Scheduling identical parallel machines involving flexible maintenance activities

Contents lists available atScienceDirect

Expert Systems With Applications

journal homepage:www.elsevier.com/locate/eswa

Scheduling identical parallel machines involving flexible maintenance activities

Chunhao Li

^a

, Feng Wang

^a

, Jatinder N.D. Gupta

^b

, Tsui-Ping Chung

^c,∗

aSchool of Business and Management, Jilin University, Changchun, China

bCollege of Business, University of Alabama in Huntsville, Huntsville, AL, USA

cSchool of Business, Northeast Normal University, Changchun, China

A R T I C L E I N F O

Keywords:

Identical parallel machines Flexible maintenance scheduling Critical machine deterioration Irregular framework Metaheuristics

A B S T R A C T

Motivated by a practical situation in chip manufacturing process, for the first time in the literature, this paper considers an identical parallel-machine scheduling problem with new flexible maintenance activities to minimize makespan where a maintenance activity is needed if and only if the machine capability has deteriorated by a critical value. To address the proposed problem, a mixed integer linear programming model and a lower bound are established. Since this problem is NP-hard, a combined constructive heuristic algorithm with six priority rules is developed. In order to improve the solution obtained by the proposed combined heuristic algorithm, an embedded learning mechanism is combined with the existing artificial immune system (AIS) algorithm to help self-adjust and modify the search direction. The effectiveness of the proposed combined constructive heuristic and the AIS algorithms is empirically tested on the randomly generated problem instances. These computational results show that the proposed AIS algorithm can generate better near-optimal solutions than several adaptations of the existing algorithms.

1. Introduction

Drivenbythetransformationandupgradingofdigitaltechnology, anincreasingnumberofmanufacturingcompanieshopetogainmore intelligentcontrolovertheirproductionprocessestoimproveefficiency and reducecosts. As an example of this phenomenon, considerthe wafer cleaning and machine maintenance scheduling problem of a leading chip manufacturer wherein the basicfundamental building blockofchips,siliconwafersundergohundredsoftediousprocessing steps. In between these steps, the wafer requires multiple cleaning operationsto remove contaminants,suchas organicfilms, particles, andmetalions(Chenetal.,2023).Thesecontaminantsareremoved from the wafer surface with a cleaning agent. As more wafers are cleaned,contaminationbuilds upin thecleaningmachine.Oncethe amountofcontaminationhasgoneoverboard,itwilldamagethewafer during cleaning. Therefore, it is critical to perform timely mainte- nanceactivitiesonthemachinetochangethecleaningagent.Because different wafers have unequal levels of contamination, the time it takes to reach the contamination accumulation threshold can vary withdifferentwafercleaningsequences.Asaresult,thetimeinterval between required maintenance activities is irregular (Chung et al., 2019).Therefore, whileschedulingjobs,decisions in suchsituations

∗ Correspondenceto:SchoolofBusiness,NortheastNormalUniversity,No.5268,Renminstreet,Changchun,130024,China.

E-mailaddresses: lich@jlu.edu.cn(C.Li),fwang20@mails.jlu.edu.cn(F.Wang),guptaj@uah.edu(J.N.D.Gupta),cpzhong@nenu.edu.cn(T.-P.Chung).

requires a simultaneousconsideration of maintenanceactivities and scheduling.

The problem of simultaneously considering job scheduling and maintenanceactivitiesoutlinedabove alsoexistsin severalotherin- dustries,suchastheautomotiveindustry(Wockeretal.,2024),steel plant (Zhang et al., 2020), and plastic industry (Fu et al., 2019).

WhileGeurtsenetal.(2023b)classifythecurrentmaintenanceplan- ning literature into four main categories according to the type of maintenanceactivity andthe timingat which theknowledge about thepreventive maintenanceinterval is known, Chung et al.(2019), Pangetal.(2021),andZhangandChen(2022)examinemaintenance scheduling based on machine contamination levels in conjunction commonlyfoundinpractice.

Toillustrate themaintenance schedulingbased on machine con- taminationlevelsandthedifferencesbetweenthisstudyandexisting researchthatsimultaneouslyconsidersjobschedulingandmaintenance activities, an example from everyday life may be helpful. We are usuallyrequiredtochangetheautomotiveoilafter6monthsor5000 kilometers of driving. However, theaging of automotive oilis also relatedtothedrivingenvironment anddriving habits.Hencetheoil changeinanautomobileisbasedontheagingoftheoilratherthan anyfixedtime-intervalormileagedriven.Translatingthisexampleto

https://doi.org/10.1016/j.eswa.2024.125722

Received13May2024;Receivedinrevisedform3November2024;Accepted4November2024 Expert Systems With Applications 263 (2025) 125722

Available online 14 November 2024

0957-4174/© 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

themanufacturingsituation,itisclearthatthemachinecontamination dependsonthejobsbeingprocessedandhencemaintenanceactivities areflexibleintime.However,asshownbytherelevantliteraturere- viewinSection2,theexitingliteraturedoesnotinvestigatescheduling withflexiblemaintenanceactivitieswithvariousjobavailabilityand processingconstraintsfoundinpractice.Therefore,thispaperstudies theidentical parallel-machineschedulingproblemwitharbitraryjob releasedatesandflexiblemaintenancebasedontheaccumulatedma- chinecontamination.Indoingso,thecontributionsofthispaperareas follows:

(1) Weintroducecontaminationaccumulation-basedflexiblemain- tenancetotheschedulingproblemforthefirsttimeonparallel machineswitharbitraryjobreleasedates.

(2) Weproposeamixed integerlinearprogrammingmodelanda newlowerboundfortheproblem.

(3) Wedesignacombinedheuristicalgorithm(CHA)withaneffec- tivedispatchingrule.TheproposedCHAcanobtainoptimalor near-optimalsolutionsforthesmall-sizeproblemwithverylittle timeconsumption.

(4) Wecombine an embeddedlearning mechanism with existing artificialimmunesystemalgorithmstohelpthealgorithmself- adjustandmodifysearchdirections.Usingtheembeddedlearn- ingmechanism,ourproposedalgorithmcandynamicallyselect theappropriatelocalsearchoperatorsandpopulationregenera- tionmethodsaccordingtotheirperformanceindices.

