Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 206230, 9 pages
http://dx.doi.org/10.1155/2015/206230

Research Article

Single Machine Group Scheduling with Position Dependent Processing Times and Ready Times

Xingong Zhang and Qiulian Xie

College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China

Received 26 January 2015; Revised 11 April 2015; Accepted 12 April 2015

Academic Editor: Chin-Chia Wu

Copyright © 2015 Xingong Zhang and Qiulian Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate a single machine group scheduling problem with position dependent processing times and ready times. The actual processing time of a job is a function of positive group-dependent job-independent positional factors. The actual setup time of the group is a linear function of the total completion time of the former group. Each job has a release time. The decision should be taken regarding possible sequences of jobs in each group and group sequence to minimize the makespan. We show that jobs in each group are scheduled in nondecreasing order of its release time and the groups are arranged in nondecreasing order of some certain conditions. We also present a polynomial time solution procedure for the special case of the proposed problem.

1. Introduction

In traditional scheduling, job processing times are assumed to be fixed and known (Pinedo [1]). However, in reality, we often encounter settings in which the job processing times vary with learning effect (Cheng et al. [2] and Lee et al. [3]), deterioration effect (Alidaee and Womer [4], Cheng et al. [5], and Yin et al. [6]), time dependent effect (Gawiejnowicz [7], Zhang et al. [8], and Yin et al. [9]), and positional effect (Rustogi and Strusevich [10, 11], Bachman and Janiak [12], and Yin et al. [13]). Rustogi and Strusevich [14] combined all kinds of effect in a paper which considered the two classical objectives: the makespan and the sum of the completion times.

In the second place, many manufactures have implemented the concept of group technology (GT. Burbidge [15]); it is conventional to schedule continuously all jobs from the same group. Group technology that groups similar products into families helps increase the efficiency of operations and decrease the requirement of facilities (Mitrofanov [16], Janiak and Kovalyov [17], and Webster and Baker [18]). In this paper, we do not assume that all maintenance periods are identical and allow each one of them to leave the processing conditions of the machine in a different state. We deal with a more general concept of a group setup time performed by the processing group. Thus, the effects that change the actual group setup time may become additionally dependent on the actual processing times of jobs before the proposed group.

Rustogi and Strusevich [10] presented real-life examples: “These general positional effects can be found in practice as well. Extending the coursework marking example above, after marking a certain number of scripts, the teacher might get tired or bored, her attention becomes less focused and each new script may even take longer to mark than the one before. We are sure our academic colleagues know this feeling, and they also know the remedy: take a break, have a cup of coffee.” However, to the best of our knowledge, only few results concerning scheduling models and problems with position dependent processing times and group technology simultaneously are known. But combining the group technology with start time dependent processing times is more common. For the case that setup time of each group is a fixed constant, Wang et al. [19] considered single machine group scheduling in which the actual processing time of a job is a general linear decreasing function of its starting time; for the makespan minimization problem and total completion time minimization problem they showed that some problems can be solved in polynomial time. Xu et al. [20] considered the single machine scheduling problems with group technology and ready times; the job processing times are described by a function which is proportional to a linear function of time; the setup times of groups are assumed to be fixed and known; it showed that minimizing the makespan with ready times can be solved in polynomial time and proposed a heuristic algorithm. J.-B. Wang and J.-J. Wang [21] continued the work of Xu et al. [20]; they considered a more general deterioration model than the group setup times and job processing times; the objective is to minimize the makespan. They showed that the problem can be solved in polynomial time when start time dependent processing times and group technology are considered simultaneously.

Motivated by the ideas of Rustogi and Strusevich [14] and J.-B. Wang and J.-J. Wang [21], we consider the single machine scheduling problem with ready times of the jobs under the group technology assumption and position dependent processing times. Our objective is to find the optimal group sequence and the optimal job sequence to minimize the makespan. The rest of the paper is organized as follows. In the next section we describe the formulation of our problem. In Section 3, we consider the solution method for minimizing makespan. In Section 4, a reduced model will be introduced and proposed a polynomial time algorithm. The conclusion is given in the last section.

2. Problem Description

The independent jobs have to be processed on a single machine. They are nonpreemptive and to be grouped into groups: . The jobs in the same group are consecutively processed as long as the job has arrived; a setup time is required if the machine switches from one group to another and all setup times are positive. Assume that each group contains a total of jobs, so that the permutation of jobs in the th group is given by , where . Let represent the ready (arrival) time of the th job in group . Depending on the choice of groups and the order in which they are performed, the actual processing time of a job , scheduled in position in group , is given as follows:

where denotes positive group-dependent job-independent positional factors of the th job in group ; we call the values deterioration factors.

Furthermore, since the number of jobs, , in each group and the total duration of each group, , are known, the actual setup time is defined as follows:

where (>0) is the deterioration rate of the group . The deterioration factors are given in the form of a collection of ordered array of numbers:

and , satisfy the following relations: when , there are

For a given schedule , let denote the completion time of job , and represent the makespan of a given schedule. Using the three-field notation schema in scheduling problems (Graham et al. [22]), the makespan minimization problem is denoted as , , , , where denote group technology.

