Research Article Balancing Lexicographic Multi-Objective

Research ArticleBalancing Lexicographic Multi-Objective AssemblyLines with Multi-Manned Stations

Talip Kellegoumlz

Industrial Engineering Department Engineering Faculty Gazi University Maltepe 06570 Ankara Turkey

Correspondence should be addressed to Talip Kellegoz tkellegozgaziedutr

Received 21 October 2015 Accepted 7 March 2016

Academic Editor Martin J Geiger

Copyright copy 2016 Talip KellegozThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In a multi-manned assembly line tasks of the same workpiece can be executed simultaneously by different workers working inthe same station This line has significant advantages over a simple assembly line such as shorter line length less work-in-processsmaller installation space and less product flow time In many realistic line balancing situations there are usually more than oneobjective conflicting with each otherThis paper presents a preemptive goal programmingmodel and some heuristicmethods basedon variable neighborhood search approach formulti-objective assembly line balancing problemswithmulti-manned stationsThreedifferent objectives are considered minimizing the total number of multi-manned stations as the primary objective minimizingthe total number of workers as the secondary objective and smoothing the number of workers at stations as the tertiary objectiveA set of test instances taken from the literature is solved to compare the performance of all methods and results are presented

1 Introduction

Assembly lines are mostly installed for producing productsin high volumes and usually include automatic materialhandling system [1] tools workers and more than oneworkpiece Therefore in fact assembly lines are designedby arranging all of their components One of most impor-tant problems in this arrangement is to group tasks intostations In this problem there are some constraints suchas precedence restrictions between task pairs and cycle timeconstraints for stations This grouping process is carried outfor optimizing some objectives by fulfilling all of restrictionsThe optimization problem of this process is called assemblyline balancing problem (ALBP)

ALBPs which are strongly NP-hard [2] can be classifiedbased on different criteria such as line layout (straightlines U lines etc) the number of models produced (singlemodel multimodel mixed model etc) duration of tasks(deterministic stochastic etc) the number of workers foreach station (traditional lines two sided lines multi-mannedlines etc) and the number of objectives considered (singleobjective multi-objective etc) The reader is referred to the

recent paper by Sivasankaran and Shahabudeen [3] for acomprehensive survey and classification of line balancingproblems Also other well-crafted surveys can be found inpapers by Lusa [4] Becker and Scholl [5] and Erel and Sarin[6]

In practice products produced in assembly lines havedifferent characteristics based on size the number of tasksdemand structure duration of tasks and so forth Thereforedifferent products may require different line structures whichare briefly mentioned above For example consider theassembly processes of a computer keyboard and a bus Itis obvious that the production of the computer keyboardrequires less number of tasks than the one of bus Alsobased on sum of task times the keyboard has smaller flowtime than the bus On the other hand more than oneworker can perform tasks on the same bus workpiece whileonly one worker is usually allowed to perform tasks onthe same keyboard workpiece Therefore for decreasing theflow time tool cost work-in-process cost space used forthe assembly line and so forth assembly lines with multi-manned workstations may be more suitable to produce largesize products like bus in practice Consider the problem with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9315024 20 pageshttpdxdoiorg10115520169315024

2 Mathematical Problems in Engineering

1 3 7 11

2 6 9

4

8

10

5

12

6 2

2

3

4

8

4

6

2

3

3

2

Figure 1 Precedence diagram for illustrative problem

Station 1 Station 2 Station 3 Station 4 Station 57 10 6 9 11 121 2 3 4 5 8

(a) Simple assembly line

1 2 3

4

5

Station 2 Station 39

10 11 12

7

8

6

Station 1

W1

W1 W1W2

W3

(b) Multi-manned assembly line

Figure 2 Optimal solutions of simple and multi-manned assembly line balancing problems

cycle time Ct = 10 and precedence diagram given in Figure 1By using the sum of task times 119905sum = 45 the lower boundvalue for the total number of stations for simple assemblyline [15] and also for the total number of workers for multi-manned assembly line [7 8] can be calculated as follows

LB1= lceil

119905sumCtrceil = lceil

45

10

rceil = 5 (1)

For simple assembly line balancing problemwithminimizingthe number of stations (SALBP1) the optimum solution isgiven in Figure 2(a) As is seen from this figure at least5 stations are required to produce the goods with desiredproperties There are 5 workpieces (and workers) in theline since there is a workpiece (a worker) at each one ofthese stations For multi-manned assembly line balancingproblem (MALBP) with minimizing the number of stationsthe optimum solution is presented in Figure 2(b) In MALBPdetermining a station for each task is necessary but notsufficient To address a MALBP solution a worker for eachtask and a sequence of tasks for each worker must also beobtained As is seen from Figure 2 the multi-manned linerequires less number of stations than the simple line Hencein multi-manned lines same products may be produced by

using less layout space work-in-process flow time and soforth

In MALBP each task is allowed to be performed by onlyone worker Also any worker performs at most one task at atime To the best of authorsrsquo knowledge the first study aboutMALBPwas done byDimitriadis [9] He defined the problemand proposed a heuristic method for solving it Then Beckeramp Scholl [7] developed a mixed integer programming (MIP)model and a branch amp bound (BB) method for optimallysolving MALBP with mounting places In the same studythey also described how the proposed BB is used as a heuristicfor large size problem instances For solving themixed-modelMALBP of real life tractor assembly line Cevikcan et al [10]proposed a heuristic method executed phase by phase in fivesteps The first three steps are devoted to balance the lineand the remaining two steps are used for sequencing modelsand transferring workers in daily operations For solvingMALBP with minimizing both number of workers andmulti-manned stations in the same order Fattahi et al [11]proposed a MIP model and a heuristic method based on antcolony optimization In their comparison study they showedtheir heuristicrsquos superiority over the heuristic by Dimitriadis[9] For solving MALBP with task times dependent on thenumber of workers at stations Yazdanparast andHajihosseini

Mathematical Problems in Engineering 3

[12] proposed a MIP model based on the model of Fattahiet al [11] Kellegoz and Toklu [8] considered MALBP withminimizing the number of workers and proposed a BBalgorithm for optimally solving them In terms of both CPUtimes and quality of feasible solutions found comparisonresults showed that their BB method has better performancethan the revised one of the BB method proposed by Beckerand Scholl [7] Kazemi and Sedighi [13] developed a MIPmodel a genetic algorithm based heuristic and a particleswarm optimization based heuristic for cost-oriented mixed-model MALBP They solved several test instances to illus-trate their methodsrsquo performance For solving MALBP withminimizing the total number of workers Kellegoz and Toklu[14] recently proposed a new MIP formulation a construc-tive heuristic based on priority rules and an improvementheuristic based on genetic algorithms By using benchmarkinstances they illustrated the performance of methods theyproposed

The purpose of this study is twofold (1) to present a goalprogramming mathematical formulation for multi-objectiveMALBP and (2) to develop metaheuristic methods basedon the variable neighborhood search (VNS) approach forsolving medium and large size instances To the best ofauthorsrsquo knowledge this study is the first attempt to solvemulti-objective MALBP by using the goal programmingapproach and VNS algorithm The objectives consideredfor MALBP in this study are as follows (1) minimizingthe total number of multi-manned stations (line length)as the primary objective (2) minimizing the total numberof workers as the secondary objective and (3) smoothingthe number of workers at stations as the tertiary objectiveThe remainder of this paper is organized as follows InSection 2 the multi-objective MALBP is defined The goalprogramming formulation for multi-objective MALBP ispresented in Section 3 Then the proposed metaheuristicalgorithms are presented in Section 4 The computationalstudy and its results are reported in Section 5 At last in Sec-tion 6 some conclusions and future research directions arediscussed

2 Multi-Objective AssemblyLine Balancing Problems withMulti-Manned Stations

A multi-objective MALBP could be described as followsThere is a set 119868 of 119899 tasks all of them have to be performed forgetting the final product Also some tasksrsquo operations cannotbe started before the completion of some of the remainingtasksrsquo operations In other words there are precedence rela-tionships within some task pairs The assembly line whichis built to produce this homogenous product has a set 119869of 119898 stations placed on a straight line Due to productrsquoscharacteristics each station is allowed to have more thanone worker However the number of workers simultaneouslyperforming different tasks at the same station cannot begreater than the value119872max Some reasons for this limitationmay be as follows [8 9 11]

(1) There are119872max tasks having no precedence relation-ship between each other and they can be performedat the same station

(2) Due to some characteristics of the product or stationsat most119872max workers are able to perform tasks on thesame workpiece

All the workers used in the line are identical and equallyequipped Each task is indivisible and has to be performedfor a predefined deterministic time by only one worker inthe line Each worker cannot perform more than one taskat a time However different workers of the same stationmay perform different tasks simultaneously In this case itis also ensured that relationships between simultaneous taskoperations are satisfied

The decision maker has three goals as follows (in order ofpriority)

Goal 1 There is no certain limitation on the line length (thenumber of stations on the line) but the line should not havemore than 119871max stations

Goal 2 There is no certain limitation on the total number ofworkers in the line but there should not bemore than119879119872maxworkers in the line

Goal 3 There is no certain limitation but the line shouldhave the same number of workers in each station in theline

It is assumed that the decision maker is not able todetermine precisely the relative importance of the goalsThushe ranks these goals from the most important (goal 1) to leastone (goal 3) As a result the decisionmaker first tries to satisfygoal 1 as close as possible Then for all alternative optimalsolutions to goal 1 he tries to find the optimal solution forgoal 2 and so forth Note that while trying to satisfy thesecond goal as close as possible it is ensured that the deviationfrom goal 1 remains at its optimal level Also while comingas close as possible to meeting goal 3 it is ensured that thedeviations from goals 1 and 2 remain at their optimal levelsThus for example while providing goal 3 it is ensured thatan additional worker that makes the second deviation worseis not used

The reason of determining target values for the firsttwo goals may be that the decision maker wants to makearrangement on his current line In this case these targetvalues may be determined by the available space and lineworkforce If there is no line active then target valuesfor these goals may be set to zero or their lower boundvalues

Smoothness index based on station times (sum of tasktimes assigned to a station) is calculated in the followingmanner [15]

SIst = radic119898

sum

119895=1

(119879119878max minus 119879119878119895)2

(2)


where 119898 is the number of opened stations in the line 119879119878119895

is the station time of station 119895 and 119879119878max is the maxi-mum station time determined as 119879119878max = max

