research article a genetic algorithm for task scheduling...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 708495 16 pageshttpdxdoiorg1011552013708495

Research ArticleA Genetic Algorithm for Task Scheduling on NoCUsing FDH Cross Efficiency

Song Chai Yubai Li Jian Wang and Chang Wu

School of Communication and Information Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China

Correspondence should be addressed to Song Chai stschaigmailcom

Received 5 August 2013 Accepted 7 November 2013

Academic Editor Hao-Chun Lu

Copyright copy 2013 Song Chai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A CrosFDH-GA algorithm is proposed for the task scheduling problem on the NoC-based MPSoC regarding the multicriterionoptimization First of all four common criterions namely makespan data routing energy average link load and workload balanceare extracted from the task scheduling problem on NoC and are used to construct the DEA DMU model Then the FDH analysisis applied to the problem and a FDH cross efficiency formulation is derived for evaluating the relative advantage among schedulesolutions Finally we introduce the DEA approach to the genetic algorithm and propose a CrosFDH-GA scheduling algorithm tofind the most efficient schedule solution for a given scheduling problemThe simulation results show that our FDH cross efficiencyformulation effectively evaluates the performance of schedule solutions By conducting comparative simulations our CrosFDH-GAproposal produces more metrics-balanced schedule solution than other multicriterion algorithms

1 Introduction

The scheduling problem has long been a research hotspotsince its proposal in the 1950s as the Job-shop schedulingproblem [1] After the computer technology emerged in the1940s the scheduling problem also found its position in thecomputer science region as the task scheduling problem onthe uniprocessor in the 1960s [2] the multiprocessor in the1970s [3] the distributed computing in the 1980s [4] andthe grid computing in the early 21st century [5] Now thechip fabrication technology has brought us to the single-chip multicore era [6] The presence of chip multiprocessor(CMP) especially theNoC (network-on-chip)-basedMPSoC[7] brings new challenges to the task scheduling algorithmdesign

In NoC solution the idea of introducing the networkinfrastructure to the chip design along with the newly arisenconcept of green communication [8] makes the goal ofscheduling algorithm change from single-objective optimiza-tion on makespan to performing optimization simultane-ously onmultiple metrics not only the traditional makespanbut also energy [9] and NoC criterions [10] and some ofthe optimizations of these metrics are even in conflict with

each other So the goal of scheduling algorithm design leansforward to balancing these multiple metrics

On the other hand DEA is a nonparametric techniquethat is used to measure the relative efficiency of multi-input multioutput DMUs (decisionmaking units) It was firstpresented byCharnes Cooper andRhodes as theCCRmodelin 1976 [11] then developed into several variations based ondifferent RTS (return-to-scale) assumptions The concept ofefficiency in DEA gives us a reasonable standard to maketrade-off between multiple metrics

In this paper a FDH DEAmodel of NoC task schedulingis constructed and a FDH cross efficiency formulation isproposed based on peer appraisal for further assessmentof the relative advantages of DMUs Then the proposedDEA approach is introduced to the genetic algorithm and aCrosFDH-GA scheduling algorithm is proposed for the taskscheduling problem on NoC to find the most efficient andbalanced schedule solution

The rest of this paper is organized as follows Section 2summarizes the related work of this paper Section 3 for-mulates the task scheduling problem on NoC the FDHand cross efficiency FDH formulation is given in Section 4Section 5 presents our CrosFDH-GA scheduling algorithms

2 Mathematical Problems in Engineering

R R R R

R

R

RRRR

R

R R R

RR

PE

PE PE PE

PEPE PE

PE

PE

PE PE

PEPE PE

PE PE

(0 0) (1 0) (2 0) (3 0)

(1 1) (2 1) (3 1)

(1 2) (2 2) (3 2)

(0 1)

(0 2)

(0 3) (1 3) (2 3) (3 3)

(a) A 4 times 4 2D mesh NoC

East Inport

FIFO

Dec

oder

Req

ReqAck

Ack

East Outport

Arbiter

Sel

WestOutport

North Outport

North Inport

South Outport

South Inport

Local OutportLocal Inport

MU

X

West Inport

full in

east in

full out

west out

(b) Microstructure of NoC node

Figure 1 Architecture of NoC-based MPSoC

simulations results and discussion are given in Section 6Section 7 concludes the paper

