SIMULTANEOUS OPTIMIZATION SCHEDULING WITH TWO AGENTS ON AN UNBOUNDED SERIAL-BATCHING MACHINE

. This paper considers a class of simultaneous optimization scheduling with two competitive agents on an unbounded serial-batching machine. The cost function of each agent depends on the completion times of its jobs only. According to whether the jobs from different agents can be processed in a common batch, compatible model and incompatible model are investigated. For the incompatible model, we consider batch availability and item availability. For each problem, we provide a polynomial-time algorithm that can find all Pareto optimal schedules. scheduling cases are presented according to the time when the jobs become available. In the case of batch availability (short for 𝑏𝑎𝑡𝑐ℎ - 𝑎𝑣𝑎𝑖𝑙 ), a job is available only when the batch to which it belongs has been processed. In the case of item availability (short for 𝑖𝑡𝑒𝑚 - 𝑎𝑣𝑎𝑖𝑙 ), a job becomes available immediately after it is completed processing (see [19]). In the paper, the objective function of agent 𝐴 is a lateness-like objective function, such as 𝐶 𝐴 max , 𝐿 𝐴 max , 𝑇 𝐴 max , 𝑊𝐶 𝐴 max and that of agent 𝐵 is a lateness-like objective function or the special total weighted completion time, such as 𝐶 𝐵 max , 𝐿 𝐵 max , 𝑇 𝐵 max , 𝑊𝐶 𝐵 max , ∑︀ 𝑤 𝐵𝑗 𝐶 𝐵𝑗 with 𝑝 𝐵𝑗 = 𝑝 or 𝑤 𝐵𝑗 = 𝑤 for any 1 ≤ 𝑗 ≤ 𝑛 𝐵 , where 𝐶 𝑋 max , 𝐿 𝑋 max , 𝑇 𝑋 max , 𝑊𝐶 𝑋 max and ∑︀ 𝑤 𝐵𝑗 𝐶 𝐵𝑗 are


