 650093, PR China Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong c Department of Statistics, Feng Chia University, Taichung, Taiwan b

a r t i c l e

i n f o

Article history: Received 31 May 2014 Received in revised form 25 October 2014 Accepted 29 October 2014 Available online 8 November 2014 Keywords: Scheduling Single machine Two agents Due date assignment

a b s t r a c t We address two scheduling problems arising when two agents (agents A and B), each with a set of jobs, compete to perform their respective jobs on a common machine, where the due dates of agent A’s jobs are decision variables to be determined by the scheduler. Specifically, the objective is to determine the optimal due dates for agent A’s jobs and the job sequence for both agents’ jobs simultaneously to minimize the total cost associated with the due date assignment and weighted number of tardy jobs of agent A, while keeping the maximum of regular functions (associated with each B-job) or the number of tardy jobs of agent B below or at a fixed threshold. We prove that both problems are NP-hard in the strong sense and develop polynomial or pseudo-polynomial solutions for some important special cases. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction

of its jobs only. Agent A wants to minimize the optimality criterion  PnA  A A A A A j¼1 aj dj þ bj U j , where for each j ¼ 1; 2; . . . ; nA ; aj is a positive

1.1. Problem definition

unit due date assignment cost of job J Aj ; bAj is a positive penalty cost

We first define the scheduling problem and introduce notations. Specially, suppose that there are two competing agents, called agents A and B, respectively. Each of them has a set of non-preemptive jobs to be processed on a common machine. All the jobs are available for processing at time zero. Agent A has to execute job

if job J Aj is tardy, and U Aj is the unit tardiness cost of job J Aj , i.e.,

set J A ¼ fJ A1 ; J A2 ; . . . ; J AnA g, whereas agent B has to execute job set

tion time of job J Bj ; j ¼ 1; 2; . . . ; nB . In addition, we assume that   B f j C Bj is computable in constant time for any possible value of   B C Bj . Examples of f j C Bj include wBj C Bj (the weighted completion   B time of job J Bj ), wBj LBj ¼ wBj C Bj  dj (the weighted lateness of job

B

J ¼

fJ B1 ; J B2 ; . . . ; J BnB g.

The jobs of the two sets are called A-jobs and

B-jobs, respectively. Let X 2 fA; Bg. Associated with each job J Xj X

A

are a job processing time pXj and a due date dj , where each dj is a decision variable to be determined by the scheduler while each B

dj is exogenously given. The total processing time of the X-jobs is denoted by PX . Without loss of generality, we assume that the job parameters are integral unless stated otherwise. Let S indicate a feasible schedule of the n ¼ nA þ nB jobs, i.e., a feasible assignment of starting times to the jobs of both agents. We use C Xj ðSÞ to denote the completion time of job J Xj and omit the argument S whenever this does not generate confusion. Each agent has a certain criterion that depends on the completion times

q

This manuscript was processed by Area Editor T.C. Edwin Cheng.

⇑ Corresponding author.

E-mail address: [email protected] (C.-C. Wu). http://dx.doi.org/10.1016/j.cie.2014.10.025 0360-8352/Ó 2014 Elsevier Ltd. All rights reserved.

A

U Aj ¼ 1 if C Aj < dj and U Aj ¼ 0 otherwise, while agent B wants to B

B

minimize the optimality criterion f max ¼ maxj¼1;2;...;nB ff j ðC Bj Þg or P B B U j , where f j ðÞ denotes a nondecreasing function of the comple-

B

J Bj ), wBj T Bj ¼ wBj maxfC Bj  dj ; 0g (the weighted tardiness of job J Bj ), and wBj U Bj (the unit tardiness cost of job J Bj ). The objective is to determine the optimal due dates for agent A’s jobs and the job sequence for both agents’ jobs simultaneously to  PnA  A A A A minimize j¼1 aj dj þ bj U j , while observing that agent B only PB B B accepts schedules such that f max 6 Q or nj¼1 U j 6 Q , where Q is a given threshold. Throughout this paper, we use the three-field notation scheme ajbjcA : cB introduced by Agnetis, Mirchandani, Pacciarelli, and Pacifici (2004) for the two-agent scheduling problems. For the a term, we use a ¼ 1 to denote the single machine A

case. For the b term, we denote dj

to indicate that each

Y. Yin et al. / Computers & Industrial Engineering 79 (2015) 148–155 A

dj ; j ¼ 1; 2; . . . ; nA , is a decision variable to be determined by the A

B

scheduler. While for the c : c term which indicates that the value of the criterion cA of agent A has to be minimized with the constraint that the value of the criterion cB of agent B will not exceed  PA  A A the fixed threshold Q, we set cA ¼ nj¼1 aj dj þ bAj U Aj and n P B Bo cB 2 f Bmax ; nj¼1 Uj . For a practical example, consider a situation in manufacturing that involves three parties: a manufacturer and its two customers (agents A and B). Agent A is a long-term customer and the manufacturer will negotiate with agent A the due dates of its jobs, while agent B is an ad hoc customer, the due dates of whose jobs are exogenously given. The manufacturer wishes to determine its production programme in such a way that all the accepted orders are completed in full and on time to satisfy agent A’s demand while keeping the objective value of agent B (with regard to its own jobs) at or below a fixed threshold. For the mutually agreed due dates for the A-jobs, the manufacturer has to bear the cost of due date assignment in order to retain the customer. The manufacturer also has the option of paying a penalty for each order that it cannot complete on time. This situation can be modeled as our problems. 1.2. Literature review Recent developments in the field of distributed decision making have led to growing interest in multi-agent scheduling, whereby different agents share a common processing resource and each agent wants to optimize its own objective related to its own set of jobs. Since the agents have conflicting objectives, they have to negotiate among themselves with regard to sharing the resources to optimize their own objectives. Scheduling problems with multiple agents have many real applications in areas such as real-time systems, integrated service networks, industrial districts, telecommunications systems, and so on. Baker and Smith (2003) and Agnetis et al. (2004) are among the pioneers that introduced the topic of multi-agent scheduling into the scheduling field, where there are two agents that compete on the use of a common processor. The criteria considered in their research include the maximum of regular functions (e.g., makespan or lateness), the total (weighted) completion time, and the number of tardy jobs. Baker and Smith (2003) focus on minimizing a linear combination of the criteria of the two agents on a single machine, while Agnetis et al. (2004) concentrate on problems in the single-machine and complex shop settings in which one agent seeks to find the best schedule for its jobs, while keeping the objective value of the other agent below or at a fixed threshold. Since then, multi-agent scheduling has received increasing attention in the last ten years. Ng, Cheng, and Yuan (2006) solve an open problem left by Agnetis et al. (2004), i.e., minimizing the total completion time of the first agent, while keeping the number of tardy jobs of the second agent below or at a fixed threshold, by establishing that it is NP-hard under high multiplicity encoding. Later on, Leung, Pinedo, and Wan (2010) generalize the results of Agnetis et al. (2004) by showing that the above unresolved problem is NP-hard in the ordinary sense, and by strengthening a complexity result that the problem of minimizing the total completion time of the first agent, while keeping the maximum of regular functions of the second agent below or at a fixed threshold, is NP-hard in the strong sense. In addition, they consider some single-machine problems where one of the criteria is total tardiness, as well as problems with preemption and release dates under different conditions. Cheng, Wu, Cheng, and Wu (2011) and Yin, Cheng, and Wu (2012) consider the two-agent scheduling problems with varying processing times, in which the actual processing time of a job gets longer if the job starts later (scheduling with deterioration effects), or gets shorter

