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Mathematical Problems in Engineering
Volume 2015, Article ID 437403, 6 pages
http://dx.doi.org/10.1155/2015/437403
Research Article

Shapley Value for Parallel Machine Sequencing Situation without Initial Order

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 27 July 2014; Accepted 5 October 2014

Academic Editor: Dongdong Ge

Copyright © 2015 Shanshan Liu and Zhaohui Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the parallel identical machine sequencing situation without initial schedule. For the situation with identical job processing time, we design a cost allocation rule which gives the Shapley value of the related sequencing game in polynomial time. For the game with identical job weight, we also present a polynomial time procedure to compute the Shapley value.

1. Introduction

We consider the -sequencing situation first introduced by Hamers et al. [1]. There are players and each owns exactly one job to be processed on parallel identical machines . Each machine can process one job at a time, and each job needs to be processed on only one machine. Job , owned by player , has a processing time and a weight . Each player takes the cost of his job which occurred in the processing system. The cost is defined as the weighted completion time. The processing is carried out according to a schedule. In a schedule the jobs are partitioned into subsets, and each is processed on one of the machines. The processing order on machine is denoted by , which is a permutation of the jobs on . A schedule is thus denoted as . Let be the completion time of job under the schedule , and the start time. Then the cost of job is , and the total cost is . There are two kinds of problems: given an initial schedule, the players cooperate to improve it; without any initial schedule, the players cooperate to reach a schedule. For both kinds of problems, a schedule is finally chosen and a payment scheme is designed in order to carry out the schedule. The payment scheme reallocates the total cost, and a player may take a cost larger or smaller than the cost of his job such that he is satisfied. In this paper we mainly focus on the -sequencing situation without initial schedule.

The solution to a sequencing situation must contain a schedule indicating how the processing is carried out, as well as an allocation scheme of the total cost. The two parts are not independent since the schedule decides the total cost to be allocated and the position of each job is an important factor of deciding the cost of the player. Finding an optimal schedule for the parallel machine scheduling problem of minimizing the total weighted completion time is NP-hard even for the two-machine case (see [2]). A natural heuristic (list heuristic) is to first put the jobs into a list and then assign them to the first idle machine one by one. The heuristic is called the algorithm if the list is formed by the nonincreasing order of the ratios of job weights and processing times. Kawaguchi and Kyan [3] proved that the algorithm has a performance ratio of . However, the algorithm is optimal for the special cases in which the jobs have identical processing time or identical weight.

Curiel et al. [4] first investigated the single machine sequencing situation with an initial schedule. They constructed a cooperative game, and gave a rule which indicates a specific core element of the game. The characteristic function value of a coalition is the maximal cost savings that the coalition can make by rearranging the jobs in the coalition without jumping over the jobs outside the coalition. Subsequently, various cost allocation schemes were presented for this sequencing situation (see [57]). Curiel et al. [8, 9] generalized the cost function of each job to an arbitrary nondecreasing function of the completion time. Some researchers investigated single machine sequencing situations with additional constraints on the jobs. Hamers et al. [10], Borm et al. [11], and Hamers et al. [12] imposed ready times, due dates, and chains precedence constraints, respectively, on the jobs. Grundel et al. [13] analyzed the single machine sequencing situation with family setup times, where the jobs within a family have an identical cost function.

Hamers et al. [1] first studied the -sequencing situation and proved the balancedness of the related sequencing game for . When , the balancedness was shown for two special cases: the identical processing time and the identical weight. Slikker [14] revisited their results and showed that the general -sequencing game needs not to be balanced.

However, in some practical situations, a clear initial schedule is not available. Klijn and Sánchez [15] first considered the sequencing game without initial schedule. In order to estimate the power of each coalition, they introduced two different cost characteristic functions: and , where stands for the minimum cost of coalition when its jobs are processed last; that is, the coalition forms the tail of a sequence, and stands for the maximum cost of coalition among all possible sequences which cannot be reduced by cooperations inside . Under these two characteristic functions, the corresponding sequencing games are both balanced. Mishra and Rangarajan [16] defined the characteristic function of coalition as the minimum total weighted completion time when the machine only processes the jobs in . In this case, the related sequencing game is no longer balanced, but its Shapley value is well formulated.

The Shapley value is a prominent cost allocation for a cooperative game (see [17]), but it is hard to compute in general. In this paper, we study the parallel identical machine sequencing situation without initial schedule and design cost allocation rules for the special cases in which the jobs have identical processing time or identical weight. The rules give the Shapley value in polynomial time.

The remainder of this paper is organized as follows. Section 2 is a preliminary section. In Section 3, we design a cost allocation rule for the -sequencing situation with identical job processing time which gives the Shapley value in polynomial time. In Section 4, we formulate the Shapley value of the -sequencing game with identical job weight.

2. Preliminaries

A -sequencing situation without initial schedule can be formulated as , where is the set of players, is the set of machines, is the vector of processing times, and is the vector of weights. Since each player owns exactly one job and each job belongs to exactly one player, we do not distinguish between the jobs and players. Then, is also the set of jobs. Also, we suppose that the jobs have been numbered in nonincreasing order of the ratios of their weights and processing times.

For the -sequencing game with respect to , we define the characteristic function as follows. For any coalition , let , where is the schedule generated by the algorithm. We denote the resulting sequencing game as . In this paper we always use this characteristic function for the -sequencing game; thus with given processing times and weights, a sequencing game can be uniquely defined. Let be the Shapley value of player . Then,where is the cardinal of coalition .

