Two-Agent Preemptive Pareto-Scheduling to Minimize Late Work and Other Criteria

: In this paper, we consider three preemptive Pareto-scheduling problems with two competing agents on a single machine. In each problem, the objective function of agent A is the total completion time, the maximum lateness, or the total late work while the objective function of agent B is the total late work. For each problem, we provide a polynomial-time algorithm to characterize the trade-off curve of all Pareto-optimal points.


Introduction
In recent decades, scheduling with two competing agents and scheduling with late-work criterion have been two hot topics in scheduling research. However, research for the combination of the two topics has not been studied extensively. One reason for this phenomenon stems from the fact that the single-machine scheduling problem for minimizing the total late work is already NP-hard when preemption is not allowed. Given the polynomial solvability of the preemptive scheduling problem for minimizing the total late work, we study the single-machine two-agent preemptive Pareto-scheduling problems with the total late work being one of the criteria.
Problem Formulation: Consider two competing agents A and B. For each agent X ∈ {A, B}, let J X = {J X 1 , J X 2 , . . . , J X n X } be the set of jobs of agent X, where J A ∩ J B = ∅ and the jobs in J X are called the X-jobs. Each job J X j has a processing time p X j > 0 and a due date d X j ≥ 0 which are integrally valued. The n = n A + n B independent jobs in J A ∪ J B need to be preemptively processed on a single machine. Let P X = ∑ n X j=1 p X j and P = P A + P B . All jobs considered in this paper are available at time zero. Our problems allow us to assume that the maximum due date of all jobs is at most P.
Let σ be a feasible schedule which assigns the jobs for processing in pieces in the interval [0, +∞). To enhance the flexibility of analysis, we allow the existence of idle times in a feasible schedule. The completion time of job J X j in σ is denoted as C X j (σ). The late work of J X j in σ, denoted Y X j (σ), is the amount of processing time of J X j after its due date d X j in σ. If Y X j (σ) = 0, J X j is called early in σ. If 0 < Y X j (σ) < p X j , J X j is called partially early in σ. If Y X j (σ) = p X j , J X j is called late in σ. The scheduling criteria related to our research are given by ∑ C A j = ∑ n A j=1 C A j (σ) (the total completion time of the A-jobs under schedule σ), L A max = max{C A j (σ) − d A j : 1 ≤ j ≤ n A } (the maximum lateness of the A-jobs in schedule σ), and ∑ j Y X j = ∑ n X j=1 Y X j (σ) (the total late work of the X-jobs under schedule σ). Then the three Pareto-scheduling problems studied in this paper are given by they presented a branch-and-bound algorithm. Zhang and Wang [24] presented a two-agent scheduling problem where the objective is to minimize the total weighted late work of agent A, while keeping the maximum cost of agent B cannot exceed a given bound U. They addressed the complexity of those problems, and presented the optimal polynomial-time algorithms or pseudo-polynomial-time algorithm to solve the scheduling problems, respectively. Zhang and Yuan [25] considered the same problem as above and further studied the three versions of the problem. Our research also uses some results in the single-machine preemptive scheduling with forbidden intervals (or maintenance activities), i.e., 1, h m |pmtn| f , where "1, h m " means that there are m forbidden intervals on the single machine and " f " is the objective function to be minimized. Without reviewing this scheduling topic in detail, we only state two known results used in our discussion. Lee [26] showed that problem 1, h m |pmtn| ∑ C j can be solved by the preemptive SPT rule in O(m + n log n) time and problem 1, h m |pmtn|L max can be solved by the preemptive EDD rule in O(m + n log n) time.
Our Contributions: In Section 2, we introduce some notations and definitions and present several important lemmas. In Section 3, we show that the trade-off curve of problem 1|pmtn| # (∑ C A j , ∑ Y B j ) can be determined in O(nn A n B ) time. In Section 4, we show that the trade-off curve of problem 1|pmtn| # (L A max , ∑ Y B j ) can be determined in O(nn A n B ) time. In Section 5, we show that the trade-off curve of problem 1|pmtn| # (∑ Y A j , ∑ Y B j ) can be determined in O(n log n) time. Finally, some concluding remarks are given in Section 6.