Introduction
There are two agents and with job sets ℱ = {︀ 1 , 2 , · · · , }︀ and ℱ = {︀ 1 , 2 , · · · , }︀ , respectively. Agents and must schedule their jobs on a common unbounded serial-batching machine and the jobs are processed in batches, where "unbounded" implies that the machine can process any number of jobs in a batch and "a batch" refers to a set of jobs which are processed jointly and contiguously. The processing time of a batch amounts to the sum of processing times of all jobs in the batch. A setup time is inserted whenever a new batch starts. According to whether the jobs from different agents can be processed in the same batch, we investigate compatible model (i.e., the jobs of two agents can be processed in a common batch, short for co) and incompatible model (i.e., the jobs of two agents cannot be processed in a common batch, short for inco). For compatible model, we suppose that the setup time equals to . For incompatible model, we suppose that the setup time that is inserted before a batch which belongs to agent is ( ∈ { , }). Moreover, in incompatible model, two kinds of batch scheduling cases are presented according to the time when the jobs become available. In the case of batch availability (short for ℎ-), a job is available only when the batch to which it belongs has been processed. In the case of item availability (short for -), a job becomes available immediately after it is completed processing (see [19]). In the paper, the objective function of agent is a lateness-like objective function, such as max , max , max , max and that of agent is a lateness-like objective function or the special total weighted completion time, such as max , max , max , max , ∑︀ with = or = for any 1 ≤ ≤ , where max , max , max , max and ∑︀ are Keywords. Serial-batching, two-agent scheduling, compatible/incompatible, batch/item availability.
the makespan, the maximum lateness, the maximum tardiness, the maximum weighted completion time (i.e., max = max {︀∑︀ : 1 ≤ ≤ }︀ ), of agent ( = or = ) and the total weighted completion time of agent respectively. Each agent's objective function depends on the completion times of its jobs only. The aim is to find all Pareto optimal schedules for the two-agents' objective functions under various scheduling environment. Here, two objective functions may on behalf of different profits of two decision-makers. Moreover, we first investigate the two problems with objective vectors (︀ max , max )︀ and (︀ max , ∑︀ )︀ , respectively, where objective vector ( 1 , 2 ) represents minimizing two criteria 1 and 2 simultaneously. Finally, we generalize the results of each problem to a class of problems.
The problems depicted above can be found in many manufacturing applications and many negotiation procedures. For example, a mill can handle orders from two types of agents. The agents' orders are interpreted as jobs to be processed. Agent expects the maximum lateness of his jobs to be as small as possible, while agent expects the maximum lateness or the total completion time of his jobs to be as small as possible. Moreover, the manufacturer is also concerned about minimizing any order delays which cause economic loss. For the purpose of meeting the needs of two agents to the maximum extent, the manufacturer needs to design some strategies to stimulate the agents to cooperate. This circumstance can be modelled as the simultaneous optimization problems under consideration, i.e., objective vector is Serial-batching scheduling problems are urged by grouped jobs' processing environment with conversion times between different groups. For example, when the machine switches from processing one batch to another batch, the machine usually need to be changed a tool or to be cleaned, which shows that the machine needs a setup time before a new batch is processed [4]. Besides, the management and technical constraints (e.g., different processing environment, etc.) lead to the compatibility and incompatibility of jobs from different agents [14,16].
For serial-batching scheduling problems with batch availability, Albers and Brucker [3] show that 1|batch-avail| ∑︀ is strongly -hard while 1|batch-avail| ∑︀ can be solved in ( log ) time (see [6]). He et al. [10,11] present an ( 2 )-time algorithm for 1|batch-avail|( max , max ) and 1|batch-avail|( max , ∑︀ ), respectively. Geng et al. [8] solve 1|batch-avail|( max , max ) in (︀ 4 )︀ time. If the number of agents is given, the single-machine scheduling problems, with item availability, to minimize the maximum lateness or the number of tardy jobs or the total weighted completion time are polynomially solvable, while all these problems become very intractable when the number of agents is a variable [17,20]. Reviews of the research on the topic are provided by Potts and Kovalyov [19] and Allahverdi [4].
The discuss on multi-agent scheduling originated from Baker and Smith [5] and Agnetis et al. [1]. Since it has been surveyed by Perez-Gonzalez and Framinan [18] and Agnetis et al. [2], we only review the results related to our study.
For two-agent constrained optimization scheduling on an unbounded serial-batching machine, Kovalyov et al. [14] investigate a series of batch availability models, in which closely related to our problems are the problems 1|batch-avail, inco, max ≤ | with ∈ {︀ max , ∑︀ }︀ . Yin et al. [21] generalize the work of [14] by adding a delivery cost for each manufacture batch; Yin et al. [22,23] study the problems in which there exist batch delivery cost and due date assignment, etc. Li et al. [15] study a series of item availability models, in which closely related to our problems are the problems 1|item-avail, inco, max ≤ | with ∈ {︀ max , ∑︀ }︀ and 1|item-avail, inco, = , max ≤ | ∑︀ . Li et al. [16] also investigated a series of job compatibility problems, in which closely related to our problems are the problems 1|batch-avail, co, max ≤ | with ∈ {︀ max , ∑︀ }︀ . For two-agent simultaneous optimization scheduling on a serial-batching machine, to the best of our knowledge, the results are very few and the solved problems are very classical (see [2] }︀ ( = or ) and ∈ {{batch-avail, inco}, {item-avail, inco}, {batch-avail, co}}. The paper is arranged as follows. In Section 2, we elaborate some preliminaries and list an overview of the results in the paper. Section 3 is dedicated to two-agent problems of minimizing objective vectors (︀ max , max )︀ and (︀ max , ∑︀ )︀ with batch availability and incompatibility. Section 4 is focused on two-agent problems of minimizing objective vectors (︀ max , max )︀ and (︀ max , ∑︀ )︀ with item availability and incompatibility. Section 5 concentrates on two-agent problems of minimizing objective vectors (︀ max , max )︀ and (︀ max , ∑︀ )︀ with batch availability and compatibility. Section 6 expands on the results in Sections 3-5. Section 7 gives a concluding remarks.

Preliminaries and overview of the results
Suppose that agents and have job sets in , where the summation notation is taken over all jobs of agent . When the jobs of two agents cannot be processed in a common batch (the model is called incompatible), an agent-dependent setup time is inserted before each new batch of agent ( ∈ { , }) is processed. When the jobs of two agents can be processed in the same batch (the model is called compatible), a setup time is inserted before each new batch. The processing time of a batch is equal to the sum of processing times of all jobs in the batch. According to the time when the jobs become available, batch availability model and item availability model are investigated. In the case of batch availability, a job is available only when the batch to which it belongs has been processed. In the case of item availability, a job is available immediately after it is processed.
The paper considers the simultaneous optimization scheduling 1| | (︀ max , max )︀ and 1| , = or = | (︀ max , ∑︀ )︀ , where ∈ {{batch-avail, inco}, {item-avail, inco}, {batch-avail, co}} and max ∈ {︀ max , max , max , max }︀ and = or . Note that "batch-avail" and "item-avail" represent that the considered problems are batch availability and item availability respectively, "inco" and "co" denote that -jobs and -jobs are incompatible and compatible respectively. The purpose is to find all Pareto optimal schedules in respect to two criteria in polynomial time for each problem.
Note that each job is available at time zero and each objective involved in the paper is regular (i.e., nondecreasing in the completion times). So there exists an optimal schedule such that all jobs (batches) are processed continuously from time zero onwards. Throughout this paper, we focus our attention on the schedules with the property.
Definition 2.1. A feasible schedule is Pareto optimal, or nondominated, with respect to the performance criteria and if there is no feasible schedule such that both ( ) ≤ ( ) and ( ) ≤ ( ), where at least one of the inequalities is strict. Besides, ( ( ), ( )) is called a Pareto optimal point corresponding to Pareto optimal schedule .
The following theorem provides a general approach, the so-called -constraint approach, for finding Pareto optimal schedules. Theorem 2.2 ( [13]). Let be the optimal value of constraint problem | ≤̂︀| (wherê︀ is a upper bound of ), and let be the optimal value of the problem | ≤ | . Then ( , ) is a Pareto optimal point for ||( , ).
Suppose that problem | ≤ | is efficiently solvable. Then the other method to find Pareto optimal schedules is unilateral -constraint approach in the following.