1le119895le119898119879119878119895

Using the same approach for multi-manned assembly linessmoothness index based on the number of workers at stationscan be calculated as follows

SIwr = radic119898

sum

119895=1

(119879119882max minus 119879119882119895)2

(3)

where 119879119882119895is the number of workers used at opened station

119895 and 119879119882max is the maximum number of used workersat a station opened in the line which is determined as119879119882max = max

1le119895le119898119879119882119895 The perfect balance is indicated

by smoothness index value 0 In this research it is assumedthat the last goal is satisfied by minimizing the worker basedsmoothness index of the line

3 Goal Programming Formulation

The mathematical model of MALBP proposed by Fattahi etal [11] is developed here to generate the goal programmingformulation of the consideredmulti-objectiveMALBP To thebest of authorsrsquo knowledge the term ldquogoal programmingrdquowas introduced by Charnes and Cooper [16] For a good

description of this approach the reader is referred to thewell-known operations research textbook by Winston [17]

31 Notations

Parameters

Ct Cycle time119905119894 Duration of task 119894119868 Set of tasks 119868 = 1 2 119899119869 Set of stations 119869 = 1 2 119899 where 119899 is a validupper bound for the number of stations [11]119870 Set of workers at a station119870 = 1 2 119872max119875(119894) Set of immediate predecessors of task 119894119875119886(119894) Set of all predecessors of task 119894

119878(119894) Set of immediate successors of task 119894119878119886(119894) Set of all successors of task 119894

120595ΩΦ Large positive numbers119871max Target value of the number of stations openedin the line119879119872max Target value of the total number of workersused in the line

Decision Variables

st119894 Start time of task 119894

119909119894119895119896=

1 if task 119894 is assigned to the worker 119896 at station 119895

0 otherwise

119910119894ℎ=

1 at the same worker if task 119894 is performed before tash ℎ

0 otherwise

(4)

119908119895 Number of workers used at station 119895

119898119908 The maximum of number of workers of stations

119898119908 = max119895119908119895

120572119895119896

=

1 if station 119895 is opened and it has (119898119908 minus 119896) workers

0 otherwise

119906119895=

1 if station 119895 is opened

0 otherwise

(5)

119897119897 Total number of stations opened in the line (linelength)

119889

+

1 119889

minus

1 Positive and negative deviations from the

target value 119871max of the number of stations opened inthe line

119889

+

2 119889

minus


target value 119879119872max of the total number of workersused in the line

119889

+

3 119889

minus


target value zero of the smoothness index value of theline (without square root)

32 Mathematical Model The proposed mixed integer goalprogramming model is as follows


321 Objective Function Consider

lexmin 119889+1 119889

+

2 119889

+

3 (6)

322 Model Constraints

General Constraints Consider

sum

119895isin119869

sum

119896isin119870

119909119894119895119896= 1 forall119894 isin 119868 (7)

Ct sdot sum119895isin119869

sum

119896isin119870

((119895 minus 1) sdot (119909ℎ119895119896minus 119909119894119895119896)) + st

ℎ+ 119905ℎle st119894

forall119894 isin 119868 ℎ isin 119875 (119894)

(8)

st119894+ 119905119894le Ct forall119894 isin 119868 (9)

stℎ+ Ψ sdot (1 minus 119909

ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot (1 minus 119910

119894ℎ) ge st119894+ 119905119894

forall119894 isin 119868 ℎ isin 119903 | 119903 isin 119868 minus (119875119886(119894) cup 119878

119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(10)

st119894+ Ψ sdot (1 minus 119909

ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot 119910

119894ℎge stℎ+ 119905ℎ


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(11)

Constraints for Goal 1 Consider

sum

119895isin119869

sum

119896isin119870

119895 sdot 119909119894119895119896le 119897119897 forall119894 isin 119868 (12)

119897119897 + 119889

minus

1minus 119889

+

1= 119871max (13)


sum

119896isin119870

119896 sdot 119909119894119895119896le 119908119895 forall119894 isin 119868 119895 isin 119869 (14)

sum

119895isin119869

119908119895+ 119889

minus

2minus 119889

+

2= 119879119872max (15)


119898119908 ge 119908119895 forall119895 isin 119869 (16)

Ω sdot 119906119895ge sum

119894isin119868

sum

119896isin119870

119909119894119895119896

forall119895 isin 119869

(17)

Φ sdot (1 minus 119906119895) + sum

119896isin119896

(119896 sdot 120572119895119896) + 119908119895ge 119898119908 forall119895 isin 119869 (18)

sum

119896isin119870

120572119895119896le 1 forall119895 isin 119869 (19)

sum

119895isin119869

sum

119896isin119870

(119896

2sdot 120572119895119896) + 119889

minus

3minus 119889

+

3= 0 (20)

Sign Restrictions Consider

119909119894119895119896isin 0 1

forall119894 isin 119868 119895 isin 119869 119896 isin 119870

(21)

119910119894ℎisin 0 1


119886(119894)) 119894 lt 119903

(22)

120572119895119896isin 0 1

forall119895 isin 119869 119896 isin 119870

(23)

119906119895isin 0 1 forall119895 isin 119869 (24)

119908119895ge 0

and integer forall119895 isin 119869(25)

st119894ge 0 forall119894 isin 119868 (26)

119889

+

1 119889

minus

1 119889

+

2 119889

minus

2 119889

+

3 119889

minus

3 119897119897 119898119908 ge 0 (27)

Constraints (7) (9)ndash(11) are directly taken from the modelwhich is proposed by Fattahi et al [11] Constraint (8) iswritten by using the clock time approach [15] The pro-posed mixed integer goal programming model has multipleobjectives given in Expression (6) The first goal is that thenumber of opened stations should not exceed predeterminedvalue 119871max As indicated earlier the variable 119889+

1corresponds

to overachievement of this target value Therefore the firstobjective aims to minimize this positive deviation Thesecond objective aims to minimize the positive deviation oftotal number of workers used in the line from predeterminedtarget value 119879119872max At last the third objective aims tominimize the positive deviation of smoothness index on thebasis of number of workers at stations (without square root)Note that the target value of smoothness index is equal tozero Constraint (7) ensures that each task is assigned toonly one worker of the line Constraint (8) ensures thatall precedence relationships among tasks are satisfied Thisconstraint can also be written as follows


sum

119896isin119870

(119895 minus 1) sdot 119909ℎ119895119896+ stℎ+ 119905ℎ

le Ct sdot sum119895isin119869

sum

119896isin119870

(119895 minus 1) sdot 119909119894119895119896+ st119894

forall119894 isin 119868 forallℎ isin 119875 (119894)

(28)

Constraint (9) cycle time constraint ensures that each taskis completed before the end of the cycle time If tasks 119894 and ℎwhich have no precedence relationship between each otherare assigned to the same worker of the same station thenone of Constraints (10) and (11) becomes active (with aninteger value Ψ ge Ct) While the activation of Constraint(10) by using 119910

119894ℎ= 1 ensures st

ℎge st119894+ 119905119894 the activation

of Constraint (11) by using 119910119894ℎ= 0 ensures st

119894ge stℎ+ 119905ℎ

Constraints (12) and (13) are used to formulate the first goalConstraint (12) assigns proper value to variable 119897119897 Constraint


(13) is the goal programming constraint of the number ofmulti-manned stations with deviational variables Having 119889+

1

with the value zero ensures that the total number of stationsis 119871max or less The set of Constraints (14) and (15) modelsthe second goal Constraint (14) assigns proper value tovariable 119908

119895 Constraint (15) with deviational variables is the

goal constraint of the total number of workers used in theline While the value of 119889+

2is reduced to zero the number

of workers used in the line decreases towards the value119879119872maxThe last set havingConstraints (16)ndash(20) correspondsto the third goal Constraint (16) ensures that the valueof 119898119908 is equal to or greater than the number of workersused in each station Constraint (17) with an integer valueΩ ge 119899 ensures that if station 119895 has at least one assignedtask (ie it is opened) then 119906

119895= 1 Constraints (18) and

(19) are used to assign proper value to each 120572119895119896

variableConstraint (18) with an integer value Φ ge 119872max is activewhen the corresponding station is opened For station 119895note that Constraint (19) does not allow more than one 120572

119895119896

variable to have the value 1 When Constraints (18) and (19)with the third objective are examined together it can beseen that all 120572

119895119896variables of unopened station 119895 will have

the value zero Constraint (20) is the goal programmingconstraint of the smoothness index on the basis of number ofworkers at stations Note that the first term of this constraintcomputes the sum of squared deviations (from themaximumof number of workers at stations 119898119908) of the number ofworkers of stations While the value of 119889+

3is reduced to

zero the smoothness index value decreases towards zeroThis constraint is written based on the fact that minimizingthe value of a positive valued function also minimizes thevalue of the square root of this function Constraints (21)ndash(25) show the integer variables and Constraints (26) and(27) indicate that all the remaining variables are nonnegativecontinuous

33 Solving the ProposedModel In the proposedmodel thereare three goals as follows (in order of importance)

Minimize1198661= 119889

+

1(Highest priority)

Minimize1198662= 119889

+

2(Mid-level priority)

Minimize1198663= 119889

+

3(Lowest priority)

(29)

This model is solved by considering one goal at a timestarting with goal 1 and terminating with goal 3 In a solutionprocess it is ensured that a lower priority goal never degradesany higher priority objective function value For doing thisin a solver the current solution is added as a new constraintto problems with lower priorities For example consider 119889lowast

1

is the optimum objective value of the first problem Thenthe constraint 119889+

1= 119889

lowast

1is added to the second problem

After solving the second problem if the optimum objectivevalue is 119889lowast

2 then both constraints 119889+

1= 119889

lowast

1and 119889+

2= 119889

lowast

2are

added to the last problem After solving the last problem theoptimum solution of the goal programming model is found[18]

For all alternative optimal solutions to the highest-level objective the solution which optimizes the sec-ond level is found and so on That is deviations fromthe goals are minimized in order of priority Thus theresulting solution is called the lexicographical minimum[19]

4 Proposed Variable NeighborhoodSearch Heuristics

VNS was introduced by Mladenovic [20] It combineslocal search with systematic changes of neighborhoodfor escaping from local optimum in the descent [21]The basic characteristic of VNS is that it searchesover several neighborhood structures in the solutionspace Hence it reduces the solution space into smallersubspaces

