Multiple-Machine Scheduling with Learning Effects and Cooperative Games

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.


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][3][4][5][6], position-based learning effects [7][8][9][10][11][12][13], combined effects of position-based effects and timedependent effects [14][15][16][17].In a scheduling problem with timedependent 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 positionbased 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 twomachine 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 multiplemachine 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 multiplemachine sequencing game with learning effects is balanced.Some concluding remarks are given in Section 5.
Let  = (  ) × be a matrix.A permutation game (, ) is defined by for all  ⊆ , where   ∈  and Π() 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.

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  = {1, 2, . . ., } denote the set of machines and let  = {1, 2, . . ., } 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   ≥ 0. The cost   of player  depends linearly on the completion time of his job; that is,   () =   , where   ≥ 0 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  0 :  →  × , and  0 () = (, ) indicates that job  is processed in position  on machine .The notation  0 −1 is the inverse mapping of  0 .
We assume that all the machines have position-based learning effects.The learning index of machine  is denoted by   , where   ≤ 0. 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 (, ,  0 , , , ), where vectors  = (  ) ∈ ,  = (  ) ∈ , and  = (  ) ∈ .
Suppose that job  is on machine ; then the starting time ( 0 , ) of job  with respect to  0 is given by and the completion time ( 0 , ) 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  0 satisfies for all ℎ ∈ , where   is the last job of machine  with respect to  0 .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  0 is denoted by ( 0 , ).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 ( 0 , ) − ( * , ) 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  0 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 ( 0 , ) and the jobs in ( 0 , ) are contained in .
The set of all admissible schedules for  is denoted by ().
Remark 1.Let  ⊆   ( 0 ) 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.

Games with Learning Effects
In this section, we restrict attention to simple multiplemachine 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  ∈ ,   ( 0 ) =  and   =   if  ∈   ( 0 ).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   ∑ −1 =1    , and the completion time, denoted by (, ), is given by For each  ⊆ , let () = ∪ ∈   ( 0 ) and let  * () denote the optimal schedule for coalition ().
Before we prove the balancedness of the corresponding -sequencing game, a machine game is defined.
Let (, ,  0 , , , ) ∈  , and let (, V) be the corresponding -sequencing game.For each  ⊆ , a machine game (, ) is defined by The 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 (  ( 0 ), V |  ( 0 ) ) are  0  -component additive games for all  ∈ , following Theorem 3.1 in [32] and Theorem 4.1 in [33], we have the following.Theorem 7. Let (, ,  0 , , , ) ∈  , , and let (, V) and (, ) be the corresponding -sequencing game and machine game, respectively.Then (, V) is balanced if and only if (, ) is balanced.