Incompatible two-agent problems with batch availability
In this section, we study the two-agent scheduling problems under the batch availability and incompatible assumption. 14]). If problem 1|batch-avail, inco, max ≤ | max is feasible, then the problem has an optimal EDD-EDD-schedule.
Step 2. Solve problem 1|batch-avail, inco, max ≤ , max ≤ ′ |− by DP1. If the current constrained problem is infeasible, then go to Step 3; otherwise we get a new schedule +1 and let := + 1 and := max ( ) (it is obvious that ≤ and ∈ ℰ ) and go to Step 4.
Proof. The validity of Algorithm PO-(︀ max , max )︀ is guaranteed by Theorem 2.2 and Lemmas 3.3 and 3.5 and the above discussion. Next, we discuss the time bound.
Proof. The validity of Algorithm PO-(︀ max , ∑︀ )︀ is guaranteed by Theorem 2.3 and Lemma 3.7 and the above discussion. Next, we discuss the time bound.

Incompatible two-agent problems with item availability
In this section, we study the two-agent scheduling problems under the item availability and incompatible assumption. Notice that under the item availability assumption, if two batches of the same agent are processed consecutively, then these two batches can be merged into a large batch with only one setup time retained (which is impossible under the batch availability assumption), i.e., the batches belonging to different agents appear alternately. We restrict our search to the schedules with this property.  15]). If problem 1|item-avail, inco, max ≤ | max is feasible, then the problem has an optimal EDD-EDD-schedule. Lemma 4.2 ([15]). If problem 1|item-avail, inco, max ≤ | ∑︀ is feasible, then the problem has an optimal EDD-SPT-schedule.
Then there exists an EDD-EDD-schedule corresponding to ( , ) by Lemma 4.1 and the definition of Pareto optimal point. Re-index the jobs such that 1 ≤ 2 ≤ · · · ≤ and 1 ≤ 2 ≤ · · · ≤ . Let = ( 1 , 2 , · · · , ) be any feasible EDD-EDD-schedule for 1|item-avail, inco| Then is a batch of agent . Further, batch −1 must be a batch of agent by the rotation of batches from the different agents. Let and denote the numbers of batches of agent and agent in the first batches 1 , 2 , · · · , , respectively. Then Similarly, all possible values of maximum lateness max belong to the set Remark. Li et al. [15] present an ( 2 2 log )-time algorithm for 1|item-avail, inco, max ≤ | max , in which the time bound is decided by sorting set with | | = ( 2 2 ). Hence we can improve the time bound to ( 2 log ) by | | = (︀ 2 )︀ .
By Lemma 4.3 and the above discussion, problem 1|item-avail, inco, max ≤ | max can be solved by solving a series of feasibility problems 1|item-avail, inco, max ≤ , max ≤ ′ |− with the the decreasing values ′ and ′ ∈ . Similarly, 1|item-avail, inco, max ≤ ′ | max can also be solved by solving a series of feasibility problems 1|item-avail, inco, max ≤ , max ≤ ′ |− with the decreasing values and ∈ . By slightly modifying Algorithm PO-(︀ max , max )︀ in Section 3.1 at the corresponding place, we derive the following result.  Notice that for any Pareto optimal point of 1|item-avail, inco| (︀ max , ∑︀ )︀ , there is an EDD-SPT-schedule corresponding to it by Lemma 4.2 and the definition of Pareto optimal point. Re-index the jobs such that 1 ≤ 2 ≤ · · · ≤ and 1 ≤ 2 ≤ · · · ≤ .