2 Related Works

The multicriterion scheduling algorithms for CMPs havebeen widely researched in recent years In [12] a schedulingalgorithm is proposed for multicore processors to avoidresource contention aswell as to reduce energy consumptionA modified genetic algorithm which incorporates bacterio-logical algorithm is proposed in [13] to maximize the systemreliability and reduce makespan In [14] the optimization ofmakespan and workload balance is addressed and a NSGA-II based schedule algorithm is proposed for multicore-basedgrid Amultiobjective evolutionary algorithm (MOEA) basedschedule heuristic is proposed for the joint optimization ofperformance energy and temperature on multicore proces-sors in [15]

As for the DEArsquos application in the field of task schedulingonmulticore a FDH-based evaluationmethod for the assess-ment of schedule heuristics is proposed in [16] Althoughboth [16] and ourwork adoptDEAFDHmodel as the analytictool ourwork introduces the concept of cross efficiency to theFDHmodel and uses FDH cross efficiency to rank schedulesMoreover the incorporation of DEA evaluation method intometaheuristic also distinguishes itself from the work in [16]

3 Problem Formulation

31 Task Model In this paper tasks are modeled usingdirected acyclic graphs (DAGs) A DAG 119866 = (119881 119864) is anacyclic graph where 119881 is the set of nodes which represent thetasks and 119864 is the set of edges in which an element 119890

119894119895denotes

the communication from task 119894 to task 119895 The edge indicatesthe precedent relation between two tasks

Each node V119894and edge 119890

119894119895are associated with a weight

denoted by119862119894and119879119894119895 respectivelyWeight119862

119894is the computa-

tional load required by a processing Element (PE) to executetask 119894 and119879

119894119895is the data transmission load between task 119894 and

task 119895 In our work both 119862119894and 119879

119894119895are presented using time

unit (cycles)

32 Network-on-Chip Hardware The target hardware is a 2Dmesh NoC-based MPSoC as illustrated in Figure 1(a) EachPE is connected to a router and routers are interconnectedwith each other through bidirection links Data is transferredthrough NoC in the form of packets

PEs are homogenous processor cores with local datacache If two consequential tasks are scheduled to the samePE the successor task reads the predecessorrsquos data directlyfrom the data cache of the PE without routing in NoC

The microstructure of a NoC router is shown in Fig-ure 1(b) The router has five Inports and Outports corre-sponding to five directions of East West North South andLocal The decoder in the Inport scans the first flit of theFIFO for any incoming packet If the decoder detects thehead flit of a packet it performs XY routing algorithm andsends request signal to the arbiter of the correspondingOutport If the arbiter receives multiple request signals thecontention is solved using Round-Robin arbitration Thegranted Inport then forwards the packet to the downstreamrouter Wormhole routing is adopted to minimize the bufferrequirement as well as the packet latency [17] The backpressure mechanism is also employed to further reduce end-to-end delay [18]

The energy model of a NoC is presented by the BitEnergy proposed in [19] Analytically the average energyconsumption of transmitting one bit from node 119894 to node 119895is calculated by

119864119894119895

bit = 119899hops sdot 119864119873bit + (119899hops minus 1) sdot 119864119871bit (1)

Mathematical Problems in Engineering 3

where 119864119873bit

and 119864119871bit

represent the energy consumed on thenode and on the link respectively and 119899hops is the number ofnodes the bit passes on its way from node 119894 to node 119895

33 Monitored Metrics In this paper four common metricsof NoC are extracted and monitored for the assessmentof schedule solution and they are makespan data routingenergy average link load [20] and workload balance [21]

Makespan (the 119872 metric) which is the amount of timerequired by a NoC to finish entire tasks in a DAG followingthe instruction of a schedule is the timemetric of a schedulewhile the data routing Energy (the 119864 metric) is the totalamount of energy dissipated in each NoC component duringthe execution of the DAG

The average Link load (the 119871 metric) is calculated byadding up all the data transmission time (in cycles) on eachlink and then dividing it by the number of links (for the NoCin Figure 1(a) there are 48 links) The 119871 metric representshow busy the NoC is during the execution and a higheraverage link load metric implies a higher possibility of linkcontentions A good schedule is supposed to reduce the 119871metric