Preliminaries
Let J A ∪ J B be the job instance to be preemptively scheduled on a single machine. The preemption assumption allows us to schedule each job in pieces. For a piece J X j of job J X j , we use p X j to denote the length (processing time) of J X j and use C X j (σ) to denote the completion time of J X j in schedule σ.
In the following, we consider the Pareto-scheduling problem 1|pmtn| We use Ω(J A , J B ) to denote the set of all Pareto-optimal points of this problem.
In a schedule σ of J A ∪ J B , each B-job J B j is partitioned into two parts: the early part J BE j (σ) and the late part J BY j (σ), where J BE j (σ) is processed before time d B j in σ and J BY j (σ) is processed after time d B j in σ. Moreover, p BE j (σ) = p B j − Y B j (σ) and p BY j (σ) = Y B j (σ) are used to denote the lengths (processing times) of J BE j (σ) and J BY j (σ), respectively. A part of length 0 is called a trivial part. We allow the existence of trivial parts to enhance flexibility in analysis. Then we have For convenience, we renumber the B-jobs such that and keep this numbering throughout this paper. Let σ B 0 = (J B 1 , J B 2 , . . . , J B n B ) which schedules the B-jobs in the EDD order described in (2). From Potts and Van Wassenhove [19], the optimal value of problem 1|pmtn| ∑ Y j on instance J B is given by T max (σ B 0 ). Then we have the following lemma.
To make our analysis operational, we now consider an integer Y * ∈ [T max (σ B 0 ), P B ] and present a procedure to schedule the B-jobs preemptively with a particular structure. This procedure imitates the algorithm in Hariri et al. [21] for solving problem 1|pmtn| ∑ w j Y j .

Algorithm 1:
For scheduling the B-jobs according to the value of Y * .

Input:
The B-jobs J B with the EDD order in (2) and an integer Y * ∈ [T max (σ B 0 ), P B ].
Step 1: Determine the minimum index j * ∈ {1, 2, . . . , n B } such that p B Step 2: Decompose the critical B-job J B j * into two parts J BE j * and J BY j * such that We call J BE j * and J BY j * the early part and the late part of J B j * , respectively, corresponding to Y * .
2) Schedule the jobs (or pieces) in J BE (Y * ) by using the algorithm in Hariri et al. [21] for solving problem 1|pmtn| ∑ Y j on instance J BE (Y * ): Beginning from time d B n B , schedule the jobs (or pieces) in J BE (Y * ) backwards in the order J B n B , J B n B −1 , . . . , J B j * +1 , J BE j * such that each job (or piece) in J BE (Y * ) is scheduled as late as possible subject to its due date. Output: The schedule σ B(Y * ) of the B-jobs.
It can be observed that Procedure(Y * ) runs in O(n) time. An objective function of the A-jobs, denoted f A , is called regular if f A is nondecreasing in the completion times of the A-jobs. Please note that ∑ C A j and L A max are regular, but ∑ Y A j is not regular since the preemptive assumption. The following lemma is critical in our discussion.
, where either f A = ∑ Y A j or f A is a regular objective function of the A-jobs. Assume that ( f * , Y * ) ∈ Ω(J A , J B ) and let σ B(Y * ) be the schedule of J B generated by Procedure(Y * ). Then there exists a Pareto-optimal schedule π corresponding to ( f * , Y * ) in which the B-jobs are scheduled in the same manner as that in σ B(Y * ) . Such a Pareto-optimal schedule π is called a Y * -standard schedule in the sequel.
Proof. Let σ be a Pareto-optimal schedule corresponding to ( f * , Y * ) such that C max (σ) is as small as possible. Then no idle exists in σ, and so, C max (σ) = P.
The late parts of B-jobs can be scheduled arbitrarily late without affecting the objective values f A and ∑ Y B j . Thus, by shifting the late parts of B-jobs in σ, we obtain a new Pareto-optimal schedule σ 1 corresponding to ( f * , Y * ) such that the following property (P1) holds for σ 1 .