Compatible two-agent problems with batch availability
In this section, we study the two-agent scheduling problems under the batch availability and compatible assumption. 16]). If problem 1|batch-avail, co, max ≤ | max is feasible, then the problem has an optimal EDD-EDD-schedule.
By Lemma 5.3 and the above discussion, problem 1|batch-avail, co, max ≤ | max can be solved by solving a series of feasibility problems 1|batch-avail, co, max ≤ , max ≤ ′ |− with the decreasing values ′ , where ′ ∈ . Similarly, 1|batch-avail, co, max ≤ ′ | max can also be solved by solving a series of feasibility problems 1|batch-avail, co, max ≤ , max ≤ ′ |− with the decreasing values , where ∈ . By slightly modifying Algorithm PO-(︀ max , max )︀ in Section 3.1 at the corresponding place, we derive the following result.
By modifying Algorithm PO-(︀ max , ∑︀ )︀ in Section 3.2 slightly on the corresponding place, we get the following result.

A class of problems with lateness-like objective function
For each of the above problems studied in Sections 3-5, there is an optimal schedule with a specific structure, such as EDD-EDD or EDD-SPT that depends on the objective functions held by the two agents. We note that the specific structure is a key to solving our problems so that our methods are applicable to a class of problems with the specific structure. We introduce two concepts in [24].
Proof. From Lemma 6.1 and EDD-like order, the proof for 1|batch-avail, inco| Proof. Due to alternate batches from the different agents in any Pareto optimal schedule, each Pareto optimal schedule of each problem contains either two batches or three batches by Lemmas 6.1 and 6.2.
Hence there is at most ( ) Pareto optimal schedules. It takes ( ) time to compute each objective vector when ∑︀ is computed in advance and all -jobs is sorted in advance. And sortingjobs needs ( log ) time and computing ∑︀ needs ( ) time. So the total time complexity is Proof. The feasibility problem 1|batch-avail, co, max ≤ |− can be solved in ( ) if the deadlines of all jobs are given in advance (see [16]). So problem 1|batch-avail, co, max ≤ , max ≤ ′ |− can be solved in ( + log ) = ( ) if ∑︀ is computed in advance and all -jobs are sorted in the nondecreasing order of their deadlines in advance by Lemma 6.1 (where all -jobs are looked as a merged large job and log is the time taken to insert the merged job into -jobs). Since problem 1|batch-avail, co, max ≤ | max can be solved by solving a series of feasibility problems 1|batch-avail, co, max ≤ , max ≤ ′ |− with varied ′ , where the varied ′ is determined by binary search on all (︀ 2 )︀ possible values of max , which needs (log ) time if all ( 2 ) possible values of max is sorted in advance, and the order of all -jobs keeps no change, problem 1|batch-avail, co, is computed in advance and all (︀ 2 )︀ possible values of max is sorted in advance and all -jobs are sorted in the nondecreasing order of their deadlines in advance.
Proof. Problem 1|batch-avail, co| (︀ max , max )︀ can be solved by solving a series of constraint problem 1|batch-avail, co, max ≤ | max with the decreasing values , where is all possible values of max .
From reference [16], max has at most (︀ 2 )︀ possible values if max is replaced by max . From Lemma 6.1, all -jobs can be treated as a merged large job with processing time ∑︀ . Thus our problem is equivalent to = 1 jobs. So max has at most (︀ 3 )︀ values in our problem. Note that the order of all -jobs (provided that they are scheduled in nondecreasing order of their deadlines) has no change in each round. So if we sort all -jobs in advance, then each time sorting all jobs needs only to insert one merged -jobs into -jobs, which takes (log ) time. And  and Pareto optimal schedules of problem 1| , = | (︀ max , ∑︀ )︀ have a similar property, i.e., their -jobs are scheduled in the EDD-like order, and all -jobs are scheduled in the SPT order for former problem and all -jobs are scheduled in the SPT-like order (i.e., 1 ≥ 2 ≥ · · · ≥ ) for latter problem. Therefore, problems 1| | (︀ max , ∑︀ )︀ and 1| , = | (︀ max , ∑︀ )︀ have the same time complexity.

Concluding remarks
In the foregoing discussion, we study two-agent simultaneous optimization scheduling, in which the objective function of agent is lateness-like objective function, such as max , max , max , max and that of agent is a lateness-like objective function or the special total weighted completion time, on an unbounded serial-batching machine. Moreover, the problems are considered under three cases: batch availability and incompatibility, item availability and incompatibility, and batch availability and compatibility. For all problems studied in the paper, we give a polynomial-time algorithm, respectively. On the one hand, our future work would be to generalize the objective vectors, for example, ( max , max ) and ( ∑︀ , ∑︀ ) and ( max , ∑︀ ). On the other hand, we can also consider the corresponding bounded cases for the problems in the paper.