Some Single-Machine Scheduling Problems with Learning Effects and Two Competing Agents

This study considers a scheduling environment in which there are two agents and a set of jobs, each of which belongs to one of the two agents and its actual processing time is defined as a decreasing linear function of its starting time. Each of the two agents competes to process its respective jobs on a single machine and has its own scheduling objective to optimize. The objective is to assign the jobs so that the resulting schedule performs well with respect to the objectives of both agents. The objective functions addressed in this study include the maximum cost, the total weighted completion time, and the discounted total weighted completion time. We investigate three problems arising from different combinations of the objectives of the two agents. The computational complexity of the problems is discussed and solution algorithms where possible are presented.


Introduction
In traditional scheduling research, it is commonly assumed that the processing times of the jobs remain unchanged throughout the scheduling horizon. However, under certain circumstances, the job processing times may become short due to learning effects in the production environment. For example, Biskup [1] points out that the repeated processing of similar tasks will improve workers' efficiency; that is, it takes workers shorter times to process setups, operate machines or software, or handle raw materials and components. In such an environment, a job scheduled later will consume less time than the same job when scheduled earlier. Jobs in such a setting are said to be under the "learning effect" in the literature.
Biskup [1] and Cheng and Wang [2] first introduce the idea of learning into the field of scheduling independently. Since then, a large body of literature on scheduling with learning effects has emerged. Examples of such studies are Mosheiov [3], Mosheiov and Sidney [4], Bachman and Janiak [5], Janiak and Rudek [6], Wang [7], and Yin et al. [8].
All the above papers consider the traditional case with a single agent. In recent years scheduling researchers have increasingly considered the setting of multiple competing agents, in which multiple agents need to process their own sets of jobs, competing for the use of a common resource and each agent has its own objective to optimize. However, there is little scheduling research in the multiagent setting in which the jobs are under learning effects. Liu et al. [17] study two models with two agents and position-dependent processing times. They assume that the actual processing time of job is + in the aging-effect model, while the actual processing time of is − in the learning-effect model, where represents the processed position of and > 0 denotes the aging or learning index. Ho et al. [18] define the actual processing time of job as = (1 − ) if it is processed at time , where denotes the normal processing time of job 2 The Scientific World Journal and ≥ 0 represents a constant such that ( 0 + ∑ =1 − min ) < 1 with min = min =1,2,..., { }. Inspired by Ho et al. [18], Yin et al. [19] consider some two-agent scheduling problems under the learning effect model proposed in Ho et al. [18], in which the objective functions for agent include the maximum earliness cost, the total earliness cost, and the total weighted earliness cost, and the objective function for agent is always the same, that is, maximum earliness cost, and the objective is to minimize the objective function of agent while keeping the objective function of agent not greater than a given level. Similar models have been further studied by Wang and Xia [20], Wang [21], and so on. For the other related two-agent works without time-dependent processing times, the reader can refer to Baker and Smith [22], Agnetis et al. [23,24], Cheng et al. [25,26], Ng et al. [27], Mor and Mosheiov [28], Lee et al. [29], Leung et al. [30], Wan et al. [31], Yin et al. [19,32], Yu et al. [33], and Zhao and Lu [34].
This study introduces a new scheduling model in which both the two-agent concept and the learning effects exist, simultaneously. We consider the following objective functions: the maximum cost, total completion time, total weighted completion time, and discounted total weighted completion time. The structural properties of optimal schedules are derived and polynomial time algorithms are developed for the problems where possible.
The remaining part of the study is organized as follows: Section 2 introduces the notation and terminology used throughout the paper. Sections 3-6 analyze the computational complexity and derive the optimal properties of the problems under study. The last section concludes the paper and suggests topics for future research.