The basic VNS procedure [22] is given in Algorithm 1 Ateach iteration of VNS search process there is an incumbentsolution which is a local optimum and VNS searches in solu-tions which are in a far neighborhood of the current incum-bent solution From the current neighborhood it randomlydetermines a solution (shaking phase) applying local searchto this solution If the resulting solution is better than the cur-rent incumbent solution then the search moves to this newsolution If not a farther neighborhood structure (after thefarthest one the nearest structure) is used to go on the searchprocess

Contrary to other metaheuristics based on local searchmethods VNS does not follow a trajectory but exploresincreasingly distant neighborhoods of the current incumbentsolution and jumps from this solution to a newone if and onlyif an improvement has been made [21] VNS heavily reliesupon the following observations [23]

(i) A local minimum with respect to one neighborhoodstructure is not necessarily a local minimum for ananother neighborhood structure

(ii) A global minimum is a local minimum with respectto all possible neighborhood structures

(iii) For many problems local minima with respect to oneor several neighborhoods are relatively close to eachother

For solving the multi-objective MALBP with VNS fol-lowing components of VNS have to be developed (1)solution coding and encoding schemes (2) the methodused for generating initial solution (3) neighborhood struc-tures used in shaking phase (4) local search procedureand (5) the method used to compare solutions (objectivefunction) Thus in the next subsections these compo-nents for solving multi-objective MALBP are described indetail

41 Representing Solutions (Coding) Solutions of the multi-objective MALBP are represented by using permutations oftasksrsquo priorities In other words for the problem with 119899 taskslet (119901

1198941

119901119894119899

119901119894119899

) be a permutation of the value 119899 Then


Initialization(i) Select the set of neighborhood structures119873

119896 119896 = 1 2 119896max that will be used in the search

(ii) Find an initial solution 119909(iii) Choose a stopping conditionMain step Repeat the following sequence until the stopping condition is met(1) Set 119896 larr 1(2) Until 119896 = 119896max repeat the following steps(a) Shaking Generate a point 1199091015840 at random from the 119896th neighborhood of 119909 (1199091015840 isin 119873

119896(119909))

(b) Local search Apply some local search method with 1199091015840 as initial solution denote 11990910158401015840 the so obtained local optimum(c)Move or not If this local optimum is better than the incumbent move there (119909 larr 119909

10158401015840) and continue the search with1198731(119896 larr 1) otherwise set 119896 larr 119896 + 1

Algorithm 1 The basic VNS procedure

Procedure - Build Solutionbegin(1) 119861119890119904119905119878119900119897 larr 119899119906119897119897(2) Create initial line configuration 119871119862119900119899119891(3) 119862119906119903119903119878119900119897 larr assign tasks to line with configuration 119871119862119900119899119891(4) if (119862119906119903119903119878119900119897 is better than 119861119890119904119905119878119900119897) then

(41) 119861119890119904119905119878119900119897 larr 119862119906119903119903119878119900119897(42) 119861119886119904119890119862119900119899119891 larr get line configuration of 119862119906119903119903119878119900119897(43) 119871119877 larr 119897119894119904119905 of 119861119890119904119905119878119900119897rsquos stations having more than one used workerend if

(5) if (119871119877 = ) then(51) return 119861119890119904119905119878119900119897end if

(6) 119904 larr first element in 119871119877(7) Delete the first element in 119871119877(8) 119871119862119900119899119891 larr 119861119886119904119890119862119900119899119891(9) Decrease the number of workers at station 119904 of 119871119862119900119899119891 by 1(10) Go to Step (3)

end

Algorithm 2 Procedure of solution construction process

the priority values for tasks 1198941 1198942 119894

119899are 1199011198941

119901119894119899

119901119894119899

respectively These priority values are used to determinewhich one of available tasks is selected for assignment ateach step This approach which is called priority basedcoding is one of commonly used structures for othertypes of assembly line balancing problems in the literature[24 25]

42 Building Solutions (Encoding) For building a solutionbased on priority values a restricted version of the construc-tive heuristic proposed byKellegoz andToklu [14] is usedTheprocedure of building solution used in this study is given inAlgorithm 2

Second step in the procedure generates an initial lineconfiguration and sets it to the current configuration 119871119862119900119899119891A line configuration is a vector of the same length as themaximumnumber of stations and the element 119894 in this vector

determines how many workers are available at the station119894 of the line The initial line configuration has the samenumber of workers at stations with the best one of solutionsin the set IS If the solution has no worker at a station thenthe corresponding element in the line configuration is setto 119872max For comparing solutions the method describedin the next subsection is used The set IS includes 119872maxsolutions and the solution in 119904th element is created by thefollowing way First a line configuration having the numberof 119904 workers in each one of 119899 stations is created As indicatedearlier 119899 is the valid upper bound value for the numberof stations Then the method described in the followingparagraph is used to assign tasks to stations In this mannerit is aimed to start the solution building process from a goodline configuration

Assigning tasks to a line with a known line configurationis carried out in Steps (2) and (3) First an empty stationwith the number of workers which is determined from the


line configuration is opened At each assignment step a taskhaving the following properties (checked in the same order)is selected (1) it is not assigned yet (2) all its predecessorsare already assigned (3) it can be completed before theend of cycle time (4) it can be started earlier than othertasks having previous properties and (5) it has greatestpriority value within tasks having previous properties Theselected task is assigned to the worker who can start theprocessing of this task earlier If more than one worker hasthe same minimum start time tie is broken by selecting theworker having the smallest index If there are unassignedtasks and none of these tasks cannot be assigned to thecurrent station then a new empty station having the num-ber of workers determined from the line configuration isopened

After getting the current solution119862119906119903119903119878119900119897 it is comparedwith the best solution 119861119890119904119905119878119900119897 found in the solution buildingprocess The method which is used to compare solutionsis presented in the next subsection If the current solutionis better than the best solution then (1) the best solutionis updated (2) the line configuration 119861119886119904119890119862119900119899119891 is createdand (3) the list 119871119877 is determined The line configuration119861119886119904119890119862119900119899119891 is determined by using the newly found bestsolution This line configuration has 119899 stations and includesthe same number of workers at each opened station of thebest solution (active part of the configuration) and 119872maxworkers at each one of other stations which is not openedin the best solution (passive part of the configuration)The list 119871119877 which includes best solutionrsquos stations whichhave more than one worker (multi-manned) is used insolution restriction process This list includes stations basedon minimum worker load-related strategy proposed by Kel-legoz and Toklu [14] According to this strategy stations in119871119877 are decreasingly ordered according to their minimumworkerrsquos load

As is seen from the solution building procedure presentedin Algorithm 2 if the list 119871119877 has no element then thesolution building process is ended If not stopped then theconfiguration restriction process is performed by using Steps(6)ndash(9) First the first element in the list 119871119877 is determined (119904)and then it is deleted from the list 119871119877 After this process thecurrent line configuration 119871119862119900119899119891 is set to the configuration119861119886119904119890119862119900119899119891 Then the procedure decreases the number ofworkers at station 119904 of 119871119862119900119899119891 by 1 By this way it is aimedto reduce the number of workers used in the line whichis the secondary objective of the considered MALBP Notethat the main function for searching the best solution basedon all objectives is performed by VNS based heuristicsproposed

For illustrating the solution construction process con-sider the priority permutation (5 12 3 10 9 1 6 2 7 11 48) for the problem instance with cycle time 10 119872max = 3and precedence diagram given in Figure 1 All three targetvalues are considered zero In this case three different lineconfigurations (a row vector with 119899 = 12 elements) areconsidered for determining initial line configuration (1) eachelement is set to the value 3 (ie 119872max) (2) each elementis set to the value 2 (ie 119872max minus 1) and (3) each elementis set to the value 1 (ie 119872max minus 2) In order to illustrate

the task assignment process the assignment process for thefirst line configuration is summarized in Table 1 Resultingsolutions for the determination process of the initial lineconfiguration are presented in Figure 3 As is seen fromthis figure resulting solutions have 3 4 and 5 stationsrespectively Also they have 7 6 and 5 workers and 4 2and 0 worker based smoothness indexes (without squareroot) respectively When the solution comparison methodgiven in the next subsection is considered it is concludedthat the best of these three solutions is the one given inFigure 3(a) This solution has 3 3 and 1 workers for stations1 2 and 3 respectively Hence the initial line configurationis determined as (3 3 1 3 3 3 3 3 3 3 3 3)

After creating the current solution and updating the bestsolution the base line configuration119861119886119904119890119862119900119899119891 is determinedas (3 3 1 3 3 3 3 3 3 3 3 3) and the list 119871119877 is formedas (Station 1 Station 2) Note that Station 3 is not added tothis list since it has only one used worker After executingSteps (6)ndash(9) of the procedure the configuration 119871119862119900119899119891 isdetermined as (2 3 1 3 3 3 3 3 3 3 3 3) and the list 119871119877 isupdated as (Station 2)The assignment process for this119871119862119900119899119891configuration and the resulting solution are given in Table 2and Figure 4(a) respectively In this case the current solutionhas the same number of stations with the best solution andless number of workers than the best solution Thereforeaccording to the solution comparison method given in thenext subsection this newly generated solution is better thanthe best solution Thus after updating the best solution thebase line configuration 119861119886119904119890119862119900119899119891 and the list 119871119877 are set to(2 3 1 3 3 3 3 3 3 3 3 3) and (Station 1 Station 2)respectively After executing Steps (6)ndash(9) of the procedurethe configuration 119871119862119900119899119891 is determined as (1 3 1 3 3 3 3 3 33 3 3) and the list 119871119877 is updated as (Station 2)The resultingsolution for this configuration is given in Figure 4(b) As isseen from this figure the resulting solution is again betterthan the best one Thus the best solution is updated Thenthe base line configuration 119861119886119904119890119862119900119899119891 and the list 119871119877 are setto (1 3 1 3 3 3 3 3 3 3 3 3) and (Station 2) respectivelyAfter executing Steps (6)ndash(9) of the procedure again theconfiguration 119871119862119900119899119891 is determined as (1 2 1 3 3 3 3 3 3 33 3) and the list 119871119877 is updated as The resulting solutionfor this configuration is given in Figure 4(c) As is seen fromthis figure the resulting solution is not better than the bestone The process is stopped since there is no element in thecurrent list 119871119877

