DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi10.1155/2020/97465389746538Research ArticleOptimization for Due-Window Assignment Scheduling with Position-Dependent WeightsWangLi-Yan1LvDan-Yang1ZhangBo1https://orcid.org/0000-0001-5753-8272LiuWei-Wei2https://orcid.org/0000-0003-2271-6459WangJi-Bo1WuChin-Chia1School of ScienceShenyang Aerospace UniversityShenyang 110136Chinasau.edu.cn2Department of ScienceShenyang Sport UniversityShenyang 110102Chinasyty.edu.cn202022720202020140520201606202022720202020Copyright © 2020 Li-Yan Wang et al.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.

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.

Natural Science Foundation of Liaoning Province2020-MS-233
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 Ji is defined by a due-window starting time di1 and a due-window finishing time di, i.e., the due-window di1,di, and the due-window size is Di=didi1. In the just-in-time (JIT) production methodology and scheduling theory, setting proper due-windows is challenging (see Gong et al. , Janiak et al. , and Geng et al. ). 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 d1,d, where di1=d1, di=d, the due-window size of all the jobs is D=dd1, and both d1 and D are decision variables. In the literature, most studies considered the CON-DW assignment method, e.g., Mosheiov and Sarig  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.  considered the batch delivery scheduling problem with an assignable common due-window on a single machine. Yin et al.  studied the single-machine scheduling problem with CON-DW assignment and batch delivery cost. Liu et al.  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  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 ) is di1=pi+q1, di=pi+q, and Di=didi1=qq1=D, where pi is the normal processing time of job Ji and q1 and D are decision variables. Wang et al.  considered the single-machine SLK-DW assignment scheduling problem with deteriorating jobs and learning effect. Ji et al.  considered the single-machine SLK-DW assignment scheduling problem with group technology. Yin et al. , Yin et al. , and Wang et al.  considered SLK-DW assignment scheduling problems with resource allocation (controllable processing time). Mor and Mosheiov  considered SLK-DW assignment proportionate flow-shop scheduling problems.

Different due-windows’ (DIF-DW) assignment method: it is assumed that the job Ji has a due-window di1,di, where di10 and di0di1di denote the starting time and finishing time of the due-window, respectively. The due-window size of the job Ji is Di=didi1, and both di1 and Di are decision variables. Wang et al.  considered DIF-DW assignment scheduling problems with deteriorating jobs and learning effect.

In a recent paper, Wang et al.  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. ). It is natural and interesting to continue the work of Wang et al.  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.  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 n jobs N˜=J1,J2,,Jn 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 Ji has a due-window di1,di, where di10 (di0) denote the starting time (finishing time) of the due-window, di1di. The due-window size of job Ji is defined by Di=didi1, and di1 and Di of all jobs are decision variables. The normal processing time of job Ji is denoted by pi (i.e., the processing time without being influenced by any factor), i=1,2,,n. For a given sequence, let Ci be the completion time of job Ji. The 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:(1)Zdi1,Di,δi=i=1nψiLδi+ψ0dδi1+ψn+1Dδi,where δi denotes the job scheduled in the ith position, ψi>0 (i=1,2,,n) denote a position-dependent weight (i.e., weight ψi does not correspond with the job but with the position in which some job is scheduled), ψ0 (ψn+1) is the unit cost of dδi1 (Dδi), Lδi is the earliness-tardiness of job Jδi (i=1,2,,n), and(2)Lδi=dδi1Cδi,for dδi1>Cδi,0,for dδi1Cσidδi,Cδidδi,for Cδi>dδi.

Using the three-field notation (Graham et al. ), the problem studied here is 1DIFDWi=1nψiLδi+ψ0dδi1+ψn+1Dδi. Wang et al.  considered single-machine scheduling problems with common due-window (CON-DW) and slack due-window (SLK-DW) assignments. They proved that the problems 1CONDWi=1nψiLδi+ψ0d1+ψn+1D and 1SLKDWi=1nψiLδi+ψ0q1+ψn+1D can be solved in Onlogn time, respectively.

3. 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.

There exists an optimal sequence such that dδi1dδiCδi.

Proof.

We consider two cases that contradict this optimal property:

Case i: if dδi1Cδi<dδi, then the total cost for job Jδi is

(3)zδi=ψ0dδi1+ψn+1dδidδi1.

We shift dδi to the left such that dδi=Cδi, and we have

(4)z˜δi=ψ0dδi1+ψn+1Cδidδi1<zδi.

Hence, Case i is not an optimal due-window assignment.