The rest of this paper is organized as follows. Section 2briefly reviews the relevant literature.Section 3 provides a detailed delin- eationof aMILPmodelanddevelopsalowerboundtoevaluatethe performanceoftheheuristicsandalgorithms.InSection4,acombined heuristicalgorithmincludingsixpriorityrulesandadispatchingrule isproposedforobtainingnearoptimalsolutions.Section5presentsan auto-revisingimmunoglobulin-basedartificialimmunesystem(A-IAIS) algorithmtoimproveinitialsolutionscreatedbythecombinedheuristic algorithm.Theresults ofcomputationalexperiments arediscussedin Section 6. Finally,Section 7concludes thepaperwith some fruitful directionsforfutureresearch.

2. Relevantliteraturereview

Since the seminal work of Graham (1966) to solve the identi- calparallel-machineschedulingproblems,schedulingwithavailability constraints has attracted significant attention during the past three decades(Iskandarnia&Harrath,2019;Kaabi&Harrath,2014;Nguyen etal.,2023).However,thesesurveysdonotincludethemaintenance activitiessuchascleaning,lubrication,inspection,repair,andoverhaul thatareperformedtoreducethefailurefrequencyofamachine.There- fore,thissectionreviewssomecloselyrelatedstudiesandpresentsthe state of theart in considering productionscheduling problemswith maintenanceactivities.

Geurtsenetal.(2023b)showthatthecurrentmaintenanceplanning literaturecanbeclassifiedintofourmaincategoriesaccordingtothe typeofmaintenanceactivityandthetimingatwhichtheknowledge aboutthepreventivemaintenanceintervalisknown.Thefirstcategory includesresearchwheremaintenancetimemayincreaseovertimeor maintenanceactivitiesmaychangethespeedofthemachine(Alfares etal.,2021;Briskornetal.,2024;Huetal.,2024;Zouetal.,2023).

Thesecond categorycontainsresearch wherea maintenanceactivity should be performedbefore the machineage reachesacertaintime value (Chen et al., 2022; Lei & Yang, 2022; Lin et al., 2023). The thirdcategoryinvolvesliteraturewherethemaintenanceactivitymust beginandendwithinatimewindow(Costa&Fernandez-Viagas,2022;

Geurtsen et al.,2023a; Lee et al., 2015; Yan et al., 2022).The last categorycontainsliteraturewherethemaintenanceactivityisexecuted accordingtotheobjective relatedtothereliabilityor availabilityof

themachines(Alietal.,2024;Anetal., 2023; Chen&Wang,2018;

Gharounetal.,2022).

Ontheotherhand,Chung etal. (2019),Pang etal.(2021),and Zhang and Chen (2022) examine maintenance scheduling based on machinecontaminationlevelscommonlyfoundinpractice.Consider- ingthat this paperstudiesthe identicalparallel-machine scheduling problem with arbitrary job release dates and flexible maintenance basedontheaccumulatedmachinecontamination,theliteraturereview in this section is dividedinto four categories where the scheduling of maintenance activities is based on (1) the machine age, (2) the preventivemaintenance policies,(3)themachinedeterioration rate- modifyingconditions,(4)apredefinedtimewindow,or(5)machine contaminationlevel.

2.1. Maintenanceschedulingbasedonmachineage

Theliterature on maintenance schedulingbased on machineage considers thesituation whereamaintenance activityshouldbe per- formedbeforethemachinereachesacertaintimevalue.Kong etal.

(2020)investigateaschedulingproblemwithdeteriorationandlearn- ingeffecton parallelmachineswhereamaintenanceactivityisper- formedafterafixednumberofbatcheshavebeenprocessed.Maoetal.

(2021) deal witha schedulingproblem ina permutation flowshop, where preventive maintenance activities are performed periodically on all machines. Kim and Kim (2021) consider an environment of unrelatedparallelmachineswheretheexacttimingofmachinebreak- downsisknowninadvance.Chenetal.(2022)exploretheproblemof schedulingjobsonidenticalparallelmachinesthatarenotcontinuously available.Theyassumethatthecontinuousprocessingtimeofparallel machines cannot exceed the predefined cumulative processing time threshold.LeiandYang(2022) giveasurveyofartificialbeecolony algorithms for scheduling jobs on unrelated parallel machines with periodicmaintenanceactivities.Souzaetal.(2022)exploreajobshop schedulingproblemwherepreventivemaintenanceactivitiesareper- formednolaterthanarecommendedage.Linetal.(2023)assumethat themachineisperiodicallyunavailableforagivenperiodoftimeand proposealgorithmsforonlineschedulingofjobsonidenticalparallel batchprocessingmachines.

2.2. Maintenanceschedulingbasedonpreventivemaintenancepolicy Apreventivemaintenancepolicyimpliesthattheintervalbetween twoconsecutivemaintenanceactivitiesisdeterminedbytheconstraints of machine reliabilityand availability.Ruiz andGarcía-Díaz (2007) considervariouspreventivemaintenance strategiestomaximizema- chine availabilityor maintain a minimum level of reliability. Chen andWang(2018)studybothrun-basedpreventivemaintenancewith reliability constraints andsetup time to minimize the makespan in parallelmachinescheduling.Zhouetal.(2021)considerasinglema- chineschedulingproblemwithfixedperiodicpreventivemaintenance and propose an efficient heuristic algorithm to minimize the total weightedcompletion time. Gharoun et al. (2022) investigate a par- allelmachineproductionenvironmentwherepreventivemaintenance activitiesareperformedbasedonmachinereliability.Anetal.(2023) consideramake-to-orderproductionenvironmentanddesignamulti- levelimperfectmaintenancemodelforeachmachine.Ifamachineis designedtoperform overhaul maintenancecycles, theyassume that themachine undergoes preventive maintenancewhenthe reliability reachesathreshold.TheworkbyAlietal.(2024)studiesajobshop schedulingproblemwherebothpreventivemaintenanceandcorrective maintenanceareconsidered.Theyalsoconsiderthesituationwherethe numberofmachinebreakdownsisknowninadvanceorthenumberof breakdownsisrandom.

Expert Systems With Applications 263 (2025) 125722

2.3. Maintenanceschedulingbasedonrate-modifyingactivities

Theliteratureonmaintenanceschedulingwithrate-modifying ac- tivities assumesmachine deterioration. Inotherwords, maintenance durationmayincreaseovertimeormaintenanceactivitiesmaychange the speed of the machine. The study by Zhao et al. (2009) is the firsttoconsidertherate-modifyingactivityontwo-parallelmachines.