(P1) The late parts of B-jobs are scheduled consecutively in the interval [P * , P * + Y * ] in an arbitrary order without idle time.
We next generate a Pareto-optimal schedule σ 2 corresponding to ( f * , Y * ) such that following property (P2) holds for σ 2 .
(P2) For every nontrivial early part J BE j (σ 2 ) and every nontrivial late part J BY k (σ 2 ) among the B-jobs, we have j ≥ k.
If σ 1 has the property (P2), we just set σ 2 = σ 1 . Otherwise, there are a nontrivial early part J BE j (σ 1 ) and a nontrivial late part J BY k (σ 1 ) among the B-jobs, such that j < k. From (2), we have d B j ≤ d B k . Let δ = min{p BE j (σ 1 ), p BY k (σ 1 )}. By exchanging an amount of length δ between J BE j (σ 1 ) and J BY k (σ 1 ) in σ 1 , we obtain a new schedule without changing the objective values but with improving in the direction we need. Repeating this procedure, we eventually obtain a new Pareto-optimal schedule σ 2 corresponding to ( f * , Y * ) such that both properties (P1) and (P2) hold for σ 2 .
Let σ 3 be the schedule obtained from σ 2 by rescheduling (if necessary) the early parts of B-jobs in the order J BE 1 (σ) ≺ J BE 2 (σ) ≺ · · · ≺ J BE n B (σ). Since the EDD property described in (2), the early parts of B-jobs in σ 2 are also early in σ 3 . Then σ 3 is a Pareto-optimal schedule corresponding to ( f * , Y * ) such that properties (P1) and (P2), and additionally, the following property (P3), hold for σ 3 .
(P3) The early parts of B-jobs are scheduled in the order J BE Since σ 3 has the three properties (P1)-(P3), from Procedure (Y * ) for generating σ B(Y * ) , we know that σ 3  Now let π be the schedule obtained from σ 3 by the following two actions: (i) from time P * , reschedule the late parts in J BY (Y * ) consecutively in the order J B 1 , J B 2 , . . . , J B j * −1 , J BY j * , and (ii) without changing the processing order of A-jobs and the processing order of the early parts of B-jobs, reschedule them such that the early parts of B-jobs are scheduled as late as possible subject to their due dates, and then, the A-jobs are scheduled as early as possible.
Clearly, in schedule π, the B-jobs are scheduled in the same manner as that in Consequently, π is a required Pareto-optimal schedule corresponding to ( f * , Y * ). The lemma follows.
From Lemmas 1 and 2, the Pareto-scheduling problem 1|pmtn| # ( f A , ∑ Y B j ) on instance J A ∪ J B can be solved by the following general approach: For each value Y * ∈ [T max (σ B 0 ), P B ], run Procedure(Y * ) to obtain the schedule σ B(Y * ) of the B-jobs. Determine the intervals occupied by the B-jobs in σ B(Y * ) and regards these intervals as forbidden intervals. The intervals which are not occupied by the B-jobs in σ B(Y * ) is called the free-time intervals. Then solve problem 1, h m |pmtn| f A on instance J A to obtain a Y * -standard schedule.
The above approach cannot be implemented in polynomial time since it enumerates all the possible choices of Y * . Therefore, in the next three sections, for f A ∈ {∑ C A j , L A max , ∑ Y A j }, we will present polynomial-time algorithms, respectively, to characterize the trade-off curves.
To this end, we set Y (0) = T max (σ B 0 ), and run Procedure(Y (0) ) to obtain the schedule σ B(Y (0) ) . Assume that the intervals occupied by the B-jobs are given by From the implementation of Procedure(Y (0) ), we have From the implementation of Procedure(Y * ) again, the set of time intervals occupied by the B-jobs in schedule σ B(Y * ) , denoted by I(σ B(Y * ) ), is given by We will write The above discussion will help us to construct the trade-off curves easily.

