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We consider a scheduling problem in which both resource dependent release times and two agents exist simultaneously. Two agents share a common single machine, and each agent wants to minimize a cost function dependent on its own jobs. The release time of each

Machine scheduling problems with multiagent have received increasing attention in recent years. Different agents share a common processing machine, and each agent wants to minimize a cost function depending on its jobs only. Scheduling with multiple agents is firstly introduced by Baker and Smith [

The scheduling problems with resource consumption have been studied for many years. Cheng and Janiak [

However, to the best of our knowledge, no work has been done on models with both aspects of resource dependent release times and multiagent in the literature. These two categories of scheduling problems have been extensively and separately researched over the last two decades. In this paper, we study the two-agent scheduling problems on a single machine with resource dependent release times, where the goal is to find a schedule that minimizes the objective function of one agent with the restriction that the objective function of the other agent cannot exceed a given bound. The problems under consideration fall into the category of scheduling problems with resource consumption and multiple agents. Such a scheduling problem commonly arises in the steel industry. Janiak [

The remainder of this paper is organized as follows. In Section

We now describe our problem formally. There are two families of independent and nonpreemptive jobs

The goal is to minimize the total amount of resource consumption

In this section, we develop an optimal polynomial time algorithm to solve the problem

Given a sequence

Thus the completion time of job

Moreover, let the maximum job completion time be denoted by

Given a sequence

Now we can define bounds for the constraint

An optimal schedule exists in which the

The resource consumption function

An optimal schedule exists in which the

The makespan of agent

Next, an algorithm to determine an optimal schedule of the problem

Algorithm

The proof of optimality is straightforward from the results of Lemmas

In this paper, we combine two important issues in scheduling that recently have received increasing attention from researchers: resource dependent release times and multiple agents. Our goal is to find a schedule for the problem of minimizing

The future research may be directed to analyze the problems with other objective functions such as minimizing the number of late jobs, the total weighted completion time and tardiness. An interesting research topic is also to analyze the scheduling problem with more than two agents or in other machine environments.

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

The authors are grateful to the editor and reviewers for their helpful comments and suggestions. This research is supported by the National Natural Science Foundation of China (Grant no. 71001074) and the Program for Liaoning Excellent Talents in University (Grant no. WJQ2013003).