43 Comparing Solutions A simple comparison method isnot applicable for comparing solutions since there is morethan one objective in the considered problem Comparisonmethod has to include all objectives with the same priorityorder Thus for comparing two different solutions (Sol

1and

Sol2) the algorithmwhich has the flowchart given in Figure 5

is used Notations 1199041and 1199042represent the number of stations

opened in Sol1and Sol

2 respectively Also notations 119898

1

and 1198982represent the total number of workers used in Sol

1

and Sol2 respectively and notations 119899119904

1and 119899119904

2represent

worker based smoothness index of Sol1and Sol

2 respectively

As is seen from Figure 5 if both solutions have the total


Table 1 Task assignment process for the line configuration (3 3 3 3 3 3 3 3 3 3 3 3)

Step Station index(man count)

Task with nopredecessor(priorities)

Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash

3 3 (3) 4 (10) 5 (9) 6(mdash) 4 6 2 Task 6 cannot be completed before the end of cycle

time

4 3 (3) 5 (9) 6 (mdash) 5 6 3 Task 6 cannot be completed before the end of cycletime

5 3 (3) 6 (mdash) 8 (mdash) 3 8 1 Tasks 6 and 8 cannot be completed before the end ofcycle time

6 6 (mdash) 7 (mdash) 8 (mdash) mdash mdash mdash(i) Tasks 6 7 and 8 cannot be completed before the end

of cycle time(ii) New station is opened

7

2 (3)

6 (1) 7 (6) 8 (2) 7 0 1 mdash8 6 (1) 8 (2) 8 0 2 mdash9 6 (1) 10 (mdash) 6 0 3 Task 10 cannot be started at time 010 9 (7) 10 (mdash) 9 4 2 Task 10 cannot be started at time 4

11 10 (11) 11 (mdash) 10 8 1 Task 11 cannot be completed before the end of cycletime

12 11 (mdash) mdash mdash mdash(i) Task 11 cannot be completed before the end of cycle

time(ii) New station is opened

13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)

5 (9) 6 (1)7 (6) 5 0 1 mdash7 6 (1) 7 (6) 8 (mdash) 7 0 2 Task 8 cannot be started at time 08 6 (1) 8 (mdash) 6 0 3 Task 8 cannot be started at time 09 8 (2) 9 (mdash) 8 3 1 Task 9 cannot be started at time 310 9 (7) 10 (mdash) 9 4 3 Task 10 cannot be started at time 4




13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10

Station 1 Station 2 Station 35 6

4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3

(a) Solution for the line configuration (3 3 3 3 3 3 3 3 3 3 3 3)

11

Station 4

121 2 3 5 8 6

9

Station 1 Station 2 Station 34 7 10

86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1

(b) Solution for the line configuration (2 2 2 2 2 2 2 2 2 2 2 2)

6Station 4

1 2 3Station 1 Station 2 Station 3 Station 5

9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1

(c) Solution for the line configuration (1 1 1 1 1 1 1 1 1 1 1 1)

Figure 3 Resulting solutions for the determination of the initial line configuration

1 2 3

11

5 8 10

127

6 9Station 1 Station 2 Station 3

4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101

Station 3Station 1 Station 2

2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2


Figure 4 Illustration of the solution building process

number of opened stations less than or equal to the targetvalue 119871max or both solutions have the same total number ofopened stations then it is decided that there is no differencebetween two solutions according to the total number ofopened stations The same approach is applied for testing

solutions based on both the total number of used workersand worker based smoothness indexes Note that the targetvalue for smoothness index is always zero and the check iscarried out in the same order with objectives If there is nodifference between two solutions based on three criteria then


Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2

Sol1 is better Sol2 is better

((s1 le Lmax)and (s2 le Lmax))or

((m1 le TMmax)and (m2 le TMmax)) or

or

Figure 5 Flowchart of procedure comparing two solutions

Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8

Figure 6 Illustration for producing a neighbor from API neighborhood

it is decided that there is no difference between two solutionsIf there is a difference based on one of three criteria then itis concluded that the solution having less value based on thiscriterion is better

44 Neighborhood Structures For generating neighbors thefollowing two methods are used in this paper

(i) adjacent pairwise interchange (API)(ii) subset randomization (SR)

In a permutation two adjacent digits are interchanged togenerate an API neighbor This process is illustrated inFigure 6There are 119899 minus 1 neighbors in API neighborhood of apermutation of the value 119899 [26]

For generating a random neighbor by using SR mech-anism a subset of digits is first selected randomly Then

unselected digits are copied to the same positions of theneighbor Last selected digits are randomly copied to emptypositions of the neighbor For SR structure it is obvious thatthere ismore than one neighborhood based on the number ofdigits selected randomly For generating a neighbor differentfrom its origin at least two digits have to be selected Let 119896and SR119896 be the number of randomly selected digits and theneighborhood structure with parameter 119896 respectively For apermutation of the value 119899 the interval of the parameter 119896is 2 le 119896 le 119899 The SR2 (119896 = 2) method generates neighborsfrom general pairwise interchange neighborhood (PI) withprobability 05 The remaining permutations are the samewith their origins Therefore for the origin solution 119909 SR2 =PIcup119909 is valid On the other hand the SR119899method generatesrandom permutations of 119899 An illustrative example for themethod SR3 is presented in Figure 7


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8

Random sequence of selected digits 9 4 12

lowast lowast lowast

Figure 7 Illustration for producing neighbor from SR3 neighborhood

API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2

Figure 8 Relationship between neighborhoods

If only 119896 minus 1 digits of a neighbor which is generated fromSR119896 has different values from its origin then this neighboris also an element of SR119896minus1 In this case this neighbor hasthe same value with its origin at only one of selected 119896 digitsLikewise if a neighbor has the same values with its origin attwo of selected 119896 digits then this neighbor is also an elementof both SR119896minus1 and SR119896minus2Thus the relationship SR119896minus1 sub SR119896 isvalidThis relationship is illustrated in Figure 8 If a neighbor119909

1015840isin SR119896 has a different value from its origin119909 at each one of 119896

digits then 1199091015840 notin SR119896minus1 is valid In this case 1199091015840 isin (SR119896minusSR119896minus1)is also valid As a result in neighborhood generation processensuring that each one of 119896 positions has different valuefrom its origin also ensures that the generated neighbor is anelement of SR119896 minus SR119896minus1

45 Variations of VNS for Multi-Objective PMALBP Forsolving large size problem instances getting a good ini-tial solution is important For these instances local searchalgorithms often require substantial amounts of runningtime Hence Hansen and Mladenovic [27] introduced a newversion of VNS called Reduced VNS (RVNS) For decreasingcomputation time RVNS heuristic omits the local search step

of the basic VNS algorithm In this paper RVNS heuristic isalso considered

Hansen et al [22] declared that decomposition mightbe worthwhile in heuristics for very large problem instancessince the performance of VNS depends in general on thelocal search subroutine used Based on this approach theyproposed a new VNS based algorithm called Variable Neigh-borhood Decomposition Search (VNDS) VNDS decom-poses the problem space by fixing all but 119896 attributes in thelocal search phase for a given solution For the consideredproblem in this paper the solution attributes are priorityvalues of tasks Thus priority values of tasks which are notchanged in the shaking phase are determined and the samepriority values are fixed in the local search phase Due tothis fixation the local search phase is carried out by usingAPI structure on a restricted part of the solution (restrictedAPI) For the SR3 neighbor solution given in Figure 7 allbut priority values of tasks 2 5 and 11 are fixed Thus localsearch phase is applied based on only these three digits Therestricted API neighborhood for this SR3 solution is given inFigure 9

In brief we developed three different heuristics basedon VNS variations for solving the multi-objective MALBP


Table 3 Properties of VNS variations

Property VNS RVNS VNDS

Initial solution(i) Generate random

(ii) Apply local search with APIstructure

(i) Generate random(ii) Apply local search with API

structure


structure

Shaking phase

SR2 minus (API cup 119909)SR3 minus SR2

SR119870maxminus SR119870maxminus1





Local search phase API mdash Restricted APIValue of the parameter 119870max max2 lceil05 sdot 119899rceil max2 lceil07 sdot 119899rceil max2 lceil08 sdot 119899rceil

5 9 3 10 4 1 6 2 7 11 12 8

Unfixed digits 9 4 12

Restricted API neighbor 1 5 4 3 10 9 1 6 2 7 11 12 8


Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor

Figure 9 Generating neighbors in the local search phase of VNDS

These heuristics are summarized in Table 3 For initiatingproposed algorithms with a good incumbent solution localsearch with API structure is applied to a solution which isgenerated randomly

As is seen fromTable 3 all methods use only one parame-ter119870max which is the top level of SRneighborhood structuresused in the heuristicrsquos shaking phase For identifying thevalue of this parameter for each method we carried outpreliminarily experiment by using different size preliminarytest instances The 119870max parameter value is set to each oneof max2 lceil119903 sdot 119899rceil values where 119899 is the number of tasks inthe problem instance and 119903 is a control parameter getting avalue from the domain 01 02 10 By using each oneof 119903 values ten different119870max values are determined based onthe size of considered instance For each parameter value foran instance each one of VNS RVNS and VNDS is run fivetimes After examining the results of each method a superiorlevel of the 119870max parameter is determined These selectedlevels are also presented in Table 3

5 Computational Study

In order to evaluate the proposed MIP and heuristicsexperiments were conducted by using benchmark instances

All heuristics were coded in C language and compiledin the programming software Microsoft Visual C 2010Express Edition MIP models of instances were solvedby using the solver IBM ILOG CPLEX (version 12510)All experiments were run on a PC with i3 predecessor6GB RAM and 64 bit Microsoft Windows 7 operatingsystem

Benchmark instances were generated by the followingway Test problems (task times and precedence relationships)with cycle times were given from the literature Three smallsize data sets are P11 P21 and P30 They have 11 21 and30 tasks and are obtained from the papers by Jackson[28] Mitchell [29] and Sawyer [30] respectively Moreoversome medium and large size data sets were also used inexperiments These data sets are P45 [31] P70 [32] P83 andP111 [33] and P148 [34] Two different levels 2 and 4 wereconsidered for the parameter 119872max (admitted maximumnumber of workers at a station) For each data set threedifferent values for the cycle time parameter were obtainedfrom the literature In the value determination process fortarget value parameters of instances it was aimed to comparethe performance of proposedmethods based on the followingmanners

(i) minimizing the number of stations the number ofworkers and smoothness index in the same order

