Abstract

In this paper, the single-machine scheduling problem is studied by simultaneously considering due-date assignment and group technology (GT). The objective is to determine the optimal sequence of groups and jobs within groups and optimal due-date assignment to minimize the weighted sum of the absolute value in lateness and due-date assignment cost, where the weights are position dependent. For the common (CON) due-date assignment, slack (SLK) due-date assignment, and different (DIF) due-date assignment, an time algorithm is proposed, respectively, to solve the problem, where is the number of jobs.

1. Introduction

In the manufacturing industry, it is well known that firms can increase production efficiency by adopting the group technology (GT). The group technology is an approach to manufacturing that seeks to improve efficiency in high-volume production by exploiting the similarities of different products and activities in their production (Neufeld et al. [1] and Yang et al. [2]). Through decades of application, people have found many advantages of the group technology. For example, changeover between different jobs in the same group is simplified, reducing the costs or time involved; jobs in the same group spend less time waiting, which results in less work-in-process inventory; jobs in the same group tend to move through production in a direct route, reducing the manufacturing lead time (see the work of Yang and Yang [3], Lu et al. [4], Yin et al. [5], Wang and Liu [6], Wang and Wang [7], Qin et al. [8], Lu et al. [9], Zhang et al. [10], Liu et al. [11], and Wang and Liang [12]).

In recent years, the problem of the due-date assignment has been closely focused on by scholars (see the work of Yin et al. [13, 14], Wang et al. [15], and Shabtay [16]). Due to the increasing interest in the Just-In-Time (JIT) system, the issue of schedule allocation is becoming more and more important in practical applications. In the classical scheduling problem, the due date is usually a given constant, while in the actual application of life, the duration of the job is not a fixed constant, but a decision variable. In order to strengthen the global competition and improve the service level for customers, the jobs are required to be processed too early or too late in the JIT system, which leads to the trouble of scheduling problems involving advance and delay costs and the expiration date of the construction period. For the early completion of the job, it means that we have to bear a certain inventory cost, while for the delayed completion of the job, we have to bear the contract penalty, and the customer’s goodwill is damaged (see the work of Li et al. [17], Liu et al. [18], Wang et al. [19]). A lot of literature deals with the problems such as the CON, SLK, and some other due-date assignment methods considering jobs. However, under the group technology, there are relatively few studies on the problem of the assignment of jobs. Li et al. [17] considered three due-date assignment methods under group technology. The objective function is to minimize the weighted sum of earliness, tardiness, due-date assignment, and completion time. For the CON, SLK, and different due-date assignments (DIFs), they proved that the problem can be solved in polynomial time, respectively. Ji et al. [20] studied the single-machine slack due-date window assignment scheduling with group technology. The objective function is to minimize the total cost including the earliness, tardiness, due window starting time, and due window size. They proved that the problem can be solved in time.

Brucker [21] considered the CON due-date scheduling problem of minimizing the total cost comprising the total weighted absolute lateness value and common due-date (CON) cost, where the weight is position dependent. He proved that the problem can be solved in a time . Liu et al. [18] considered the SLK due-date assignment scheduling, and the goal is to minimize the total cost comprising the total weighted absolute lateness value and common flow allowance (SLK) cost, where the weight is a position-dependent weight. They proved that the problem can be solved in a time . Wang et al. [19] studied the scheduling problems of single machine resource allocation in job-dependent learning effects. Under the linear and convex resource consumption functions, they proved that the CON and SLK due-date assignment problems can be solved in polynomial time, respectively. Sun et al. [22] considered single-machine scheduling problems on resource allocation, group technology, and learning effect. Under the linear and convex resource consumption functions, they proved that the SLK due-date assignment problem can be solved in polynomial time.

