a memetic algorithm for multiskill resource-constrained...

Research ArticleA Memetic Algorithm for Multiskill Resource-ConstrainedProject Scheduling Problem under Linear Deterioration

Huafeng Dai 12 andWenming Cheng12

1School of Mechanical Southwest Jiaotong University 610031 Chengdu China2Technology and Equipment of Rail Transit Operation andMaintenance Key Laboratory of Sichuan Province 610031 Chengdu China

Correspondence should be addressed to Huafeng Dai annmyswjtueducn

Received 7 May 2019 Accepted 20 June 2019 Published 4 July 2019

Academic Editor Giovanni Berselli

Copyright copy 2019 Huafeng Dai and Wenming Cheng This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper proposes a general variable neighborhood search-based memetic algorithm (GVNS-MA) for solving the multiskillresource-constrained project scheduling problem under linear deterioration Integrating a solution recombination operator and alocal optimization procedure the proposed GVNS-MA is assessed on two sets of instances and achieves highly competitive resultsOne set of benchmark instances is commonly used in the literature where the capability of the proposed algorithm to find highquality solutions is demonstrated compared with the state-of-the-art algorithms in the literatureThe other set revises the formerthrough incorporating the linear deterioration effect Two key components of the proposed algorithm are investigated to confirmtheir critical role to the success of the proposed method

1 Introduction

Scheduling is a form of decision-making that plays an essen-tial role in manufacturing and service industries A functionthat assigns tasks to resources to complete the project canbe formulated as an informal definition of the schedulingproblem However such simplification is not enough to coverall real-world applications with a variety of complex environ-ments Take the task duration for example the processingtime of any task is fixed and constant in classical schedulingtheory and system [1] while a hypothesis that the actualduration of the task is a linearly nondecreasing function ofits starting time is put forward by Gupta and Gupta [2] incontrast to the cases found in traditional literature Fromthen on many approaches concerning various schedulingtypes accompanying with the phenomenon denominated asdeterioration effect [3] constantly spring up

Scheduling with deterioration is very common Forinstance in traditional manufacturing industries like thefurniture some tasks need to be processed by a carpenterand the carpenter may lower his machining speed graduallydue to fatigue In this case the later a task is handled the

longer the time it needs to complete For another examplein steel rolling mills ingots need to be heated to requiredtemperature before rolling Heating time relies on the ingotsrsquocurrent temperature which depends upon the time it hasbeen waiting to be heated It is because the ingot cools downconsequently requiring more heating time in the period ofwaiting Similar cases often occur inmanufacturing financialmanagement steel production medicine treatment and soon Scheduling deteriorating jobs was first considered byBrowne and Yechiali [4] who assumed that task process-ing times are nondecreasing start time dependent linearfunctions Since then researchers have devoted considerableefforts to this area and lots of remarkable papers have beenpublished Alidaee and Womer [5] as well as Cheng et al[6]made the comprehensive overviews of existing schedulingproblems with various deteriorating mechanisms

In addition to changes in the definition of duration timein terms of research hotpot in the field of scheduling projectscheduling problem (PSP) provided a set of precedence-constrained tasks to be scheduled aiming at minimizinga given objective Furthermore tasks need to compete forscarce resources additionally in the resource-constrained

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9459375 16 pageshttpsdoiorg10115520199459375

2 Mathematical Problems in Engineering

project scheduling problem (RCPSP) which make it possiblethrough a better adaption to apply in production planningproject management or manufacturing etc Ultimately moreadaptable multiskill resource-constrained project schedulingproblem (MS-RCPSP) gives each resource a set of capabilitiesand each resource can bid for being assigned tasks

Scheduling with deterioration and MS-RCPSP are tworesearch hotspots of scheduling area and these two subfieldsare not absolutely separate In terms of most tasks of actualMS-RCPSP deterioration effects existed for the reason offatigued (humans ormachines)However there is no researchconcentrated on the integration of MS-RCPSP and deteriora-tion

In this paper we pick the deterioration mechanism aslinear deterioration As for the researched MS-RCPSP withlinear deterioration which is dubbed MS-RCPSPLD thedifferences compared to the scheduling problems with lineardeterioration in the reported literature can be explainedhere Foremost the resource or the machine in the usuallyresearched scheduling problem with deterioration mostly isrestricted to be single-mode which means that it owns onlyone capability and can execute just a kind of task

Moreover as theRCPSP is proven to beNP-hard [7] thereis no optimal solution that could be computed in polynomialtime And this paper also demonstrates that the MS-RCPSPand the MS-RCPSPLD are NP-hard because they are moregeneral problems compared with the single-mode RCPSPHence methods that find feasible solutions which maynot achieve global optimal but can obtained in acceptabletime are built by researchers In such cases soft computingmethods are used mostly heuristics and metaheuristics

Within the metaheuristics group of methodologies itis noticed that memetic algorithm (MA) as a quite simpleapproach gives promising results in the fields of both com-puter science and operational research [8] MA representsone of the recent growing areas of research in evolutionarycomputation and is introduced by Moscato [9] inspired byboth Darwinian principles of natural evolution and Dawkinsrsquonotion of a meme With MA the traits of Universal Darwin-ism are more appropriately captured in particular dealingwith areas of evolutionary algorithms that marry otherdeterministic refinement techniques for solving optimizationproblems [10] As a general framework MA provides thesearch with desirable trade-off between intensification anddiversification through the combined use of a crossoveroperator to generate new promising solutions and a localoptimization procedure to locally improve the generatedsolutions [11 12]