Case ii: if Cδi<dδi1dδi, then the total cost for job Jδi is

(5)zδi=ψidδi1Cδi+ψ0dδi1+ψn+1dδidδi1.

We shift dδi1 and dδi to the left such that dδi1=dδi=Cδi, and we have(6)z˜δi=ψ0Cδi<zδi.

Hence, Case ii is not an optimal due-window assignment.

To summarise, we have dδi1dδiCδi.

Lemma 2.

For a given sequence δ, the optimal due-window locations dδi1 and dδi for job Jδi can be obtained as follows:

When minψi,ψ0,ψn+1=ψi, then set dδi1=dδi=0

When minψi,ψ0,ψn+1=ψ0, then set dδi1=dδi=Cδi

When minψi,ψ0,ψn+1=ψn+1, then set dδi1=0 and dδi=Cδi

Proof.

When minψi,ψ0,ψn+1=ψi and dδi1=dδi=0, we have zδi=ψiCδi

From Lemma 1, we consider the following two cases:

Case i: if dδi1Cδidδi, then the total cost for job Jδi is z˜δi=ψ0dδi1+ψn+1dδidδi1ψidδi1+ψidδidδi1=ψidδiψiCδi=zδi

Case ii: if dδi1dδiCδi, then the total cost for job Jδi is z˜δi=ψiCδidδi+ψ0dδi1+ψn+1dδidδi1ψiCδidδi+ψidδi1+ψidδidδi1=ψiCδi=zδi

To summarise, if minψi,ψ0,ψn+1=ψi, then set dδi1=dδi=0.

Similarly, cases (2) and (3) can be proved.

Lemma 3.

For a given sequence δ, the optimal due-window locations dδi1 and dδi for job Jδi can be obtained as follows:

When ψi=ψ0<ψn+1, then set dδi1=dδi=Cδj, where j=0,1,,i

When ψi=ψn+1<ψ0, then set dδi1=0,dδi=Cδj, where j=0,1,,i

When ψ0=ψn+1<ψi, then set dδi1=Cδj and dδi=Cδi, where j=0,1,,i

When ψ0=ψn+1=ψi, then set dδi1=Cδj1 and dδi=Cδj2, where j1=0,1,,i and j2=j1,j1+1,,i

Proof.

The proof is similar to the proof of Lemma 2.

Lemma 4.

The optimal sequence of the problem 1DIFDWi=1nψiLδi+ψ0dδi1+ψn+1Dδi can be obtained by sequencing the jobs in a nondecreasing order of pi, i.e., the smallest processing time (SPT) first rule.

Proof.

From Lemmas 13, the objective function i=1nψiLδi+ψ0dδi1+ψn+1Dδi can be transformed into the following three cases: (1) i=1nψiCδi; (2) ψ0Cδi; and (3) ψn+1Cδi. For all the three cases, it is easy to verify (by the pairwise interchange method) that sequencing the jobs in a nondecreasing order of pi is optimal.

Let A=iminψi,ψ0,ψn+1=ψi,i=1,2,,niψi=ψ0<ψn+1,i=1,2,,niψi=ψn+1<ψ0,i=1,2,,niψi=ψ0=ψn+1,i=1,2,,n, B=iminψi,ψ0,ψn+1=ψ0,i=1,2,,niψ0=ψn+1<ψi,i=1,2,,n, and C=iminψi,ψ0,ψn+1=ψn+1,i=1,2,,n; then,(7)Zdi1,Di,δ=i=1nψiLδi+ψ0dδi1+ψn+1Dδi=iAψij=1ipδj+iBψij=1ipδj+iCψij=1ipδj=i=1nψij=1ipδj=i=1npδij=inψi=i=1nλipδi,where(8)ψi=ψi,iA,ψ0,iB,ψn+1,iC.

And(9)λi=j=inψi.

Remark 1.

Obviously, λi=j=inψi is a decreasing function on i; from Hardy et al. , the optimal sequence can be obtained by the SPT rule, and it is the same as Lemma 4.

From Lemmas 14, a polynomial-time algorithm can be proposed for the 1DIFDWi=1nψiLδi+ψ0dδi1+ψn+1Dδi problem.

<bold>Algorithm 1:</bold>

Step 1: obtain the optimal sequence by the SPT rule (see Lemma 4)

Step 2: calculate the completion time of each job under the optimal sequence, and determine the optimal due-window locations dδi1 and dδi for each job according to Lemmas 2 and 3

Step 3: obtain the optimal due-window size by setting Dδi=dδidδi1 (i=1,2,,n), and calculate the objective function i=1nψiLδi+ψ0dδi1+ψn+1Dδi by equation (7)

Theorem 1.