149

if the job is scheduled later (scheduling with learning effects). Li and Yuan (2012) consider the two-agent scheduling problems on a common unbounded parallel batching machine, where job families are incompatible (jobs from different families cannot be processed in the same batch) or compatible (all jobs can be processed in the same batch). Mor and Mosheiov (2010) consider the two-agent scheduling problem in which the objective is to minimize the maximum (or total weighted) earliness cost of one agent while keep the maximum earliness cost of the other agent cannot exceed a fixed threshold. Nong, Cheng, and Ng (2011) study the problem of minimizing total cost of the two agents, where the cost of the first agent is the maximum weighted completion time of its jobs while the cost of the second agent is the total weighted completion time of its jobs. Wan, Vakati, Leung, and Pinedo (2010) consider the two-agent scheduling problems with controllable processing times, where the processing times of jobs of one agent are compressible at additional cost. Yin, Cheng, Cheng, Wu, and Wu (2012) consider the single-machine scheduling problems with two agents and assignable due dates in which the due dates are viewed as given variables and must be assigned to individual jobs. Agnetis, Pacciarelli, and Pacifici (2007), Cheng, Ng, and Yuan (2008), Cheng, Ng, and Yuan (2006) consider the scheduling problems with more than two agents. For recent results on multi-agent scheduling, the reader may refer to the survey paper of Perez-Gonzalez and Framinan (2014). This paper is also related to the scheduling problem with due date assignment. In many classic scheduling problems, most researchers treat the due dates as given constants. In many real-life situations such as just-in-time production and logistics, however, due dates need to be negotiated rather than simply dictated by the customers. Setting long due dates facilitates production planning and reduces the tardiness cost, but will be detrimental to revenue generation due to the possible need for price reduction and/or loss of customer goodwill. On the other hand, setting short due dates has the opposite effects. It is thus crucial to set appropriate due dates to minimize a total cost function that embraces all the relevant cost factors. This is why increasingly number of recent literature regards due date assignment as part of the scheduling process. Early research on due date assignment in single-machine scheduling is conducted by Panwalkar, Smith, and Seidmann (1982) and Seidmann, Panwalkar, and Smith (1981). Panwalkar et al. (1982) consider the constrained version where the scheduler must decide on a common due date for all the jobs (this due date assignment method is usually referred to as common due date assignment method), while Seidmann et al. (1981) study the unrestricted case where each job can have a different due date (this due date assignment method is usually referred to as unrestricted due date assignment method). These two studies have inspired extensive research on due date assignment in scheduling. Extensive reviews of scheduling models with due date assignment are provided by Cheng and Gupta (1989), Gordon, Proth, and Chu (2002), Gordon, Proth, and Chu (2002), Gordon, Proth, and Strusevich (2004), Gordon, Strusevich, and Dolgui (2012), and Lauff and Werner (2004). Therefore, we review only the works that are directly related to this paper. Shabtay and Steiner (2006)invesP  tigate the single-agent problem 1jdj j nj¼1 aj ðdj  Aj Þ þ bj U j with unrestricted due date assignment, where Aj is the acceptable lead P time for job J j , and show that the problem 1jdj j j   n ¼ 1 aj ðdj  Aj Þ þ bj U j is NP-hard  in the strong sense and that P the problem 1jdj j nj¼1 adj þ bj U j can be solved in Oðn4 Þ time.  P  However, whether the problem 1jdj j nj¼1 aj dj þ bj U j is NP-hard or polynomially solvable is still open. Koulamas (2010) further 2 develops a faster Oðn Þ dynamic programming algorithm for the  Pn  problem 1jdj j j¼1 adj þ bj U j . Yin, Liu, Cheng, Wu, and Cheng  P  (2013) address the problem 1jdj j nj¼1 adj þ bj U j with pastsequence-dependent (p-s-d) delivery times and position-dependent

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learning effects and show that it can be solved in Oðn3 Þ time. Single-machine scheduling problems with unrestricted due date assignment and controllable job-processing times are considered by Shabtay and Steiner (2007), Shabtay and Steiner (2008), where the processing time of a job can be controlled by allocating an additional resource that may speed up job processing. In their models, the job processing time is either a bounded linear function or a convex function of the amount of resource allocated to the P processing of the job. For the problem of minimizing j   ¼ 1n adj þ bj U j þ v j uj þ dC max , where uj is the amount of resource allocated to job J j ; v j is the cost for one unit of resource allocated to job J j , and C max ¼ maxj¼1;...;n C j is the maximal completion time, Shabtay and Steiner (2007) show that it can be solved in Oðn4 Þ time for both the linear and convex resource consumption functions; Pn  while for the problem of minimizing j¼1 adj þ bEj þ cT j þv j uj Þ þ dC max , where Ej ¼ maxfdj  C j ; 0g is the earliness and T j ¼ maxfC j  dj ; 0g is the tardiness of job J j , respectively, Shabtay and Steiner (2008) develop efficient algorithms that require Oðn4 Þ and Oðn log nÞ time for the linear and convex resource consumption function, respectively. Shabtay (2010) deals with the batch delivery scheduling with unrestricted due date assignment to minimize a criterion that includes earliness, tardiness, holding, due date assignment and delivery costs, where the jobs are to be delivered in batches to the downstream customer and the cost per delivery is fixed. He shows that the problem is NP-hard and develops polynomial time algorithms for two special cases. Li, Ng, and Yuan (2011) consider a single-machine scheduling problem with unrestricted due date assignment under a group technology environment, where the jobs of customers are classified into groups according to their production similarities in advance and all jobs of the same group are required to be processed contiguously on the machine. They propose an Oðn log nÞ algorithm to determine the due date assignment strategy and job schedule so as to minimize a criterion that includes earliness, tardiness, due date assignment and flow time costs. Although the literature on multi-agent scheduling and due date assignment has grown rapidly, to the best of our knowledge, no paper has combined the two sub-fields into a unified framework. Specifically, we focus on the innovative scheduling model with unrestricted due date assignment in the framework of the twoagent setting. The rest of the paper is organized as follows: In Sections 2 and 3, we analyze the computational complexity of each of the problems by establishing that both problems are NP-hard in the strong sense and provide polynomial-time or pseudo-polynomial-time solutions for some important cases, respectively. The last section concludes the paper and suggests some future research topics.   nA A A B 2. Problem 1jdj j+j¼1 aAj dj þ bAj U Aj : f max
A

