Abstract
Multiple-machine scheduling problems with position-based learning effects are studied in this paper. There is an initial schedule in this scheduling problem. The optimal schedule minimizes the sum of the weighted completion times; the difference between the initial total weighted completion time and the minimal total weighted completion time is the cost savings. A multiple-machine sequencing game is introduced to allocate the cost savings. The game is balanced if the normal processing times of jobs that are on the same machine are equal and an equal number of jobs are scheduled on each machine initially.
1. Introduction
In a single-machine scheduling problem, a finite number of jobs are processed on a machine. Jobs are characterized by normal processing times, weights, due-dates, release times, and so on. The objective is to minimize a given function, such as the sum of the completion times, the sum of the weighted completion times, the maximal lateness, and the makespan.
Scheduling problems with learning effects were first introduced by Biskup [1], who described the actual processing time of a job as a decreasing power function of its position. Since then, several classes of learning effects in scheduling problems have been studied, namely, time-dependent learning effects [2–6], position-based learning effects [7–13], combined effects of position-based effects and time-dependent effects [14–17]. In a scheduling problem with time-dependent learning effects, the actual processing time of a job depends on the total processing times of jobs already processed, while, in a scheduling problem with position-based learning effects, the actual processing time of a job is a function of its position. In a scheduling problem with learning effects, the later a job starts, the shorter its actual processing time is. Learning effects have also been introduced to multiple-machine scheduling problems [8–10, 12, 15]. The main aims of the above works are optimal schedules, algorithms, and complexities of the algorithms. We refer to [18] for a review of scheduling with learning effects.
Cooperative games arising from scheduling or sequencing situations are called sequencing games. Curiel et al. [19] considered a single-machine sequencing situation, in which each job belongs to an agent (a player) and there is an initial order of jobs. The weight of a job is the cost per unit time of the agent. The optimal order minimizes the total cost which is the sum of the weighted completion times. The difference between the initial total cost and the minimal total cost can be seen as cost savings of all agents. This begs the question: How to allocate the cost savings among the agents fairly? Curiel et al. tackled this problem by introducing sequencing games. Sequencing games have been extended by many researchers. The extensions focus on the allocation rules [20, 21], the number of jobs [22, 23], the admissible rearrangements of jobs [24, 25], the initial order of jobs [26, 27], the family or batch sequencing situations [28, 29], the ready times [30], and the due-dates [31]. Convexity and balancedness of the corresponding sequencing games are the main goals of these studies as convex games and balanced games have nice properties.
Studies on the sequencing games with multiple machines are also found in the literatures. Hamers et al. [32] and Slikker [33] proposed a multiple-machine scheduling in which each player has one job and each job is processed on one machine only; the corresponding sequencing game is balanced if the normal processing times are equal or the weights are equal. Calleja et al. [34] considered a two-machine scheduling in which each player has two jobs and each job must be processed on each machine; it is shown that the corresponding sequencing game is balanced. Slikker [35] extended the model in [34] to sequencing games with machines and jobs.
In all the above mentioned models, the processing times are constant. van Velzen [36] studied a sequencing game with controllable processing times, where the processing times can be reduced to crashed processing times; the corresponding sequencing games are balanced but need not be convex.
In this paper, a multiple-machine scheduling problem with position-based learning effects is studied. In our model, each player has one job to be processed on one machine only, there is an initial schedule of the jobs, and the learning index depends on the machine. The actual processing time of a job is not a constant but a power function of its position. To achieve the optimal schedule, the jobs should be rearranged, and a rearrangement results in some changes of the processing times. The processing time of a job decreases if the job moves to the back of its queue; it increases if the job moves to the front of its queue. A multiple-machine sequencing game is introduced to allocate the cost savings. The game is balanced if and only if a related machine game is balanced. If in the initial schedule the normal processing times of jobs that are on the same machine are equal and each machine has an equal number of jobs, then the related machine game is balanced; furthermore, the multiple-machine sequencing game is balanced. To the best of our understanding, sequencing games have not been studied so far.
The rest of the paper is organized as follows. In Section 2 some preliminaries are recalled. In Section 3 a multiple-machine scheduling problem with learning effects is considered, and a cooperative game is defined on this scheduling problem. In Section 4 it is shown that a simple multiple-machine sequencing game with learning effects is balanced. Some concluding remarks are given in Section 5.