They assume that the processing times of jobs depend on whether thejobisprocessed beforeoraftertheactivity.TheworkinAlfares etal.(2021)examinesatwo-machineschedulingproblemwithvariable maintenance activitiesandproposes avariable neighborhoodsearch algorithm. Pang etal. (2021) studythe problemof unrelated paral- lelmachineswithperiodic maintenanceactivities,assumingthatthe processing timeandthe amount of job-induceddirt are determined bytheassignedmachine.Zouetal.(2023)investigatethescheduling of identicalparallel machines,whereeachmachine hasatmost one deteriorating maintenance activity. Ahmadi et al. (2024) explore a parallelsystemschedulingproblemwheremaintenanceactivitiesare scheduledaccordingtothemeanresidualtimeindexandthenumberof failedcomponents.Huetal.(2024)consideraschedulingproblemfor parallelmachineswithdeterioratingjobsandmaintenanceactivities.

They assume that the numberof available maintenanceactivities is limited.

2.4. Maintenanceschedulingbasedontimewindow

The literature on maintenance scheduling in a flexible window examines thesituation wherethemaintenance operationcouldstart andfinishinatimewindow.ThepioneeringstudybyKolenandKroon (1992)considersafixedjobschedulingproblematanairport,where maintenance activities are tobe performed at a fixed start time, a fixed finishtime, andafixed aircrafttype. LeiandLiu(2020)then investigate the problem of scheduling unrelated parallel machines.

Each maintenanceactivity shouldbe performed within apredefined timewindow.Chenetal.(2021)assumethatthecontinuousprocessing timeofamachinecannotexceedthemaintenancethreshold.Yanetal.

(2022)exploreadynamicflexiblejobshopschedulingproblemwhere maintenanceactivitiesareperformedwithinflexibletimewindows.In the workof Costa andFernandez-Viagas (2022),maintenance activ- ities are carried out within a specifictime window.Geurtsen et al.

(2023a)consideraschedulingproblemforunrelatedparallelmachines, whereeach machinerequiresonlyonemaintenanceactivityandthe maintenance activitycan be performed flexiblywithin a giventime window.

2.5. Maintenanceschedulingbasedonmachinecontaminationlevels Theflexible maintenancediscussedin thispaper islike changing theoilinanautomobilebasedontheagingoftheoil,notonafixed timeintervalor mileage.Basedon thesameresearchbackground of flexiblemaintenance,Chungetal.(2019)studyonlyasinglemachine schedulingandignorethesituationwherewafercleaningisperformed inidenticalparallelbathsinawetstation(Kimetal.,2013).Pangetal.

(2021) study the parallel machine maintenancescheduling problem based on contamination accumulationand assume that the amount of contamination in the job is predetermined by the cleaning ma- chine.Similarly,whenstudyingthemachinemaintenancescheduling problem affected by contamination accumulation, Zhang and Chen (2022)assumethatalljobshaveidenticalreleasetimes.However,these assumptionsdonotalwaysholdtrueinpractice.

From the above relevant literature review, it is clear that the schedulingproblemofflexiblemaintenancebasedonthecontamina- tionaccumulationonparallelmachineswitharbitraryjobreleasedates studiedinthispaperhasnotbeenaddressedintheliterature.Therefore, itisnecessaryandusefultoconsidertheproposedproblemespecially intheimplementationofanyintelligentmachinemanagement.

Table1 Notations.

Notations Problemparameters 𝑛 Thenumberofjobs 𝑚 Thenumberofmachines 𝑊 Amaintenanceactivity

𝑇 Thepermittedvalueoftheaccumulationofcontamination 𝑝_𝑖 Theprocessingtimeofeachjobwhere𝑖=1,2,…,𝑛

𝑡_𝑖 Theamountofcontaminationafterprocessingagivenjob𝑖where 𝑖=1,2,…,𝑛

𝑟_𝑖 Thereleasedateofeachjobwhere𝑖=1,2,…,𝑛 𝑤 Thefixedprocessingtimeofeachmaintenanceactivity Decisionvariables

𝑥_𝑖𝑗𝑘 Equals1ifjob𝑖isassignedtoposition𝑗onmachine𝑘and0otherwise 𝑦_𝑗𝑘 Equals1ifmaintenanceactivityistakenimmediatelyfollowingposition

𝑗onmachine𝑘and0otherwise Decisiondependentvariables

𝑀 Averylargepositivenumber

𝑎_𝑗𝑘 Theaccumulationofthecontaminationproducedbythejobswhichare sequencedbetweenthelatestmaintenanceactivityandthejobin position𝑗onmachine𝑘

𝑆𝑇_𝑗𝑘 Thestartingtimeformachine𝑘atposition𝑗 𝐶𝑇_𝑗𝑘 Thecompletiontimeformachine𝑘atposition𝑗 𝐶_𝑘 Thecompletiontimeofmachine𝑘where𝑘=1,2,…,𝑚

3. Problemformulationandlowerbound

This section defines the problem, formulates the mixed integer linearprogrammingmodeltooptimallysolvethedefinedproblem,and developsalowerboundthatisusefulinevaluatingtheeffectivenessof theproposedalgorithms.Todoso,Table1definesthenotationstorep- resentproblemparameters,decisionvariables,anddecisiondependent variablesusedinthispaper.

3.1. Problemdefinition

Considerthescenariowhereaset𝑁 ={1,2,…,𝑛}of 𝑛jobsisto beprocessedonaworkcenterwithagivenset𝑀 =(1,2,…,𝑚)of𝑚 identicalparallelmachineswithoutpreemption.Eachjobrequiresonly asingleoperationthatisperformedwithoutinterruptiononanyoneof the𝑚machines.Amachinecanprocessonlyonejobatanygiventime.

Associatedwitheachjob𝑖∈𝑁 isitsreleasedate𝑟_𝑖,processingtime (includingits setuptime) 𝑝_𝑖, andthemachinecontaminationcaused byitsprocessing𝑡_𝑖.Whenthecumulativecontaminationonamachine reachesagivencriticalvalue𝑇,amaintenanceactivity(𝑊)ofduration 𝑤timeunitsisrequiredbeforethatmachinecanprocessanyjob(i.e.,a machinecannotprocessanyjobduringthemaintenanceactivity).