Finally the workload Balance (the 119861metric) is defined tobe the inverse coefficient of variant of the total workload oneach processor as shown in (2) The loadprc(119899) is the actualload on processor 119899 and the loadave is the average load The119861metric reflects the load balance of the processors

workload balance =loadave

(sum119899(loadave minus loadprc (119899))

2

)

12

(2)

4 DEA Evaluation of Schedules

41 A Brief Review of DEA Data envelopment analysis(DEA) is a nonparametric technique that is widely used tomeasure the relative efficiency among many-input many-output decision-making units (DMUs) which in our contextare the schedules The efficiency of a DMU is defined as theweighted sum of its output divided by the weighted sum ofits input The essence of DEA is that it allows each DMU tochoose a particular set of weight coefficients which favors itsown efficiency under the constraint that the efficiencies of allDMUs calculated by this set of coefficients do not exceed 1A DMU is ldquoefficientrdquo if the efficiency calculated by DEA is 1otherwise the DMU is marked as ldquoinefficientrdquo The followingLinear Programing (LP) problem is the CCR model of DEA(CCR multiplier form)

max 119885 = 119906119879

sdot 119910119894

st

V sdot 119909119894= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899

V 119906 ge 0

(3)

where 119909119894= (119909

11989411199091198942

sdot sdot sdot 119909119894119898)119879

isin 119877119898 and 119910

119894=

(1199101198941

1199101198942

sdot sdot sdot 119910119894119904)119879

isin 119877119904 are the input and out-

put vectors of DMU 119894 V = (V1

V2

sdot sdot sdot V119898) and

119906 = (11990611199062

sdot sdot sdot 119906119904) are the coefficient (multiplier)

vectors of the inputs and outputs 119899 is the number of DMUsand the objective function 119885 is the efficiency of DMU 119894

The result of applying DEA is a classification amongDMUs as efficient group or inefficient group The efficientDMUs forman efficient frontier on themulti-inputmultiout-put space that envelops all inefficient DMUs The projectionof inefficient DMUon the efficient frontier is the hypotheticalefficient unit which is a linear combination of the efficientDMUs An inefficient DMU can also be converted to an effi-cient DMU by proportionally scaling down by the value of itsefficiency in the inputs and maintaining its original outputsThis interpretation of DEA efficiency is corresponding to theenvelop form of DEA which is the dual problem of (3) (CCRenvelop form)

min120579120582

119885 = 120579

st

120579119909119894minus 119883 sdot 120582 ge 0

119910119894minus 119884 sdot 120582 le 0

120582 ge 0

(4)

where 119883 = (11990911199092

sdot sdot sdot 119909119899) and 119884 =

(11991011199102

sdot sdot sdot 119910119899) are the input and output matrices

120582 = (12058211205822

sdot sdot sdot 120582119899)119879

isin 119877119899 is a nonnegative vector and

120579 is the efficiency of DMU 119894The way that the efficient frontier is generated differen-

tiates between DEA models which imply different returnsto scale assumptions There are four basic returns to scaleassumption constant returns to scale (CRS) correspondingto the CCR model [11] variable returns to scale (VRS)corresponding to the BCC model [22] increasing returns toscale and decreasing returns to scale

In this paper we focus on a special case of DEA namelythe free disposal hull (FDH) [23] In VRS FDH formulationof DEA each DMU is evaluated by comparing itself to otherDMUs on a one-on-one basis and a DMU is consideredefficient only when no other DMU dominates it

In most cases the idea of DEA that let DMU specifyits own weight coefficients to show its maximum advantageis desirable However in some extreme scenes a DMU canldquocheatrdquo a high efficiency score by weighting a single input ora single output and setting the rest weight coefficients closeto 0 This can happen when some DMUs have a particularlysmall input or particularly large output in our words theseDMUs have unbalanced metrics and these ldquomavericksrdquoneed to be depreciated Moreover although DEA effectivelydiscriminates between efficient and inefficient DMUs it doesnot further assess the relative advantages among the efficientones

One solution to the above questions is to introduce crossefficiency in the efficiencymeasurementThe concept of crossefficiency corresponding to the simple efficiency implied byoriginal DEA is the peer-appraisal equivalent of DEArsquos self-appraisal process The cross efficiency of a certain DMU isthe efficiency value calculated by using weight coefficientsderived by other DMUs If a DMU is a maverick or hasunbalanced metrics then its cross efficiency value derivedfrom other DMUrsquos coefficients is not likely to be high