In next stages to make MA more efficient it needs tobe supported by other metaheuristics eg general variableneighborhood search (GVNS) proposed by Hansen et al[13] GVNS is a simple and effective metaheuristic forcombinatorial optimization yield through systematic changeof neighborhoods within a refinement local search It canavoid becoming mired in local optima and develop the searchdirection to an excellent solution ultimately by means ofexploring new regions throughout the whole solution spaceThe GVNS has attracted attentions and shows remarkableperformance compared to other basic variable neighborhood

search (VNS) variants Up to now various optimization prob-lems have taken the GVNS to pursue better solutions suchas the single-machine total weighted tardiness problem withsequence dependent setup times [14] the one-commoditypickup-and-delivery traveling salesmanproblem [15] and thevehicle routing problem [16]

As far as we know there is no published work on solvingthe MS-RCPSP by the GVNS-MA By natural extension itmakes sense to solve the MS-RCPSPLD with the GVNS-MAThemain goal of this paper is to get an insight to the problemintegrating the MS-RCPSP with linear deterioration HowGVNS can be effectively hybridized with MA to solve MS-RCPSP and MS-RCPSPLD is examined in the GVNS-MAThe proposed approach is demonstrated to show its effective-ness in comparison to other state-of-the-art methods

The remaining part of this paper is organized as followsIn Section 2 a short summary of existing publications ispresented Section 3 gives the description of the problemunder consideration In Section 4 the proposedGVNS-MA iselaborated Section 5 shows performed experiments and theirresults before concluding the paper in Section 6 finally

2 Related Work

21 Deterioration Effect There are many practical situationsthat any delay or waiting in starting time of a task may causeto increase its processing time and two main deteriorationmechanisms are discussed

Besides usual linear deterioration mechanism discussedin this paper situations in which taskrsquos processing times arerepresented by step functions characterized by a sharp changein processing time at the deadline points [17ndash22] also arefairly common Sundararaghvan and Kunnathur [23] firstlyconsidered the corresponding single-machine schedulingproblem for minimizing the sum of the weighted completiontimes while Mosheiov [17] provided some simple heuristicsfor these NP-hard problems tominimize themakespan NextJeng and Lin [24] introduced a branch and bound algorithmfor the single-machine problem with the same goal Theproblem was extended by Cheng et al [19] to the case withparallel machines where a variable neighborhood searchalgorithm (VNS) was proposed

As for the linear deterioration mechanism Mosheiov[25] considered the case that processing times increase ata common rate and job weights are proportional to theirnormal processing time He demonstrated that the optimalschedule is Λ-shaped if the optimal objective is to minimizethe total weighted completion time on a single machineMoreover simple linear deterioration where jobs have a fixedjob-dependent growth rate but no basic processing time wasfurther considered [17] Ji and Cheng [26] discussed corre-sponding methods refer to the parallel-machine schedulingand gave a fully polynomial-time approximation schemewhereas a single-machine scheduling problem with lineardeterioration was studied by Jafari and Moslehi [27] Wuet al [28] Wang and Wang [29] with respective goals ofminimizing the number of tardy jobs and earliness penaltiesas well as total weighted completion time More studies and

Mathematical Problems in Engineering 3

discussions about linear deteriorating scheduling problemswere shown in Bachman and Janiak [30] Oron [31] Wangand Wang [29] Wu et al [28]

22 MS-RCPSP For the sake of its practicality RCPSP hasreceived more and more attentions [32] U and Kusiak [33]Babiceanu et al [34] and Coban [35] deal with the RCPSPin a dynamic real time way To obtain particular schedulesat some point metaheuristics are the most used techniquesfor solving RCPSP including Tabu search [36 37] SimulatedAnnealing [38 39] or Genetic Algorithm [40] SwarmIntelligence metaheuristics [41] are also effective for solvingrelated issues For example Particle Swarm Optimization[42] Bee Colony Optimization [43] or the popular AntColony Optimization [44 45] Although classical RCPSP isdeeply investigated and numerous methods could be easilycompared using PSPLIB instances it is not same for MS-RCPSP There are not too many researches paying attentionto MS-RCPSP

Myszkowski et al [46] defined new problem MS-RCPSPand benchmark to compare the effectiveness of examinedapproaches Next a hybrid ant colony optimization approach(HAC) [47] which links classical heuristic priority and theworst solutions stored by ants to update pheromone valuewas proposed To research this problem more thoroughlya greedy randomized adaptive search procedure (GRASP)[48] a hybrid differential evolution and greedy algorithm[49] and a Co-Evolutionary algorithm [50] are developed toimprove the quality of solutions Before this period Alanzi etal [51] proposed a lower bound using a linear programmingscheme for the RCPSP to solve the new extended MS-RCPSPmodelwhile Santos and Tereso [52] developed a filtered beamas a bonus for early completion into account Zheng et al [53]presented a teaching-learning-based optimization algorithmwith a task-resource list-based encoding scheme combiningthe task list and the resource list and a left-shift decodingscheme where the balance between global exploration andlocal exploitation to achieve satisfactory performances wasmainly stressed Dai et al [54] did the related work on theintegration of MS-RCPSP and proposed step structures aswell as two mutation operators