Giventheabovescenario,theproblemconsideredinthispaperis oneoffindingtheassignmentandschedulingofall𝑛jobsonall𝑚ma- chinestominimizemakespan.Followingthethreefieldrepresentation ofschedulingproblems(Lawleretal.,1993),werepresenttheproposed problemas𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_max.Intheabovenotation,𝑃_𝑚repre- sents𝑚identicalmachinesinparallel.𝑟_𝑖referstothereleasedateand

∑𝑡_𝑖≤ 𝑇 ensures thatanaccumulationofthecontamination𝑡_𝑖(after processingthejob𝑖)isnomorethanthecriticalvalue𝑇.𝑓𝑝𝑎means thatmaintenanceactivitiesareflexibleandoccurperiodically(Chen, 2008).𝐶_maxistheobjectivefunctiontominimizethemakespan.Since the𝑃∥𝐶_maxproblemisknowntobeisNP-hard(Lenstraetal.,1977), theproposed𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_max problemasanextensionofthe 𝑃 ∥ 𝐶_max problemisalsoNP-hard.Givenits NP-hardnatureandthe fact that no existing solution procedures exist tosolve the studied problem, it is appropriateto formulateit as a mixed integer linear programmingmodel,todeveloplowerboundsonthemakespan,and toproposeheuristicalgorithmstosolvethe𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_max problemconsideredinthispaper.Theheuristicalgorithmsareneeded Expert Systems With Applications 263 (2025) 125722

as the existing MILPoptimization procedures arelimited tosolving onlysmall-sizedproblemcontainingnomorethan13jobswhereasthe realisticpracticalscenarioscontainlotmorethan13jobs.

3.2. Mixedintegerlinearprogrammingmodel

InspiredbythemodelinPangetal.(2021),amixedintegerlinear programming (MILP)modelof the𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_max problem studiedinthispaperisasfollows:

𝑀𝑖𝑛 𝐶_max (1)

Subjectto:

∑𝑛

𝑗=1

∑𝑚

𝑘=1

𝑥_𝑖𝑗𝑘 =1 𝑖=1,2,…,𝑛

(2)

∑𝑛

𝑖=1

𝑥_𝑖𝑗𝑘≤1 𝑘=1,2,…,𝑚;𝑗=1,2,…,𝑛

(3) 𝑎_1𝑘=

∑𝑛

𝑖=1

𝑡_𝑖𝑥_𝑖1𝑘 𝑘=1,2,…,𝑚

(4) 𝑎_{(𝑗−1)𝑘}+

∑𝑛 𝑖=1

𝑡_𝑖𝑥_𝑖𝑗𝑘≤𝑎_𝑗𝑘+𝑀𝑦_{(𝑗−1)𝑘} 𝑘=1,2,…,𝑚;𝑗=2,…,𝑛 (5)

∑𝑛

𝑖=1

𝑡_𝑖𝑥_𝑖𝑗𝑘≤𝑎_𝑗𝑘+𝑀(1−𝑦_{(𝑗−1)𝑘}) 𝑘=1,2,…,𝑚;𝑗=2,…,𝑛 (6) 𝑎_𝑗𝑘≤𝑇 𝑘=1,2,…,𝑚;𝑗=1,2,…,𝑛 (7) 𝑆𝑇_𝑗𝑘≥

∑𝑛 𝑖=1

𝑟_𝑖𝑥_𝑖𝑗𝑘 𝑘=1,2,…,𝑚;𝑗=1,2,…,𝑛 (8) 𝑆𝑇_𝑗𝑘≥𝐶𝑇_{(𝑗−1)𝑘}+𝑦_{(𝑗−1)𝑘}𝑤 𝑘=1,2,…,𝑚;𝑗=2,3,…,𝑛 (9) 𝐶𝑇_𝑗𝑘≥𝑆𝑇_𝑗𝑘+

∑𝑛

𝑖=1

𝑝_𝑖𝑥_𝑖𝑗𝑘 𝑘=1,2,…,𝑚;𝑗=1,2,…,𝑛 (10)

∑𝑛

𝑖=1

𝑥_𝑖𝑗𝑘≤

∑𝑛

𝑖=1

𝑥_{𝑖(𝑗−1)𝑘} 𝑘=1,2,…,𝑚;𝑗=2,3,…,𝑛 (11) 𝐶_max≥𝐶𝑇_𝑗𝑘 𝑘=1,2,…,𝑚;𝑗=1,2,…,𝑛 (12) Theobjectivefunction(1)representstheminimizationofmakespan.

Constraints(2)and(3)ensurethateach jobisassignedtoa unique positiononauniquemachineandeachpositiononamachineperforms no more than one job. Constraint (4) illustrates the contamination produced bythe first jobon each machine. Constraints (5)and(6) are therestrictionsof the contaminationaccumulationbetween two adjacent maintenance activities. Constraint (7) guarantees that the contaminationaccumulationdoesnot exceed theallowablevalue𝑇. Constraint(8)specifiesthatthestarttimeforprocessingjobatthe𝑗th positiononmachine𝑘willnotbelessthanthereleasetimeofthejob 𝑖assignedtothatposition.Constraint(9)specifiesthatthestarttime forprocessingjobatthe𝑗thpositiononmachine𝑘willbegreaterthan thecompletiontimeofalljobsassignedtomachine𝑘throughposition 𝑗−1plusanyneededmaintenancetime.Constraint(10)specifiesthat thecompletion timeatthe𝑗th positionon machine𝑘 shouldnot be

Table2

ProblemdataforExample1.

Job𝑖 1 2 3 4 5 6 7 8 9 10

𝑝_𝑖 5 1 8 3 8 6 6 7 4 2

𝑡_𝑖 4 3 4 1 4 4 3 3 7 5

𝑟_𝑖 4 9 0 2 7 5 15 7 12 0

less than thesum of thestart time andthe processing timeof the jobassigned tothatposition. Constraint(11) ensures that there are no empty machiningpositions between jobs assignedtoa machine.

Constraint(12)definesthemakespanwhichisequaltothemaximum completiontimeofthemachine.

Example1. Anexamplewith𝑛=10,𝑚=2,𝑤=10,and𝑇=10with problemparametersdetailedinTable2isusedtoillustratethesolution fromtheMILPmodel.AfterexecutingthemodelinILOGOPL3.5.1, theoptimalmakepsanis35whenthesequenceofprocessingjobson machines1and2are{10,4,1,𝑊,2,7,5}and{3,6,𝑊,9,8},respectively where𝑊 isrepresentsthemaintenanceactivity.Thisoptimalsolution is shown graphically in Fig.1. The computational timerequired to optimallysolvethisexampleprobleminstanceis0.86s.

3.3. Lowerbound

ThefollowingProposition1establishesthelowerbound(LB)ofthe 𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_maxproblemconsideredinthispaper.

Proposition1. 𝐿𝐵=max{min_𝑖∈𝑛{𝑟_𝑖}+(∑𝑛

𝑖=1𝑝_𝑖+𝑤×max{⌈∑𝑛 𝑖=1𝑡_𝑖∕𝑇⌉

− 𝑚,0})∕𝑚,max_𝑖∈𝑛{𝑟_𝑖+𝑝_𝑖}}isalowerboundforthe𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||

𝐶_maxproblem.