1jdj j

NP-hard in the strong sense, but a special case of interest is polynomially solvable.

(2) all the jobs are processed consecutively without idle time and the first job starts at time 0; (3) all the tardy A-jobs are processed after the early A-jobs and the B-jobs in any order.

Proof. (1) The proof is similar to that of Lemma 7 in Shabtay and Steiner (2006). A (2) As a consequence of (1), if dj ¼ C Aj , job J Aj should be schedA uled as an early job; and if dj ¼ 0, job J Aj should be scheduled as a tardy job. Now assume that the first job in a schedule S starts at time t0 > 0. We construct a new schedule S0 from S by reducing the starting times of all the jobs and the due dates of the early A-jobs by t0 time units. Clearly, schedule S0 remains feasible and the objective value of S0 is smaller than that of S because we move the B-jobs backward and reduce the due dates of the early A-jobs. Next assume that there exists an optimal schedule S with an idle interval of length t between consecutive jobs J ½k and J ½kþ1 ; k ¼ 1; 2; . . . ; n  1. We shift job J ½kþ1 and any job that follows it earlier by t time units to obtain a new schedule S0 . Carrying out an analysis analogous to the above analysis, we see that schedule S0 is at least as good as S, as required. (3) Let S be an optimal schedule that does not fulfill property (3). Consider construction a new schedule S0 by moving all the tardy A-jobs to the end of the schedule while keeping the due dates for the A-jobs are identical in the two schedules. Clearly, schedule S0 remains feasible and the objective value of S0 is not larger than that of S because we move the early A-jobs and the B-jobs backward. h As

a

consequence of Proposition 2.1, the problem  aAj dAj þ bAj U Aj : f Bmax 6 Q reduces to the problem of findPnA A A ing a schedule to minimize j¼1 f j ðC j Þ, while keeping the maxi-

A P A 1jdj j nj¼1



mum of regular functions of agent B below or at a fixed PA A A B f j ðC j Þ : f max 6 Q or threshold Q, denoted by 1jj nj¼1 P A P B A A 1jdj j Jj 2NA aAj C j þ Jj 2NA bj : f max 6 Q , where E

( A

f j ðC Aj Þ ¼

T

a

A A j Cj

bAj

A A j Cj

if a

6 bAj ;

ð1Þ

otherwise; A

A

j ¼ 1; 2; . . . ; nA , and NAE ¼ fJAj jC Aj 6 dj g and N AT ¼ fJAj jC Aj > dj g denote the sets of early A-jobs and tardy A-jobs, respectively. This implies the following result.   A P A Proposition 2.2. The problem 1jdj j nj¼1 aAj dAj þ bAj U Aj : f Bmax 6 Q is P A A A B A P equivalent to either 1jj nj¼1 f j ðC j Þ : f max 6 Q or 1jdj j J 2NA aAj C Aj þ j E P B A J 2NA bj : f max 6 Q . j

T

In view of Proposition 2.2, the following result holds. A

Theorem 2.3. The problem 1jdj j

PnA  j¼1



aAj dAj þ bAj U Aj : f Bmax 6 Q is

NP-hard in the strong sense even when bAj ¼ b for j ¼ 1; . . . ; nA with

2.1. Analysis and complexity

an appropriately large b. We start with establishing some structural properties of the optimal schedule. 



A

Proof. Consider an instance of the problem 1jdj j

Proposition 2.1. For the problem 1jdj j aAj dj þ bAj U Aj : f max 6 Q ,

B bAj U Aj Þ:f max 6Q

there exists an optimal schedule that fulfills the following properties: A A (1) dj ¼ C Aj if aAj C Aj 6 bAj and dj ¼ 0 otherwise for j ¼ 1; 2; . . . ; nA ;

For

A

A

B

PnA  j¼1

aAj dAj þ

where bj ¼ b P ðP A þ P B ÞmaxJA 2JA aAj for j¼1;2;...;nA . j

such an instance, by Proposition 2.2, the problem   PA A A B A P A 1jdj j nj¼1 aAj dAj þbAj U Aj : f Bmax 6 Q reduces to 1jj nj¼1 aj C j : f max 6 Q ,

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which is known to be NP-hard in the strong sense (see Leung et al. (2010)), as required. h 2.2. A polynomially solvable case In this subsection we consider a special case of   A PnA A B 1jdj j j¼1 aAj dj þ bAj U Aj : f max 6 Q where aAj ¼ a for j ¼ 1; . . . ; nA and the A-jobs have reversely agreeable penalty costs, i.e., pAi 6 pAj implies bAi P bAj for all A-jobs J Ai and J Aj . The second condition is reasonable since it reflects the situation in which smaller jobs incur higher tardiness costs because customers normally do not expect that small jobs will be tardy and they impose disproportionately high penalties on small jobs when they become tardy. This condition also covers the case where pAj ¼ p and the case bAj ¼ b for j ¼ 1; 2; . . . ; nA . The following proposition establishes a crucial property of the optimal schedule for the problem under consideration. Proposition

2.4. For

the

problem

A

1jdj j

PnA  j¼1

D ¼ bAl þ bAh  bAh  bAl ¼ 0: Case 2: aðt þ pAh Þ 6 bAh < aðt þ pAl þ P þ pAh Þ and aðt þ pAl Þ > bAl . In this case, the difference between the objective values of schedules S and S0 is

D ¼ bAl þ bAh  aðt þ pAh Þ  bAl P 0: Case 3: aðt þ pAh Þ 6 bAh < aðt þ pAl þ P þ pAh Þ and aðt þ pAl Þ 6 b A l < aðt þ pAl þ P þ pAh Þ. In this case, the difference between the objective values of schedules S and S0 is