3 Problem Statement

To define two considered MS-RCPSP and MS-RCPSPLD aswell asmake them clear the problem description and amixedinteger programming model can be described as followsLet 119866(119881 119864) be a task-on-node network consisting of a set119881 of nodes denoting task 119894 (119894 isin 119881) and a set 119864 of edgesrepresenting the precedence relationships between a pair oftasks (119894 119895) (119894 119895) isin 119864 Specifically each pair (119894 119895) isin 119864 meansthe task 119894 precedes 119895 In other words task 119895 cannot start untiltask 119894 finished The duration of each task 119894 in MS-RCPSP is aprior known and constant while the value of duration inMS-RCPSPLD turning out to hinge on the basic processing time119886119894 and deteriorating rate ℎ119894 can be calculated by the function119889119894 = 119886119894 + 119878119894 lowastℎ119894 where 119878119894 shall be the earliest available time ofassigned resource 119896 119894 isin 119881119896 To perform any task 119894 the specific

skill 119902119894 also is required To make the 119899 tasks completed aset of 119870 of 119898 renewable resources which unassigned taskscan compete for when resources are idle will be providedResources differ inmastered skills and let119876119896 denote the skillscovered by resource 119896 Naturally a subset 119881119896 of 119881 also isavailable to incorporate all tasks that can be processed byresource 119896 (119902119894 isin 119876119896 119894 isin 119881119896) In same way a subset ofresources 119870119894 including all resources which can be utilizedto handle task 119894 is obtained Moreover each resource canperform at most one task at a time and a task can be executedby at most one resource simultaneously The objective is tominimize the makespan (the completion time of all tasks)

Based on the above descriptionwe formulate the problemas a 0-1 integer programming model Firstly the binaryvariables 119909119895119896 receive value 1 if task 119895 is assigned to resource119896 and 0 otherwise As for the binary variables 119910119894119895 are set to 1if task 119894 is scheduled to precede task 119895 0 otherwiseThen MS-RCPSP and MS-RCPSPLD can be formulated by objectivefunction (1) subject to constraints (2) to (9)

119872119894119899119894119898119894119911119890 119862119898119886119909 (1)

subject to 119865119894 = 119878119894 + 119889119894 forall119894 isin 119881 (2)

sum119896isin119870119895

119909119895119896 = 1 forall119895 isin 119881 (3)

119909119895119896 = 0 forall119895 notin 119870119895 (4)

119904119895 minus 119904119894 ge 119889119894 forall (119894 119895) isin 119864 (5)

119910119894119895 + 119910119895119894 ge 119909119894119896 + 119909119895119896 minus 1

forall (119894 119895) isin 119864 119896 isin 119870(6)

119878119895 ge 119878119894 + 119889119894 minus 119872(1 minus 119910119894119895)

forall119894 = 119895 119894 119895 isin 119881(7)

119862119898119886119909 ge 119865119894 forall119894 isin 119881 (8)

119909119895119896 119910119894119895 isin 0 1

119904119895 ge 0

forall119894 isin 119881 119895 isin 119881(9)

In the above mode objective function (1) denotes theoptimal direction of MS-RCPSP and MS-RCPSPLD Equa-tion (2) defines the finishing time of any task Constraint (3)restricts exactly one resource to a specific task among all avail-able resources Constraint (4) respects the skill constraintsbetween resources and tasks Constraint (5) highlights theprecedence relationship between a pair of nodes (119894 119895) inother words task 119895 can start only after 119894 is finished Constraint(6) shows logical relationships between the assignment vari-ables and sequencing variables That is when resource 119896 issimultaneously assigned to task 119894 and task 119895 the sequencebetween two tasksmust be determined either 119894 precedes 119895 or 119895precedes 119894 Constraint (7) is the big-119872 formulation to enforce


the relationship between the sequencing variable and thecontinuous starting time variable Constraint (8) calculatesthe makespan of project while Constraint (9) regulars thedomains of the variables

4 GVNS-Based Memetic Algorithm

In this section we describe in detail the general solutionmethodology and the supporting procedures in the followingsubsections

41 Search Space and Evaluation Function For a givenproblem the GVNS-MA searches a space Ω composed of allpossible assignments respecting skill constraints in any orderof tasks including both legal and illegal configurations Thesize of the search space Ω is bounded by 119874(119898119899 lowast 2119899)

Based on preceding notations and depictions to evaluatethe quality of a candidate solution 119904 isin Ω we adopt anevaluation function which is defined as 119891(119904) induced by 119904

119891 (119904) = 119898119886119909119894119898119894119911119890 119865119894 119894 isin 119881 + 119872 lowast sum(119894119895)isin119864

120575 (119894 119895) (10)