Proof. Assumingthatalljobs areassignedtoasinglemachine, the minimumnumberofmaintenanceactivitiesis⌈∑𝑛

𝑖=1𝑡_𝑖∕𝑇⌉

.Considering thatthesumofthejobprocessingtimeis∑𝑛

𝑖=1𝑝_𝑖andtheprocessing timeofamaintenanceactivityis𝑤,theminimumcompletiontimefor thesinglemachineequals∑𝑛

𝑖=1𝑝_𝑖+𝑤×⌈∑𝑛 𝑖=1𝑡_𝑖∕𝑇⌉

.Furthermore,with 𝑚identicalparallelmachinesavailable,alljobsshouldbedividedinto 𝑚sequences.Sinceamaintenanceactivityshouldberemovedifitisat theendofasequence,theminimumnumberofmaintenanceactivities ismax{⌈∑𝑛

𝑖=1𝑡_𝑖∕𝑇⌉

−𝑚,0}.Thus,theminimumtotalprocessingtimeof maintenanceactivitiesequals𝑤×max{⌈∑𝑛

𝑖=1𝑡_𝑖∕𝑇⌉

−𝑚,0}.Takingthe sumofjobprocessingtimeandtheminimumjobreleasedatesintocon- sideration,therefore,𝐿𝐵1=min_𝑖∈𝑛{𝑟_𝑖}+(∑𝑛

𝑖=1𝑝_𝑖+𝑤×max{⌈∑𝑛 𝑖=1𝑡_𝑖∕𝑇⌉

− 𝑚,0})∕𝑚 is a lowerbound for the𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_max problem.

Sincetheproposedproblemisanextensionofthe𝑃_𝑚||𝑟_𝑖||𝐶_maxproblem, 𝐿𝐵2 = max_𝑖∈𝑛{𝑟_𝑖+𝑝_𝑖}, of𝑃_𝑚||𝑟_𝑖||𝐶_max isanotherlower boundof the proposedproblem.

Thus,thelowerbound(LB)ofthe𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_maxproblem consideredinthis paperis𝐿𝐵 = max(𝐿𝐵1,𝐿𝐵2)= max{min_𝑖∈𝑛{𝑟_𝑖}+ (∑𝑛

𝑖=1𝑝_𝑖+𝑤×max{⌈∑𝑛 𝑖=1𝑡_𝑖∕𝑇⌉

−𝑚,0})∕𝑚,max_𝑖∈𝑛{𝑟_𝑖+𝑝_𝑖}}. ■

Example 2. The problem of Example 1 is used to illustrate the proposed LB. min_𝑖∈𝑛{𝑟_𝑖} = 0 and ∑𝑛

𝑖=1𝑝_𝑖 = 50 are obtained.Since

⌈∑𝑛 𝑖=1𝑡_𝑖∕𝑇⌉

= 4 > 𝑚, min_𝑖∈𝑛{𝑟_𝑖}+(∑𝑛

𝑖=1𝑝_𝑖+𝑤×max{⌈∑𝑛 𝑖=1𝑡_𝑖∕𝑇⌉

− 𝑚,0})∕𝑚 = (50+10×2)∕2 = 35 can be calculated. Considering the valueof max_𝑖∈𝑛{𝑟_𝑖+𝑝_𝑖} = 21, thelowerbound isobtained by𝐿𝐵= max{35,21} = 35.Forthisexample,thelowerbound isequaltothe minimummakespanfoundbyusingtheMILPmodel.

4. Proposedcombinedheuristicalgorithm

Thebasicideaoftheproposedcombinedheuristicalgorithm(CHA) forsolvingthe𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_maxproblemistoobtainajoblist byapriorityruleandthenassignjobstotheidenticalparallelmachines byadispatchingrule.

Expert Systems With Applications 263 (2025) 125722

Fig.1. ThesequenceofprocessingjobsontwoparallelmachinesinExample1.

Algorithm SPR:SixPriorityRules

ListCP: Alist𝜎isobtainedbynon-increasingorderofjob’scontaminations,i.e.,𝑡_𝜎(1)≥𝑡_𝜎(2),…,≥𝑡_𝜎(𝑛).Tiesarebrokeninfavorofajobwithalongerprocessing time.Iftwojobshaveidenticalcontaminationsandprocessingtimes,theonewiththeearlierreleasedateisgivenhigherpriority.

ListCR: Alist𝜎isobtainedbynon-increasingorderofjob’scontaminations,i.e.,𝑡_𝜎(1)≥𝑡_𝜎(2),…,≥𝑡_𝜎(𝑛).Tiesarebrokeninfavorofajobwithanearlierrelease date.Iftwojobshaveidenticalcontaminationsandreleasedates,theonewiththelongerprocessingtimeisgivenhigherpriority.

ListPC: Alist𝜎isobtainedbysortingjobsinnon-increasingorderoftheirprocessingtimes,i.e.𝑝_𝜎(1)≥𝑝_𝜎(2),…,≥𝑝_𝜎(𝑛).Tiesarebrokeninfavorofajobwitha largercontamination.Iftwojobshaveidenticalprocessingtimesandcontaminations,theonewiththeearlierreleasedateisgivenhigherpriority.

ListPR: Alist𝜎isobtainedbysortingjobsinnon-increasingorderoftheirprocessingtimes,i.e.𝑝_𝜎(1)≥𝑝_𝜎(2),…,≥𝑝_𝜎(𝑛).Tiesarebrokeninfavorofajobwith anearlierreleasedate.Iftwojobshaveidenticalprocessingtimesandreleasedates,theonewiththelargercontaminationisgivenhigherpriority.

ListRC: Alist𝜎isobtainedbynon-decreasingorderofjob’sreleasedates,i.e.,𝑟_𝜎(1)≤𝑟_𝜎(2),…,≤𝑟_𝜎(𝑛).Tiesarebrokeninfavorofajobwithalargercontamination.

Iftwojobshaveidenticalreleasedatesandcontaminationlevels,theonewiththelongerprocessingtimeisgivenhigherpriority.

ListRP: Alist𝜎isobtainedbynon-decreasingorderofjob’sreleasedates,i.e.,𝑟_𝜎(1)≤𝑟_𝜎(2),…,≤𝑟_𝜎(𝑛).Tiesarebrokeninfavorofajobwithalongerprocessing time.Iftwojobshaveidenticalreleasedatesandprocessingtimes,theonewiththelargercontaminationisgivenhigherpriority.