D ¼ aðt þ pAl Þ þ bAh  aðt þ pAh Þ  bAl ¼ ðapAl  bAl Þ  ðapAh  bAh Þ > 0: Case 4: aðt þ pAh Þ 6 bAh < aðt þ pAl þ P þ pAh Þ and aðt þ p A l þ P þ pAh Þ 6 bAl . In this case, the difference between the objective values of schedules S and S0 is

D ¼ aðt þ pAl Þ þ bAh  aðt þ pAh Þ  aðt þ pAl þ P þ pAh Þ P aðt þ pAl Þ þ bAh  aðt þ pAh Þ  bAl



¼ ðapAl  bAl Þ  ðapAh  bAh Þ > 0:

adAj þ bAj U Aj :

B

f max 6 Q , if the A-jobs have reversely agreeable penalty costs, then there exists an optimal schedule in which the A-jobs are scheduled according to nondecreasing order of apAj  bAj .

Case 5: aðt þ pAl þ P þ pAh Þ 6 bAh and aðt þ pAl Þ > bAl . In this case, the difference between the objective values of schedules S and S0 is

D ¼ bAl þ aðt þ pAl þ P þ pAh Þ  aðt þ pAh Þ  bAl ¼ aðpAl þ PÞ > 0: Proof. Consider an optimal schedule S where the A-jobs are not scheduled according to nondecreasing order of apAj  bAj . Let J Al and J Ah be the first pair of jobs such that apAl  bAl > apAh  bAh . Then pAl P pAh ; bAl 6 bAh and at least one of the two inequalities is strict since the A-jobs have reversely agreeable penalty costs. In this schedule, job J Al is scheduled earlier, followed by a set of B-jobs (say, p0 ), and then job J Ah , i.e., S ¼ ðp; J Al ; p0 ; J Ah ; p00 Þ, where p and p00 are partial job sequences. Furthermore, let t denote the completion time of the last job in p and P represent the total processing time of the jobs in p0 . Consider construction a new schedule S0 from S by interchanging jobs J Al and J Ah while leaving the other jobs unchanged in S, i.e., S0 ¼ ðp; J Ah ; p0 ; J Al ; p00 Þ. We claim that:

Case 6: aðt þ pAl þ P þ pAh Þ 6 bAh and aðt þ pAl Þ 6 bAl < aðt þ p A l þ P þ pAh Þ. In this case, the difference between the objective values of schedules S and S0 is

D ¼ aðt þ pAl Þ þ aðt þ pAl þ P þ pAh Þ  aðt þ pAh Þ  bAl P aðt þ pAl Þ þ aðt þ pAl þ P þ pAh Þ  aðt þ pAh Þ  aðt þ pAl þ P þ pAh Þ ¼ aðpAl  pAh Þ P 0: Case 7: aðt þ pAl þ P þ pAh Þ 6 bAh and aðt þ pAl þ P þ pAh Þ 6 bAl . In this case, the difference between the objective values of schedules S and S0 is

D ¼ aðt þ pAl Þ þ aðt þ pAl þ P þ pAh Þ  aðt þ pAh Þ

(1) Schedule S0 is feasible. It is obvious that the completion time of any job in p and p00 is identical in both S and S0 . Since pAl P pAh , we have C Ah ðS0 Þ ¼ t þ pAh 6 t þ pAl ¼ C Al ðSÞ, implying that the completion time of any job in p0 is earlier in S0 . It folB

B

lows that f j ðC Bj ðS0 ÞÞ 6 f j ðC Bj ðSÞÞ for any j ¼ 1; 2; . . . ; nB . So

 aðt þ pAl þ P þ pAh Þ ¼ aðpAl  pAh Þ P 0: Thus, for any case, we have D P 0, as required. So repeat doing this for all the A-jobs not sequenced according to the specified rule yields the result. h

0

schedule S is feasible since every job in S is feasible. (2) Schedule S0 is not a worse schedule than S. By Eq. (1), we have

A

f l ðC Al ðSÞÞ ¼ A f h ðC Ah ðSÞÞ

 ¼

A

f h ðC Ah ðS0 ÞÞ ¼ A



f h ðC Al ðSÞÞ ¼

aðt þ pAl Þ bAl

if aðt þ pAl Þ 6 bAl ; otherwise;

aðt þ pAl þ P þ pAh Þ bAh





aðt þ pAh Þ bAh

if aðt þ pAh Þ 6 bAh ; and otherwise;

aðt þ pAl þ P þ pAh Þ bAl

if aðt þ pAl þ P þ pAh Þ 6 bAh ; otherwise;

if aðt þ pAl þ P þ pAh Þ 6 bAl ; otherwise:

Now, by Proposition 2.2, there are seven cases to consider. Case 1: aðt þ pAh Þ > bAh and aðt þ pAl Þ > bAl . In this case, the difference between the objective values of schedules S and S0 is

Now, for each B-job J Bj , let us define a deadline DBj such that B 6 Q for C Bj 6 DBj and f j ðC Bj Þ > Q for C Bj > DBj . (If the inverse B function f j ðÞ is available, the deadlines can be computed in constant time; otherwise this requires logarithmic time.)

B f j ðC Bj Þ

Proposition

2.5. For

the

problem

A

1jdj j

PnA  j¼1



aAj dAj þ bAj U Aj :

B

f max 6 Q , there exists an optimal schedule in which the B-jobs are scheduled according to nondecreasing order of DBj . Proof. Consider an optimal schedule S where the B-jobs are not scheduled according to non-decreasing order of DBj . Let J Bl and J Bh be the first pair of jobs, such that DBl > DBh . In this schedule, job J Bl is scheduled earlier followed by a set of A-jobs (say, p), and then job J Bh , i.e., S ¼ ðp; J Bl ; p0 ; J Bh ; p00 Þ, where p and p00 are partial job sequences. Consider constructing a new schedule S0 from S by extracting job J Bl , reinserting it just after job J Bh while leaving the other jobs unchanged in S, i.e., S0 ¼ ðp; p0 ; J Bh ; J Bl ; p00 Þ. By Lemma 2.1, the completion time of job J Bh in S is equal to that of job J Bl in S0 , i.e., C Bl ðS0 Þ ¼ C Bh ðSÞ, and therefore the completion time of any