120575 (119894 119895) =

1 (119894 119895) isin 119864 119878119895 lt 1198651198940 otherwise

forall119894 isin 119881 119895 isin 119881 (11)

where 119872 is a large positive constant such that 119872 997888rarr infin as119899 997888rarr infin The first part of (10) represents the completiontime of the last unfinished task and the last part is anaugmented penalty function wheresum(119894119895)isin119864 120575(119894 119895) denotes thedegree violation to precedence relationships If 120575(119894 119895) =0 holds for any (119894 119895) isin 119864 it demonstrates the solutionis feasible corresponding to a legal configuration and itsevaluation function will all depend on the first part equalto the completion time of the schedule When a probleminstance admits no solution able to satisfy 120575(119894 119895) = 0 forall(119894 119895) isin119864 the search space of GVNS-MA is empty and no feasiblesolution can be found Given two solutions 1199041015840 and 11990410158401015840 1199041015840is better than 11990410158401015840 if 119891(1199041015840) le 119891(11990410158401015840) This statement impliesan assumption that a better solution has fewer precedenceconstraint violations

42Methodology and General Procedure Let119875 denote a pop-ulation of 119901 candidate configurations Let 119904119887 119904119908 represent thebest solution attainable so far and the worst solution in 119875 (interms of the evaluation function in Section 41) respectivelyLet119875119886119894119903119878119890119905 be a set of solution pairs (119904119894 119904119895) initially composedof all possible pairs in 119875 Next the proposed GVNS-MA canbe described as depicted in Algorithm 1

GVNS-MA first builds an initial population 119875 including119901 candidate configurations by the procedures in Section 43Then the algorithmenters into awhile loopwhich constitutesthe main part of the GVNS-MA On each new generationthe subsequent operations are executed In the first place aconfiguration pair (119904119894 119904119895) is taken at random and deleted fromPairSet Next GVNS-MA builds with a crossover operator(see Section 44) a new configuration 119904119888 (the offspring)After that the offspring is used as a starting point to further

(1) Input Problem instance I the size of population(119901) the depth of GVNS 120572

(2) Output the best configuration found during thesearch

(3) 119875 = 1199041 119904119901 larr997888 Population initialization(119868) lowastSec-tion 43lowast

(4) 119875119886119894119903119878119890119905 larr997888 (119904119894 119904119895) 1 le 119894 lt 119895 le 119901(5) while 119875119886119894119903119878119890119905 = do(6) Isolate a solution pair (119904119894 119904119895) isin 119875119886119894119903119878119890119905 ran-

domly(7) 119875119886119894119903119878119890119905 larr997888 119875119886119894119903119878119890119905 (119904119894 119904119895)(8) 119904119888 larr997888 crossOver(119904119894 119904119895) lowastSection 44lowast(9) 119904 larr997888 GVNS Operator(119904119888 120572) lowastSection 45lowast(10) if 119891(119904) le 119891(119904119908) then(11) UpdatePopulation(119904 119875 119875119886119894119903119878119890119905 119891(119904)) lowastSec-

tion 46lowast(12) end if(13) end while(14) Return 119904119887 larr997888 arg min 119891(119904119894) 119894 = 1 119901

Algorithm 1 Main sketch of the proposed GVNS-MA

improve by the GVNS operator (see Section 45) Finally ifthe improved offspring 119904 is better than the worst solution 119904119908in 119875 it is used to update the population 119875 and PairSet Thedetailed update operations are described in Section 46 Thewhile loop continues until PairSet becomes empty At the endof the while loop the algorithm terminates and returns thebest configuration 119904119887 found during the search Note that thedepth of GVNS 120572 represents a maximum number betweentwo iterationswithout improvement regarded as the stoppingcriterion of the GVNS operator

43 Population Initialization In order to build the initialpopulation (in Algorithm 2) the construction operator togenerate a new solution is executed 3 times 119901 times From thescratch a new configuration is constructed as follows Theoperator starts from assigning each task with a randomresource satisfying skill constraint Subsequently all tasksassigned to same resource are sequenced randomly at the endof previous task-to-resource phase Then for each generatedsolution the GVNS operator with the evaluation function119891(119904) (see Section 41) is used to optimize it to a local optimumand the obtained 119901 best configurations are selected to formthe initial population The detailed procedures are describedin Algorithm 2

44 The Crossover Operator Within a memetic algorithmthe crossover operator is another essential ingredient whosemain goal is to bring the search process to new promisingsearch regions to diversify the search In this paper theoffspring of two parent configurations 1199041 = (1198601 1198791198761) 1199042 =(1198602 1198791198762) shows as 119904119900 = (119860119900 119879119876119900) To inherit the advantagesof parent solutions the tasks assigned to same resource in 11990411199042 are given priority to keep the task-to-resource assignmentunchanged As for the remain unassigned tasks and 1198791198760


(1) Input The set 119881 = V1 V2 V119899 of 119899 tasks the set 119870 =1198961 1198962 119896119898 of 119898 renewable resources skill constraintsprecedence relationships and the size of population (119901)

(2) Output The best 119901 configurations(3) for pop = 1 3119901 do(4) Set RA = 119881(5) while 119877119860 = do(6) Choose V119894 randomly from 119877119860 remove V119894 from 119877119860

randomly isolate 119896119895 isin 119870119894mdash the set covering all re-sources can perform V119894 and record this assignmentas 119860(V119894) = 119895