Two widely accepted crossefficiency formulations theaggressive and benevolent formulations were proposedin [24] based on CCR Model Both formulations add asecondary goal to the normal DEA efficiency calculation(maximizing reference DMUrsquos efficiency) the aggressiveformulation minimizes target DMUrsquos crossefficiency whilethe benevolent formulation maximizes its efficiency Giventhat the simple DEA efficiency of DMU 119894 is 119864

119894119894 then the

crossefficiency of DMU 119895 evaluated by DMU 119894 is defined by(CCR crossefficiency benevolent formulation)

max 119864119895119894= 119906119879

sdot 119910119895

st

V sdot 119909119895= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899

119864119894119894sdot V sdot 119909119894minus 119906 sdot 119910

119894= 0

V 119906 ge 0

(5)

42 FDHDEA Evaluation of Schedules In order to apply dataenvelopment analysis to the schedule evaluation the multi-input multioutput DMUmodel needs to be defined using theschedule metrics namelymakespan (119872) routing energy (119864)average link load (119871) and workload balance (119861) proposed inSection 3

The classification of metrics as inputs and outputs followsa simple ldquorule of thumbrdquo [16] If the value of a metric islarger-is-better then it is an output otherwise the metric isan input As a result of this classification the DMUmodel ofour schedule evaluation is as follows 119909 = (119872 119864 119871)

119879 arethe inputs and 119910 = 119861 is the output

Moreover in the rest of this paper the term ldquoschedulerdquo isreferring to a schedule that is discriminable using the four-metric classification If two schedules have identical metricsthey are regarded as the same schedule at least they are notdiscriminable under current metrics

With the schedulingDMUmodel the observed schedulesare defined as follows

Definition 1 (schedule set) The observed schedules or theschedule setΓ = 120574

1 120574

119894 120574

119899 = (119909

1 1199101) (119909

119894 119910119894)

(119909119899 119910119899) is the set of 119899 observed schedules where each

schedule 120574119894is defined by the input vector 119909

119894= (119872119894119864119894119871119894)119879

and output scalar 119910119894= 119861119894

Another concept that is relevant to DEA is the possi-ble production set The possible production set is the spaceenveloped by the efficient frontier on the multi-input mul-tioutput space Normally the possible production set isunknown in the DEA and needs to be constructed using theobserved DMUs The FDH possible production set postulateswere proposed in [23] Here under our context we restatethese postulates as the following axiom

Axiom 1 (possible schedule set) The possible schedule set(PSS) of a schedule set Γ satisfies the following

Postulate I PSS contains all schedules in Γ

Postulate II PSS contains unobserved schedule 120574119894if

(i) 120574119895= (119909119895 119910119895) isin Γ and 119909

119894= (119872119894119864119894119871119894) ge 119909119895or

(ii) 119910119894= 119861119894le 119910119895

Postulate II is the free disposal postulate which suggestsa free disposal hull of Γ [25] Together with the deterministPostulate I they define the FDH possible schedule set ofour schedule efficiency analysis Now we formally introduceFDH DEA to the schedule evaluation and define the efficientschedule as follows

Definition 2 (efficient schedule and efficient schedule set)In a schedule set Γ a schedule 120574

119894is called efficient if the

optimization problem (6) has an optimal solution of 119885lowast = 1otherwise 120574

119894is inefficientThe set of all efficient schedules in Γ

is called the efficient schedule set denoted by Γ119890 and likewise

the inefficient schedule set is denoted by Γ119894

min120579120582119904+119904minus

119885 = 120579 minus 120576 sdot (1 sdot 119904+ + 119904minus)

st

120579119909119894minus 119883 sdot 120582 minus 119904

+

= 0119910119894minus 119884 sdot 120582 + 119904

minus

= 0

1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899

119904+

119904minus

ge 0

(6)

where 119909119894= (119872

119894119864119894119871119894)119879 and 119910

119894= 119861119894are the input vector

and output scalar of 120574119894 119899 is the number of schedules in Γ

119883 = (11990911199092

sdot sdot sdot 119909119899) and 119884 = (119910

11199102

sdot sdot sdot 119910119899) are

the input and output matrices of Γ 120582 = (12058211205822

sdot sdot sdot 120582119899)119879

is a binary vector in 119877119899 119904+ and 119904minus are the nonnegative slackvariables representing input excess and output shortfall 120576is a non-Archimedean infinitesimal constant and 120579 is theefficiency of 120574

