Abstract
This paper considers a single-machine due-window assignment scheduling problem with position-dependent weights, where the weights only depend on their position in a sequence. The objective is to minimise the total weighted penalty of earliness, tardiness, due-window starting time, and due-window size of all jobs. Optimal properties of the problem are given, and then, a polynomial-time algorithm is provided to solve the problem. An extension to the problem is offered by assuming general position-dependent processing time.
1. Introduction
Conventionally, in scheduling theory, due-windows are job-dependent either if they are dictated by the customer (i.e., given constants) or they are decision variables (i.e., due-window assignment). A due-window for job is defined by a due-window starting time and a due-window finishing time , i.e., the due-window , and the due-window size is . In the just-in-time (JIT) production methodology and scheduling theory, setting proper due-windows is challenging (see Gong et al. [1], Janiak et al. [2], and Geng et al. [3]). In the literature, three very popular due-window assignment methods are studied: Common due-window (CON-DW) assignment method (Liman et al. [4, 5]): all jobs are assigned a common due-window, i.e., all the jobs have a common due-window , where , , the due-window size of all the jobs is , and both and are decision variables. In the literature, most studies considered the CON-DW assignment method, e.g., Mosheiov and Sarig [6] addressed a minmax CON-DW assignment problem, the objective of which is to minimise the largest cost among earliness, tardiness, due-window starting time, and due-window size. They proved that the single-machine and two-machine flow-shop problems can be solved in polynomial time. They also proved that the cases of parallel identical machines and uniform machines are NP-hard. Yin et al. [7] considered the batch delivery scheduling problem with an assignable common due-window on a single machine. Yin et al. [8] studied the single-machine scheduling problem with CON-DW assignment and batch delivery cost. Liu et al. [9] considered the single-machine CON-DW assignment scheduling problem with deteriorating jobs. For the weighted sum of earliness, tardiness, and due-window location penalty minimization, they proposed a polynomial-time algorithm to solve the problem. Wang and Wang [10] considered the single-machine resource allocation scheduling problem with learning effect and CON-DW assignment. Slack due-window (SLK-DW) assignment method (Mosheiov and Oron [11]) is , , and , where is the normal processing time of job and and are decision variables. Wang et al. [12] considered the single-machine SLK-DW assignment scheduling problem with deteriorating jobs and learning effect. Ji et al. [13] considered the single-machine SLK-DW assignment scheduling problem with group technology. Yin et al. [14], Yin et al. [15], and Wang et al. [16] considered SLK-DW assignment scheduling problems with resource allocation (controllable processing time). Mor and Mosheiov [17] considered SLK-DW assignment proportionate flow-shop scheduling problems. Different due-windows’ (DIF-DW) assignment method: it is assumed that the job has a due-window , where and denote the starting time and finishing time of the due-window, respectively. The due-window size of the job is , and both and are decision variables. Wang et al. [12] considered DIF-DW assignment scheduling problems with deteriorating jobs and learning effect.
In a recent paper, Wang et al. [18] considered CON-DW and SLK-DW assignment methods with position-dependent weights, i.e., the weight does not correspond with the job but with the position in which some job is scheduled. They proved that both these due-window assignment methods with position-dependent weights can be solved in polynomial time, respectively. “The scheduling with due-window assignment has many real-world applications. For example, the due-window might reflect an assembly environment in which the components of the product should be ready within a time interval in order to avoid staging delays or a shop where several jobs constitute a single customer’s order. It is clear that a wide due-window increases the supplier’s production and delivery flexibility. However, a large due-window and delaying job completion reduce the supplier’s competitiveness and customer service level” (Yang et al. [19]). It is natural and interesting to continue the work of Wang et al. [18] but study the DIF-DW assignment scheduling problem with position-dependent weights. The contributions of this paper are given as follows: (1) the structural properties of scheduling problems are derived; (2) the total weighted penalty of earliness, tardiness, due-window starting time, and due-window size of all jobs’ minimization can be solved in polynomial time; and (3) it is further extended the model to the case with general position-dependent processing time. We refer the reader to the survey of Janiak et al. [2] on the scheduling problems with (CON-DW, SLK-DW, and DIF-DW) due-windows.
The remainder of the paper is organized as follows. In Section 2, we formulate the problem. Section 3 gives some results and an optimal policy for the proposed problem. An extension of the proposed problem is given in Section 4. Finally, the conclusion and future work are given.
2. Problem Description
A set of jobs needs to be processed on a single machine. All the independent jobs are available at time zero, and preemption is not allowed. For a given sequence, we assume that job has a due-window , where () denote the starting time (finishing time) of the due-window, . The due-window size of job is defined by , and and of all jobs are decision variables. The normal processing time of job is denoted by (i.e., the processing time without being influenced by any factor), . For a given sequence, let be the completion time of job . The aim is to find the optimal starting time of the due-windows, the size of the due-windows, and the sequence of jobs such that the following measure is minimized:where denotes the job scheduled in the th position, () denote a position-dependent weight (i.e., weight does not correspond with the job but with the position in which some job is scheduled), () is the unit cost of (), is the earliness-tardiness of job (), and
Using the three-field notation (Graham et al. [20]), the problem studied here is . Wang et al. [18] considered single-machine scheduling problems with common due-window (CON-DW) and slack due-window (SLK-DW) assignments. They proved that the problems and can be solved in time, respectively.