(7) end while(8) for 119895 = 1 119898 do(9) Generate a random task sequence 119879119876119895 for candidate

tasks assigned to 119896119895(10) end for(11) Calculate the objective value 119891(119860TQ) with the evalu-

ation function 119891(119904) (see Section 41)(12) 119904119887119901119900119901 larr997888 GVNS operator(119891(119860TQ) 120572) lowastSection 45lowast(13) end for(14) Sort the 3119901 configurations in an ascending sort of evalua-

tion function values and return the first 119901 solutions as theinitial population

Algorithm 2 Population initialization

they will be determined by same methods in Section 43Analogously we apply the GVNS operator (see Section 45)to the offspring to finally gain the candidate for furtherpopulation updatingThe principle of this operator is detailedin the procedure crossover (Algorithm 3)

45 GVNS Operator This section discusses the local opti-mization phase of GVNS MA a key part of memetic algo-rithm Its function ensures an intensified search to locatehigh quality local optima from any starting point Here wedesign a generable variable neighborhood search (GVNS)heuristic as the local refinement procedure which showsgood performance compared to other variable neighborhoodsearch variants in terms of local search capability

Given three neighborhood structures 1198731 11987321198733 and aninitial solution (1199040) our GVNS operator does the refinementas follows Attention the quality of any solution is evaluatedas depicted in Section 41 To start with the sequential orderSO which determines the applying sequence of these neigh-borhood structures is at random generated For exampleassuming that the sequence SO equals (3 1 2) the searchbegins from1198733 and ends at1198731 at the given iteration For eachneighborhood structure a new local optima 11990410158401015840 is obtainedby applying the corresponding local search procedures to theincumbent solution 1199041015840 set at 1199040 at the beginning of GVNSprocedure If 11990410158401015840 is better than 1199041015840 1199041015840 is updated with thenew solution 11990410158401015840 accepted as a descent to continue the localsearch for current neighborhood otherwise the search turnsto the next neighborhood structure in SO One iterationterminates until the last neighborhood structure in SO isexplored and then the search goes on with the next iteration

until the stopping criteria is met ie the best solution 1199041015840has not improved for 120572 consecutive iterations The generalsketch of GVNS operator is described in Algorithm 4 andthe neighborhoods employed as well as the technique tocalculate objective value rapidly are depicted in the followingsubsections

451 Move and Neighborhood Three neighborhoods119873119896 (119896 = 1 3) are adopted in GVNS MA The neighbor-hood 1198731 is defined by the swap move operator which swapstwo tasks processed by same resource and keeps previoustask-to-resource assignment unchanged As such given asolution 119904 the swap neighborhood 1198731(119904) of 119904 is composedof all possible configurations that can be applied with theswap move to 119904 The neighborhood 1198732 is designed on thebase of reversion which reverses all the tasks incorporatedinto two designated random tasks of one resource As forthe neighborhood 1198733(119904) it is designed by the alter moveoperator which alters assigned resource from the originalto another resource equipped with demanded skill for oneselected randomly task with a random position in the tasksequence of given resource To efficiently assess the qualityof any neighborhood solution we devise a rapid evaluationtechnique for neighborhood solutions which is committedgreatly to the computational efficiency of the GVNS-MA

452 Rapid Evaluation Mechanism Our rapid evaluationtechnique to neighborhood solutions realizes through effec-tively calculating the move value (Δ119891) which identifies thechange in the evaluation function 119891 (see Section 41) ofeach possible move applicable to the incumbent solution


(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I

(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned

resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the

same way as Algorithm 2 lowastSection 43lowast

Algorithm 3 Procedure crossover

(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572

(2) Output The current best solution 119904119887 found during GVNSprocess

(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41

(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not

improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three

neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-

determined order specified by SO update 1199041015840 if a betterconfiguration is attained

(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887

Algorithm 4 GVNS operator

119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated

For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved

by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)

46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an


Table 1 Settings of important parameters

Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107

to precedence constraint

excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet

5 Computational Experiments and Results

This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set

51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54

52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out

Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872

is set at 1000 for the first experimental group and 1 times 107 forthe second

53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature

Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation

119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905

119891119901119903119890119887times 100 (12)

where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution

Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that


Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic

instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)

100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050

the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050

54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et

al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness

Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively

Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the


three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated

Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms

Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs

First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate

55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework

551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain

Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas

Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations

119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)

respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other

The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()

However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other

To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks


Table3Summaryandcomparis

onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45


Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]

instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922

100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880

These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA

552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS

algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can


Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54


100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298

visually detect the gap between the current algorithm and thebest

Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP

6 Conclusions

The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has

a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR

The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance

Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan


Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]

instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()

100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583

Science and Technology Program (nos 2019YFG0300 no2019YFG0285)