119894

Optimization problem (6) is aMixed-Integer Programing(MIP)The binary vector 120582 and the constraint 1sdot120582 = 1 enforcea one-on-one comparison between target schedule 120574

119894and all

the schedules in Γ to search for a reference schedule 120574119903that

minimizes 120579The first two constraints of problem (6) can be simplified

to (7) by removing the slack variables Obviously the problemhas feasible solutions when 119897 = 119894 and 120579 = 1 in this situationFrom this point on if a reference schedule 120574

119903is found with

120579 lt 1 that means that 120574119903produces output 119910

119903at least as same

as 120574119894 with the inputs nomore than 120579119909

119894 which is a scale-down

from 119909119894 Then this makes 120574

119894inefficient

120579119909119894ge 119909119897

119910119894le 119910119897 119897 = 1 2 119899

(7)

However only 120579 = 1 is not enough for a schedule to beefficient Consider an efficient schedule 120574 = (119872 119864 119871 119861)

and an inefficient schedule 120574 = (119872 + Δ119898

119864 119871 119861) whichis distinguished from 120574 by a small input excess Δ

119898in the

makespan the constraint (7) holding for 120574 also holds for120574 thus 120579 = 1 for 120574 The slacks variables 119904+ and 119904minus areintroduced to remove these schedules with input excess andoutput shortfall The nonzero 119904+ and 119904minus force the objectivefunction in (6) to be less than 1


Definition 1 implies the dominancePareto optimality ofan efficient schedule A dominant schedule is defined asfollows

Definition 3 (dominant schedule) A schedule 120574119886is said to

dominate 120574119887if

(I) each component of 119909119886= (119872

119886119864119886119871119886)119879 is not

greater than that of 119909119887

(II) 119910119886= 119861119886is not less than 119910

119887

(IIIa) at least of component of 119909119886is less than the corre-

sponding one of 119909119887(dominates in input) or

(IIIb) 119910119886is greater than 119910

119887(dominates in output)

Then the relationship between efficient schedule anddominant schedule is given inTheorem 4

Theorem 4 (dominance and efficiency) In a schedule set Γa schedule 120574

119894is called efficient if and only if no other schedule

dominates it

Proof If Part Suppose there is a schedule 120574119895that dominates

120574119894The efficiency of 120574

119894is calculated by solving the following

optimization problem

min120579120582119904+119904minus

119885 = 120579 minus 120576 sdot (119904+

119872+ 119904+

119864+ 119904+

119871+ 119904minus

119861)

st

120579 sdot 119872119894minus (11987211198722sdot sdot sdot 119872

119899) sdot 120582 minus 119904

+

119872= 0

120579 sdot 119864119894minus (11986411198642sdot sdot sdot 119864119899) sdot 120582 minus 119904

+

119864= 0


119899) sdot 120582 minus 119904

+

119871= 0

119861119894minus (11986111198612sdot sdot sdot 119861119899) sdot 120582 + 119904

minus

119861= 0

sum120582119897= 1 120582

119897isin 0 1 119897 = 1 2 119899

(8)

Let120582119895= 1 and 120579 = 1 then constraints of (8) are converted

to

119872119894minus119872119895minus 119904+

119872= 0

119864119894minus 119864119895minus 119904+

119864= 0

119871119894minus 119871119895minus 119904+

119871= 0

119861119894minus 119861119895+ 119904minus

119861= 0

(9)

From the domination relation we have 119872119894ge 119872

119895

119864119894ge 119864119895 119871119894ge 119871119895 and 119861

119894le 119861119895 and at least one of above

inequations holds strictly This means at least one of 119904+119872 119904+119864

119904+

119871 and 119904minus

119861in (4) is not zero So the objective function119885 in (8)

has a feasible solution of 1198851015840 = 1 minus 120576 sdot (119904+119872+ 119904+