2. Preliminaries
A cooperative game is denoted by , where is the set of players and is the characteristic function such that .
Let be a cooperative game and an order of the players. A coalition is connected if for all , and with it holds that . The game is a -component additive game if(1) for all ,(2) for all with ; that is, is superadditive,(3) for all , where is the set of maximally connected components of with respect to . A coalition is maximally connected if is connected and is not connected for all .
The core of a cooperative game is defined bywhere .
For a cooperative game and a coalition , the subgame is defined by for all .
Let be a matrix. A permutation game is defined byfor all , where and is the set of all permutations of .
It is well known that the component additive games and the permutation games are balanced [37, 38]; thus their cores are not empty.
3. Multiple-Machine Scheduling with Learning Effects and Sequencing Games
In a multiple-machine scheduling problem, there are machines and players. Each player has one job to be processed on one of the machines, every job can be processed by any machine, and there is an initial schedule of jobs. Let denote the set of machines and let denote the set of players. With an abuse of notation we denote the job of player by itself. The normal processing time of job is denoted by . The cost of player depends linearly on the completion time of his job; that is, , where is the weight of job . The objective is to minimize the total cost which is the sum of the weighted completion times of the jobs.
The initial schedule of jobs is denoted by , and indicates that job is processed in position on machine . The notation is the inverse mapping of .
We assume that all the machines have position-based learning effects. The learning index of machine is denoted by , where . Let be the actual processing time of job if it is scheduled in position on machine ; then
A multiple-machine scheduling with learning effects as described above is denoted by , where vectors , , and .
The ordering on is defined as follows. For , , if and only if and . It indicates that jobs and are on the same machine and follows . Note that and are the first component and the second component of , respectively. Let and .
For each , the set represents the jobs that are on machine with respect to , and is the number of jobs on machine ; that is, . In addition, denotes the initial processing order of the jobs that are on machine ; that is, implies that job is in position on machine .
Suppose that job is on machine ; then the starting time of job with respect to is given byand the completion time of job is the sum of its processing time and the waiting time; that is,
We assume that, in the initial schedule, the starting time of the last job of each machine is less than or equal to the completion times of the last jobs of other machines. Formally, for each , the initial schedule satisfies for all , where is the last job of machine with respect to . This assumption implies that in the initial schedule the last job of a machine cannot make any profit by joining the end of a queue of any other machine.
The cost of job with respect to schedule is denoted by . For each coalition , the total cost of is the sum of the costs of jobs contained in ; that is,Clearly, there is an optimal schedule such that the total cost is minimal. The maximal cost savings can be seen as a profit of all players.
The maximal cost savings of a coalition depend on the set of admissible schedules of this coalition. A schedule is admissible for with respect to if it satisfies the following conditions.(1)For , jobs and which are on the same machine can be switched only if the jobs that are between and belong to .(2)For , jobs and that are on different machines can be switched only if both the jobs in and the jobs in are contained in .The set of all admissible schedules for is denoted by .
For multiple-machine scheduling problems with learning effects , a corresponding -sequencing game is defined bywhere is the optimal schedule for coalition . Obviously, for all .
Remark 1. Let and the last job . To achieve the optimal schedule , the queue of machine should be rearranged. After the rearrangement, the last job (it may be different from ) can make a profit by joining the end of a queue of another machine, although this will not occur in the initial schedule. The reason is that the starting time of the last job of the new queue may increase. The following example illustrates this.
Example 2. Let be a two-machine scheduling problem with learning effects, where , , , , and . The initial schedule is , , and ; it satisfies and . Consider the coalition ; we have .
Schedule is given by , , and ; thus .
Schedule is , , and ; then . The schedules , , and are depicted in Figure 1. Obviously, the optimal schedule for is . The cost increases if the initial schedule is changed to , but there is a profit if the last job of machine 1 in schedule joins the end of the queue of machine 2.
4. Balancedness of Simple -Sequencing Games with Learning Effects
In this section, we restrict attention to simple multiple-machine scheduling problem with learning effects. A multiple-machine scheduling problem with learning effects is simple if the normal processing times of the jobs that are on the same machine are equal and an equal number of jobs are scheduled on each machine initially. Without loss of generality, we assume that, for all , and if . The set of such simple multiple-machine scheduling problems is denoted by .