According to this study, we consider due-date assignment and scheduling problems with group technology. Under the assumption of group technology, the jobs are classified into groups by exploiting the similarities of different products and activities in their production. In this paper, we proceed to the study Brucker [21], Liu et al. [18], and Wang et al. [19], which is an extension of their work from considering the CON, SLK, and DIF due-date assignment scheduling problems with group technology and position-dependent weights. We organize the rest of the paper as follows: the problem is formulated in Section 2, several results of the optimal solution are introduced in Section 3, and the conclusions are summarized in Section 4.

2. Problem Formulation

In the study, the problem can be formally described as follows: there are independent non-preemption jobs grouped into groups, i.e., . A single machine and all the jobs are available at time zero, and the single machine can handle one job at a time and preemption is not allowed. Let the number of jobs in group be , i.e., , where denotes the job of group , , and . Jobs in the same group are processed consecutively and do not need setup times. Let denote the processing time of job and be the sequence-independent machine setup time incurred before the process of the first job of group . Each job has a due date , which is assignable according to one of the following three due-date assignment methods:(1)The common (CON) due-date assignment in which all jobs of group are assigned the same due date, i.e., for and (2)The common flow allowance (slack, SLK) assignment in which the due-date for all jobs of group are assigned an equal flow allowance that is equal to its processing time plus the common flow allowance, i.e., for and (3)The different (DIF) due-date assignments in which the due date for all jobs of group are assigned a different due date with no restrictions ( and )

Let be the completion time of job . The goal is to determine the due date (i.e., the CON , the SLK , and the DIF ) and an optimal sequence such that the following objective functions are to be minimized:where is the nonnegative weight of the th position in group (i.e., position-dependent weight, ), denotes the unit cost of ( and ), , is the due-date assignment vector under schedule , and is the lateness of job . Note that , where is the earliness of job , is the tardiness of job , and represents the job that is in the th position of group in for . By using the three-field notation (see Graham et al. [23]), the problems can be denoted byrespectively, where denotes the group technology.

3. Main Results

Lemma 1. For each of the three due-date assignment methods, there exists an optimal schedule with zero machine idle times.

Proof. It is omitted due to simplicity.

Therefore, it is convenient to introduce a dummy job of group for with processing time and weight which is always scheduled at time 0, i.e., . Consequently, for any given job schedule , the completion times can be calculated by the following equation:where denotes the starting time of group in schedule for .

Lemma 2. If the sequence of groups is fixed, for the CON and SLK due-date assignment methods, there exists an optimal sequence with the property that () coincides with the job completion times of group for .

Proof. Let the sequence of groups be fixed. Consider an optimal sequence in .(a)First, we prove that the common due-date of group coincides with the completion time of a job of group . Assume that ; therefore, If , we have If , we have Let , , then we can obtain that Thus, if and otherwise.(b)Assume that ; therefore,where .
If , we haveIf , we haveLet and ; then, we haveThus, if and otherwise.
From (a) and (b), this implies that () is equal to the job completion times of group for .

Lemma 3. For the CON due-date assignment, if the sequence of groups is fixed, for a given schedule , there exists an optimal schedule of group in which , where is a median for the sequence and and .

Proof. By using the technique of small perturbations, from Lemma 2, we assume that . Applying (7) and (8) to the cases of moving the common due-date units of time to the left (right), we have and .

Lemma 4. For the SLK due-date assignment, if the sequence of groups is fixed, for a given schedule , there exists an optimal schedule of group in which , where is a median for the sequence and and .

Proof. It is similar to the proof of Lemma 3.

Lemma 5. For the DIF due-date assignment method, there exists an optimal sequence such that .

Proof. We consider the case that contradicts this optimal property. Consider an optimal sequence in , if , and then, the total cost for job isWe shift to the left such that , and we haveHence, the case is not an optimal due-date assignment. Lemma 5 is proved.

Lemma 6. For the DIF due-date assignment method, let be a given sequence, and the optimal due-date for job can be obtained as follows:

Proof. From Lemma 5, for job , we haveObviously, if , should be 0, if should be . Hence, the total cost for job iswhere .