119864+ 119904+

119871+ 119904minus

119861) lt 1

and schedule solution 120574119894is not efficient

Only If Part Suppose schedule 120574119894is not efficient then there

must exist a set of 120579lowast 120582lowast 119904+lowast and 119904minuslowast that makes the optimal

value of (8) be 119885lowast lt 1 Assuming 120582119895= 1 in 120582lowast from (8) we

have

120579lowast

sdot 119872119894= 119872119895+ 119904+lowast

119872

120579lowast

sdot 119864119894= 119864119895+ 119904+lowast

119864

120579lowast

sdot 119871119894= 119871119895+ 119904+lowast

119871

119861119894= 119861119895minus 119904minuslowast

119861

(10)

The optimal value of 119885lowast = 120579lowast minus 120576 sdot (119904+lowast119872+ 119904+lowast

119864+ 119904+lowast

119871+ 119904minuslowast

119861) lt 1

suggests that (I) 120579lowast = 1 and at least one of the 119904+lowast and 119904minuslowast isnot zero or (II) 120579lowast lt 1 Either of the above situations impliesthat schedule 120574

119894is dominated by 120574

119895

Theorem 4 relates the relatively abstract concept ofFDH efficiency to the concept of dominance Moreover thefollowing two corollaries are deduced fromTheorem 4

Corollary 5 In a schedule set Γ if a schedule 120574119894satisfies

(1) 119872119894lt 119872119897 or

(2) 119864119894lt 119864119897 or

(3) 119871119894lt 119871119897 or

(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)

then it is in the efficient schedule set Γ119890

Proof If 120574119894satisfies one of the above conditions then there is

no schedule dominating 120574119894 From Theorem 4 120574

119894is efficient

Corollary 5 points out that the schedule with the smallestmakespan or the smallest energy consumption or the smallestqueuing time or the best workload balance is an efficientschedule

Corollary 6 In a schedule set Γ removing any inefficientschedule from Γ does not change the elements in Γ

119890

Proof The schedules in Γ119890are dominant ones and removing

any inefficient schedule will not change the dominant posi-tion of the elements in Γ

119890 Thus Γ

119890remains unchanged

43 FDH Cross Evaluation of Schedules In this section across evaluation process is proposed based on peer-appraisalFDH DEA for further assessment of schedules DEA calcu-lates the efficiency of a DMU by allowing the DMU to choosea scenario that is best for itself Although this basis is plausiblein most situations some ldquomaverickrdquo DMUs especially theDMUs with a single small input or a single large output mayldquocheatrdquo DEA to achieve high score by valuing its only strengthand depreciating other metrics These unbalanced DMUsmust be devaluated during further assessment Moreover anassessment of relative advantages among efficient DMUs isalso required

FDHmodel is aMIP problem in nature In order to deriveits peer-appraisal variation the dual problem of the FDHDEA in (6) is needed to construct the formulation Normally


it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in

min120579119897120582119904+119904minus

119885 = sum

119897

120579119897+ 120576 sdot sum

119897

(1 sdot 119904+119897+ 119904minus

119897)

st

120579119897119909119894minus 119909119897sdot 120582119897minus 119904+

119897= 0 119897 = 0 1 119899

(119910119894minus 119910119897) sdot 120582119897+ 119904minus

119897= 0 119897 = 0 1 119899

1 sdot 120582 = 1120582 119904+

119897 119904minus

119897ge 0

(11)

The dual problem of (11) is (FDH multiplier form equiv-alent)

max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899

119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(12)

LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906

119897 V119897) is calculated by

119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized

V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to

(119906119897sdot 119910119894minus 119911 sdot V

119897sdot 119909119894) minus (119906

119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)

which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911

Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows

Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574

119894is 119864119894119894 the FDH

cross efficiency 119864119895119894of 120574119895evaluated by 120574

119894is the optimal value

of 119911 in

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(14)

Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864

119894119894(primary goal) and under

this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between

FDH efficiency and FDH cross efficiency

Theorem 8 The cross efficiency of schedule 120574119894evaluated by

itself is its FDH efficiency

Proof Assume that the cross efficiency of schedule 120574119894evalu-

ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)

is 119864The calculation of 119864

119894119894is solving the following LP

max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899


119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909

119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(15)

Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of


119894)

= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899

(16)

where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864

119894119894

Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by

schedule 120574119894does not exceed the value of its simple efficiency 119864

119895119895

Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574

119895rsquos FDH efficiency

using (12)

