This paper investigates a single-machine two-agent scheduling problem to minimize the maximum costs with position-dependent jobs. There are two agents, each with a set of independent jobs, competing to perform their jobs on a common machine. In our scheduling setting, the actual position-dependent processing time of one job is characterized by variable function dependent on the position of the job in the sequence. Each agent wants to fulfil the objective of minimizing the maximum cost of its own jobs. We develop a feasible method to achieve all the Pareto optimal points in polynomial time.

Scheduling theory is very useful in industrial applications and provides much guidance in the real world. The traditional scheduling problems have only one agent to optimize its own objective. With the development of modern technology, more and more scheduling models occur. Once entering the 21st century, the existing one-agent scheduling models can not meet with the requirement of the time. Particularly, in the big data period, the internet must handle various instructions of online net citizens and meet with their different preferences. Under such environments, multiagent scheduling models appear. The ultimate theoretical results of multiagent scheduling problems are to get all the Pareto optimal points. Such a problem is usually called as

Of multiagent problems, the two-agent problems are obviously most simple and most representative. Agnetis et al. [

In the paper, we study the two-agent PP scheduling model introduced in [

There are two agents, agent

The first job is processed at time zero.

The processing of the jobs of

The machine processes the jobs of

From the conditions, any schedule is consistent with a sequence of

Let

From Lemma

Let

If problem

In this section, we will describe our algorithm to show that the PP problem

We can apply Algorithm

Algorithm

Suppose

We complete the proof.

In the following, we state a very popular method to get a Pareto optimal point of problem

Algorithm

For each schedule

For any two Pareto optimal points

Since the cost function

We complete the proof.

Algorithm

It is obvious that Algorithm

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Natural Sciences Foundation (Grant no. 20142BAB211017) of Jiangxi Province and the School Subject (Grant no. 06162015) of Jiangxi University of Finance and Economics.