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Y. Yin et al. / Computers & Industrial Engineering 79 (2015) 148–155

job in p and p00 in S is not affected. Since S is feasible, it follows that C Bl ðS0 Þ ¼ C Bh ðSÞ 6 DBh < DBl , and so job J Bl is feasible in S0 . Moreover, each job in p and job J Bh are scheduled earlier in S0 , implying that their actual processing times are smaller in S0 and so their completion times are earlier in S0 , and thus they remain feasible. Therefore, schedule S0 is still feasible and optimal. Thus, repeat doing this for all B-jobs not sequenced according to the non-decreasing order of DBj will yield the theorem. h

Else go to step 4; Endif Endif Step 4: If h P 1 or k P 1, then go to Step 2. Else output the sequence S ¼ ðSð1Þ; Sð2Þ; . . . ; SðnA þ nB ÞÞ, the optimal due dates of

From Proposition 2.4, we know that there exists an optimal schedule in which the early A-jobs are scheduled according to nonde-

A

A

A jobs, (d½1 ; . . . ; d½nA  ), and the optimal objective value F.

creasing order of apAj  bAj , while Proposition 2.5 indicates that there exists an optimal schedule in which the B-jobs are scheduled

Endif

A

in nondecreasing order of DBj . Noting also that f j ðC Aj Þ is a regular scheduling objective function, which is also a nondecreasing function of DBj , we should assign the B-jobs as late as possible in any optimal schedule. Based on these results, we can develop an algorithm   A P A for the case of the problem 1jdj aj nj¼1 adAj þ bAj U Aj : f Bmax 6 Q where the A-jobs have reversely agreeable penalty costs as follows: In each step, we select a job among the unassigned jobs to be processed last. Let T be the sum of the processing times of the unassigned jobs. Any unassigned B-job with the maximum Dj value such that T 6 Dj should be assigned to complete at T. If there is no such B-job, we assign the A-job with the maximum apAj  bAj value. If during the course of executing the algorithm all the A-jobs have been assigned and no B-job can be assigned last, we conclude that the instance is unfeasible. The algorithm can be described in detail as follows: Algorithm 1. Step 1: Set T ¼ PA þ P B ; k ¼ nA ; h ¼ nB ; l ¼ nA þ nB , and F ¼ 0; sort the A-jobs in nondecreasing order of apAj  bAj such that

apA½1  bA½1 6 apA½2  bA½2 6 . . . 6 apA½nA   bA½nA  ; and calculate the deadline for each B-job JBj

Theorem 2.6. If the A-jobs have reversely penalty costs, Algorithm 1 A P A 1jdj j nj¼1

generates an optimal schedule for the problem   adAj þ bAj U Aj : f Bmax 6 Q in OðnA log nA þ nB log nB Þ time.

Proof. The correctness is guaranteed by Propositions 2.1 and 2.2, B and Propositions 2.4 and 2.5. Suppose that each f j ðÞ value can be computed in constant time. The two sorting procedures in Step 1 require OðnA log nA Þ and OðnB log nB Þ, respectively. In each execution of Steps 2 and 3, there is only one candidate from each job set. Thus, the overall computational complexity of the algorithm is given as OðnA log nA þ nB log nB Þ. h Remark 2.7. If pAj ¼ p for j ¼ 1; 2; . . . ; nA or bAj ¼ b for j ¼ 1; 2; . . . ; nA , then the A-jobs have reversely agreeable penalty P A costs, which implies that both problems 1jdj ; pAj ¼ pj j     P A B A A ¼ 1nA adj þ bAj U Aj : f max 6 Q and 1jdj j nj¼1 adAj þ bU Aj : f Bmax 6 Q can be solved in OðnA log nA þ nB log nB Þ time by Theorem 2.6. Now we present an example to illustrate the working of Algorithm 1.

from f j ðDBj Þ ¼ Q and sort them in nondecreasing order such that

Example

DB½1 6 DB½2 6 . . . 6 DB½nB  ;

2, respectively, and f j ¼ C Bj  dj for j ¼ 1; . . . ; nB . We see that

B

B

If T 6 DB½h , then set SðlÞ ¼

an

instance

with

B

Step 2: If h P 1, then J B½h ; l

2.8. Consider

nA ¼ 5; nB ¼ 4; a ¼ 2; Q ¼ 3; bAj ; pAj ; pBj ; dj as given in Tables 1 and B

apA1  bA1 < apA2  bA2 < apA3  bA3 < apA4  bA4 < apA5  bA5 ; ¼ l  1; h ¼ h  1; T ¼ T 

pB½h ,

and go

to step 2; Else go to step 3; Endif Else go to step 3; Endif Step 3: If k P 1, then set SðlÞ ¼ J A½k ; l ¼ l  1 and k ¼ k  1; If aT 6 bA½k then A

set F ¼ F þ aT; d½k ¼ T, T ¼ T  pA½k , and go to step 4; else A

set F ¼ F þ bA½k , d½k ¼ 0; T ¼ T  pA½k , and go to step 4; Elseif h P 1 output the result that the instance is not feasible;

B

and the DBj values can be calculated from DBj  dj ¼ Q and given in Table 2. Using Algorithm 1 to solve the instance, we have the following results. Step 1: Since T ¼ PA þ PB ¼ 210 > 203 ¼ DB4 and aT ¼ 420 > 200 ¼ bA5 , set Sð9Þ ¼ JA5 ; F ¼ bA5 ¼ 200, dA5 ¼ 0, and T ¼ T  pA5 ¼ 180. Step 2: Since T ¼ 180 < 203 ¼ DB4 , set Sð8Þ ¼ J B4 and T ¼ T  pB4 ¼ 145. Step 3: Since T ¼ 145 < 153 ¼ DB3 , set Sð7Þ ¼ J B3 and T ¼ T  pB3 ¼ 120. Step 4: Since T ¼ 120 > 108 ¼ DB2 and aT ¼ 240 > 230 ¼ bA4 , set A Sð6Þ ¼ J A4 ; F ¼ F þ bA4 ¼ 430; d4 ¼ 0, and T ¼ T  pA4 ¼ 95. Table 1 Data of A-jobs. Job J Aj

J A1

pAi bai

JA2

J A3

JA4

J A5

10

15

20

25

30

310

305

250

230

200

153

Y. Yin et al. / Computers & Industrial Engineering 79 (2015) 148–155 A

f h ðC Ah ðSÞÞ ¼ aC Ah ðSÞ ¼ aðt þ pAl þ P þ pAh Þ 6 bAh :

Table 2 Data of B-jobs.