References

[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012

[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988

[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008

[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990

[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999

[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004

[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983

[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005

[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989


[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011

[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009

[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011

[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997

[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012

[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012

[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012

[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995

[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003

[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012

[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015

[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014

[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009

[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994

[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004

[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991

[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008

[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012

[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007

[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010

[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000

[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008

[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999

[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002

[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005

[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013

[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017

[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016

[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017

[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018

[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018

[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009

[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018

[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear


inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015

[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017

[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017

[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015

[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015

[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016

[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018

[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017

[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010

[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011

[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017

[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018

[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


project scheduling problem (RCPSP) which make it possiblethrough a better adaption to apply in production planningproject management or manufacturing etc Ultimately moreadaptable multiskill resource-constrained project schedulingproblem (MS-RCPSP) gives each resource a set of capabilitiesand each resource can bid for being assigned tasks

Scheduling with deterioration and MS-RCPSP are tworesearch hotspots of scheduling area and these two subfieldsare not absolutely separate In terms of most tasks of actualMS-RCPSP deterioration effects existed for the reason offatigued (humans ormachines)However there is no researchconcentrated on the integration of MS-RCPSP and deteriora-tion

In this paper we pick the deterioration mechanism aslinear deterioration As for the researched MS-RCPSP withlinear deterioration which is dubbed MS-RCPSPLD thedifferences compared to the scheduling problems with lineardeterioration in the reported literature can be explainedhere Foremost the resource or the machine in the usuallyresearched scheduling problem with deterioration mostly isrestricted to be single-mode which means that it owns onlyone capability and can execute just a kind of task

Moreover as theRCPSP is proven to beNP-hard [7] thereis no optimal solution that could be computed in polynomialtime And this paper also demonstrates that the MS-RCPSPand the MS-RCPSPLD are NP-hard because they are moregeneral problems compared with the single-mode RCPSPHence methods that find feasible solutions which maynot achieve global optimal but can obtained in acceptabletime are built by researchers In such cases soft computingmethods are used mostly heuristics and metaheuristics

Within the metaheuristics group of methodologies itis noticed that memetic algorithm (MA) as a quite simpleapproach gives promising results in the fields of both com-puter science and operational research [8] MA representsone of the recent growing areas of research in evolutionarycomputation and is introduced by Moscato [9] inspired byboth Darwinian principles of natural evolution and Dawkinsrsquonotion of a meme With MA the traits of Universal Darwin-ism are more appropriately captured in particular dealingwith areas of evolutionary algorithms that marry otherdeterministic refinement techniques for solving optimizationproblems [10] As a general framework MA provides thesearch with desirable trade-off between intensification anddiversification through the combined use of a crossoveroperator to generate new promising solutions and a localoptimization procedure to locally improve the generatedsolutions [11 12]

In next stages to make MA more efficient it needs tobe supported by other metaheuristics eg general variableneighborhood search (GVNS) proposed by Hansen et al[13] GVNS is a simple and effective metaheuristic forcombinatorial optimization yield through systematic changeof neighborhoods within a refinement local search It canavoid becoming mired in local optima and develop the searchdirection to an excellent solution ultimately by means ofexploring new regions throughout the whole solution spaceThe GVNS has attracted attentions and shows remarkableperformance compared to other basic variable neighborhood

search (VNS) variants Up to now various optimization prob-lems have taken the GVNS to pursue better solutions suchas the single-machine total weighted tardiness problem withsequence dependent setup times [14] the one-commoditypickup-and-delivery traveling salesmanproblem [15] and thevehicle routing problem [16]

As far as we know there is no published work on solvingthe MS-RCPSP by the GVNS-MA By natural extension itmakes sense to solve the MS-RCPSPLD with the GVNS-MAThemain goal of this paper is to get an insight to the problemintegrating the MS-RCPSP with linear deterioration HowGVNS can be effectively hybridized with MA to solve MS-RCPSP and MS-RCPSPLD is examined in the GVNS-MAThe proposed approach is demonstrated to show its effective-ness in comparison to other state-of-the-art methods

The remaining part of this paper is organized as followsIn Section 2 a short summary of existing publications ispresented Section 3 gives the description of the problemunder consideration In Section 4 the proposedGVNS-MA iselaborated Section 5 shows performed experiments and theirresults before concluding the paper in Section 6 finally

2 Related Work

21 Deterioration Effect There are many practical situationsthat any delay or waiting in starting time of a task may causeto increase its processing time and two main deteriorationmechanisms are discussed

Besides usual linear deterioration mechanism discussedin this paper situations in which taskrsquos processing times arerepresented by step functions characterized by a sharp changein processing time at the deadline points [17ndash22] also arefairly common Sundararaghvan and Kunnathur [23] firstlyconsidered the corresponding single-machine schedulingproblem for minimizing the sum of the weighted completiontimes while Mosheiov [17] provided some simple heuristicsfor these NP-hard problems tominimize themakespan NextJeng and Lin [24] introduced a branch and bound algorithmfor the single-machine problem with the same goal Theproblem was extended by Cheng et al [19] to the case withparallel machines where a variable neighborhood searchalgorithm (VNS) was proposed

