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A new maximum lateness scheduling model in which both cooperative games and variable processing times exist simultaneously is considered in this paper. The job variable processing time is described by an increasing or a decreasing function dependent on the position of a job in the sequence. Two persons have to cooperate in order to process a set of jobs. Each of them has a single machine and their processing cost is defined as the minimum value of maximum lateness. All jobs have a common due date. The objective is to maximize the multiplication of their rational positive cooperative profits. A division of those jobs should be negotiated to yield a reasonable cooperative profit allocation scheme acceptable to them. We propose the sufficient and necessary conditions for the problems to have positive integer solution.

In this paper, we introduce a new maximum lateness scheduling model in which both cooperative games and variable processing times exist simultaneously. There are many situations where one person is not able to undertake all the jobs alone in a large project and two persons need to cooperate in order to complete a project. Each person offers a single machine to process jobs. A division of those jobs should be negotiated to yield a reasonable cooperative profit allocation scheme acceptable to them. In many manufacturing processes, the processing times of jobs may be dependent on their positions in the sequence. This phenomenon is called

Scheduling problems with two-person cooperative games have received increasing attention in recent years. Jin et al. [

Next, we will present a brief review of scheduling problems with variable processing times as follows. Gawiejnowicz [

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

The problems of maximum lateness scheduling on two-person cooperative games with variable processing times are described as follows. There is a set of jobs

Given a solution for the two-person cooperative games, we use

The problems of maximum lateness scheduling on two-person cooperative games with variable processing times and common due date can be described by the three-field notation of Jin et al. [

We consider the two-person cooperative games on maximum lateness scheduling problems

Let

Hence, we prove

Likewise, we can show that

The profit functions are as follows:

Thus, the two cooperative profit functions are the quadratic functions of positive integer

When the discriminant

Under the condition of

The sufficient and necessary condition for the problem

If the problem has a solution, the two cooperative profit functions

We set

We have the following theorem from Theorem

Under the condition of expression (

If

If

If

when

when

The problem size is

Furthermore, we consider the following numerical example.

For the problem

We have

For the problem

The profit functions are as follows.

Thus, the two cooperative profit functions are the quadratic functions of positive integer

When the discriminant

Under the condition of

In consideration of clear expression, we set

The sufficient and necessary condition for the problem

If the problem has a solution, the two cooperative profit functions

We get

Thus, we obtain

We have the following Theorem

Under the condition of expression (

If

If

If

when

when

In this paper, we introduce an aging effect and a learning effect into the cooperative games on scheduling. Two persons have to cooperate in order to process a set of jobs. The objective is to maximize the multiplication of their rational positive cooperative profits. We show that two scheduling problems

For future research, it is interesting to extend the problems to involve other position-dependent processing times scheduling model. Another interesting future research direction is to analyze the problems with other objective functions such as minimizing the number of late jobs, the total weighted completion time, and tardiness.

The authors declare that they have no conflicts of interest.

This research is supported by the Key Program of Social Science Foundation of Liaoning Province (Grant no. L15AGL013), the Natural Science Foundation of Liaoning Province (Grant no. 201602545), the National Natural Science Foundation of China (Grant no. 71001074), the Science Research Foundation of Department of Education of Liaoning Province (Grant no. W2015310), and the Humanity and Social Science Research Project of Department of Education of Liaoning Province (Grant no. ZJ2015037).