#### Abstract

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.

#### 1. Introduction

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* “aging effect”* or* “learning effect”* in the literature. In the aging effect, the processing time of a job increases as a function of its position in the sequence, while, in the learning effect, the processing time of a job decreases as a function of its position in the sequence. In this paper, we address maximum lateness scheduling on two-person cooperative games with variable processing times. To the best of our knowledge, no work has been done on maximum lateness scheduling models combining two aspects of cooperative games and variable processing times.

Scheduling problems with two-person cooperative games have received increasing attention in recent years. Jin et al. [1] consider the two-person cooperative games on makespan scheduling. The case of jobs with same processing time is addressed and each party’s processing cost is determined by their minimized makespan. Dou et al. [2] consider the two-person cooperative games on total completion time scheduling where the processing time of each job is the same. Gu et al. [3] discuss the two-person Nash Bargaining problem with changeable processing time, where the processing cost is defined as their minimized maximum flow time. Jin [4] establishes a mathematical model of the problem where two persons process a batch of jobs by cooperation, where job processing time is a linear function of its starting time. Jin [5] studies two-person cooperative games problem on total completion time scheduling with changeable processing time. Gu et al. [6] consider the scheduling problem on two-person cooperative games, where the processing cost is defined as their minimized maximum latency. Liu et al. [7] study the problem of two-person cooperative games on minimizing the total (weighted) number of tardy jobs and on minimizing the total (weighted) completion time. Chen [8] studies an integer-valued cooperative games model on scheduling. Tang et al. [9] propose a new topic on dual relationships between job games and machine games and provide the extensive reviews of research on scheduling games. The literature mentioned above investigates the cooperative games on scheduling; however, they do not consider the maximum lateness scheduling with variable processing times.

Next, we will present a brief review of scheduling problems with variable processing times as follows. Gawiejnowicz [10] is the first time to introduce the position dependent on a number of executed jobs on a single processor. Biskup [11] is the first to consider the learning effect in a scheduling problem. Cheng and Wang [12] consider a single machine scheduling problem in which the job processing times will decrease as a result of learning. Mosheiov [13] proves that flow-time minimization with a learning effect on parallel identical machines has a polynomial time solution. Bachman and Janiak [14] investigate some single machine scheduling problems, where job processing times are defined by functions dependent on their positions in the sequence. Liu et al. [15] consider the single machine scheduling problems with two agents and position-dependent processing times. Rudek [16] provides the computational complexity results of scheduling problems with position-dependent job processing times, where the boundary between polynomially solvable and strongly NP-hard cases is given. J. B. Wang and J. J. Wang [17] investigate flowshop scheduling problems with a general exponential learning effect. Zhou and Zhang [18] consider the cooperative games based on multiple-machine scheduling problems with learning effects. Zhang et al. [19] propose a group scheduling model with deteriorating and learning effect on a single machine. The aging effect is mentioned for the first time by Mosheiov [13] during analysis of the learning effect. Janiak and Rudek [20] prove that the makespan minimization problems on a single machine with release dates and some special cases of the given aging effect are NP-hard. Yang et al. [21] consider single machine scheduling and slack due date assignment problems simultaneously with the position-dependent aging effect and deteriorating maintenance. Rudek [22] analyzes the single machine maximum lateness minimization scheduling problem with processing time-based aging effects. Ji et al. [23] consider a single machine due date assignment scheduling problem with job-dependent aging effects and a deteriorating maintenance activity. Choi [24] considers a two-agent single machine scheduling problem with linear position-based aging effects and job-dependent aging ratios. Zeng et al. [25] consider the problem of bidirectional nonnegative deep model and its optimization in learning. The difference between our model and the above scheduling models is that we incorporate the important factor of cooperative games in the model.

The remainder of this paper is organized as follows. In Section 2, we describe the proposed problems. In Section 3, we discuss the two-person cooperative games on maximum lateness scheduling problems with an aging effect and a learning effect, respectively. Section 4 gives some concluding remarks.

#### 2. Problem Description

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 that has to be processed by two persons. Each job should be processed only once without interruption. All jobs are available for processing at time zero. Two persons, each with a single machine, cooperate to complete the processing of these jobs. The due date of each job is denoted by , and all jobs have a common due date ; that is, . The processing time of job is a function dependent on its position in a sequence. We consider two special models of job variable processing times: (1) an* aging effect * and (2) a* learning effect *, , , where is the normal processing time, denotes a constant* aging ratio* in model (1) and a constant* learning ratio* in model (2). Since job processing time is positive value, it is assumed that for the learning effect model. The normal processing time and the* aging/learning ratio *are identical for all jobs.