As for the linear deterioration mechanism Mosheiov[25] considered the case that processing times increase ata common rate and job weights are proportional to theirnormal processing time He demonstrated that the optimalschedule is Λ-shaped if the optimal objective is to minimizethe total weighted completion time on a single machineMoreover simple linear deterioration where jobs have a fixedjob-dependent growth rate but no basic processing time wasfurther considered [17] Ji and Cheng [26] discussed corre-sponding methods refer to the parallel-machine schedulingand gave a fully polynomial-time approximation schemewhereas a single-machine scheduling problem with lineardeterioration was studied by Jafari and Moslehi [27] Wuet al [28] Wang and Wang [29] with respective goals ofminimizing the number of tardy jobs and earliness penaltiesas well as total weighted completion time More studies and





3 Problem Statement




119872119894119899119894119898119894119911119890 119862119898119886119909 (1)


sum119896isin119870119895

119909119895119896 = 1 forall119895 isin 119881 (3)

119909119895119896 = 0 forall119895 notin 119870119895 (4)


119910119894119895 + 119910119895119894 ge 119909119894119896 + 119909119895119896 minus 1

forall (119894 119895) isin 119864 119896 isin 119870(6)

119878119895 ge 119878119894 + 119889119894 minus 119872(1 minus 119910119894119895)

forall119894 = 119895 119894 119895 isin 119881(7)

119862119898119886119909 ge 119865119894 forall119894 isin 119881 (8)

119909119895119896 119910119894119895 isin 0 1

119904119895 ge 0

forall119894 isin 119881 119895 isin 119881(9)









120575 (119894 119895) (10)

120575 (119894 119895) =


forall119894 isin 119881 119895 isin 119881 (11)





























































119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905

119891119901119903119890119887times 100 (12)






100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050















119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)







onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018












3 Problem Statement




119872119894119899119894119898119894119911119890 119862119898119886119909 (1)


sum119896isin119870119895

119909119895119896 = 1 forall119895 isin 119881 (3)

119909119895119896 = 0 forall119895 notin 119870119895 (4)


119910119894119895 + 119910119895119894 ge 119909119894119896 + 119909119895119896 minus 1

forall (119894 119895) isin 119864 119896 isin 119870(6)

119878119895 ge 119878119894 + 119889119894 minus 119872(1 minus 119910119894119895)

forall119894 = 119895 119894 119895 isin 119881(7)

119862119898119886119909 ge 119865119894 forall119894 isin 119881 (8)

119909119895119896 119910119894119895 isin 0 1

119904119895 ge 0

forall119894 isin 119881 119895 isin 119881(9)









120575 (119894 119895) (10)

120575 (119894 119895) =


forall119894 isin 119881 119895 isin 119881 (11)





























































119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905

119891119901119903119890119887times 100 (12)






100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050















119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)







onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018















120575 (119894 119895) (10)

120575 (119894 119895) =


forall119894 isin 119881 119895 isin 119881 (11)





























































119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905

119891119901119903119890119887times 100 (12)






100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050















119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)







onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018























































119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905

119891119901119903119890119887times 100 (12)






100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050















119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)







onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018











100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050















119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)







onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018

















119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051

times 100 (13)

and

119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922

119879119886V1198921times 100 (14)







onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018










onon

thes

etof

45newgeneratedinsta

nces

with

119899=10

0of

GVN

S-MA

GVN

S-119872

119860 ℎG

VNS-

119872119860 119886ℎand

theT

Sheuristic[54]