(ii) minimizing the number of workers and smoothnessindex in the same order due to given target value ofthe number of stations

(iii) minimizing smoothness index due to given targetvalues of the number of stations and workers

The method which was used to determine target values ispresented in the next subsections For an instance with 119899tasks each heuristic was run for 3 sdot 119899 CPU seconds Alsofor every experimental instance each heuristic was applied5 times Each execution of MIP models has a time limit of3600CPU seconds

51 Minimizing the Number of Stations the Number ofWorkers and Smoothness Index in the Same Order A targetvalue for a variable means that the decision maker desires


Table 4 Results for small and medium size instances for minimizing the number of stations the number of workers and smoothness indexin the same order

Problem Ct 119872max VNS result RVNS result VNDS result MIP result MIP CPU

P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45

57 2 6 10 2 6 10 2 6 10 2 6int mdash mdash 360024 mdash mdash4 5 10 7 5 10 7 5 10 7 5 10int mdash 1214 347907 mdash

110 2 3 6 0 3 6 0 3 6 0 3 6int mdash 31587 328437 mdash4 3 6 0 3 6 0 3 6 0 3 6int mdash 21503 338543 mdash


P70

176 2 12 22 2 12 22 2 12 216 24 No mdash mdash 360027 mdash mdash4 9 228 62 9 23 56 9 224 66 70int mdash mdash 360136 mdash mdash

364 2 6 10 2 6 10 2 6 10 2 No mdash mdash 360027 mdash mdash4 4 108 28 4 11 1 4 11 1 No mdash mdash 360032 mdash mdash

468 2 4 8 0 4 8 0 4 8 0 No mdash mdash 360025 mdash mdash4 3 9 0 3 88 02 3 9 0 3 9int mdash 116864 243265 mdash

aMean number of stationsbMean number of workerscMean smoothness index (without square root)intInteger solution was found within a given CPU time limitNo no solution was found within a given CPU time limitmdash phase was not performed since previous phases had consumed the given CPU time

this variable to have a value not greater than the targetvalue In this case for minimization of a variable there isno difference between values equal to or smaller than thetarget value From this point on for minimizing the numberof stations the number of workers and smoothness index inthe same order all target values are set to zero Results forall heuristics and MIP models are given together in Table 4Results of heuristics in this table are presented based onmean values Note that results of MIP models are also equalto deviation values since all target values are equal to zeroFor each instance heuristic results are compared with each

other based on the comparison method given in Figure 5As is seen from Table 4 there is no difference between allmethods for instances of P11 and P21 since all heuristicsfound optimal solutions For one instance (with Ct = 41 and119872max = 4) in the problem group P30 the MIP formulationwas not able to find the optimal solution Also the remaininginstances (P45 and P70) could not be solved by using theMIP formulation For example for the instance P45 withCt = 57 and119872max = 2 the MIP formulation was only ableto find an integer solution for goal 1 within the given CPUtime limit Hence the remaining phases for the remaining


Table 5 Results for large size instances for minimizing the number of stations the number of workers and smoothness index in the sameorder

Problem Ct 119872max VNS result RVNS result VNDS result

P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02

aMean number of stationsbMean number of workerscMean smoothness index (without square root)

goals were not carried out (sign ldquominusrdquo) For the instance P45with Ct = 57 and 119872max = 4 the MIP formulation wasable to find the optimum solution for goal 1 within 1214 CPUseconds Therefore phase 2 was started In phase 2 only aninteger solution could be found within the given CPU timelimit (1214 + 347907 = 360047CPU seconds) Hence thelast phase was not carried out The overall evaluation of theresults found in this group is as follows All heuristics wereable find optimal solutions for 14 instances whose optimalsolutions were found by the MIP formulation There are 8instances for which heuristics showed different performanceFor 2 of them (25) VNS was able to find better solutionsthan at least one of other heuristics RVNS and VNDSfound better solutions for 4 (50) and 3 (375) instancesrespectively

Results for large size instances are presented in Table 5There are 18 instances in this group All heuristics foundthe same solution for each of 7 instances For one of theremaining 11 instances VNS was able find better solutionthan at least one of other two heuristics (9) On the otherhand RVNS found better solutions for 10 instances (91)Lastly VNDS has better performance than at least one ofother heuristics for 5 instances (46) For these 5 instancesnote that RVNS found the same solutions with VNDSHence VNDS outperformed only VNS for this group ofinstances

52 Minimizing the Number ofWorkers and Smoothness Indexin the Same Order due to Given Target Value of the Number

of Stations As stated earlier there is no difference betweenvalueswhich are in the desired interval determined by a targetvalue for a variable For example if the target value for thenumber of stations is set 4 then values 2 3 and 4 are notsuperior against each other On the other hand in this casethese values are superior against values greater than 4 and thevalue 5 is superior against values greater than 5 and so forthFor comparing the performance of heuristics according tothe minimization of the number of workers and smoothnessindex in the same order a target value of the number ofstations for each instance was determined Care has beentaken to ensure that all heuristics could reach target valuesHence for an instance a target value of the number stationswas set to the maximum of 15 values found in Section 51Note that each of three heuristics was applied 5 times for aninstance Due to goals of minimizing the number of workersand smoothness index target values for these goals were setto zero

The results of the computational experiments for smallandmedium size instances are presented in Table 6 As is seenfrom this table instances having more than 30 tasks couldnot be solved by the MIP formulation For instances in setsP11 P21 and P45 there is no difference between heuristicsaccording to their results For the problem group P30 VNSand VNDS produced better solutions for one instance andtwo instances respectively A similar observation may bedone for the problem group P70 since VNDS emerges asthe best method for this group Shortly for 6 medium size


Table 6 For a given target value of the number of stations results for small andmedium size instances forminimizing the number of workersand smoothness index in the same order

Problem Ct 119872maxTarget VNS result RVNS result VNDS result MIP

stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45

57 2 6 6 10 2 6 10 2 6 10 2 0 6 10int mdash 25377 334647 mdash4 5 5 10 7 5 10 7 5 10 7 0 5 10int mdash 7391 352656 mdash



P70

176 2 12 12 22 2 12 22 2 12 22 2 No No mdash mdash 360029 mdash mdash4 9 9 228 108 9 23 6 9 228 10 0 9 22int mdash 49719 310408 mdash




instances for which heuristics showed different performanceVNDS produced better solutions than at least one of othertwo heuristics

For large size instances heuristicsrsquo results are given inTable 7 For 7 of 18 instances in this group all three heuristicsobtained the same results For 9 of the remaining 11 instances(82) RVNS heuristic outperformed at least one of other twoheuristics

53 Minimizing Smoothness Index due to Given Target Valuesof the Number of Stations and Workers The same method

with the one for generating target values for the num-ber of stations was used to obtain target values for thenumber of workers The target value for the number ofworkers for an instance was set to the maximum of 15values obtained by heuristics for the number of workers inSection 51

The results for small and medium size instances arepresented in Table 8The following observations can bemadefrom this table By using the MIP formulation instanceswith target values may be optimally solved more easily sincemore instances are solved optimally Also in this groupthere are only 4 instances for which heuristics showed


Table 7 For a given target value of the number of stations results for large size instances for minimizing the number of workers andsmoothness index in the same order

Problem Ct 119872maxTarget

stationsVNS result RVNS result VNDS result

P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04


different performance For 3 of these 4 instances (75) RVNSobtained better solutions than other two heuristics For theremaining instance VNDS was superior against other twoheuristics

Similar observations can be made for large size instanceswhose results are given in Table 9 As is seen from thistable VNS was not able to outperform other heuristicseven for a single instance On the other hand for 10instances at least one of other two heuristics outperformedVNS For these 10 instances RVNS obtained the same(for 3 instances) or better results (for 5 instances) thanVNDS For the remaining 2 instances VNDS outperformedRVNS

6 Conclusions and Future Works

In this research multi-objective MALBP is considered Firsta MIP model based on goal programming approach isdeveloped Then after defining the problem three differ-ent variations of VNS algorithm are proposed to solvemulti-objective MALBP heuristically Experiment study isperformed based on benchmark instances ranging in sizefrom small to large for different target values of problemparameters

Based on well-known goal programming approach thisstudy is the first one attempting to define the line balanc-ing problem with multi-manned stations and to proposesolution methods to solve it After analyzing the results ofexperimental study it has been seen that only small sizeinstances could be solved optimally On the other hand if theinstance has target values attainable then its optimal solutionmay be found more easily For small size instances it hasbeen seen that there is no significant difference between pro-posed VNS variations However for medium and especiallylarge size instances RVNS mostly outperforms other twoheuristics

A possible future research direction is to develop math-ematical programming models for other goal such as taskbased smoothness index and cost Also to the best ofauthorsrsquo knowledge there is no study on the MALBPwhich has layout different from straight layout such asU line and parallel line Therefore there is a need todevelop solution algorithms for MALBP having different linelayouts

Competing Interests

The author declares that there are no competing interests


Table 8 For given target values of the number of stations andworkers results for small andmedium size instances forminimizing smoothnessindex


stations workers 119889

+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45

57 2 6 10 6 10 2 6 10 2 6 10 2 0 No 6 No mdash 25324 334686 mdash4 5 10 5 10 7 5 10 7 5 10 7 0 0 510No 7368 10618 342032

110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70

176 2 12 22 12 22 2 12 22 2 12 22 2 No mdash No mdash mdash 360027 mdash mdash4 9 23 9 23 6 9 23 52 9 23 94 0 0 9 23 No 4966 12128 298248

364 2 6 10 6 10 2 6 10 2 6 10 2 No mdash No mdash mdash 360027 mdash mdash4 4 12 4 12 0 4 12 0 4 12 0 0 0 4 12 0 39805 6635 7541




Table 9 For given target values of the number of stations and workers results for large size instances for minimizing smoothness index

Problem Ct 119872maxTarget VNS result RVNS result VNDS result

stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References

[1] E Gurevsky O Hazır O Battaıa and A Dolgui ldquoRobustbalancing of straight assembly lines with interval task timesrdquoJournal of the Operational Research Society vol 64 no 11 pp1607ndash1613 2013

[2] H Gokcen and E Erel ldquoBinary integer formulation for mixed-model assembly line balancing problemrdquo Computers and Indus-trial Engineering vol 34 no 2ndash4 pp 451ndash461 1998