Algorithm 1 solves the problem 1DIFDWi=1nψiLδi+ψ0dδi1+ψn+1Dδi in Onlogn time.

Proof.

Optimality can be guaranteed by Lemmas 14. In Algorithm 1, Step 1 needs Onlogn time by the SPT rule; Steps 2 and 3 can be performed in On time. Thus, the total time for Algorithm 1 is Onlogn.

In order to illustrate Algorithm 1 for the problem 1DIFDWi=1nψiLδi+ψ0dδi1+ψn+1Dδi, we present the following instance.

Results of the optimal due-window location.

Job JδiJob Jδidδi1dδi
Jδ1J10dδi1=0dδ1 = 0
Jδ2J1dδ21=Cδ2=29dδ2=Cδ2=29
Jδ3J5dδ31=0dδ3 = 0
Jδ4J2dδ41=Cδ4=66dδ4=Cδ4=66
Jδ5J7dδ51Cδii=0,1,2,3,4,5 i.e., dδ510,14,29,46,66,87dδ5=dδ51
Jδ6J4dδ61=Cδ6=111dδ6=Cδ6=111
Jδ7J8dδ71=Cδ7=136dδ7=Cδ7=136
Jδ8J3dδ81=0dδ8 = 0
Jδ9J9dδ91=Cδ9=189dδ9=Cδ9=189
Jδ10J6dδ101=0dδ10 = 0
Example 1.

The data are as follows: n=10,p1=15,p2=20,p3=26,p4=24,p5=17,p6=28,p7=21,p8=25,p9=27,p10=14, ψ0=14,ψ1=7,ψ2=20,ψ3=12,ψ4=24,ψ5=14,ψ6=22,ψ7=15,ψ8=8,ψ9=19,ψ10=12, and ψ11=50.

Now, we can solve the problem 1DIFWi=1nψiLδi+ψ0dδi1+ψn+1Dδi according to Algorithm 1 as follows:

Step 1: according to Lemma 4, the optimal sequence is δ=J10,J1,J5,J2,J7,J4,J8,J3,J9,J6

Step 2: for the optimal sequence δ=J10,J1,J5,J2,J7,J4,J8,J3,J9,J6, the completion time of all jobs is C10=14,C1=29,C5=46,C2=66,C7=87,C4=111,C8=136,C3=162,C9=189, and C6=217, and the optimal due-window locations dδi1 and dδi for each job are given in Table 1

Step 3: the optimal due-window sizes are Dδi=0 (i=1,2,,10), λ1=123, λ2=116, λ3=102, λ4=90, λ5=76, λ6=62, λ7=48, λ8=34, λ9=26, and λ10=12, and the objective function is Z=i=1nψiLδi+ψ0dδi1+ψn+1Dδi=i=1nλipδi=13202.

4. An Extension

In this section, the problem 1DIFDWi=1nψiLδi+ψ0dδi1+ψn+1Dδi is extended to a setting of general position-dependent processing time. Let piA be the actual processing time of Ji; under the general position-dependent processing time setting, the actual processing time of Ji is piA=θi,r if it is assigned to position r, i,r=1,,n. Thus, the input for the problem contains a matrix of n×n job-position values. Biskup  introduced a job-independent learning effect model in which piA=θi,r=pirα, where α0 is the learning index (see also Wang et al. ). Mosheiov and Sidney  introduced job-dependent learning effects, i.e., piA=θi,r=pirαi, where αi0 is the job-dependent learning index of job Ji. Wang et al.  introduced truncated job-dependent learning effects, i.e., piA=θi,r=pimaxrαi,β, where 0<β<1 is a truncation parameter. We refer the reader to the survey of Azzouz et al.  on scheduling problems with learning effects.

From (7), we have(10)Zdi1,Di,δ=i=1nψiLδi+ψ0dδi1+ψn+1Dδi=i=1nλiθi,r,where λi are given by (9).

From (10), the optimal sequence of the problem 1DIFDW,piA=θi,ri=1nψiLδi+ψ0dδi1+ψn+1Dδi can be obtained by solving the following assignment problem:(11)Mini=1nr=1nλrθi,rxir,s.t.i=1nxir=1,r=1,,n,r=1nxir=1,i=1,,n,where λr,r=1,,n, are given by (9), and(12)xir=1,if job Ji is assigned to position r,0,otherwise.

Based on the above analysis, the solution procedure of the problem 1DIFDW,piA=θi,ri=1nψiLδi+ψ0dδi1+ψn+1Dδi can be summarized as follows.

