Abstract
This paper considers a singlemachine duewindow assignment scheduling problem with positiondependent weights, where the weights only depend on their position in a sequence. The objective is to minimise the total weighted penalty of earliness, tardiness, duewindow starting time, and duewindow size of all jobs. Optimal properties of the problem are given, and then, a polynomialtime algorithm is provided to solve the problem. An extension to the problem is offered by assuming general positiondependent processing time.
1. Introduction
Conventionally, in scheduling theory, duewindows are jobdependent either if they are dictated by the customer (i.e., given constants) or they are decision variables (i.e., duewindow assignment). A duewindow for job is defined by a duewindow starting time and a duewindow finishing time , i.e., the duewindow , and the duewindow size is . In the justintime (JIT) production methodology and scheduling theory, setting proper duewindows is challenging (see Gong et al. [1], Janiak et al. [2], and Geng et al. [3]). In the literature, three very popular duewindow assignment methods are studied: Common duewindow (CONDW) assignment method (Liman et al. [4, 5]): all jobs are assigned a common duewindow, i.e., all the jobs have a common duewindow , where , , the duewindow size of all the jobs is , and both and are decision variables. In the literature, most studies considered the CONDW assignment method, e.g., Mosheiov and Sarig [6] addressed a minmax CONDW assignment problem, the objective of which is to minimise the largest cost among earliness, tardiness, duewindow starting time, and duewindow size. They proved that the singlemachine and twomachine flowshop problems can be solved in polynomial time. They also proved that the cases of parallel identical machines and uniform machines are NPhard. Yin et al. [7] considered the batch delivery scheduling problem with an assignable common duewindow on a single machine. Yin et al. [8] studied the singlemachine scheduling problem with CONDW assignment and batch delivery cost. Liu et al. [9] considered the singlemachine CONDW assignment scheduling problem with deteriorating jobs. For the weighted sum of earliness, tardiness, and duewindow location penalty minimization, they proposed a polynomialtime algorithm to solve the problem. Wang and Wang [10] considered the singlemachine resource allocation scheduling problem with learning effect and CONDW assignment. Slack duewindow (SLKDW) 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 singlemachine SLKDW assignment scheduling problem with deteriorating jobs and learning effect. Ji et al. [13] considered the singlemachine SLKDW assignment scheduling problem with group technology. Yin et al. [14], Yin et al. [15], and Wang et al. [16] considered SLKDW assignment scheduling problems with resource allocation (controllable processing time). Mor and Mosheiov [17] considered SLKDW assignment proportionate flowshop scheduling problems. Different duewindows’ (DIFDW) assignment method: it is assumed that the job has a duewindow , where and denote the starting time and finishing time of the duewindow, respectively. The duewindow size of the job is , and both and are decision variables. Wang et al. [12] considered DIFDW assignment scheduling problems with deteriorating jobs and learning effect.
In a recent paper, Wang et al. [18] considered CONDW and SLKDW assignment methods with positiondependent 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 duewindow assignment methods with positiondependent weights can be solved in polynomial time, respectively. “The scheduling with duewindow assignment has many realworld applications. For example, the duewindow 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 duewindow increases the supplier’s production and delivery flexibility. However, a large duewindow 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 DIFDW assignment scheduling problem with positiondependent 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, duewindow starting time, and duewindow size of all jobs’ minimization can be solved in polynomial time; and (3) it is further extended the model to the case with general positiondependent processing time. We refer the reader to the survey of Janiak et al. [2] on the scheduling problems with (CONDW, SLKDW, and DIFDW) duewindows.
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 duewindow , where () denote the starting time (finishing time) of the duewindow, . The duewindow 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 duewindows, the size of the duewindows, and the sequence of jobs such that the following measure is minimized:where denotes the job scheduled in the th position, () denote a positiondependent 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 earlinesstardiness of job (), and
Using the threefield notation (Graham et al. [20]), the problem studied here is . Wang et al. [18] considered singlemachine scheduling problems with common duewindow (CONDW) and slack duewindow (SLKDW) 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 duewindow 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 duewindow assignment.
To summarise, we have .
Lemma 2. For a given sequence , the optimal duewindow 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 duewindow 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 polynomialtime algorithm can be proposed for the problem.

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 duewindow locations and for each job are given in Table 1 Step 3: the optimal duewindow sizes are (), , , , , , , , , , and , and the objective function is .
4. An Extension
In this section, the problem is extended to a setting of general positiondependent processing time. Let be the actual processing time of ; under the general positiondependent 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 jobposition values. Biskup [22] introduced a jobindependent learning effect model in which , where is the learning index (see also Wang et al. [23]). Mosheiov and Sidney [24] introduced jobdependent learning effects, i.e., , where is the jobdependent learning index of job . Wang et al. [25] introduced truncated jobdependent 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.

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 jobdependent processing time is given in Table 2.
5. Conclusion and Future Work
This study addressed the duewindow (DIFDW) assignment scheduling problem under the consideration of positiondependent weights. The goal is to determine the optimal sequence, the optimal duewindow location, and size such that the total penalty (including the earliness, tardiness, duewindow starting time, and duewindow 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 positiondependent processing time, and the polynomialtime solution was provided. Further extensions are considering the above problems in the setting of machine flowshop and identical (unrelated) parallel machines (Hsu and Liao [27]), studying the scheduling with twoagent resourcedependent release time (Liu and Duan [28]), or investigating scheduling with ratemodifying 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 (2020MS233).