[3] P Sivasankaran and P Shahabudeen ldquoLiterature review ofassembly line balancing problemsrdquo International Journal ofAdvancedManufacturing Technology vol 73 no 9ndash12 pp 1665ndash1694 2014

[4] A Lusa ldquoA survey of the literature on the multiple or parallelassembly line balancing problemrdquo European Journal of Indus-trial Engineering vol 2 no 1 pp 50ndash72 2008

[5] C Becker and A Scholl ldquoA survey on problems and methodsin generalized assembly line balancingrdquo European Journal ofOperational Research vol 168 no 3 pp 694ndash715 2006

[6] E Erel and S C Sarin ldquoA survey of the assembly line balancingproceduresrdquo Production Planning and Control vol 9 no 5 pp414ndash434 1998

[7] C Becker andA Scholl ldquoBalancing assembly lines with variableparallel workplaces problem definition and effective solutionprocedurerdquo European Journal of Operational Research vol 199no 2 pp 359ndash374 2009

[8] T Kellegoz and B Toklu ldquoAn efficient branch and bound algo-rithm for assembly line balancing problems with parallel multi-mannedworkstationsrdquoComputers andOperations Research vol39 no 12 pp 3344ndash3360 2012

[9] S G Dimitriadis ldquoAssembly line balancing and group workinga heuristic procedure for workersrsquo groups operating on the same

product and workstationrdquo Computers and Operations Researchvol 33 no 9 pp 2757ndash2774 2006

[10] E Cevikcan M B Durmusoglu and M E Unal ldquoA team-oriented design methodology for mixed model assembly sys-temsrdquo Computers and Industrial Engineering vol 56 no 2 pp576ndash599 2009

[11] P Fattahi A Roshani and A Roshani ldquoA mathematical modeland ant colony algorithm for multi-manned assembly linebalancing problemrdquo International Journal of Advanced Manu-facturing Technology vol 53 no 1ndash4 pp 363ndash378 2011

[12] V Yazdanparast and H Hajihosseini ldquoMulti-manned produc-tion lines with labor concentrationrdquo Australian Journal of Basicand Applied Sciences vol 5 no 6 pp 839ndash846 2011

[13] A Kazemi andA Sedighi ldquoA cost-orientedmodel for balancingmixed-model assembly lines withmulti-manned workstationsrdquoInternational Journal of Services and Operations Managementvol 16 no 3 pp 289ndash309 2013

[14] T Kellegoz and B Toklu ldquoA priority rule-based constructiveheuristic and an improvement method for balancing assemblylines with parallel multi-manned workstationsrdquo InternationalJournal of Production Research vol 53 no 3 pp 736ndash756 2015

[15] A Scholl Balancing and Sequencing of Assembly Lines Physica-Verlag Heidelberg Germany 1999

[16] A Charnes and W W Cooper Management Model and Indus-trial Application of Linear Programming John Wiley amp SonsNew York NY USA 1961

[17] W L Winston Operations Research Applications and Algo-rithms Duxbury Press Belmont Calif USA 3rd edition 1994

[18] H A TahaOperations Research An Introduction Pearson NewJersey NJ USA 8th edition 2007


[19] Y Wilamowsky S Epstein and B Dickman ldquoOptimizationin multiple-objective linear programming problems with pre-emptive prioritiesrdquo Journal of the Operational Research Societyvol 41 no 4 pp 351ndash356 1990

[20] N Mladenovic ldquoA variable neighborhood algorithmmdasha newmetaheuristic for combinatorial optimizationrdquo in Abstractsof Papers Presented at Optimization Days p 112 MontrealCanada 1995

[21] P Hansen and N Mladenovic ldquoVariable neighborhood searchprinciples and applicationsrdquo European Journal of OperationalResearch vol 130 no 3 pp 449ndash467 2001

[22] P Hansen N Mladenovic and D Perez-Britos ldquoVariableneighborhood decomposition searchrdquo Journal of Heuristics vol7 no 4 pp 335ndash350 2001

[23] P Hansen N Mladenovic and J A M Perez ldquoVariableneighbourhood search methods and applicationsrdquo 4OR vol 6no 4 pp 319ndash360 2008

[24] R K Hwang H Katayama and M Gen ldquoU-shaped assemblyline balancing problem with genetic algorithmrdquo InternationalJournal of Production Research vol 46 no 16 pp 4637ndash46492008

[25] U Ozcan and B Toklu ldquoMultiple-criteria decision-making intwo-sided assembly line balancing a goal programming anda fuzzy goal programming modelsrdquo Computers amp OperationsResearch vol 36 no 6 pp 1955ndash1965 2009

[26] M L Pinedo Scheduling Theory Algorithms and SystemsPrentice Hall New Jersey NJ USA 2008

[27] P Hansen and N Mladenovic ldquoAn Introduction to variableneighborhood searchrdquo in Metaheuristics Advances and Trendsin Local Search Paradigms for Optimization S Voss S MartelloI H Osman and C Roucairol Eds pp 433ndash458 KluwerAcademic Dordrecht The Netherlands 1998

[28] J R Jackson ldquoA computing procedure for a line balancingproblemrdquoManagement Science vol 2 no 3 pp 261ndash271 1956

[29] J Mitchell ldquoA computational procedure for balancing zonedassembly linesrdquo Research Report 6-94801-1-R3 WestinghouseResearch Laboratories Pittsburgh Pa USA 1957

[30] J F H Sawyer Line Balancing Machinery and Allied ProductsInstitute Washington DC USA 1970

[31] M D Kilbridge and L Wester ldquoA heuristic method of assemblyline balancingrdquo Journal of Industrial Engineering vol 12 pp292ndash298 1961

[32] F M Tonge A Heuristic Program of Assembly Line BalancingPrentice Hall Englewood Cliffs NJ USA 1961

[33] A L Arcus An analysis of a computer method of sequencingassembly line operations [PhD dissertation] University ofCalifornia 1963

[34] J J Bartholdi ldquoBalancing twondashsided assembly lines a casestudyrdquo International Journal of Production Research vol 31 no10 pp 2447ndash2461 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


1 3 7 11

2 6 9

4

8

10

5

12

6 2

2

3

4

8

4

6

2

3

3

2

Figure 1 Precedence diagram for illustrative problem

Station 1 Station 2 Station 3 Station 4 Station 57 10 6 9 11 121 2 3 4 5 8

(a) Simple assembly line

1 2 3

4

5

Station 2 Station 39

10 11 12

7

8

6

Station 1

W1

W1 W1W2

W3

(b) Multi-manned assembly line

Figure 2 Optimal solutions of simple and multi-manned assembly line balancing problems

cycle time Ct = 10 and precedence diagram given in Figure 1By using the sum of task times 119905sum = 45 the lower boundvalue for the total number of stations for simple assemblyline [15] and also for the total number of workers for multi-manned assembly line [7 8] can be calculated as follows

LB1= lceil

119905sumCtrceil = lceil

45

10

rceil = 5 (1)

For simple assembly line balancing problemwithminimizingthe number of stations (SALBP1) the optimum solution isgiven in Figure 2(a) As is seen from this figure at least5 stations are required to produce the goods with desiredproperties There are 5 workpieces (and workers) in theline since there is a workpiece (a worker) at each one ofthese stations For multi-manned assembly line balancingproblem (MALBP) with minimizing the number of stationsthe optimum solution is presented in Figure 2(b) In MALBPdetermining a station for each task is necessary but notsufficient To address a MALBP solution a worker for eachtask and a sequence of tasks for each worker must also beobtained As is seen from Figure 2 the multi-manned linerequires less number of stations than the simple line Hencein multi-manned lines same products may be produced by

using less layout space work-in-process flow time and soforth

In MALBP each task is allowed to be performed by onlyone worker Also any worker performs at most one task at atime To the best of authorsrsquo knowledge the first study aboutMALBPwas done byDimitriadis [9] He defined the problemand proposed a heuristic method for solving it Then Beckeramp Scholl [7] developed a mixed integer programming (MIP)model and a branch amp bound (BB) method for optimallysolving MALBP with mounting places In the same studythey also described how the proposed BB is used as a heuristicfor large size problem instances For solving themixed-modelMALBP of real life tractor assembly line Cevikcan et al [10]proposed a heuristic method executed phase by phase in fivesteps The first three steps are devoted to balance the lineand the remaining two steps are used for sequencing modelsand transferring workers in daily operations For solvingMALBP with minimizing both number of workers andmulti-manned stations in the same order Fattahi et al [11]proposed a MIP model and a heuristic method based on antcolony optimization In their comparison study they showedtheir heuristicrsquos superiority over the heuristic by Dimitriadis[9] For solving MALBP with task times dependent on thenumber of workers at stations Yazdanparast andHajihosseini
















SIst = radic119898

sum

119895=1

(119879119878max minus 119879119878119895)2

(2)




1le119895le119898119879119878119895


SIwr = radic119898

sum

119895=1

(119879119882max minus 119879119882119895)2

(3)








31 Notations

Parameters




Decision Variables


119909119894119895119896=


0 otherwise

119910119894ℎ=


0 otherwise

(4)



119898119908 = max119895119908119895

120572119895119896

=


0 otherwise

119906119895=


0 otherwise

(5)


119889

+

1 119889

minus



119889

+

2 119889

minus



119889

+

3 119889

minus






lexmin 119889+1 119889

+

2 119889

+

3 (6)



sum

119895isin119869

sum

119896isin119870

119909119894119895119896= 1 forall119894 isin 119868 (7)


sum

119896isin119870


ℎ+ 119905ℎle st119894


(8)



ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot (1 minus 119910

119894ℎ) ge st119894+ 119905119894


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(10)

st119894+ Ψ sdot (1 minus 119909

ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot 119910

119894ℎge stℎ+ 119905ℎ


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(11)


sum

119895isin119869

sum

119896isin119870


119897119897 + 119889

minus

1minus 119889

+

1= 119871max (13)


sum

119896isin119870


sum

119895isin119869

119908119895+ 119889

minus

2minus 119889

+

2= 119879119872max (15)


119898119908 ge 119908119895 forall119895 isin 119869 (16)


119894isin119868

sum

119896isin119870

119909119894119895119896


(17)

Φ sdot (1 minus 119906119895) + sum

119896isin119896


sum

119896isin119870

120572119895119896le 1 forall119895 isin 119869 (19)

sum

119895isin119869

sum

119896isin119870

(119896

2sdot 120572119895119896) + 119889

minus

3minus 119889

+

3= 0 (20)