Model Formulation
The problem investigated in this paper can be formally described as follows. Suppose that there are two agents and , each of whom has a set of nonpreemptive jobs. The two agents compete to process their jobs on a common machine. Agent has to process the job set = { 1 , 2 , . . . , }, while agent has to process the job set = { 1 , 2 , . . . , }.
where ≥ 0 denotes job's starting time and ≥ 0 represents constant such that ( 0 + ∑ ∈ ∪ − min ) < 1, where min = min ∈ ∪ { } (see Ho et al. [18] for details). Given a feasible schedule of the = + jobs, we use ( ) to denote the completion time of job and omit the argument whenever this does not cause confusion. The makespan of agent is max = max =1,2,..., { }. For each job , let (⋅) be a nondecreasing function. In this case, the maximum cost is defined as max = max =1,2,..., { ( )}. The objective function of agent considered in this paper includes the following: max (maximum cost), ∑ (total completion time), ∑ (total weighted completion time), and ∑ (1 − − ) (discounted total weighted completion time).
Using the three-field notation scheme | | introduced by Graham et al. [35], the problems considered in this paper are denoted as follows: Note that all the objective functions involved in the considered problems are regular; that is, they are nondecreasing in the job completion times. Hence there is no benefit in keeping the machine idle.

Problem
In this section we address the problem 1 | = (1 − ) | max : max ≤ and show that it can be solved optimally in polynomial time. We first develop some structural properties of optimal schedules for the problem which will be used in developing the algorithm.

then there exists an optimal schedule such that is scheduled last and there is no optimal schedule where an A-job is scheduled last.
Proof. Assume that is an optimal schedule where thejob ℎ is not scheduled in the last position. Let denote the set of jobs scheduled prior to job ℎ . We construct from a new schedule by moving job ℎ to the last position and leaving the other jobs unchanged in . Then, the completion times of the jobs processed before job ℎ in are the same as that in since there is no change for any job preceding ℎ in . The jobs belonging to are scheduled earlier, so their completion times are smaller in by Lemma 1. It follows that ( ( )) ≤ ( ( )) for any job in , where ∈ { , }. By the assumption that ℎ ( ) ≤ , job ℎ is feasible in , so schedule is feasible and optimal, as required.
For each -job , let us define a deadline such that is available, the deadlines can be evaluated in constant time; otherwise, this requires logarithmic time). Proof. Assume that is an optimal schedule where the -jobs are not scheduled according to the nondecreasing order of . Let and ℎ be the first pair of jobs such that > ℎ .
In this schedule, job is processed earlier; then a set ofjobs, denoted as , are consecutively processed and then job ℎ . In addition, denote by the set of jobs processed after job ℎ . We construct from a new schedule by extracting job , reinserting it just after job ℎ and leaving the other jobs unchanged in schedule . Then the completion times of the jobs processed prior to job in are the same as that in . By Lemma 1, the completion time of job ℎ in equals that of job in ; that is, ( ) = ℎ ( ), so the completion times of the jobs belonging to are identical in both and . Since is feasible, it follows that ( ) = ℎ ( ) ≤ ℎ < , so job is feasible in . The -jobs and job ℎ are scheduled earlier in , implying that their actual processing times are smaller in , so their completion times are earlier in , and thus they remain feasible. Therefore, schedule is feasible and optimal.
Thus, repeating doing this procedure for all the -jobs not sequenced according to nondecreasing order of completes the proof. Proof. Assume that is an optimal schedule where the -job with the smallest cost ℎ , that is, ℎ ( ) = min ∈ { ( )}, is not processed in the last position. By the hypothesis, the last job in schedule is an -job, say . This means ℎ ( ) < ( ). In this schedule, job ℎ is scheduled earlier. Let denote the set of jobs scheduled after job ℎ and prior to job . We construct from a new schedule by extracting job ℎ , reinserting it just after job , and leaving the other jobs unchanged in schedule . There is no change for any job preceding ℎ in . We claim the following.
(1) Schedule is feasible. First, the completion times of the jobs processed prior to job ℎ in are the same as that in . Since the jobs belonging to are scheduled earlier in , their actual processing times are smaller in , so their completion times are earlier in . It follows that ( ( )) ≤ ( ( )) for any job in , where ∈ { , }, as required.
(2) Schedule is a better schedule than . By Lemma 1, the completion time of the last job in equals that of the last job ℎ in ; that is, ( ) = ℎ ( ) = .
Thus, to prove that is better than , it suffices to show that Summing up the above analysis, our algorithm for problem 1 | = (1 − ) | max : max ≤ can be formally described as in Algorithm 1.
Step 1 requires a sorting operation of the -jobs, which takes ( log ) time.