3. The Solution Method

In the following section, we will give the solution method so that the single machine minimization scheduling problem with deteriorating jobs and ready times can be solved under certain conditions. Firstly, we consider that all jobs can be processed in one group; that is, .

Lemma 1. For the problem , , where meets (3), the optimal job sequence can be obtained by sequencing the jobs in nondecreasing order of .

Proof. Suppose that and are two job sequence, where and denote a partial sequence (note that and may be empty), and the difference between and is a pairwise interchange of two adjacent jobs and . In addition, the completion time of the last job of in sequence is denoted by . Then, the completion times of job and under are

Similarly, the completion times of jobs and under are

Suppose that , based on (6) and (8), we have

Based on the formula (3) , , and the values of , , and , we divide these values into three cases as follows.

(1); that is, jobs and have arrived at time ; then

(2); that is, job has arrived before time , and has not arrived at time ; then

(3); that is, jobs and have not arrived at time ; then

In conclusion, we have . Repeating this interchange argument, an optimal schedule can be obtained by sequencing the jobs in nondecreasing order of .

Next, we consider the problem , , ; assume that denotes the completion time of the th group and is satisfied in group . Then we have

where

Theorem 2. For the problem , , , , where satisfies formula (3), and satisfy formula (4). An optimal schedule can be obtained by the conditions as follows.
If (1) jobs in each group are scheduled in nondecreasing order of , that is,

and if (2) the groups are arranged in nondecreasing order of

where

where , .

Proof. In the same group, the result of (1) can be easily obtained by Lemma 1. Next, we consider the case in item (2). Let and be a pairwise interchange of two adjacent groups and , that is, and , where and are partial sequences. Further, we assume that denotes the completion time of the last job in . To show dominates , it suffices to show that . Under , using (13), we obtain that the completion time of group is

and the completion time of the group is

Under , the completion time of the group is

and the completion time of the group is

Suppose that

that is

Based on (19) and (21), we have

Therefore ; this completes the proof.

4. A Reduced Model

In this section, we explore a single machine model, which can be expressed as a special case of the general problem , , , ; that is, the actual setup time of group is a constant (>0). The main purpose of this section is to give a streamlined conclusion and algorithm. Assume that denotes the completion time of the th group and is satisfied in the group . Then we have

where

and , . Then can be written as

Theorem 3. For the problem , , , the satisfies formula (3), and satisfy formula (4); the optimal schedule satisfies the following.

(1)Jobs in each group are scheduled in nondecreasing order of ; that is,

(2)The groups are arranged in nondecreasing order of , where

where , .

Proof. Similar to Theorem 2, using the two exchange methods, under , based on (27), we can obtain the completion time of

and the completion time of

Under , the completion times of and are

Supposing that , based on (31) and (32), we have

Therefore ; this completes the proof.

For the problem , , , , we provide an algorithm based on the result of Theorem 3 as follows.

Algorithm 4.
Step 1. Jobs in each group are scheduled in nondecreasing order of ; that is,

Step 2. Let

where , .
Calculate and , .
Step 3. Groups are scheduled in nondecreasing order of .
Obviously, the complexity of obtaining the optimal job sequence within a certain group is and that of obtaining the optimal group sequence is . It is easy to show that . Hence, the complexity of Algorithm 4 is at most . In addition, we demonstrate Algorithm 4 by the following example.

Example 5. There are eight jobs divided into three groups to be processed on a single machine.
Let , and , , , , , ; , , , , , , , , , ; , , , , , , , , , , .

Solution. According to Algorithm 4, we solve Example 5 as follows.

Step 1. The optimal job sequence in , , is

Step 2. Calculating and , we compute the following values for each group:

, that is, the jobs from the first to continuous processing in group , :

, that is, the jobs from the first to continuous processing in group , :

, that is, the jobs from the second to continuous processing in group , .

Step 3. Since , hence, the optimal group sequence is .

Therefore, the optimal schedule is , and the optimal value of the makespan is 90.2.

5. Conclusion

In this paper we have considered the scheduling problem with group technology and position dependent processing times, for the case that group setup times are linearly related to the completion time of the former group and the job processing times are the general nondecreasing function of the positional factors. We showed that the makespan minimization problem with ready times can be solved under certain conditions. A reduced model can be considered as special case of our general model. Furhermore, we present an time algorithm to solve the proposed problem.

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and anonymous referees for their helpful comments on earlier version of this paper. This work was supported by National Natural Science Foundation of China (11401065), the China Postdoctoral Science Foundation funded project (2013M540698 and 2014T70854), The Chongqing Municipal Science and Technology Commission of Natural Science Fund Projects (cstc2014jcyjA00003), and the Key Foundation of Chongqing Normal University (2011XLZ05).

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