4.1. Sixpriorityrules

Let {𝜎(1),𝜎(2),…,𝜎(𝑛)}bea list𝜎 isobtainedbyusing apriority rule.Let𝑣_𝑘betheaccumulationofcontaminationbetweentwoadja- centmaintenanceactivitiesonmachine𝑘.Aconstraint𝑡_𝜎(𝑗)+𝑣_𝑘≤ 𝑇 is satisfied whena job 𝜎(𝑗) is assigned tomachine 𝑘. Similarly, let 𝛥_𝜎(𝑗)=𝑇−𝑡_𝜎(𝑗)−𝑣_𝑘betheremainingpermittedvalueofthecontam- ination beforethenextmaintenanceactivityisperformed. Toobtain thelist𝜎,CuiandLu(2017) proposeanearliestreleasedate-longest processingtime(ERD-LPT)ruletosolveasinglemachinewithflexible maintenanceandjobs’releasedates.Bytakingthejobs’contamination intoconsideration,weproposethefollowingsixpriorityrules,named List CP,List CR,ListPC, ListPR, List RC,andList RPwhereC, P, andRindicatecontaminationtime,processingtime,andreleasedate respectivelyandthesecondletterinthelistnameisthetie-breaking ruleused.

4.2. Dispatchingrule

Let𝑆_𝑘bethesequenceofjobsandmaintenanceactivitiesprocessed onmachine𝑘∈𝑀.Basedononeoftheabovelists,jobsareassigned tomachinestakingintoaccountthecontaminationaccumulationcon- straint.Thestepsofthedispatchingruleareasfollows.

Using the six priority rules and the above dispatching rule, six heuristicalgorithms,namedCPD,CRD,PCD,PRD,RCD,andRPD,are proposedtosolve𝑃_𝑚||𝑟_𝑖,∑𝑡_𝑖≤𝑇,𝑓𝑝𝑎||𝐶_max.

Example3. ForthejobdataconsideredinExample1,supposethata ListRPwith𝜎={3,10,4,1,6,5,8,2,9,7}isgiven.Byapplyingtheabove dispatchingrule,thejobsequencesonthetwoparallelmachinescan beobtained.Usingthesimilarprocesstofindfivemoreschedulesusing thealgorithmsCPD,CRD,PCD,PRD,RCD,andRPD,resultsinthesix solutionsshowninTable3below.AlgorithmsRCDandRPDgenerate optimalscheduleswithaminimummakespanof35.

4.3. Combinedheuristicalgorithm

Sincenoneoftheabovesixheuristicalgorithmstheoreticallydom- inateothersintheireffectiveness tofindabetterschedule, thesesix algorithmsarecombined togenerate a better schedule. Specifically, the Combined Heuristic Algorithm (CHA) generates six initial lists basedonthesixpriorityrulesandobtainsthecorrespondingmakespan using the dispatchingrule. Then, the minimum makespanis output as thesolution of theCHA. The steps of this proposed CHA are as follows.

Expert Systems With Applications 263 (2025) 125722

Algorithm D:DispatchingRule

Step0. Let𝜎={𝜎(1),𝜎(2),…,𝜎(𝑛)}bealistofunscheduledjobs,𝑢=𝑛,andforeachmachine𝑘∈𝑀,𝑣𝑘=0,𝐶𝑘=0,and𝑆𝑘=∅.

Step1. If𝜎=∅,gotoStep5;otherwise,selectfirstjob𝜎(1)in𝜎andmachine𝑘withthesmallest𝐶_𝑘.If𝑣_𝑘+𝑡_𝜎(1)>𝑇 enterStep2;otherwiseassignjob𝜎(1)to machine𝑘,delete𝜎(1)from𝜎.Let𝑆_𝑘=(𝑆𝑘,𝜎(1)),𝑢=𝑢−1,𝑣_𝑘=𝑣_𝑘+𝑡_𝜎(1),𝐶_𝑘=max{𝐶𝑘+𝑝_𝜎(1),𝑟_𝜎(1)+𝑝_𝜎(1)},andrepeatStep1.

Step2. Calculate𝛥_𝜎(𝑗)=𝑇−𝑡_𝜎(𝑗)−𝑣𝑘for𝑗=2,…,𝑢.Ifthereisatleastonejob𝑗with𝛥_𝜎(𝑗)≥0,enterStep3;otherwiseassign𝑊tomachine𝑘,𝑣𝑘=0,𝐶𝑘=𝐶𝑘+𝑤, andreturntoStep1.

Step3. Ifthereisatleastonejob𝑗with𝑟_𝜎(𝑗)≤𝐶_𝑘,enterStep4;otherwiseassign𝑊 tomachine𝑘.Let𝑆_𝑘=(𝑆𝑘,𝑊),𝑣_𝑘=0,𝐶_𝑘=𝐶_𝑘+𝑤,andreturntoStep1.

Step4. Findajob𝜎(𝑗)withthesmallest𝛥_𝜎(𝑗)and𝑟_𝜎(𝑗)≤𝐶𝑘.Assignjob𝜎(𝑗)tomachine𝑘,delete𝜎(𝑗)from𝜎.Let𝑆𝑘=(𝑆𝑘,𝜎(𝑗)),𝑢=𝑢−1,𝑣𝑘=𝑣𝑘+𝑡_𝜎(𝑗), 𝐶_𝑘=max{𝐶𝑘+𝑝_𝜎(𝑗),𝑟_𝜎(𝑗)+𝑝_𝜎(𝑗)},andreturntoStep1.

Step5. Removethemaintenanceactivityifitisperformedinthelastpositiononeachmachine,update𝑆_𝑘foreachmachine𝑘∈𝑀,andSTOP.Inthesolution obtained,𝑆_𝑘isthesequenceofjobsandmaintenanceactivitiesassignedtomachine𝑘∈𝑀withmakespan𝐶_max=max_𝑘∈𝑀(𝐶_𝑘).

Table3

ThesolutionsofheuristicalgorithmsforExamples3and4.