The First Problem
In this section, we consider problem 1|pmtn| By the job-exchanging argument, we can verify that the A-jobs must be scheduled in the SPT order in every Pareto-optimal schedule. Thus, in this section, we renumber the A-jobs by the SPT order such Then we only consider the schedules in which the A-jobs are scheduled in the order Then the set of forbidden intervals (occupied by the B-jobs) is given by (5) and the A-jobs are preemptively scheduled in the order J A 1 ≺ J A 2 ≺ · · · ≺ J A n A from time 0 in the free-time intervals as early as possible. Thus, there are m − i * + 1 forbidden intervals and the first forbidden interval in σ is given by If P A ≤ τ * , then all the A-jobs are scheduled before the first forbidden interval h i * in σ. In this case, we have no further action.
In general, suppose that P A > τ * . Then at least one A-job completes after Please note that when the schedule σ is given, After that, the value e(σ) defined in (6) can be determined in O(n A ) time. Finally, the value θ(σ) can be determined by its definition in (7) in constant time. Then we have the following lemma. For each Y ∈ [Y * , Y * + θ(σ)], let σ be the Y-standard schedule. Then σ is obtained from σ by shifting the first Y − Y * units of h i * to the last forbidden interval and then moving the A-jobs in {J A k(σ) , J A k(σ)+1 , . . . , J n A } left to eliminate the idle times accordingly. This means that Assume that the total completion time of A-jobs in σ is C. According to Lemma 2, (C, Y) is a Pareto-optimal point. In the following, we consider the trade-off curve between (C * , Y * ) and (C, Y). For convenience, point (C, Y) is simply called point Y.
We will show that the trade-off curve for Y ∈ [Y * , Y * + θ(σ)) is a line segment. However, the point Y * + θ(σ) may have the singularity.
. When we change σ to σ , no crucial A-jobs are moved left across their corresponding nearest forbidden intervals in σ . As a result, compared with σ, each of the completion times of the n A − k(σ) We use ∆(σ) to denote the total length of all the nearest forbidden intervals corresponding to the κ(σ) crucial A-jobs in σ, i.e., The following lemma is only used to display the singularity of point Y * + θ(σ).
we have the following three statements.
Proof. The correctness of Algorithm 2 is guaranteed by Lemmas 2 and 4. We estimate the time complexity of the algorithm in the following. The preprocessing procedure runs in O(n A log n A + n B log n B ) time. Each of Steps (1.1) and (1.2) runs in O(n B ) time, so Step 1 runs in O(n B ) time. After Step 1, the algorithm has K iterations. In each iteration, either one forbidden interval is eliminated or at least one A-job is moved left across its corresponding nearest forbidden interval. Since m ≤ n B , we have K = O(n A n B ). The above discussion establishes the O(nn A n B )-time complexity of Algorithm 2.
Preprocessing: Renumber the A-jobs such that p A 1 ≤ p A 2 ≤ · · · ≤ p A n A and renumber the B-jobs Step 1: Do the following: . . , m, as described in (3). Then regard h 1 , h 2 , . . . , h m as forbidden intervals which will be updated in the implementation of the algorithm. We take the convention that the forbidden intervals are just occupied by the B-jobs. Step 2: Do the following: . . , h m are the forbidden intervals and the A-jobs are preemptively scheduled in the order n A (and so, every A-job) is scheduled before the first forbidden interval h t i in σ i , then set K = i and go to Step 4. (In this case, we have obtained the whole trade-off curve.) If J A n A completes after h t i in σ i , then go to Step 3. (In this case, we use J k(σ i ) to denote the first A-job completing after h t i in σ i .) Step 3: Do the following: (3.1) Calculate the values k(σ i ) and θ(σ i ).
Step 4: Output the trade-off curve It can be observed that the total interruption time (i.e., the number of interruptions) of all the jobs in each schedule generated by Algorithm 2 is upper bounded by 1 + m ≤ 1 + n B .