If job is in position on machine k, then the actual processing time of job is , the starting time is , and the completion time, denoted by , is given by
For each , let and let denote the optimal schedule for coalition .
Lemma 3. If , then for all .
Proof. In the initial schedule , the starting time of the last job of machine is , and the completion time of the last job of machine is since each machine has jobs. According to assumption (6), we have for all .
In the following, we will show that there is a contradiction if .
Assume that . Then there is a machine such that . According to the admissible rearrangement, the last job of machine with respect to is in coalition . In , the completion time of the last job of machine satisfies since machine has at most jobs, and the starting time of the last job of machine satisfies as machine has at least jobs. If joins the end of the queue of machine , the cost of will decrease since ; furthermore, the cost will decrease. It is contrary to the fact that is minimal.
Thus for all .
Lemma 4. Let and let be the corresponding -sequencing game. Then, for each , the subgame is a -component additive game.
Proof. Obviously, is superadditive, and for all . It is necessary to prove that .
It follows from Lemma 3 that, in the optimal schedule , the jobs in are not allowed to join the ends of other queues; thus and the subgame is a single-machine sequencing game.
For each , the cost of job with respect to is denoted by and is given by . Following from (9), it holds thatFor any coalition , where is the optimal order for .
Suppose ; then and are connected, , and are not connected. The optimal orders of and are denoted by and , respectively. Thuswhere the third equality follows from (10). Since jobs in are not allowed to join the coalition and vice versa, we have if and if . HenceThe proof follows exactly in the same way for .
Remark 5. The subgame need not be a -component additive game if there is an such that .
Example 6. Let , , , , and , and the initial order is given by for all . Take the coalition such that . It follows thatand . Obviously, is not a -component additive game since .
Although the position of job 4 is unchanged, its completion time decreases if jobs 2 and 3 are switched. In other words, switching of jobs 2 and 3 results in an “extra” profit for job 4.
Before we prove the balancedness of the corresponding -sequencing game, a machine game is defined.
Let and let be the corresponding -sequencing game. For each , a machine game is defined byThe machine game is defined on the set of machines. The value is the cost savings that the machines in can attain if they cooperate.
Since subgames are -component additive games for all , following Theorem 3.1 in [32] and Theorem 4.1 in [33], we have the following.
Theorem 7. Let , and let and be the corresponding -sequencing game and machine game, respectively. Then is balanced if and only if is balanced.
The following lemma shows that the coalition value can be rewritten as a value of a permutation game.
Lemma 8. If and is the corresponding -sequencing game, then there is a permutation game such thatfor each .
Proof. For convenience, renumber the machines in such that . Note thatFor , , let if ; then we haveFor each schedule , there is a permutation such that for all . For each permutation , there is a schedule satisfyingwhere is such that . Thuswhere the first equality follows from (19) and the second equality follows under the assumption .
Arising from , we have
Theorem 9. If , then the corresponding -sequencing game is balanced.
Proof. Following from (22), we haveSince the permutation game is balanced, the core is not empty, and there is a vector . Let for all ; thenwhere the inequality follows from the fact that is a core element of the game . If , then . Thus , the core of the game is not empty, and is balanced.
The balancedness of the -sequencing game follows from Theorem 7 directly.
5. Conclusions
Cooperative games based on multiple-machine scheduling problems with learning effects are investigated, where the processing times are not constants. A necessary and sufficient condition for the balancedness of the corresponding sequencing games is that the related machine games are balanced. Furthermore, the sequencing games are balanced if normal processing times of the jobs that are on the same machine are equal and each machine has an equal number of jobs to be processed.
If the normal processing times of jobs that are on the same machine are different, then the corresponding subgames which are defined on one machine are not component additive games, and Theorem 7 does not hold. The balancedness of general sequencing games with learning effects in which the normal processing times are different will be studied in the future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Science Foundation of China (71201089, 71371030, and 71401003), Specialized Research Fund for the Doctoral Program of Higher Education (20111101110036), and Yichang Science and Technology Research and Development Project (A201230225).