It follows that bAh P aðt þ pAl þ P þ pAh Þ > aðt þ pAh Þ ¼ C Ah ðS0 Þ, i.e., J Ah is

Job JBj

JB1

J B2

JB3

JB4

pBi

30

20

25

35

di

B

90

105

150

200

DBi

93

108

153

203

A

an early job in S0 , and so f h ðC Ah ðS0 ÞÞ ¼ aC Ah ðS0 Þ. For job J Al , we have two cases to consider. Case 1: aðt þ pAl þ P þ pAh Þ 6 bAl . In this case, J Al is an early job in A S0 , and so f l ðC Al ðS0 ÞÞ ¼ aC Al ðS0 Þ. The difference between the objective values of schedules S and S0 is

Step 5: Since T ¼ 95 < 108 ¼ DB2 , set Sð5Þ ¼ J B2 and T ¼ T  pB2 ¼ 75. Step 6: Since T ¼ 75 < 93 ¼ DB1 , set Sð4Þ ¼ J B1 and T ¼ T  pB1 ¼ 45. Step 7: Since aT ¼ 90 < 250 ¼ bA3 , set Sð3Þ ¼ J A3 ; F ¼ F þ aT ¼ 520, A d3 ¼ 45, and T ¼ T  pA3 ¼ 25. Step 8: Since aT ¼ 50 < 305 ¼ bA2 , set Sð2Þ ¼ J A2 ; F ¼ F þ aT ¼ 570, A d2 ¼ 25, and T ¼ T  pA2 ¼ 10. Step 9: Since aT ¼ 20 < 310 ¼ bA1 , set Sð1Þ ¼ J A1 ; F ¼ F þ aT ¼ 590, A d1 ¼ 10, and T ¼ T  pA1 ¼ 0. It follows that the optimal job sequence is (J A1 ; J A2 ; J A3 ; J B1 ; J B2 ; J A4 ; J B3 ; J B4 ; J A5 ), the optimal due dates of the AA A A A A (d1 ; d2 ; d3 ; d4 ; d5 )

are (10; 25; 45; 0; 0), and the optimal jobs objective value is 590.

In A A j dj

a

this þ

bAj U Aj

A

plexity of the problem and develop a pseudo-polynomial-time dynamic programming for the case where aAj ¼ a for j ¼ 1; 2; . . . ; nA . 3.1. Analysis and complexity The following proposition provides some structural properties of the optimal schedule for the problem under consideration. A

A



Proposition 3.1. For the problem 1jdj j j¼1 aAj dj þ bAj U Aj : PnB B j¼1 U j 6 Q , there exists an optimal schedule that fulfills the following properties: A

D ¼ aðC Al ðSÞ þ C Ah ðSÞ  C Ah ðS0 ÞÞ  bAl P aðC Al ðSÞ þ C Ah ðSÞ  C Ah ðS0 Þ  C Al ðS0 ÞÞ ¼ aðpAh  pAl Þ > 0: Thus, in both cases, we have D > 0, as required. So repeat doing this for all the A-jobs not sequenced according to the SPT rule yields the result. h

A

(1) dj ¼ C Aj if aAj C Aj 6 bAj and dj ¼ 0 otherwise for j ¼ 1; 2; . . . ; nA ; (2) all the early jobs are processed consecutively without idle time such that the first early job starts at time 0 and all the tardy jobs are scheduled after the early jobs in any order; (3) the early A-jobs are scheduled according to the shortest processing time (SPT) order if aAj ¼ a for j ¼ 1; . . . ; nA , and the early Bjobs are scheduled according to the earliest due date (EDD) order. Proof. The proof of (1), (2) and the second part of (3) is analogous to that of Propositions 2.1 and 2.5. We only prove the first part of (3). We adopt the same notations as that in the proof of Proposition 2.4. Consider an optimal schedule S where the early A-jobs are not scheduled according to nondecreasing order of pAj . Let J Al and J Ah be the first pair of jobs such that J Al ; J Ah 2 N AE but pAl > pAh . In this schedule, job J Al is scheduled earlier, followed by a set of B-jobs (say p0 ), and then job JAh , i.e., S ¼ ðp; JAl ; p0 ; JAh ; p00 Þ. Consider constructing a new schedule S0 from S by interchanging jobs J Al and J Ah while leaving the other jobs unchanged in S. Analogous to the proof of Proposition 2.4, the completion time of every B-job is earlier in S0 , implying that schedule S0 is feasible since S is feasible. In what follows, we prove that S0 is not a worse schedule than S. Since both J Al and J Ah are early, by Eq. (1), we have

PnA 



P B B A A A A : nj¼1 Uj 6 j dj þ bj U j j¼1 P Pn B B nA A A Q is equivalent to either 1jj j¼1 f j ðC j Þ : j¼1 U j 6 Q or P P B B A P 1jdj j J 2NA Aj C Aj þ J 2NA bAj : nj¼1 U j 6 Q , where j j E T A

PnA

section we consider the problem 1jdj j j¼1  P B : nj¼1 U Bj 6 Q . We analyze the computation com-

Pn A 

Case 2: aðt þ pAl þ P þ pAh Þ > bAl . In this case, J Al is a tardy job in S0 , A and so f l ðC Al ðS0 ÞÞ ¼ bAl . The difference between the objective values of schedules S and S0 is

Analogous to Proposition 2.2, we obtain the following result.

  nA nB A A 3. Problem 1jdj j+j¼1 aAj dj þ bAj U Aj : +j¼1 U Bj
D ¼ aðC Al ðSÞ þ C Ah ðSÞ  C Ah ðS0 Þ  C Al ðS0 ÞÞ ¼ aðpAh  pAl Þ > 0:

Proposition 3.2. The problem 1jdj j

a

a

( A

f j ðC Aj Þ ¼

aAj C Aj

if aAj C Aj 6 bAj ;

bAj

otherwise;

j ¼ 1; 2; . . . ; nA . We now address the complexity issue of the problem PB B A P A 1jdj j nj¼1 aAj dAj þ bAj U Aj : nj¼1 Uj 6 Q A

Theorem 3.3. The problem 1jdj j

PnA  j¼1

 P nB

aAj dAj þ bAj U Aj :