Problem
Step 2. Proof. Assume that is an optimal schedule where -jobs are not scheduled in the WSPT order. Let and ℎ be the first pair of jobs such that / > ℎ / ℎ . Then ≥ ℎ and ≤ ℎ due to the fact that the -jobs have reversely agreeable weights. Assume that, in schedule , job starts its processing at time ; then a set of -jobs are consecutively processed and then job ℎ . In addition, let denote the set of jobs processed after job ℎ . We construct a new scheduling from by swapping jobs and ℎ and leaving the other jobs unchanged. We conclude the following.
(1) Schedule is feasible. By Lemma 1, the completion time of job ℎ in equals that of job in ; that is, ( ) = ℎ ( ), so the completion times of the jobs belonging to are identical in both and . Since ≥ ℎ , we have ℎ ( ) = + ℎ (1− ) ≤ + (1− ) = ( ). Hence the -jobs are scheduled earlier in , implying that their actual processing times are smaller in , so their completion times are earlier in . Hence ( ( )) ≤ ( ( )) for any job in , as required.
Thus, repeating this swapping argument for all the -jobs not sequenced in the WSPT order yields the theorem.
Based on the results of Propositions 2, 3, and 6, our algorithm to solve the problem 1 | = (1 − ) | ∑ : max ≤ for the case where the -jobs have reversely agreeable weights can be formally described as in Algorithm 2. Step 1. Set = ( 0 − 1/ ) ∏ =1 (1 − ) + 1/ , = , ℎ = , and ∑ = 0; sort the -jobs according to the non-decreasing order of / , that is, calculate the deadlines of the -jobs from ( ) = and renumber them according to the non-decreasing order such that [1] Step 2. Therefore, the overall time complexity of the algorithm is indeed ( log + log ). Proof. We adopt the same notation as that used in the proof of Proposition 6. Assume that

Problem
Since -jobs have reversely agreeable weights, we have ≥ ℎ and ≤ ℎ . Then by the proof of Proposition 6, we know that ℎ ( ) ≤ ( ), ( ) = ℎ ( ), and ( ) = ( ) for all the other jobs ∈ /{ , } and that schedule is feasible. To show that is better than , it is sufficient to show that In fact, since ∈ (0, 1), ℎ ( ) ≤ ( ), and ( ) = ℎ ( ), we have Hence, . Therefore, is not worse than . Thus, repeating this swapping argument for all the -jobs not sequenced in the WDSPT order yields the theorem.
Based on the above analysis, our algorithm to solve the problem 1 | = (1 − ) | ∑ (1 − − ) : max ≤ for the case where the -jobs have reversely agreeable weights can be described as in Algorithm 3. Proof. The proof is analogous to that of Theorem 7.

Conclusions
This paper introduced a new scheduling model on a single machine that involves two agents and learning effects simultaneously. We studied the problem of finding an optimal schedule for agent , subject to the constraint that the maximum cost of agent does not exceed a given value. We derived the optimal structural properties of optimal schedules and provided polynomial time algorithms for the problem 1 | = (1 − ) | max : max ≤ . We also showed that the problems 1 | = (1 − ) | ∑ : max ≤ and 1 | = (1 − ) | ∑ (1 − − ) : max ≤ can also be solved in polynomial time under certain agreeable conditions. Future research may consider the scheduling model with more than two agents.