Heuristic Initiallist Sequences Completiontimes Makespan

CPD {9,10,3,5,6,1,8,7,2,4} 𝑆₁={9,2,𝑊,6,8,7};𝑆₂={10,3,4,𝑊,5,1} 𝐶₁=46;𝐶₂=36 46 CRD {9,10,3,1,6,5,8,2,7,4} 𝑆₁={9,8,𝑊,5,2,7};𝑆₂={10,3,4,𝑊,1,6} 𝐶₁=48;𝐶₂=34 48 PCD {3,5,8,6,7,1,9,4,10,2} 𝑆₁={3,8,2,𝑊,7,1,𝑊,10}𝑆2={5,6,4,𝑊,9} 𝐶₁=49;𝐶₂=38 49 PRD {3,5,8,6,7,1,9,4,10,2} 𝑆₁={3,8,4,𝑊,7,9,𝑊,2};𝑆₂={5,6,𝑊,1,10} 𝐶₁=49;𝐶₂=38 49 RCD {10,3,4,1,6,5,8,2,9,7} 𝑆₁={10,4,1,𝑊,5,2,7};𝑆₂={3,6,𝑊,8,9} 𝐶₁=35;𝐶₂=35 35*

RPD {3,10,4,1,6,5,8,2,9,7} 𝑆₁={3,6,𝑊,8,9};𝑆₂={10,4,1,𝑊,5,2,7} 𝐶₁=35;𝐶₂=35 35*

CHA:CombinedHeuristicAlgorithm

Step1. Let𝐶_max(𝑋𝐷)bethemakespanoftheschedulegeneratedbyheuristicXD,whereXDisoneofthesixheuristicalgorithmsCPD,CRD,PCD,PRD,RCD, orRPD.

Step2. 𝐶_max(CHA)=min{𝐶_max(𝐶𝑃𝐷),𝐶_max(𝐶𝑅𝐷),𝐶_max(𝑃𝐶𝐷),𝐶_max(𝑃𝑅𝐷),𝐶_max(𝑅𝐶𝐷),𝐶_max(𝑅𝑃𝐷)}isselectedasthefinalsolution.

Example4. ConsideragainthesamedetailgiveninExample1.Job schedulesandfinalsolutionsbysixheuristicalgorithmsarepresented inTable3.Thus,theminimummakespanis35,whichisalsothesame asthatobtainedbytheproposedMILPmodelinExample1.

5. Theauto-revisingimmunoglobulin-basedartificialimmunesys- temalgorithm

An immunoglobulin-based artificial immune system algorithm (IAIS) involvesfivecomponents:encoding anddecoding, somaticre- combination, somatic hypermutation, isotype switching, and elimi- nation (Chung & Liao, 2013). In theliterature, IAIS has been used to solve severalproblems and shows better performancethan some algorithms with fixed frameworks (Li et al., 2022; Liu & Chung, 2017).InIAIS,theexploratoryandexploitativeoperatorsareselected randomly inthedynamic operatorselection. However,Davis(1989) revealsthatvariationoperatorswithhighexploratorycapabilityshould be employed in the earlystage, while variation operators used for local fine-tuning are favored in the later stage. Xing et al. (2006) propose a dynamic operator selection method, where the selection probabilitiesofdifferentoperatorsarecalculatedbytheirperformance.

An operator with a higher probability score is more likely to be selected as theoperator in thesubsequent evolution.Following this idea,Fialhoetal.(2008,2010)attempttotackletheoperatorselection problemwithadaptiveparametercontrolmethodsandshow thatthe choiceof thebestoperator toapply shouldbe continuouslyadapted whilesolvingagivenproblem.Consolietal.(2016)usetwodifferent rewardmeasuresincreditassignmentanddynamicallyselectoperators throughouttheoptimizationprocessofthealgorithm.Inspiredbythe ideaofthecreditassignmentusedintheadaptiveoperatorselection, some researchers(Karimi-Mamaghan etal., 2022;Tianet al., 2022) attempttousefitnessimprovementasarewardfordifferentoperators inmachinelearning.Theyshowthehigheffectivenessofthislearning mechanisminsolvingmanyoptimizationproblems.

ToimprovetheIAISalgorithm,alearningmechanismisnecessary.

Itcanrecordtheperformanceofeachoperatorusedduringthesearch process. Using of alocal search operatorwith abetter performance

before exploringthe searchspace hasahigher chanceto be anap- propriate operator in the next generation. Therefore,by combining IAISwiththelearningmechanism,wenowproposeanauto-revising immunoglobulin-basedartificialimmunesystem(A-IAIS)algorithm.To describe the proposed A-IAIS algorithm, we first briefly discuss its components.

5.1. Encodinganddecoding

The purpose of encoding is togenerate an initial population of solutionstobeimproved.Forthispurpose, encodingcompriseslym- phocytesthatarerecognizedbytheirreceptors,i.e.,cellsurface-bound antibodies(Parham,2015).Areceptorrepresentsasequenceof all𝑛 jobs.Inthe proposed A-IAISalgorithm, each possible sequence of 𝑛 jobs (receptor)is representedbyalist with𝑛jobs generatedby the sixpriorityrulesdescribedearlierinSection4.Thus,thereareatleast 𝐴=6receptorsinA-IAIS.Additionalrequiredreceptorsarerandomly generated.

Thedispatchingrule,i.e.,ProcedureD,proposedinSection4isused asthe decodingruleto obtaintheassignment andsequenceof jobs andmaintenanceactivitiestoeachmachine(calledaschedule)andto calculatetheobjectivefunctionvalueofthereceptor.

5.2. Somaticrecombination

Insomaticrecombinationoftheadaptiveimmunity,receptorsare expressed as a resultof combinations of gene segments. A receptor withthesmallestmakespanin thecurrent generationis accepted as thestandardreceptor. Forotherreceptors,anumber𝑅of 𝑛jobsare randomlyselectedandtheninsertedintothesequencepositionswhere theyareinthestandardreceptor.

Example5. ForthejobdataconsideredinExample1,supposethat weconsideraselectedreceptor{7,5,9,2,8,6,3,4,10,1}andastandard receptor{3,10,4,1,6,5,8,2,9,7}.For𝑅=2numberofjobs,assumethat jobs9and1arerandomlyselected.Jobs9and1areinpositions9and 4respectivelyinthestandardreceptor.Therefore,weinsertjobs1and 9insequencepositions4and9oftheselectedreceptortoobtainanew receptor{7,5,2,1,8,6,3,4,9,10}showninFig.2.

Expert Systems With Applications 263 (2025) 125722

Fig.2. SomaticrecombinationforExample5.

5.3. Somatichypermutation

Theinitialproductionoflymphocytesisalwaysimmature.There- fore,themainfunctionofsomatichypermutationistofindthepossible positionof external antigens.In ordertosearch abroader area,the inversemutationisusedhere.Forareceptor,let𝑖and𝑗betworan- domlychosenpositions.Then,anewreceptorisacquiredbyinverting thesequenceofjobsbetween(andincluding)thepositions𝑖and𝑗.The inversemutationisnotallowedif|𝑖−𝑗|<2.