3. Main Results
Obviously, there exists an optimal sequence without any machine idle time between the processing of jobs, and the first job in the sequence starts at time zero.
Lemma 1. There exists an optimal sequence such that .
Proof. We consider two cases that contradict this optimal property: Case i: if , then the total cost for job is We shift to the left such that , and we have Hence, Case i is not an optimal due-window assignment. Case ii: if , then the total cost for job isWe shift and to the left such that , and we haveHence, Case ii is not an optimal due-window assignment.
To summarise, we have .
Lemma 2. For a given sequence , the optimal due-window locations and for job can be obtained as follows:(1)When , then set (2)When , then set (3)When , then set and
Proof. (1)When and , we have From Lemma 1, we consider the following two cases: Case i: if , then the total cost for job is Case ii: if , then the total cost for job is To summarise, if , then set .
Similarly, cases (2) and (3) can be proved.
Lemma 3. For a given sequence , the optimal due-window locations and for job can be obtained as follows:(1)When , then set , where (2)When , then set , where (3)When , then set and , where (4)When , then set and , where and
Proof. The proof is similar to the proof of Lemma 2.
Lemma 4. The optimal sequence of the problem can be obtained by sequencing the jobs in a nondecreasing order of , i.e., the smallest processing time (SPT) first rule.
Proof. From Lemmas 1–3, the objective function can be transformed into the following three cases: (1) ; (2) ; and (3) . For all the three cases, it is easy to verify (by the pairwise interchange method) that sequencing the jobs in a nondecreasing order of is optimal.
Let , , and ; then,whereAnd
Remark 1. Obviously, is a decreasing function on ; from Hardy et al. [21], the optimal sequence can be obtained by the SPT rule, and it is the same as Lemma 4.
From Lemmas 1–4, a polynomial-time algorithm can be proposed for the problem.
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Theorem 1. Algorithm 1 solves the problem in time.
Proof. Optimality can be guaranteed by Lemmas 1–4. In Algorithm 1, Step 1 needs time by the SPT rule; Steps 2 and 3 can be performed in time. Thus, the total time for Algorithm 1 is .
In order to illustrate Algorithm 1 for the problem , we present the following instance.
Example 1. The data are as follows: , .
Now, we can solve the problem according to Algorithm 1 as follows: Step 1: according to Lemma 4, the optimal sequence is Step 2: for the optimal sequence , the completion time of all jobs is , and the optimal due-window locations and for each job are given in Table 1 Step 3: the optimal due-window sizes are (), , , , , , , , , , and , and the objective function is .
4. An Extension
In this section, the problem is extended to a setting of general position-dependent processing time. Let be the actual processing time of ; under the general position-dependent processing time setting, the actual processing time of is if it is assigned to position , . Thus, the input for the problem contains a matrix of job-position values. Biskup [22] introduced a job-independent learning effect model in which , where is the learning index (see also Wang et al. [23]). Mosheiov and Sidney [24] introduced job-dependent learning effects, i.e., , where is the job-dependent learning index of job . Wang et al. [25] introduced truncated job-dependent learning effects, i.e., , where is a truncation parameter. We refer the reader to the survey of Azzouz et al. [26] on scheduling problems with learning effects.
From (7), we havewhere are given by (9).
From (10), the optimal sequence of the problem can be obtained by solving the following assignment problem:where , are given by (9), and
Based on the above analysis, the solution procedure of the problem can be summarized as follows.
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Theorem 2. Algorithm 2 solves the problem in time.
Proof. Optimality is guaranteed by Lemmas 1–3 and the above analysis. In Algorithm 2, Step 1 needs time by the SPT rule; Steps 2 and 3 can be performed in time. Thus, the total time for Algorithm 2 is . In order to illustrate Algorithm 2 for the problem , we present the following instance.
Example 2. The data are as follows: , . The job-dependent processing time is given in Table 2.
5. Conclusion and Future Work
This study addressed the due-window (DIF-DW) assignment scheduling problem under the consideration of position-dependent weights. The goal is to determine the optimal sequence, the optimal due-window location, and size such that the total penalty (including the earliness, tardiness, due-window starting time, and due-window size of all jobs) is minimized. It was proved that the problem can be solved in polynomial time. The proposed model was also extended to the general position-dependent processing time, and the polynomial-time solution was provided. Further extensions are considering the above problems in the setting of -machine flow-shop and -identical (unrelated) parallel machines (Hsu and Liao [27]), studying the scheduling with two-agent resource-dependent release time (Liu and Duan [28]), or investigating scheduling with rate-modifying activity under deterioration effect (Xue and Zhang [29]).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Liaoning Province (2020-MS-233).