andPR

[55]B

estresultsare

indicatedin

italic

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-10-26-15

165572

6598

13899

963646

6492

66871

63274

6392

33092

661967

6379

730698

62632

63629

29422

100-10-26-15

260

1396

618416

28607

608971

625436

79283

587145

613934

37496

580557

615474

23879

572794

615156

34826

100-10-26-15

3172856

184175

31236

1646

96

175513

59776

160283

16936

22225

166221

17124

17541

163898

172228

3461

100-10-47-91

7008

72053

40304

6974

471232

10018

69445

70098

20666

67645

6915

731499

69673

7039

36314

100-10-47-92

700352

739287

39112

703439

721152

70563

672221

715138

2417

9672304

6959

20614

693553

702369

25114

100-10-47-93

188206

195206

4816

818358

190863

93463

174162

184389

30008

177323

183958

19383

173708

183368

41432

100-10-48-15

165868

6914

13991

567093

6890

88879

164

851

65526

3491

663253

66595

22686

6479

86610

226662

100-10-48-15

2638747

659809

42043

621357

658747

9116

161867

651609

4116

1590432

625462

27097

618507

6346

3233655

100-10-48-15

3170999

174232

29659

171864

178061

73395

159497

166887

23395

161483

166366

3216

9163501

170273

27056

100-10-64-91

70527

7572

53770

371049

7692

773024

6872

27092

931287

6884

71215

26024

67894

69682

2775

6100-10-64-92

705806

744775

39415

7040

32738141

6279

4663729

711114

3013

968421

719153

22806

696727

736833

3991

8100-10-64-93

193844

206206

3115

32006

42

2044

03

108918

189324

19445

35415

180842

200842

3104

193844

199996

31871

100-10-64-15

170119

71497

34626

71075

7279

683543

69346

70022

39097

6691

69112

21249

70042

7079

617116

100-10-64-15

2690964

744775

3899

8672578

72823

61684

6606

14685085

22491

62823

682957

3178

4626331

666418

2376

100-10-64-15

3193844

218506

31871

194247

204235

71318

1806

92

1860

08

25336

187149

192556

45438

185585

190957

25296

100-20-22-15

119089

19355

29631

1936

19739

5517

618883

19118

2222

18572

1898

21438

1878

18965

2474

2100-20-22-15

24615

847731

36854

46032

46714

6792

645371

46008

28455

45218

46627

2892

645833

4633

2312

4100-20-22-15

326328

2798

128528

27212

2697

36972

92597

26419

2812

25563

25918

17531

25575

26254

21352

100-20-46-15

126243

2673

33424

26214

26631

60354

25862

26329

4119

925511

25896

1215

325476

26006

17466

100-20-46-15

260

647

66244

34408

59367

63596

71878

5796

16165

17216

56542

5942

26696

5465

59285

22313

100-20-46-15

332421

3490

937523

33539

3417

977634

31695

32909

16433

31651

33263

29116

32841

33349

2791

100-20-47-91

19007

1975

929027

1916

319685

5776

318864

1912

28311

1874

719269

2732

18455

18892

30731

100-20-47-92

50591

53481

37119

47839

51484

102173

4803

49319

3549

46278

4876

435418

44966

47893

24241

100-20-47-93

30802

31827

4572

630631

31365

6995

829437

30352

57846

27712

29458

29737

28399

2879

927559

100-20-65-15

19013

492446

26413

89865

90543

6795

788801

90095

17505

86826

89518

15052

87305

89338

2843

100-20-65-15

2253899

2606

7628718

250126

253267

5876

3244762

250316

4099

7243449

24751

17659

242353

244944

2573

5100-20-65-15

3110

567

1115886

27491

109874

113685

53418

105215

109224

2513

8108595

111243

1591

1104829

106776

1412

6100-20-65-91

19113

1978

341248

1895

719548

79561

18369

18898

30561

18697

1914

61896

18694

19265

3097

9100-20-65-92

46242

4776

865469

4593

447443

58112

4495

746229

1768

44719

45985

22324

45593

46591

30606

100-20-65-93

28018

28776

4894

327512

2810

868532

2719

2772

124279

26587

27691

3101

26455

27845

2473

100-5-22-151

500988

514285

2897

8510285

505098

84526

486586

499318

43529

494364

500988

18444

498567

510285

22335

100-5-22-152

1193440

1227080

4319

4119

2300

1210150

61537

1118610

1199248

24322

1184940

1219514

20664

1138490

1226780

26656

100-5-22-153

663947

700818

5691

2661553

792627

7997

660

6322

652627

10639

592509

6279304

2619

8643856

6597526

17267

100-5-46

-151

675739

749763

3679

7652793

701457

137862

63503

679663

5075

8636903

687941

29888

649164

671982

36202

100-5-46

-152

1404

680

1453430

58595

1443529

1470970

108595

1298430

1370870

26564

1315940

1360

830

23536

1332210

1399014

2417

6100-5-46

-153

795664

870313

50687

7977713

863567

6891

1740178

796512

81792

4755866

7786062

16809

752568

80128

1876

3100-5-48-9

1563787

586832

31214

5666

27

78386

84807

55206

560407

2415

55206

563664

25018

554299

56657

2495

2100-5-48-9

21284230

1315720

3614

51289811

1364

650

71286

1222350

1285020

2795

8116

3860

1259332

2812

21233010

1268572

34659

100-5-48-9

3684299

692453

45335

66785

717279

29393

8657279

67622

4561

6604

26

670917

41286

64822

667213

828446


Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018









Table3Con

tinued

insta

nces

ITS

PRGVN

S-MA

GVN

S-119872

119860 ℎGVN

S-119872

119860 aℎ

119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)119891 119887119890119904119905

119891 119886V119892

119879119894119898119890

(119904)100-5-64

-151

581916

627374

55661

576589

618167

102415

5544

26

579234

3516

7546543

590352

29678

56942

580175

56619

100-5-64

-152

120944

01261510

31296

1190830

1257643

123533

1131450

1183014

40595

1118430

1151812

46305

1047250

1158556

3337

100-5-64

-153

642267

709602

3498

8634651

688889

89403

612857

6657858

3798

5626771

660614

44591

1624753

674996

261623

100-5-64

-91

550231

577524

37365

5544

81

566747

98693

528177

5404

36

3393

530748

543895

48396

515984

536747

2993

2100-5-64

-92

1214340

1271610

40553

1183479

1236750

11992

711140

60115

5584

41523

11640

101201126

40287

1121650

1159384

27286

100-5-64

-93

610356

648765

36395

6151502

632323

87448

604586

6210378

31363

594514

6171502

30543

595191

623223

2774

9119861

119890119904119905

00

00

1514

1716

1415

119879119900119905119886

11989745

4545

4545

4545

4545

45




100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018











100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880







100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018











100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298



6 Conclusions





Data Availability




Acknowledgments





100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018











100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583


References


































































Journal of












Journal of







Volume 2018






Volume 2018
































































Journal of












Journal of







Volume 2018






Volume 2018








a memetic algorithm for multiskill resource-constrained...

Documents