Optimization for Due-Window Assignment Scheduling with Position-Dependent Weights

/is paper considers a single-machine due-window assignment scheduling problemwith position-dependentweights, where theweights only depend on their position in a sequence./e 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.


Introduction
Conventionally, in scheduling theory, due-windows are jobdependent either if they are dictated by the customer (i.e., given constants) or they are decision variables (i.e., duewindow assignment). A due-window for job J i is defined by a due-window starting time d 1 i and a due-window finishing time d i ′ , i.e., the due-window [d 1 i , d i ′ ], and the due-window size is In the just-in-time (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 due-window (CON-DW) assignment method (Liman et al. [4,5]): all jobs are assigned a common duewindow, i.e., all the jobs have a common due-window the due-window size of all the jobs is D � d ′ − d 1 , and both d 1 and D 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. ey proved that the single-machine and two-machine flow-shop problems can be solved in polynomial time. ey 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 duewindow 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] where p i is the normal processing time of job J i and q 1 and D 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 J i has a due-window denote the starting time and finishing time of the due-window, respectively. e due-window size of the job J i is D i � d i ′ − d 1 i , and both d 1 i and D i 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. ey proved that both these due-window assignment methods with position-dependent weights can be solved in polynomial time, respectively. " e 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 positiondependent weights. e 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 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 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. e 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.

Problem Description
A set of n jobs N � J 1 , J 2 , . . . , J n 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 J i has a due-window and D i of all jobs are decision variables. e normal processing time of job J i is denoted by p i (i.e., the processing time without being influenced by any factor), i � 1, 2, . . . , n. For a given sequence, let C i be the completion time of job J i . e aim is to find the optimal starting time of the due-windows, the size of the due-windows, and the sequence of jobs δ such that the following measure is minimized: where δ(i) denotes the job scheduled in the ith position, ψ i > 0 (i � 1, 2, . . . , n) denote a position-dependent weight (i.e., weight ψ i does not correspond with the job but with the position in which some job is scheduled), ψ 0 (ψ n+1 ) is the unit cost of d 1 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. ey proved that the problems

Main Results
Obviously, there exists an optimal sequence δ * without any machine idle time between the processing of jobs, and the first job in the sequence starts at time zero.

Lemma 1.
ere exists an optimal sequence such that Proof. We consider two cases that contradict this optimal property: We shift d δ(i) ′ to the left such that d δ(i) ′ � C δ(i) , and we have 2 Discrete Dynamics in Nature and Society Hence, Case i is not an optimal due-window assignment.
We shift d 1 δ(i) and d δ(i)

′
to the left such that Hence, Case ii is not an optimal due-window assignment.
To summarise, we have d 1 For a given sequence δ, the optimal due-window locations d 1 δ(i) and d δ(i) ′ for job J δ(i) can be obtained as follows: From Lemma 1, we consider the following two cases: (2) and (3) can be proved. □ Lemma 3. For a given sequence δ, the optimal due-window locations d 1 δ(i) and d δ(i) ′ for job J δ(i) can be obtained as follows: e proof is similar to the proof of Lemma 2. □

Lemma 4.
e optimal sequence of the problem can be obtained by sequencing the jobs in a nondecreasing order of p i , i.e., the smallest processing time (SPT) first rule.
Proof. From Lemmas 1-3, the objective function n i�1 ψ i L δ(i) + ψ 0 d 1 δ(i) + ψ n+1 D δ(i) can be transformed into the following three cases: (1) n i�1 ψ i C δ(i) ; (2) ψ 0 C δ(i) ; and (3) ψ n+1 C δ(i) . For all the three cases, it is easy to verify (by the pairwise interchange method) that sequencing the jobs in a nondecreasing order of p i is optimal. Let where Discrete Dynamics in Nature and Society And □ Remark 1. Obviously, λ i � n j�i ψ i ′ is a decreasing function on i; 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 Proof. Optimality can be guaranteed by Lemmas 1-4. In Algorithm 1, Step 1 needs O(n log n) time by the SPT rule; Steps 2 and 3 can be performed in O(n) time. us, the total time for Algorithm 1 is O(n log n).

An Extension
In this section, the problem 1|DIF − DW| n i�1 ψ i L δ(i) + ψ 0 d 1 δ(i) + ψ n+1 D δ(i) is extended to a setting of general position-dependent processing time. Let p A i be the actual processing time of J i ; under the general position-dependent processing time setting, the actual processing time of J i is if it is assigned to position r, i, r � 1, . . . , n. us, the input for the problem contains a matrix of (n × n) jobposition values. Biskup [22] introduced a job-independent learning effect model in which p A i � θ(i, r) � p i r α , where α ≤ 0 is the learning index (see also Wang et al. [23]). Mosheiov and Sidney [24] introduced job-dependent learning effects, i.e., p A i � θ(i, r) � p i r α i , where α i ≤ 0 is the job-dependent learning index of job J i . Wang et al. [25] introduced truncated job-dependent learning effects, i.e., p A i � θ(i, r) � p i max r α i , β , where 0 < β < 1 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 have (10) where λ i are given by (9). From (10), the optimal sequence of the problem 1|DIF − DW, can be obtained by solving the following assignment problem: where λ r , r � 1, . . . , n, are given by (9), and x ir � 1, if job J i is assigned to position r, Based on the above analysis, the solution procedure of the problem can be summarized as follows.
Proof. Optimality is guaranteed by Lemmas 1-3 and the above analysis. In Algorithm 2, Step 1 needs O(n 3 ) time by the SPT rule; Steps 2 and 3 can be performed in O(n) time.
us, the total time for Algorithm 2 is O(n 3 ). In order to illustrate Algorithm 2 for the problem 1|DIF − DW,

Conclusion and Future Work
is study addressed the due-window (DIF-DW) assignment scheduling problem under the consideration of position-dependent weights. e 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. e 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 m-machine flow-shop and m-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
e authors declare that they have no conflicts of interest.