119909119894119895119896isin 0 1


(21)

119910119894ℎisin 0 1


119886(119894)) 119894 lt 119903

(22)

120572119895119896isin 0 1

forall119895 isin 119869 119896 isin 119870

(23)

119906119895isin 0 1 forall119895 isin 119869 (24)

119908119895ge 0


st119894ge 0 forall119894 isin 119868 (26)

119889

+

1 119889

minus

1 119889

+

2 119889

minus

2 119889

+

3 119889

minus

3 119897119897 119898119908 ge 0 (27)


1corresponds



sum

119896isin119870

(119895 minus 1) sdot 119909ℎ119895119896+ stℎ+ 119905ℎ


sum

119896isin119870

(119895 minus 1) sdot 119909119894119895119896+ st119894


(28)





119894ge stℎ+ 119905ℎ




1









119895119896




3is reduced to



Minimize1198661= 119889

+

1(Highest priority)

Minimize1198662= 119889

+


Minimize1198663= 119889

+

3(Lowest priority)

(29)


1


1= 119889

lowast




1= 119889

lowast

1and 119889+

2= 119889

lowast

2are












1198941

119901119894119899

119901119894119899






119896(119909))








end



119899are 1199011198941

119901119894119899

119901119894119899














1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























SIst = radic119898

sum

119895=1

(119879119878max minus 119879119878119895)2

(2)




1le119895le119898119879119878119895


SIwr = radic119898

sum

119895=1

(119879119882max minus 119879119882119895)2

(3)








31 Notations

Parameters




Decision Variables


119909119894119895119896=


0 otherwise

119910119894ℎ=


0 otherwise

(4)



119898119908 = max119895119908119895

120572119895119896

=


0 otherwise

119906119895=


0 otherwise

(5)


119889

+

1 119889

minus



119889

+

2 119889

minus



119889

+

3 119889

minus






lexmin 119889+1 119889

+

2 119889

+

3 (6)



sum

119895isin119869

sum

119896isin119870

119909119894119895119896= 1 forall119894 isin 119868 (7)


sum

119896isin119870


ℎ+ 119905ℎle st119894


(8)



ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot (1 minus 119910

119894ℎ) ge st119894+ 119905119894


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(10)

st119894+ Ψ sdot (1 minus 119909

ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot 119910

119894ℎge stℎ+ 119905ℎ


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(11)


sum

119895isin119869

sum

119896isin119870


119897119897 + 119889

minus

1minus 119889

+

1= 119871max (13)


sum

119896isin119870


sum

119895isin119869

119908119895+ 119889

minus

2minus 119889

+

2= 119879119872max (15)


119898119908 ge 119908119895 forall119895 isin 119869 (16)


119894isin119868

sum

119896isin119870

119909119894119895119896


(17)

Φ sdot (1 minus 119906119895) + sum

119896isin119896


sum

119896isin119870

120572119895119896le 1 forall119895 isin 119869 (19)

sum

119895isin119869

sum

119896isin119870

(119896

2sdot 120572119895119896) + 119889

minus

3minus 119889

+

3= 0 (20)


119909119894119895119896isin 0 1


(21)

119910119894ℎisin 0 1


119886(119894)) 119894 lt 119903

(22)

120572119895119896isin 0 1

forall119895 isin 119869 119896 isin 119870

(23)

119906119895isin 0 1 forall119895 isin 119869 (24)

119908119895ge 0


st119894ge 0 forall119894 isin 119868 (26)

119889

+

1 119889

minus

1 119889

+

2 119889

minus

2 119889

+

3 119889

minus

3 119897119897 119898119908 ge 0 (27)


1corresponds



sum

119896isin119870

(119895 minus 1) sdot 119909ℎ119895119896+ stℎ+ 119905ℎ


sum

119896isin119870

(119895 minus 1) sdot 119909119894119895119896+ st119894


(28)





119894ge stℎ+ 119905ℎ




1









119895119896




3is reduced to



Minimize1198661= 119889

+

1(Highest priority)

Minimize1198662= 119889

+


Minimize1198663= 119889

+

3(Lowest priority)

(29)


1


1= 119889

lowast




1= 119889

lowast

1and 119889+

2= 119889

lowast

2are












1198941

119901119894119899

119901119894119899






119896(119909))








end



119899are 1199011198941

119901119894119899

119901119894119899














1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1le119895le119898119879119878119895


SIwr = radic119898

sum

119895=1

(119879119882max minus 119879119882119895)2

(3)








31 Notations

Parameters




Decision Variables


119909119894119895119896=


0 otherwise

119910119894ℎ=


0 otherwise

(4)



119898119908 = max119895119908119895

120572119895119896

=


0 otherwise

119906119895=


0 otherwise

(5)


119889

+

1 119889

minus



119889

+

2 119889

minus



119889

+

3 119889

minus






lexmin 119889+1 119889

+

2 119889

+

3 (6)



sum

119895isin119869

sum

119896isin119870

119909119894119895119896= 1 forall119894 isin 119868 (7)


sum

119896isin119870


ℎ+ 119905ℎle st119894


(8)



ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot (1 minus 119910

119894ℎ) ge st119894+ 119905119894


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(10)

st119894+ Ψ sdot (1 minus 119909

ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot 119910

119894ℎge stℎ+ 119905ℎ


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(11)


sum

119895isin119869

sum

119896isin119870


119897119897 + 119889

minus

1minus 119889

+

1= 119871max (13)


sum

119896isin119870


sum

119895isin119869

119908119895+ 119889

minus

2minus 119889

+

2= 119879119872max (15)


119898119908 ge 119908119895 forall119895 isin 119869 (16)


119894isin119868

sum

119896isin119870

119909119894119895119896


(17)

Φ sdot (1 minus 119906119895) + sum

119896isin119896


sum

119896isin119870

120572119895119896le 1 forall119895 isin 119869 (19)

sum

119895isin119869

sum

119896isin119870

(119896

2sdot 120572119895119896) + 119889

minus

3minus 119889

+

3= 0 (20)


119909119894119895119896isin 0 1


(21)

119910119894ℎisin 0 1


119886(119894)) 119894 lt 119903

(22)

120572119895119896isin 0 1

forall119895 isin 119869 119896 isin 119870

(23)

119906119895isin 0 1 forall119895 isin 119869 (24)

119908119895ge 0


st119894ge 0 forall119894 isin 119868 (26)

119889

+

1 119889

minus

1 119889

+

2 119889

minus

2 119889

+

3 119889

minus

3 119897119897 119898119908 ge 0 (27)


1corresponds



sum

119896isin119870

(119895 minus 1) sdot 119909ℎ119895119896+ stℎ+ 119905ℎ


sum

119896isin119870

(119895 minus 1) sdot 119909119894119895119896+ st119894


(28)





119894ge stℎ+ 119905ℎ




1









119895119896




3is reduced to



Minimize1198661= 119889

+

1(Highest priority)

Minimize1198662= 119889

+


Minimize1198663= 119889

+

3(Lowest priority)

(29)


1


1= 119889

lowast




1= 119889

lowast

1and 119889+

2= 119889

lowast

2are












1198941

119901119894119899

119901119894119899






119896(119909))








end



119899are 1199011198941

119901119894119899

119901119894119899














1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lexmin 119889+1 119889

+

2 119889

+

3 (6)



sum

119895isin119869

sum

119896isin119870

119909119894119895119896= 1 forall119894 isin 119868 (7)


sum

119896isin119870


ℎ+ 119905ℎle st119894


(8)



ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot (1 minus 119910

119894ℎ) ge st119894+ 119905119894


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(10)

st119894+ Ψ sdot (1 minus 119909

ℎ119895119896) + Ψ sdot (1 minus 119909

119894119895119896) + Ψ sdot 119910

119894ℎge stℎ+ 119905ℎ


119886(119894)) 119894 lt 119903 119895 isin 119869 119896 isin 119870

(11)


sum

119895isin119869

sum

119896isin119870


119897119897 + 119889

minus

1minus 119889

+

1= 119871max (13)


sum

119896isin119870


sum

119895isin119869

119908119895+ 119889

minus

2minus 119889

+

2= 119879119872max (15)


119898119908 ge 119908119895 forall119895 isin 119869 (16)


119894isin119868

sum

119896isin119870

119909119894119895119896


(17)

Φ sdot (1 minus 119906119895) + sum

119896isin119896


sum

119896isin119870

120572119895119896le 1 forall119895 isin 119869 (19)

sum

119895isin119869

sum

119896isin119870

(119896

2sdot 120572119895119896) + 119889

minus

3minus 119889

+

3= 0 (20)


119909119894119895119896isin 0 1


(21)

119910119894ℎisin 0 1


119886(119894)) 119894 lt 119903

(22)

120572119895119896isin 0 1

forall119895 isin 119869 119896 isin 119870

(23)

119906119895isin 0 1 forall119895 isin 119869 (24)

119908119895ge 0


st119894ge 0 forall119894 isin 119868 (26)

119889

+

1 119889

minus

1 119889

+

2 119889

minus

2 119889

+

3 119889

minus

3 119897119897 119898119908 ge 0 (27)


1corresponds



sum

119896isin119870

(119895 minus 1) sdot 119909ℎ119895119896+ stℎ+ 119905ℎ


sum

119896isin119870

(119895 minus 1) sdot 119909119894119895119896+ st119894


(28)





119894ge stℎ+ 119905ℎ




1









119895119896




3is reduced to



Minimize1198661= 119889

+

1(Highest priority)

Minimize1198662= 119889

+


Minimize1198663= 119889

+

3(Lowest priority)

(29)


1


1= 119889

lowast




1= 119889

lowast

1and 119889+

2= 119889

lowast

2are












1198941

119901119894119899

119901119894119899






119896(119909))








end



119899are 1199011198941

119901119894119899

119901119894119899














1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1









119895119896




3is reduced to



Minimize1198661= 119889

+

1(Highest priority)

Minimize1198662= 119889

+


Minimize1198663= 119889

+

3(Lowest priority)

(29)


1


1= 119889

lowast




1= 119889

lowast

1and 119889+

2= 119889

lowast

2are












1198941

119901119894119899

119901119894119899






119896(119909))








end



119899are 1199011198941

119901119894119899

119901119894119899














1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119896(119909))








end



119899are 1199011198941

119901119894119899

119901119894119899














1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















1and





1


1


1and 119899119904

2represent


2 respectively






Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Selectedtask

Starttime

Manindex Notes

1

1 (3)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time





7

2 (3)