5.4. Performance-basedisotypeswitching

Inthehumanimmunesystem,therearethreeisotypeimmunoglob- ulinoperators,i.e.,IgA,IgE,andIgG.ForA-IAIS,therefore,thereare alsothreeoperators.Eachoperatorhasadifferentfunctionandcanbe usedtogeneratethemostresponsivereceptors.IgAcanpenetratethe bloodstreamandprotectotherinfectedtissues.Theinsertionmutation isusedherebecauseitcansearchforaremoteneighborofthecurrent receptor. Fora givenreceptor, let𝑖 and𝑗be therandomly selected joband thesequenceposition. Then,a newreceptoris obtained by inserting job 𝑖 to sequence position 𝑗. As a flexible operator, IgG can accessforeignantigensinthedamagedandinfectedtissues.The pairwise mutationis used in IgGbecause itcan perform a fastand deepsearchforthereceptor’sneighbor.Forareceptor,let𝑖and𝑗be thetworandomlyselectedsequencepositions.Then,anewreceptoris generatedbyswappingthetwojobsatsequencepositions𝑖and𝑗.

Unliketheothertwooperators,IgEcanidentifyanddestroyanti- gens.Thus,bothpairwiseandinsertionmutationsareusedinIgEto escapefrom localoptima. Forthereceptor𝑎, let𝑖and𝑗 bethetwo randomlyselected sequencepositions.Anewreceptor𝑎^′ is obtained byswapping thetwojobsat sequencepositions𝑖and𝑗. Then,let𝑖^′ and𝑗^′betherandomlyselectedjobandsequencepositioninreceptor 𝑎^′.Anewreceptorisobtainedbyinsertingjob𝑖^′atposition𝑗^′.

In theIAIS-based algorithms, isotypeswitching is repeatedfor a fixednumberofiterations,withthethreeoperatorsIgA,IgG,andIgE randomlyselectedwithequalprobability.FortheproposedA-IAIS,IgA, IgG, andIgE areselectedwith different probabilities basedon their performance.𝑁_𝐼𝑔𝑋,whereIgXcan beIgA,IgG,orIgE,is definedas an indexthat can show thechange in performance.If theobjective functioninundertakingIgXisbetterthanthecurrentbestone,𝑁_𝐼𝑔𝑋 = 𝑁_𝐼𝑔𝑋+1.Threeindiceshavethesamevalueintheinitialgeneration.At eachiteration,theselectionprobabilityofeachoperatoriscalculated as 𝑆_𝐼𝑔𝑋 = 𝑁_𝐼𝑔𝑋∕(𝑁_𝐼𝑔𝐴+𝑁_𝐼𝑔𝐺+𝑁_𝐼𝑔𝐸). According tothe selection probability𝑆_𝐼𝑔𝑋,anappropriateoperatorisselectedusingtheroulette- wheelselectionmethod.Thus,anoperatorwithabetterperformance beforewillhaveahigherchancetobeselectedforthenextiteration.

5.5. Auto-revisedregenerationmethods

TheproposedA-IAISisanautorevisingalgorithmwherethebasic ideaisthatduringthesearchprocess,exploringthesearchspacemay beneededinearliergenerationsbutexploitingthebettersolutionmay beneededin latergenerations.Toadaptthesearchmechanism, two methods,namedrandomregenerationmethodandelitebasedregen- erationmethod areproposedtoregenerateinitialsolutions.Thefirst method,the randomregeneration method, i.e.,thecomponent named elimination, is from IAIS where a receptor with the best objective functionisretainedandotherreceptorsarerandomlyregenerated.

Thesecond method is anelite-based regeneration method that can record better receptors in A-IAIS. An elite set with 𝐴 receptors is initializedbycopyingthereceptorsintheinitialpopulation.Theelite setisdynamicallyupdatedduringthesearchprocess.Thatis,ifthere arenewlygeneratedreceptorswithasmallerobjectivefunctionthan theelitereceptors,thentheworseelitereceptorsarereplacedbynew ones.

Inthecurrentgeneration,onejobofanelitereceptorisrandomly selectedandinsertedintoallpossiblepositions.Sincethereare𝑛jobs inareceptor,𝑛candidatereceptorsincludingtheelitereceptoritself areobtained.Oneofthesecandidatereceptorswithabetterobjective functionisselectedasthenewreceptor.Repeatingtheabovestepsfor eachofthe𝐴elitereceptors,initialsolutionsofthenextgenerationare obtained.

5.6. Auto-revisedselectionstrategy

Ideally, the random regeneration method should have a higher selectionprobabilitywhenexploring thesearchspace.Similarly, the elitebasedregenerationmethodshouldhaveahigherselectionprob- abilitywhenexploitingthepromisingregions.Therefore,theselection probabilitiesofthesetwomethodsareusedtodecidewhichoneshould beusedinthecurrentgeneration.

Intheinitialgeneration,twomethodshavethesameperformance indices𝑁¹

𝑚𝑒𝑡ℎ𝑜𝑑and𝑁²

𝑚𝑒𝑡ℎ𝑜𝑑withthesameselectionprobabilities𝑆¹

𝑚𝑒𝑡ℎ𝑜𝑑

and𝑆²

𝑚𝑒𝑡ℎ𝑜𝑑 calculatedby𝑆^𝑥

𝑚𝑒𝑡ℎ𝑜𝑑 =𝑁^𝑥

𝑚𝑒𝑡ℎ𝑜𝑑∕(𝑁_{𝑚𝑒𝑡ℎ𝑜𝑑}¹ +𝑁²

𝑚𝑒𝑡ℎ𝑜𝑑)where 𝑥 = 1 or 2. If the current best receptor is changed in the current generation,theperformanceindexoftheusedmethodisincreasedby 1.Thus,𝑁^𝑥

𝑚𝑒𝑡ℎ𝑜𝑑 =𝑁^𝑥

𝑚𝑒𝑡ℎ𝑜𝑑+1andtheselectionprobabilityofmethod 𝑥will be higher thanbefore in thenext generation.Based ontheir selectionprobabilities,thesetwomethodsarerandomlyselectedusing theroulette-wheelselectionmethod.

5.7. ProposedA-IAISalgorithm

Theembeddedlearningmechanism usedin theA-IAIS algorithm ensuresthatappropriateisotypeimmunoglobulinoperatorsandpopu- lationregeneration methodsareselectedbasedon theirperformance indicesduringthevariousstagesofthealgorithm.Basedonthecom- ponentdescriptionsabove,theA-IAISalgorithmrepeatsthefollowing stepsuntilthelowerboundoratimelimitof𝑛∕10secondsisreached.

6. Computationalresults

Thissectionreportsthecomputationalresultsfromtheexperimen- tal tests used to evaluate the effectiveness of the proposed MILP Expert Systems With Applications 263 (2025) 125722