The Second Problem
In this section, we consider problem 1|pmtn| # (L A max , ∑ Y B j ) on instance J A J B . By the job-shifting argument, we can verify that for each Pareto-optimal point, there is a corresponding Pareto-optimal schedule in which the A-jobs are scheduled in the EDD order. Thus, in this section, we renumber the A-jobs by the EDD order such that d A 1 ≤ d A 2 ≤ · · · ≤ d A n A . Then we only consider the schedules in which the A-jobs are scheduled in the order J A 1 ≺ J A 2 ≺ · · · ≺ J A n A . For a point (L * , Y * ) ∈ Ω(J A , J B ), let σ be a Y * -standard schedule of J A ∪ J B . Then the set of forbidden intervals (occupied by the B-jobs) is given by (5) and the A-jobs are preemptively scheduled in the order J A 1 ≺ J A 2 ≺ · · · ≺ J A n A from time 0 in the free-time intervals. Thus, there are m − i * + 1 forbidden intervals and the first forbidden interval in σ is given by If P A ≤ τ * , then all the A-jobs are scheduled before the first forbidden interval h i * in σ. In this case, we have no further action.
In general, suppose that P A > τ * . Then at least one A-job completing after h i * in σ. Let J A k(σ) be the first A-job which completes after h i * in σ. Then, there are totally n A − k(σ) + 1 A-jobs completing after h i * in σ. Let called the nearest forbidden interval corresponding to desired A-job J A j in σ. We further define and In the case that no A-job completes before interval h i * , we define BL(σ) = −∞ and ∆(σ) = +∞. Moreover, we define Please note that when the schedule σ is given, the values C A j (σ) and L A j (σ), j = 1, 2, . . . , n A , and L * can be determined in O(n A ) time. Then the set J A c (σ) and the interval indices After that, the value λ(σ), BL(σ) and ∆(σ) can be determined in O(n A ) time. Finally, the value ϑ(σ) can be determined by its definition in (12) in constant time. Then we have the following lemma. Lemma 6. Given the Y * -standard schedule σ in advance, the values ∆(σ) and ϑ(σ) can be determined in O(n A ) time.
We will show that the trade-off curve for Y ∈ [Y * , Y * + ϑ(σ)) is a line segment. Discussion for the singularity of the point Y * + ϑ(σ) will be omitted. This does not affect the characterization of the trade-off curve.
. When we change σ to σ , no desired A-jobs are moved left across their corresponding nearest forbidden intervals in σ . As a result, compared with σ, each desired job must move forward Y − Y * units and the other A-jobs move forward at least Y − Y * units in σ . Thus, the desired A-jobs in σ are also critical A-jobs in σ . Then we have L−L * Y−Y * = −1, as required.

Theorem 2. Algorithm 3 generates the trade-off curve of 1|pmtn| # (L
Step 1: Do the following: Step 2: Do the following: . . , h m are the forbidden intervals and the A-jobs are preemptively scheduled in the order J A 1 ≺ J A 2 ≺ · · · ≺ J A n A as early as possible. Determine the values , then set t i+1 := t i + 1; and if ϑ(σ i ) < δ t i (σ i ), then set t i+1 := t i and h t i+1 := [τ 2 ] (which is obtained from interval h t i by deleting the first ϑ(σ i ) units.) (3.4) Set i := i + 1. Go to Step 2.
Proof. Correctness of Algorithm 3 is guaranteed by Lemmas 2 and 7. The time complexity can be estimated by the similar way of Theorem 1 by putting Lemma 6 in discussion.
It can be observed that the total interruption time (i.e., the number of interruptions) of all the jobs in each schedule generated by Algorithm 3 is upper bounded by 1 + m ≤ 1 + n B .

The Third Problem
In this section, we consider problem We renumber the A-jobs by the EDD order such that d A Thus, we only need to consider the trade-off curve of problem 1|pmtn| # (∑ Y A j , ∑ Y B j ) on instance J A ∪ J B . We first establish a nice property for problem 1|pmtn| ∑ Y j in the following lemma. Lemma 8. Let J = {J 1 , J 2 , . . . , J n } be a job instance of problem 1|pmtn| ∑ Y j . Let U be a subset of J such that there is a schedule of instance J such that all the jobs in U are early. Then there is an optimal schedule of problem 1|pmtn| ∑ Y j on instance J such that all the jobs in U are early.