The Shapley value can also be characterized by a collection of properties or axioms: efficiency, anonymity, dummy player property, and additivity. Efficiency means the costs taken by the players sum up to the total cost incurred, which is . A player is called dummy player if he contributes no cost when joining any coalition. The dummy player property assures that the Shapley value of a dummy player is zero. In the -sequencing game, if the processing time or weight of some job is zero, then its owner is a dummy player. Anonymity means that the Shapley value is not influenced by the numbering way of the players, and renumbering will not change the value. In the -sequencing game, if two jobs have the same weight and processing time, then their Shapley values are the same. Additivity means that if the characteristic functions , , and satisfy , then .

In general, it is hard to compute the Shapley value according to the definition. We present a cost allocation rule for the -sequencing situation with identical processing time, which gives the Shapley value for the respective sequencing game in polynomial time. And for the -sequencing game with identical weight, we also obtain a method to compute the Shapley value in polynomial time.

3. -Sequencing Situation with Identical Processing Time

In this section, we focus on the -sequencing situation with identical processing time. Let for any . Consider a job list . Note that job is on the th position of the list. For job , let be the set of jobs listed before , and the set of jobs listed after . Given the job list , a schedule can be obtained by list heuristic. We define the cost allocation rule with respect to the job list such that player takes the costwhere is the processing cost of job , is the waiting cost of job , and is the waiting cost of job . Note that in a schedule , the waiting cost of job is , while the processing cost is unrelated with . When , we have

Next we prove that formula (3) gives the Shapley value for the -sequencing situation with identical processing time. First we consider the case with identical processing time and identical weight; that is, and for any .

Lemma 1. For related to the -sequencing situation with and for any , the Shapley value of any player is , where .

Proof. It is easy to see the schedule generated by the algorithm is an optimal processing order for the scheduling problem. The total cost of this schedule isNote that all jobs have the same processing time and the same weight. By anonymity property of the Shapley value, each player shares the same cost. Thus, the Shapley value of any player is .

Lemma 2. For related to the -sequencing situation with and for any , the cost allocated to any player by rule is the Shapley value.

Proof. By Lemma 1, we need only to prove that for any job list and any job , where . Suppose that job is on the th position of list ; that is, . We have When , it holds that . Thus, Let and . If , then for . Thus, . If , we have This completes the proof.

The following theorem shows that the rule under the job list generates the Shapley value for the -sequencing game with identical processing time.

Theorem 3. For related to the -sequencing situation with for any , the Shapley value of player is equal to .

Proof. We prove the theorem by induction on the number of jobs of the -sequencing situation. By the algorithm, when , each machine processes at most one job; that is, there is no waiting cost for any job; thus the Shapley value of player satisfies .
When , we calculate the Shapley value by its definition. Note that holds for all satisfying , and . We have that is, the theorem holds for too.
Next we suppose that the conclusion holds for the -sequencing situation with jobs and consider the situation with jobs. We first construct two auxiliary sequencing games and . Note that the jobs in the original game have the same processing time and the weights . We suppose that the jobs in and have also the processing time . For , we let all jobs have the same weight , and for , we let the jobs be of weights . It is easy to verify that . Then we have since the Shapley value satisfies the additivity property.
From Lemma 2, the Shapley value of can be calculated by formula (3). That is, In , player is a dummy player since his job is of zero weight and contributes nothing in any coalition. By the dummy player property, . Let stand for the sequencing game in which there are only players and is the restriction of on . For player , the Shapley value in is the same as in . According to the induction hypothesis, we have for . Applying the additivity of the Shapley value, we obtain So the conclusion holds.

4. -Sequencing Situation with Identical Weight

In this section, we consider the -sequencing situation in which all jobs have the same weight , but their processing times may be different. We denote the sequencing situation as . The algorithm generates an optimal schedule for this problem. We calculate the Shapley value of the related -sequencing game by decomposing the game into games with some dummy players.

Lemma 4. Let , , and be three -sequencing situations with the same weight , and , and . Then, for the respective sequencing games , , and , one has .

Proof. It is easy to see that, according to the algorithm, the jobs of the same number have the same position for , , and . For any coalition , this is also true. Consider a schedule produced by the algorithm. Suppose that job is processed in the th position on . Let be the set of jobs processed before on according to . For , , and , we have respectively. Since for any and there are exactly jobs in , we get . This relation holds for any job , so .

By Lemma 4, we decompose the -sequencing game into and . Denote as . Then we decompose into and , where the processing times of the jobs in and are given by and , respectively. Further, we decompose into and such that the processing times of the jobs in and are and . Repeating the process, we finally decompose into , where the processing times of the jobs in are given by By Lemma 4, we have . And by the additivity of the Shapley value, we have for any . By Lemma 1, we have For , the first players in are dummy players, so Let and be the restriction of on . It holds that for . However, the Shapley value of can be calculated by Lemma 1. Thus, we haveand the following theorem holds.

Theorem 5. For related to the -sequencing situation with for any , the Shapley value of any player can be computed in polynomial time.

5. Conclusions

In this paper, we investigate the parallel identical machine sequencing situation without initial schedule. We characterize the Shapley value for two special cases in which the jobs have identical processing time or identical weight. However, it is difficult to compute the Shapley value for the general problem with characteristic function we defined before.

We may study the general problem in the following way: choose a feasible schedule, and reach stable cost allocation under the very schedule. And we can use heuristic algorithms such as genetic algorithm or simulate anneal arithmetic to get a final schedule the players agree to. The procedure of deciding the final schedule and the related cost allocation among the players can be done in succession. With a given schedule we do cost allocation machine by machine, which means players whose jobs are on the same machine share the cost which occurred on the very machine. We can design a genetic algorithm to choose the final schedule among all feasible schedules, and for each feasible schedule its fitness function value is the total cost. The cost allocation on each machine is then just like the classic single machine sequencing game, and a cost allocation for each player can be calculated in polynomial time.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is supported by the National Natural Science Foundation of China under Grant no. 11171106.

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