<bold>Algorithm 2:</bold>

Step 1: solve assignment problem (11) to obtain the optimal sequence

Step 2: calculate the completion time of each job under the optimal sequence, and determine the optimal due-window locations dδi1 and dδi for each job according to Lemmas 2 and 3

Step 3: obtain the optimal due-window size by setting Dδi=dδidδi1 (i=1,2,,n), and calculate the objective function i=1nψiLδi+ψ0dδi1+ψn+1Dδi by assignment problem (11)

Theorem 2.

Algorithm 2 solves the problem 1DIFDW,piA=θi,ri=1nψiLδi+ψ0dδi1+ψn+1Dδi in On3 time.

Proof.

Optimality is guaranteed by Lemmas 13 and the above analysis. In Algorithm 2, Step 1 needs On3 time by the SPT rule; Steps 2 and 3 can be performed in On time. Thus, the total time for Algorithm 2 is On3. In order to illustrate Algorithm 2 for the problem 1DIFDW,piA=θi,ri=1nψiLδi+ψ0dδi1+ψn+1Dδi, we present the following instance.

Example 2.

The data are as follows: n=8, ψ0=14,ψ1=8,ψ2=18,ψ3=12,ψ4=24,ψ5=10,ψ6=20,ψ7=15,ψ8=7, and ψ9=21. 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 m-machine flow-shop and m-identical (unrelated) parallel machines (Hsu and Liao ), studying the scheduling with two-agent resource-dependent release time (Liu and Duan ), or investigating scheduling with rate-modifying activity under deterioration effect (Xue and Zhang ).

Date of Example 2.

Jir
12345678
J1671156221321
J2814781117126
J39111332710128
J41722191059137
J5168151411171314
J6181731148231520
J7151218198162113
J81317241618161315

Step 1: by (9), we have λ1=93, λ2=85, λ3=71, λ4=59, λ5=45, λ6=35, λ7=21, and λ8=7. According to assignment problem (11), the optimal sequence is δ=J3,J5,J2,J1,J6,J4,J8,J7.

Step 2: for the optimal sequence δ=J3,J5,J2,J1,J6,J4,J8,J7, the completion time of all jobs is C3=9,C5=17,C2=24,C1=29,C6=37,C4=46,C8=59, and C7=72, and the optimal due-window locations dδi1 and dδi for each job are given in Table 3.

Step 3: the optimal due-window sizes are Dδi=0 (i=1,2,,8), and the objective function is Z=i=1nψiLδi+ψ0dδi1+ψn+1Dδi=3348.

Results of the optimal due-window location.

Job JδiJob Jδidδi1dδi
Jδ1J3dδ11=0dδ1 = 0
Jδ2J5dδ21=Cδ2=17dδ2=Cδ2=17
Jδ3J2dδ31=0dδ3 = 0
Jδ4J1dδ41=Cδ4=29dδ4=Cδ4=29
Jδ5J6dδ51=0dδ5=0
Jδ6J4dδ61=Cδ6=46dδ6=Cδ6=46
Jδ7J8dδ71=Cδ7=59dδ7=Cδ7=59
Jδ8J7dδ81=0dδ8 = 0
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).