B j¼1 U j

6Q

bAj

is NP-hard in the strong sense even when ¼ b for j ¼ 1; 2; . . . ; nA with an appropriately large b. Furthermore, the problem   P A P A B 1jdj j nj¼1 adAj þ bAj U Aj : nj¼1 U Bj 6 Q is NP-hard even when bAj ¼ b for j ¼ 1; 2; . . . ; nA with an appropriately large b. A P A Proof. Consider an instance of the problem 1jdj j nj¼1   P B aAj dAj þ bAj U Aj : nj¼1 U Bj 6 Q where bj ¼ b P ðP A þ P B ÞmaxJA 2JA aAj j

for j ¼ 1; 2; . . . ; nA . For such an instance, by Proposition 3.2, the   P A P A A P A B problems 1jdj j nj¼1 aAj dAj þ bAj U Aj : nj¼1 U Bj 6 Q and 1jdj j nj¼1   P P A A A PnB B B adAj þ bAj U Aj : nj¼1 U Bj 6 Q reduce to 1jj nj¼1 aj C j : j¼1 U j 6 Q PnA A PnB B and 1jj j¼1 aC j : j¼1 U j 6 Q , which are known to be NP-hard in the strong sense and NP-hard in ordinary the sense (See Theorem 1 in Leung et al. (2010) and Theorem 2 in Ng et al. (2006)), respectively. h 3.2. Pseudo-polynomial-time algorithm for the problem   P A PnA A A A B : nj¼1 U Bj 6 Q j¼1 adj þ bj U j

1jdj j

A P A 3.3 indicates that the problem 1jdj j nj¼1  P B adAj þ bAj U Aj : nj¼1 U Bj 6 Q is NP-hard. In this subsection, we



Theorem

f l ðC Al ðSÞÞ ¼ aC Al ðSÞ ¼ aðt þ pAl Þ 6 bAl

present a pseudo-polynomial-time algorithm for the problem   P A P A B adAj þ bAj U Aj : nj¼1 U Bj 6 Q , establishing that it is 1jdj j nj¼1

and

NP-hard in the ordinary sense.

A

154

Y. Yin et al. / Computers & Industrial Engineering 79 (2015) 148–155

8 Fði  1; j; t  pAi ; kÞ þ at; > > > > > < Fði  1; j; t; kÞ þ bA ; i Fði; j; t; kÞ ¼ min B > > Fði; j  1; t  pj ; kÞ þ f ði; j; t; kÞ; > > > : Fði; j  1; t; k  1Þ:

In what follows, we say that a schedule S for the problem   P A P A B 1jdj j nj¼1 adAj þ bAj U Aj : nj¼1 U Bj 6 Q is regular if S sequences the early A-jobs according to the SPT order and the early B-jobs according to the EDD order. By Proposition 3.1, there is an optimal sche  P A P A B dule for the problem 1jdj j nj¼1 adAj þ bAj U Aj : nj¼1 U Bj 6 Q such that S is regular. Now suppose that A-jobs are re-numbered according to the SPT order and B-jobs are re-numbered according to the EDD order, respectively. We next present a recursion relation that can be used to design a pseudo-polynomial-time dynamic programming algo  P A P A B rithm for 1jdj j nj¼1 adAj þ bAj U Aj : nj¼1 U Bj 6 Q . Let ði; j; t; kÞ be a

The boundary conditions are Fði; j; t; kÞ ¼ þ1 when i < 0; j < 0, P P t < 0; t > il¼1 pAl þ jl¼1 pBl ; k < 0, or k > j.

state encoding a partial solution for the jobs fJ A1 ; J A2 ; . . . ; J Ai g and

FðnA ; nB ; t; Q Þ which can be computed in OðnA n2B ðP A þ P B ÞÞ time.

Theorem 3.4. The optimal solution value of the problem   P A PnA A A A B : nj¼1 U Bj 6 Q is given as F  ¼ min06t6PA þPB j¼1 adj þ bj U j

1jdj j

fJ B1 ; J B2 ; . . . ; J Bj g, where t measures the completion time of the last early job, and k (0 6 k 6 j) stands for the number of tardy B-jobs. Let Fði; j; t; kÞ be the value of an optimal solution for the problem   P A P A B adAj þ bAj U Aj : nj¼1 U Bj 6 Q for the state ði; j; t; kÞ and let 1jdj j nj¼1 Sði; j; t; kÞ be any regular schedule in state ði; j; t; kÞ with solution value Fði; j; t; kÞ. Since all early A-jobs and early B-jobs among P P fJ A1 ; J A2 ; . . . ; J Ai ; J B1 ; J B2 ; . . . ; J Bj g are completed by time il¼1 pAl þ jl¼1 pBl , so Fði; j; t; kÞ is defined for i ¼ 1; . . . ; nA ; j ¼ 1; . . . ; nB ; k P P ¼ 0; . . . ; j and t ¼ 0; 1; . . . ; il¼1 pAl þ jl¼1 pBl . Moreover, we set Fði; j; t; kÞ ¼ þ1 if no such schedule exists. Then the regular schedule Sði; j; t; kÞ can be constructed by taking one of the following four decisions in a previous state. and

J Ai

(1) Job is the last early job: in this case Sði; j; t; kÞ is obtained from regular schedule Sði  1; j; t  pAi ; kÞ and Fði; j; t; kÞ ¼ Fði  1; j; t  pAi ; kÞ þ at. (2) Job J Ai is tardy: in this case Sði; j; t; kÞ is obtained from regular schedule Sði  1; j; t; kÞ and Fði; j; t; kÞ ¼ Fði  1; j; t; kÞ þ bAi . (3) Job J Bj is the last early job: in this case Sði; j; t; kÞ is obtained from regular schedule Sði; j  1; t  pBj ; kÞ and Fði; j; t; kÞ ¼ Fði; j  1; t  pBj ; kÞ. (4) Job J Bj is tardy: in this case Sði; j; t; kÞ is obtained from regular schedule Sði; j  1; t; k  1Þ and Fði; j; t; kÞ ¼ Fði; j  1; t; k  1Þ.

(1) If t >

then

J Bj

cannot be the last early job. So either

J Ai

is

t. So either one of J Ai and J Bj is the last early job or J Ai is a tardy job or J Bj is a tardy job. Based on the above analysis, we develop a dynamic programming recursion for the following two cases.

( Fði; 0; t; kÞ ¼ min

f ði; j; t; kÞ ¼

þ

Pj

B l¼1 pl ,

Proof. The proof is analogous to that of Proposition 2.4.

