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.

We consider the

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 [

Curiel et al. [

Hamers et al. [

However, in some practical situations, a clear initial schedule is not available. Klijn and Sánchez [

The Shapley value is a prominent cost allocation for a cooperative game (see [

The remainder of this paper is organized as follows. Section

A

For the

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

In general, it is hard to compute the Shapley value according to the definition. We present a cost allocation rule for the

In this section, we focus on the

Next we prove that formula (

For

It is easy to see the schedule generated by the

For

By Lemma

When

The following theorem shows that the rule

For

We prove the theorem by induction on the number of jobs of the

When

Next we suppose that the conclusion holds for the

From Lemma

In this section, we consider the

Let

It is easy to see that, according to the

By Lemma

For

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.

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

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