Proof. We first prove the result for problem 1|pmtn| ∑ Y j without maintenance intervals by induction on |U |. The result holds trivially if |U | = 0.
Inductively, suppose that |U | = k ≥ 1, U = {J j 1 , J j 2 , . . . , J j k }, d j 1 ≤ d j 2 ≤ · · · ≤ d j k , and there is a feasible schedule of instance J such that all the jobs in U are early. Moreover, the result holds for every proper subset of U (the induction hypothesis).
Since U \ {J j k } is a proper subset of U , from the induction hypothesis, there is an optimal schedule π of problem 1|pmtn| ∑ Y j on instance J such that all the k − 1 jobs J j 1 , J j 2 , . . . , J j k−1 are early in π. Since all the jobs in U are early in some feasible schedule, we have p j 1 + p j 2 + · · · + p j k ≤ max{d j 1 , d j 2 , . . . , d j k } = d j k . This implies that all the k − 1 jobs J j 1 , J j 2 , . . . , J j k−1 are completed by time d j k in π and at least p j k units of time in the interval [0, d j k ] are not occupied by the k − 1 jobs J j 1 , J j 2 , . . . , J j k−1 .
If C j k (π) ≤ d j k , i.e., J j k is early in π, then π is a required optimal schedule. Suppose in the following that C j k (π) > d j k . Then there is a certain index i ∈ {0, 1, . . . , p j k } such that for job J j k , the first i unit pieces J j k ,1 (π), J j k ,2 (π), . . . , J j k ,i (π) are early in π and the last p j k − i unit pieces J j k ,i+1 (π), . . . , J j k ,p j k (π) are late in π. Let S be the time space which consists of the last p j k − i units of time in the interval [0, d j k ] that are not occupied by the k − 1 jobs J j 1 , J j 2 , . . . , J j k−1 and the i unit pieces J j k ,1 (π), J j k ,2 (π), . . . , J j k ,i (π) of J j k . Let T be the time space which consists of the p j k − i units of time that are occupied by the p j k − i unit pieces J j k ,i+1 (π), . . . , J j k ,p j k (π) of J j k . Let σ be the schedule of J obtained from π by exchanging the subschedules in S and in T . Then J j k is early in σ. Moreover, ∑ Y j (σ) ≤ ∑ Y j (π) − |T | + |S| = ∑ Y j (π), implying that σ is also optimal. Now all the jobs in U = {J j 1 , J j 2 , . . . , J j k } are early in σ. Consequently, σ is an optimal schedule of problem 1|pmtn| ∑ Y j on instance J such that all the jobs in U are early. The result follows by the induction principle.
We next use Lemma 8 to prove the following useful lemma. Lemma 9. Let J = {J 1 , J 2 , . . . , J n } be a job instance of problem 1|pmtn| ∑ Y j . Let π be a schedule of the jobs in J . Then there is an optimal schedule σ of problem 1|pmtn| ∑ Y j on instance J such that Y j (σ) ≤ Y j (π) for j = 1, 2, . . . , n.
Proof. For each j ∈ {1, 2, . . . , n}, we partition J j into two parts J j and J j such that p j = p j − Y j (π) is the early work of J j in π, p j = Y j (π) is the late work of J j in π, and d j = d j = d j . Let J = {J j , J j : j = 1, 2, . . . , n}. Let π be the schedule of J which is obtained from π by just regarding the early part of J j in π as job J j and regarding the late part of J j in π as job J j . Then all the jobs in {J j : j = 1, 2, . . . , n} are early in π . According to Lemma 8, there is an optimal schedule σ of problem 1|pmtn| ∑ Y j on instance J such that all the jobs in {J j : j = 1, 2, . . . , n} are early. Since the preemptive assumption, the two instances J and J have no essential difference for problem 1|pmtn| ∑ Y j , σ is an optimal schedule of problem 1|pmtn| ∑ Y j on instance J . The result follows by noting that Y j (σ) = Y j (σ) ≤ p j = Y j (π) for j = 1, 2, . . . , n.