GongH.ZhangB.PengW.Scheduling and common due date assignment on a single parallel-batching machine with batch deliveryDiscrete Dynamics in Nature and Society20152015746439010.1155/2015/4643902-s2.0-84928792890JaniakA.JaniakW. A.KrysiakT.KwiatkowskiT.A survey on scheduling problems with due windowsEuropean Journal of Operational Research2015242234735710.1016/j.ejor.2014.09.0432-s2.0-84920645208GengX.-N.WangJ.-B.BaiD.Common due date assignment scheduling for a no-wait flowshop with convex resource allocation and learning effectEngineering Optimization20195181301132310.1080/0305215x.2018.15213972-s2.0-85055552732LimanS. D.PanwalkarS. S.ThongmeeS.Determination of common due window location in a single machine scheduling problemEuropean Journal of Operational Research1996931687410.1016/0377-2217(95)00181-62-s2.0-0030212471LimanS. D.PanwalkarS. S.ThongmeeS.Common due window size and location determination in a single machine scheduling problemThe Journal of the Operational Research Society19984991007101010.2307/3010176MosheiovG.SarigA.Minmax scheduling problems with a common due-windowComputers & Operations Research20093661886189210.1016/j.cor.2008.06.0012-s2.0-56549128360YinY.ChengT. C. E.HsuC.-J.WuC.-C.Single-machine batch delivery scheduling with an assignable common due windowOmega201341221622510.1016/j.omega.2012.06.0022-s2.0-84867017496YinY.ChengT. C. E.WangJ.WuC.-C.Single-machine common due window assignment and scheduling to minimize the total costDiscrete Optimization2013101425310.1016/j.disopt.2012.10.0032-s2.0-84871460607LiuJ.WangY.MinX.Single-machine scheduling with common due-window assignment for deteriorating jobsJournal of the Operational Research Society201465229130110.1057/jors.2013.62-s2.0-84892168000WangJ.-B.WangM.-Z.Single-machine due-window assignment and scheduling with learning effect and resource-dependent processing timesAsia-Pacific Journal of Operational Research2014315145003610.1142/s02175959145003652-s2.0-84928417779MosheiovG.OronD.Job-dependent due-window assignment based on a common flow allowanceFoundations of Computing and Decision Sciences2010353185195WangJ.-B.LiuL.WangC.Single machine SLK/DIF due window assignment problem with learning effect and deteriorating jobsApplied Mathematical Modelling20133718-198394840010.1016/j.apm.2013.03.0412-s2.0-84883449262JiM.ChenK.GeJ.ChengT. C. E.Group scheduling and job-dependent due window assignment based on a common flow allowanceComputers & Industrial Engineering201468354110.1016/j.cie.2013.11.0172-s2.0-84891424683YinY.ChengT. C. E.WuC.-C.ChengS.-R.Single-machine due window assignment and scheduling with a common flow allowance and controllable job processing timeJournal of the Operational Research Society201365111310.1016/j.cie.2013.05.0032-s2.0-84879305921YinY.WangD.-J.ChengT. C. E.WuC.-C.Bi-criterion single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing timesJournal of the Operational Research Society20166791169118310.1057/jors.2016.142-s2.0-85010955798WangD.YinY.ChengT. C. E.A bicriterion approach to common flow allowances due window assignment and scheduling with controllable processing timesNaval Research Logistics (NRL)2017641416310.1002/nav.217312-s2.0-85017202692MorB.MosheiovG.Minsum and minmax scheduling on a proportionate flowshop with common flow-allowanceEuropean Journal of Operational Research2017254236037010.1016/j.ejor.2016.03.0372-s2.0-84979490189WangJ.-B.ZhangB.LiL.BaiD.FengY.-B.Due-window assignment scheduling problems with position-dependent weights on a single machineEngineering Optimization202052218519310.1080/0305215x.2019.15774112-s2.0-85063465351YangD.-L.LaiC.-J.YangS.-J.Scheduling problems with multiple due windows assignment and controllable processing times on a single machineInternational Journal of Production Economics20141509610310.1016/j.ijpe.2013.12.0212-s2.0-84896842867GrahamR. L.LawlerE. L.LenstraJ. K.KanA. H. G. R.Optimization and approximation in deterministic sequencing and scheduling: a surveyAnnals of Discrete Mathematics1979528732610.1016/s0167-5060(08)70356-x2-s2.0-74849101820HardyG. H.LittlewoodJ. E.PolyaG.Inequalities1967Cambridge, UKCambridge University PressBiskupD.Single-machine scheduling with learning considerationsEuropean Journal of Operational Research1999115117317810.1016/s0377-2217(98)00246-x2-s2.0-0345201636WangJ.-B.XuJ.YangJ.Bi-criterion optimization for flow shop with a learning effect subject to release datesComplexity2018201812914951010.1155/2018/91495102-s2.0-85062729126MosheiovG.SidneyJ. B.Scheduling with general job-dependent learning curvesEuropean Journal of Operational Research2003147366567010.1016/s0377-2217(02)00358-22-s2.0-0037449223WangX.-R.WangJ.-B.JinJ.JiP.Single machine scheduling with truncated job-dependent learning effectOptimization Letters20148266967710.1007/s11590-012-0579-02-s2.0-84893704957AzzouzA.EnnigrouM.SaidL. B.Scheduling problems under learning effects: classification and cartographyInternational Journal of Production Research201756312010.1080/00207543.2017.13555762-s2.0-85045942361HsuC.-L.LiaoJ.-R.Two parallel-machine scheduling problems with function constraintDiscrete Dynamics in Nature and Society202020206271709510.1155/2020/2717095LiuP.DuanL.A note on two-agent scheduling with resource dependent release times on a single machineDiscrete Dynamics in Nature and Society20152015450329710.1155/2015/5032972-s2.0-84928808356XueP.ZhangY.Single-machine scheduling with upper bounded maintenance time under the deteriorating effectDiscrete Dynamics in Nature and Society20132013675625110.1155/2013/7562512-s2.0-84878710951