Pi

A l¼1 pl

þ

Pj

B l¼1 pl ,

and 0 6 k. Then we

the early jobs in fJ B1 ; J B2 ; . . . ; J Bj g, and k (0 6 k 6 j) stands for the number of tardy B-jobs. Let Fði; j; PEBj ; kÞ be the value of an optimal solu  P A P A B tion for the problem 1jdj j nj¼1 adAj þ bU Aj : nj¼1 U Bj 6 Q for the state ði; j; PEBj ; kÞ. Denote by P Ai the total processing time of the jobs P in fJ A1 ; J A2 ; . . . ; J Ai g, i.e., P Ai ¼ il¼1 pAl . By definition, we set Fði; j; PEBj ; kÞ ¼ þ1 if no such schedule exists. Then we can write a dynamic programming recursion for the following two cases. Case 1: 1 6 i; j ¼ 0; PEB0 ¼ 0, and 0 6 k. Then we have

( Fði; 0; 0; kÞ ¼ min

and 0 6 k. Then we

t>

otherwise:

Fði  1; 0; 0; kÞ þ aPAi ; Fði  1; 0; 0; kÞ þ b:

Case 2: 1 6 i; j ¼ 0; 0 6 PEB0 6 Then we have

( B dj ;

h

As a consequence of Propositions 3.1 and 3.5, we develop a recursion relation that can be used to design a pseudo-polynomial-time dynamic programming algorithm for   P A P A B 1jdj j nj¼1 adAj þ bU Aj : nj¼1 U Bj 6 Q as follows: Let ði; j; PEBj ; kÞ be a

Fði  1; 0; t; kÞ þ bAi :

þ1; if 0;

A l¼1 pl

Fði  1; 0; t  pAi ; kÞ þ at;

Case 2: 1 6 i; 1 6 j; 0 6 t 6 have

(

Pi

  A P A Proposition 3.5. For the problem 1jdj j j ¼ 1nA adj þ bU Aj : PnB B j¼1 U j 6 Q , there exists an optimal schedule in which the A-jobs are arranged in the SPT order.

fJ B1 ; J B2 ; . . . ; J Bj g, where PEBj measures the total processing time of

the last early job or J Ai is a tardy job or J Bj is a tardy job. B (2) If t 6 dj , we still allow J Bj to be a tardy job by the definition of

Case 1: 1 6 i; j ¼ 0; 0 6 t 6 have

In the sequel, we further address a special case of the problem where aAj ¼ a and bAj ¼ b for j ¼ 1; 2; . . . ; nA . The following result demonstrates the structure properties of the optimal schedule.

state encoding a partial solution for the jobs fJ A1 ; J A2 ; . . . ; J Ai g and

When j P 1, we observe the following two facts. B dj ,

Proof. The correctness is guaranteed by Propositions 3.1 and 3.2 and the principles underlying dynamic programming. We now address the time complexity of the algorithm. There are at most nA n2B ðPA þ P B Þ states in Fði; j; t; kÞ. Computing each Fði; j; t; kÞ takes constant time, so computing all of them needs OðnA n2B ðPA þ PB ÞÞ time, while computing F  takes OðP A þ PB Þ time. Thus, the overall computational complexity of the algorithm is given as OðnA n2B ðP A þ P B ÞÞ. h

f ði; j; PEBj ; kÞ

¼

þ1; if 0;

Pj

B l¼1 pl ,

and 0 6 k, and k P 0. B

PAi þ PEBj > dj ;

otherwise:

Y. Yin et al. / Computers & Industrial Engineering 79 (2015) 148–155

Fði; j; PEBj ; kÞ ¼ min

8 > Fði  1; j; PEBj ; kÞ þ aðPAi þ PEBj Þ; > > > > > > < Fði  1; j; PEBj ; kÞ þ b; > > Fði; j  1; PEBj  pBj ; kÞ þ f ði; j; PEBj ; kÞ; > > > > > : Fði; j  1; PEBj ; k  1Þ:

The boundary conditions are Fði; j; PEBj ; kÞ ¼ þ1 when i < 0, P j < 0; PEBj < 0; PEBj > jl¼1 pBl ; k < 0, or k > j. A P A Theorem 3.6. The optimal solution value of the problem 1jdj j nj¼1   P B adAj þ bU Aj : nj¼1 U Bj 6 Q is given as F  ¼ min06PEBn 6PB FðnA ; nB ; B

PEBnB ; Q Þ which can be computed in OðnA n2B P A Þ time. Proof. The correctness is guaranteed by Propositions 3.1 and 3.5, and the principles underlying dynamic programming. We now address the time complexity issue. The values PAi (i ¼ 1; 2; . . . ; nA ) can be determined in OðnA Þ time initially. There are at most nA n2B PA states in Fði; j; PEBj ; kÞ. Computing each Fði; j; PEBj ; kÞ takes constant time, so computing all of them needs OðnA n2B P A Þ time, while computing F  needs OðPB Þ time. Thus, the overall computational complexity is dominated by the former quantity. h 4. Conclusions This paper proposes an innovative scheduling model that considers two competing agents and unrestricted due date assignment simultaneously. The objective is to find the optimal due dates for the A-jobs and the job sequence for both agents’ jobs simultaneously to minimize a certain criterion of agent A with the restriction that the objective value of agent B cannot exceed a given  A P A bound Q. We show that both the problems 1jdj j nj¼1 aAj dAj þ  P P B A B U Bj 6 Q are bAj U Aj Þ : f max 6 Q and 1jdj j j ¼ 1nA aAj d Aj þ bAj U Aj Þ : nj¼1 NP-hard in the strong sense. We also present a polynomial-time   A P A adAj þ bAj U Aj : f Bmax 6 Q where algorithm for the problem 1jdj j nj¼1 the A-jobs have reversely agreeable penalty costs, and a pseudopolynomial dynamic programming algorithm for the problem   P A P A B 1jdj j nj¼1 adAj þ bAj U Aj : nj¼1 U Bj 6 Q . For further research, it would be interesting to consider the common due date assignment method, which may render the problem harder to solve given the common due date constraint, and extend the problem to the multi-machine setting incorporating a variety of real-world operating characteristics.

Acknowledgments We are grateful to the Editor, Associate Editor, and two anonymous referees for their constructive comments on earlier versions of our paper. This research was supported in part by the National Natural Science Foundation of China (No. 71301022); the Natural Science Foundation of Jiangxi of China (No. 20132BAB201010); the Science Foundation of Education Committee of Jiangxi of China (No. GJJ13458); in part by the National Science Council (NSC) of Taiwan under Grant No. NSC 102–2221-E-035–070-MY3; and in part by Ministry of Science and Technology (MOST) of Taiwan under Grant No. MOST 103–2410-H-035–022-MY2.

155

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