Let Y AB be the optimal value of problem 1|pmtn| ∑ Y j on instance J A ∪ J B . We have the following lemma.
Proof. Let σ be a Pareto-optimal schedule of problem 1|pmtn| The optimality of σ implies that ∑ Y j (σ ) = Y AB . From the property of Pareto-optimal point, we can obtain that The result follows.
B be the optimal value of problem 1|pmtn|Y B j on instance J B . Recall that Y AB is the optimal value of problem 1|pmtn| ∑ Y j on instance J A ∪ J B . From Hariri et al. [21], Y Please note that Y A is the minimum total late work of A-jobs among all Pareto-optimal points and Y (0) B is the minimum total late work of B-jobs among all Pareto-optimal points. Thus, from Lemma 10, the trade-off curve is the line segment A ). So, the overall complexity to obtain the trade-off curve is given by O(n log n).
It can be observed that the total interruption time (i.e., the number of interruptions) of all the jobs in each Pareto-optimal schedule is upper bounded by max{n A , n B } + 2.
Let us consider the job instance I 3 displayed in Table 3. The trade-off curve of problem 1|pmtn| # (∑ Y A j , ∑ Y B j ) on instance I 3 is shown in Figure 3. Table 3. The instance I 3 .
(ii) Generate the Y (0) -standard schedule σ 0 of J A ∪ J B in which h 1 , h 2 , h 3 , h 4 are the forbidden intervals and the A-jobs are preemptively scheduled by running the algorithm in Hariri et al. [21] for solving problem 1|pmtn| ∑ Y j in the free-time intervals not occupied by the B-jobs in σ B(Y 0 ) . Determine the value Y A (σ 0 ) = ∑ Y A j (σ 0 ) = 7. Then σ 0 is a Pareto-optimal schedule corresponding to (7,1) ∈ Ω. (iii) By using the same method as (i) and (ii), we schedule A-jobs first. Generate the schedule σ A(Y (0) ) , where Y (0) = T max (σ A 0 ) = 2. Generate the schedule σ 1 of J A ∪ J B in which the B-jobs are preemptively scheduled by running the algorithm in Hariri et al. [21] for solving problem 1|pmtn| ∑ Y j in the free-time intervals not occupied by the A-jobs in σ A(Y (0) ) . Then σ 1 is a Pareto-optimal schedule corresponding to (2,6) ∈ Ω. From Lemma 10, σ 1 is the final Pareto-optimal schedule. Then the trade-off curve is just the line segment in the interval [1,6], which satisfies Y A = −Y B + 8 as displayed in Figure 3.

Conclusions
This paper considers three preemptive Pareto-scheduling problems with two competing agents on a single machine. Two agents compete to perform their respective jobs on a common single machine and each agent has his own criterion to optimize. In each problem, the goal of agent A is to minimize the total completion time, the maximum lateness, or the total late work while agent B wants to minimize the total late work. For each problem, we provide a polynomial-time algorithm to characterize the trade-off curve of all Pareto-optimal points.
Late-work criterion can be met in all cases where the penalty imposed on a solution depends on the number of tardy units of jobs performed in a system. For example, in production planning where the manufacturer is concerned with minimizing any order delays which cause financial loss, in control systems where the accuracy of control procedures depends on the amount of information provided as their input, in agriculture where performance measures based on due dates, and so on. In the case where two criteria need to be minimized, the trade-off curve results an ideal solution. Once the trade-off curve is characterized, decision makers can make decisions as needed.
For the future research, the trade-off curve of the problem 1|pmtn| # (∑ C A j , ∑ w j Y B j ) or 1|pmtn| # (L A max , ∑ w j Y B j ) is worthy of study. Since the existence of precedence constraints on scheduling problems reflects real-life problems, it is also worthy to study the two-agent problems with precedence constraints. Another interesting future research direction is to investigate fairness issues when the total late work is one of the criteria in two-agent scheduling problems.