Given a solution for the two-person cooperative games, we use to denote the completion time for job . If we use and to denote each person’s subset of jobs, then , . For person , processing a job per unit time will bring unit profit; the processing cost is defined as the minimum value of the maximum lateness of jobs, that is, min , and then the profit function of person is defined by . If two persons cannot achieve a cooperation agreement, let denote the th individual’s cost; then there should be the inequalities of the form , (). Hence, the cooperative profit of person is formulated by , the overall profit of two persons is measured by the product of and , and our objective is to maximize this product.

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. [1] of the forms and , respectively. The first-field 2 represents two-person cooperative games and each person has one machine. The second field describes job characteristics, that is, the structures of the processing time and the due date. The third field represents the optimization goal.

#### 3. Main Results

We consider the two-person cooperative games on maximum lateness scheduling problems and in Sections 3.1 and 3.2, respectively.

##### 3.1. Problem

Let be the number of jobs in subset assigned to person 1, and let be the number of jobs in subset assigned to person 2, . Next, we will show that for person 1 by induction.Suppose for job ; then we consider job :

Hence, we prove , which is a fixed constant for all permutations of the jobs.

Likewise, we can show that for person 2 by induction. , which is a fixed constant for all permutations of the jobs.

The profit functions are as follows: Since the profit functions have positive values, it should hold that ,

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

When the discriminant of and the discriminant of are smaller than zero, we do not need to study this problem. So we only consider the solution when

Under the condition of , can be expressed as the following form: where

Theorem 1. *The sufficient and necessary condition for the problem to have positive integer solution is Furthermore, if the problem has a solution, then the optimal solution is certainly in closed interval , where , .*

*Proof. *If the problem has a solution, the two cooperative profit functions and must be of positive value. We have and from . We obtain and from . Since , , and , we have . Hence, we have . Since is a integer, we set , . Thus, , . So the problem to have a solution equivalent to the closed interval has at least one positive integer , that is, . Therefore, the sufficient and necessary condition for the problem to have positive integer solution is .

We set as a four-time polynomial in and the function can be expressed as . So is a cubic polynomial in and have three roots. Under the conditions of (6) and (9), we know that there are three roots to . Without loss of generality, let us set them to be and their relationship can be defined as . So , , and are the three extreme points of and we can get . Since the coefficient of four times polynomial about expression (7) is positive, the coefficient of cubic polynomial is positive and is the only extreme point and maximum value point. Therefore, the optimal solution of the problem is certainly in closed interval .

We have the following theorem from Theorem 1.

Theorem 2. *Under the condition of expression (9), there is an optimal solution for the problem , and the result can be expressed as or . The optimal solution is as follows:*(I)*If and , then ;*(II)*If and , then ;*(III)*If and , when , ; when , .*

*Remark 3. *The problem size is (1) and the optimal solution can also be found in a constant time (1).

Furthermore, we consider the following numerical example.

*Example 4. *For the problem , we assume that , , , , , , , and .

*Solution*. The two cooperative profit functions are

We have , , , , , and . So we obtain , . Three roots to are , , and , respectively. So we obtain or . Because , we have .

##### 3.2. Problem

For the problem , we can calculate similarly the makespan for persons 1 and 2 by induction.

, which is a fixed constant for all permutations of the jobs.

, which is a fixed constant for all permutations of the jobs.

The profit functions are as follows.Since the profit functions have positive values, it should hold that ,

Thus, the two cooperative profit functions are the quadratic functions of positive integer of the formSince the profit functions have positive values, it should hold that ,

When the discriminant of and the discriminant of are smaller than zero, we do not need to study this problem. So we only consider the solution when

Under the condition of , can be expressed as the following form: where

In consideration of clear expression, we set , .

Theorem 5. *The sufficient and necessary condition for the problem to have positive integer solution is Furthermore, if the problem has a solution, then the optimal solution is certainly in closed interval , where , , , , , and .*

*Proof. *If the problem has a solution, the two cooperative profit functions and must be of positive value. We have from . Since , we have ; that is, . Then, we know through the comparing between and . So we have .

We get from . Because , so we have . Since , then we have .

Thus, we obtain and . Since and* k* is a integer, so we set , , , , , and . Then we obtain , , , , and from the expressions of , , , and . Hence, , . Therefore, the sufficient and necessary condition for the problem to have integer solution is . Similarly, we can prove that the optimal solution is certainly in closed interval .

We have the following Theorem 6 that is similar to Theorem 2. If we prove , we obtain that is the only extreme point of in and it is also the maximum value point of .

Theorem 6. *Under the condition of expression (17), there is an optimal solution for the problem , and the result can be expressed as or . The optimal solution is as follows: *(I)*If and , then ;*(II)*If and , then ;*(III)*If and , when , then ; when , then .*

#### 4. Conclusions

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 and are polynomially solvable. The sufficient and necessary conditions for the problems to have positive integer solution are proposed.

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.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

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).