13 3 (3) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash




Selectedtask

Starttime

Manindex Notes

1

1 (2)

1 (5) 1 0 1 mdash

2 2 (12) 3 (3) 4 (10) 5(9) 2 6 1 mdash


time




6

2 (3)





13 3 (1) 11 (4) 11 0 1 mdash14 12 (8) 12 6 1 mdash


1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 7 10


4 8 9 11 12

6 8 8

9

3 4 6 6 9

0 10 0 10

0 10W1

W1W2

W3

W1

W2

W3


11

Station 4

121 2 3 5 8 6

9


86 8 3 6

80 10

0 100 102 0 103W1

W2

W1

W2

W1W1


6Station 4


9 11 124 5 8 7 1086 8 4 7 0 100 100 10 0 104 0 1066 9

W1W1 W1 W1 W1



1 2 3

11

5 8 10

127


4

6 96 8

4

3 6 8

0 10

0 100 10 W1

W2

W3

W1

W2

W1


4 8 9

12

7

115 6 101


2 36 96 8 3

7 94

8 0 10

0 10

0 10W1

W2

W3

W1 W1


Station 3

1 2 394 8

1110

Station 1 Station 2

57 12

6

Station 4

6 983

7 946 8 0 10 0 10

0 100 10 W1

W1W1W1

W2






Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

No difference

Yes (no difference)

Yes (no difference)

Yes (no difference)

No

No

No

No

NoNo

Yes

Yes

Yes

(s1 = s2)

(m1 = m2)

((ns1 = 0)and (ns2 = 0))

(ns1 = ns2)

s1 lt s2

m1 lt m2

ns1 lt ns2




or


Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 12 3 9 10 1 6 2 7 11 4 8









Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Permutation 5 12 3 10 9 1 6 2 7 11 4 8

Neighbor 5 9 3 10 4 1 6 2 7 11 12 8




API cup x

x

SR3

SR3

SR2

SR4

SR3minus SR2















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















structure


structure

Shaking phase








5 9 3 10 4 1 6 2 7 11 12 8




Nei

ghbo

r 1

Nei

ghbo

r 2

SR3 neighbor
















P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












P11

7 2 6a 8b 4c 6 8 4 6 8 4 6 8 4 016 014 0334 5 9 10 5 9 10 5 9 10 5 9 10 014 011 008

10 2 4 5 3 4 5 3 4 5 3 4 5 3 014 011 0284 3 6 5 3 6 5 3 6 5 3 6 5 019 011 009

21 2 2 3 1 2 3 1 2 3 1 2 3 1 011 009 0224 2 3 1 2 3 1 2 3 1 2 3 1 016 017 02

P21

14 2 7 8 6 7 8 6 7 8 6 7 8 6 062 075 0374 7 8 6 7 8 6 7 8 6 7 8 6 087 052 089

21 2 4 6 2 4 6 2 4 6 2 4 6 2 092 064 0584 4 6 2 4 6 2 4 6 2 4 6 2 218 194 086

35 2 3 3 0 3 3 0 3 3 0 3 3 0 067 117 024 3 3 0 3 3 0 3 3 0 3 3 0 147 194 033

P30

25 2 8 14 2 8 14 2 8 14 2 8 14 2 91 5694 1364 8 14 2 8 14 2 8 14 2 8 14 2 1064 12227 63

30 2 7 12 2 7 12 2 66 12 12 6 12 0 10349 8065 0894 6 12 10 6 12 8 6 12 10 6 12 0 1666 28222 6254

41 2 48 82 14 48 8 16 5 8 2 4 8 0 174197 35852 10054 4 9 3 4 88 36 4 9 3 4 8 No 2652 59438 297915

P45




P70










P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












P83

5048 2 10a 16b 4c 10 16 4 10 16 44 9 172 12 9 174 144 9 174 132

6842 2 76 124 28 74 126 22 76 124 284 7 13 58 7 13 54 7 13 76

7571 2 62 11 14 6 11 1 6 11 14 6 11 1 6 11 1 6 11 1

P111

5755 2 158 284 32 156 282 3 158 282 344 122 294 36 12 294 422 12 30 416

8847 2 11 18 4 11 18 4 11 18 44 10 182 274 10 18 212 10 18 22

10743 2 84 15 18 8 15 1 8 15 14 7 156 284 7 154 244 7 154 252

P148

434 2 7 14 0 7 14 0 7 14 04 5 14 132 44 14 24 48 14 16

626 2 5 10 0 5 10 0 5 10 04 3 10 2 3 10 2 3 10 2

805 2 4 8 0 4 78 02 4 8 04 26 8 06 2 8 0 22 8 02










stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












stations 119889

+

1Result CPU

P11

7 2 6 6a 8b 4c 6 8 4 6 8 4 0 6 8 4 012 016 0334 5 5 9 10 5 9 10 5 9 10 0 5 9 10 016 011 009

10 2 4 4 5 3 4 5 3 4 5 3 0 4 5 3 011 016 0284 3 3 6 5 3 6 5 3 6 5 0 3 6 5 009 013 011

21 2 2 2 3 1 2 3 1 2 3 1 0 2 3 1 013 013 0224 2 2 3 1 2 3 1 2 3 1 0 2 3 1 008 019 02

P21

14 2 7 7 8 6 7 8 6 7 8 6 0 7 8 6 059 075 0384 7 7 8 6 7 8 6 7 8 6 0 7 8 6 222 053 092

21 2 4 4 6 2 4 6 2 4 6 2 0 4 6 2 094 066 0564 4 4 6 2 4 6 2 4 6 2 0 4 6 2 257 201 084

35 2 3 3 3 0 3 3 0 3 3 0 0 3 3 0 061 117 0174 3 3 3 0 3 3 0 3 3 0 0 3 3 0 236 175 028

P30

25 2 8 8 14 2 8 14 2 8 14 2 0 8 14 2 1312 5616 1374 8 8 14 2 8 14 2 8 14 2 0 8 14 2 604 1223 629

30 2 7 7 12 2 7 12 2 66 12 12 0 7 12int mdash 755 359252 mdash4 6 6 12 10 6 12 10 6 12 10 0 6 12 0 1637 305252 6728

41 2 5 5 82 18 48 82 14 46 8 12 0 5 8 0 1261 46786 42624 4 4 88 24 4 9 3 4 9 3 0 4 8 No 1094 63906 295007

P45




P70














P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













P83

5048 2 10 10a 16b 4c 10 16 4 10 16 44 9 9 174 136 9 176 134 9 176 134

6842 2 8 8 12 4 8 12 4 8 12 44 7 7 13 36 7 128 38 7 13 8

7571 2 7 64 11 18 6 11 1 62 11 144 6 6 11 3 6 11 1 6 11 1

P111

5755 2 16 158 282 34 16 28 4 16 286 344 13 13 29 332 13 288 232 13 286 398

8847 2 11 11 18 4 11 18 4 11 18 44 10 10 18 36 10 18 204 10 18 212

10743 2 9 88 15 26 84 15 18 9 15 34 7 7 158 168 7 154 252 7 16 154

P148

434 2 7 7 14 0 7 14 0 7 14 04 5 5 14 8 5 14 1 5 14 32

626 2 5 5 10 0 5 10 0 5 10 04 3 3 10 2 3 10 2 3 10 2

805 2 4 4 8 0 4 8 0 4 8 04 3 26 8 06 2 8 0 24 8 04








Competing Interests






+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+

1119889

+

2Result CPU

P11

7 2 6 8 6a 8b 4c 6 8 4 6 8 4 0 0 6 8 4 014 008 0334 5 9 5 9 10 5 9 10 5 9 10 0 0 5 9 10 016 008 009

10 2 4 5 4 5 3 4 5 3 4 5 3 0 0 4 5 3 013 011 0314 3 6 3 6 5 3 6 5 3 6 5 0 0 3 6 5 008 008 009

21 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 011 008 0234 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1 009 008 019

P21

14 2 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 058 027 0394 7 8 7 8 6 7 8 6 7 8 6 0 0 7 8 6 222 083 091

21 2 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 092 028 0584 4 6 4 6 2 4 6 2 4 6 2 0 0 4 6 2 262 059 089

35 2 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 058 161 0174 3 3 3 3 0 3 3 0 3 3 0 0 0 3 3 0 206 12 028

P30

25 2 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 1303 081 1374 8 14 8 14 2 8 14 2 8 14 2 0 0 8 14 2 602 457 63

30 2 7 12 7 12 2 68 12 16 7 12 2 0 0 7 12 0 755 129 8184 6 12 6 12 8 6 12 10 6 12 6 0 0 6 12 0 1512 13 6239

41 2 5 9 5 9 1 5 9 1 5 9 1 0 0 5 8 0 1173 136 142484 4 9 4 9 3 4 88 24 4 9 3 0 0 4 8 0 1017 296 297737

P45


110 2 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 5395 228 2234 3 6 3 6 0 3 6 0 3 6 0 0 0 3 6 0 9522 764 2346

184 2 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 No 3922 52708 303384 2 3 2 3 1 2 3 1 2 3 1 0 0 2 3 1int 6023 260169 93855

P70








stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












stations workers

P83

5048 2 10 16 10a 162b 38c 10 16 4 10 16 44 9 18 9 18 13 9 18 126 9 18 122

6842 2 8 12 8 12 4 8 12 4 8 12 44 7 13 7 13 76 7 13 1 7 13 76

7571 2 7 11 66 11 22 64 11 18 64 11 184 6 11 6 11 1 6 11 1 6 11 1

P111

5755 2 16 29 16 29 3 16 29 3 158 29 264 13 29 13 29 156 13 29 148 13 29 23

8847 2 11 18 11 18 4 11 18 4 11 18 44 10 18 10 18 212 10 18 204 10 18 284

10743 2 9 15 84 15 18 82 15 14 88 15 264 7 17 7 17 94 7 17 56 7 17 78

P148

434 2 7 14 7 14 0 7 14 0 7 14 04 5 14 5 14 114 5 14 1 5 14 1

626 2 5 10 5 10 0 5 10 0 5 10 04 3 10 3 10 2 3 10 2 3 10 2

805 2 4 8 4 8 0 4 8 0 4 8 04 3 8 3 8 1 2 8 0 22 8 02


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Balancing Lexicographic Multi-Objective