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(17)

Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)


Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows

Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by

119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894

is the FDH cross efficiency of 120574119895evaluated by

120574119894using (14) The average cross efficiency of DMU 119894 is

the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864

119894= sum119899

119897=1119864119894119897119899 The most efficient

schedule (120574MES) is the schedule with the largest average crossefficiency

The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system

Corollary 11 The average cross efficiency 119864119895of a schedule 120574

119895

which is defined in Definition 10 is not greater than its simpleefficiency 119864

119895119895

Proof Using the result of previous theorem the cross effi-ciency 119864

119895119894of schedule 120574

119895evaluated by arbitrary schedule 120574

119894is

not greater than the 119864119895119895 Thus 119864

119895 which is the average value

of 119864119895119894 is not greater than the 119864

119895119895

Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem

Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule

Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574

120572= (119909

120572 119910120572) be the most

efficient schedule of Γ where 119909120572= (119872 119864 119871)

119879 is theinput vector and 119910

120572= 119861 is the output scalar Let 119911lowast

120572be

the cross efficiency of schedule 120574120572evaluated by schedule 120574

119894

and 119906lowast119897= 119906lowast

119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are

the optimal coefficients Let 120574120573be a dominant schedule of

120574120572First assume that 120574

120573dominates 120574

120572in output (balance) and

120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=

119861 + Δ119861 has a small increment in balance Now we calculate


119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1

119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573

= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V

119897sdot 119909119897+ 119911120573

= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)

Compared with the calculation of the cross efficiency ofschedule 120574

120572evaluated by schedule 120574

119894 it is easy to verify that

120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast

119897 Vlowast119897 119897 = 1 2 119899 is a feasible

solution of the above LP Then the optimal value of 119911120573is not

less than 119911lowast120572+ 119906lowast

119897119861sdot max119897(Δ119861) which is greater than 119911lowast

120572 That

means the cross efficiency of schedule 120574120573evaluated by an

arbitrary schedule 120574119894is greater than the 120574

120572 which contradicts

the assumption of 120574120572being the most efficient schedule

Then assume that 120574120573dominates 120574

120572in input Without

loss of generality we assume that 120574120573

dominates 120574120572

inmakespan and 120574

120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=

(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan

Then the cross efficiency of schedule 120574120573evaluated by schedule

120574119894is calculated by solving the following LP

max119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V

119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)

minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0

119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)

= 119906119897sdot (119910119897minus 119910119894)

minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)

+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)

+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast

119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max

119897(Vlowast119897119872)) By substituting V1015840

119897 1199061015840119897 and

1199111015840

120573in (20) it is easy to verify that they are a feasible solution

of (20) Then the optimal solution of the optimal value of 119911120573

is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax

119897(Vlowast119897119872)) which contradicts


Finally if 120574120573dominates 120574

120572in multiple metrics say 120574

120573=

(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=

(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)

can be constructed to prove that 120574120572is not the most efficient

schedule using the previous results

Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel


Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in

Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too

In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let

119909119897= (119872

119897119864119897119871119897)119879 and 119910

119897= 119861119897be the input vector and

output scalar and let 119911lowast 119906lowast119897= (119906lowast

119897119872119906lowast

119897119864119906lowast

119897119871) and Vlowast

119897be the

optimal value for (12)Under the new unit system the inputs and outputs now

are the original values multiplied by conversion coefficientsdenoted by 1199091015840

119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879

and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =

1205741015840

119897= (1199091015840

119897 1199101015840

119897) | 119897 = 1 2 119899 obviously we have a feasible

solution of 1015840 = 119911lowast 1015840119897= (119906lowast

119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and

V1015840119897= Vlowast119897120573 which is transformed from the original problem

Therefore we have the optimal value of new problem 1199111015840lowast

ge

1015840 Suppose 1199111015840lowast gt

1015840 and the weight coefficients are1199061015840lowast

119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast

119897under this circumstance

Converting theweight coefficients to the original unit systemwe have

119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which

also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast

119897 and V

119897are also a feasible solution of

original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =

1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in

Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner

5 Efficient Scheduling Using CrosFDH-GA

51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation

For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906

119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000

and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs

Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times

In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of

Bottom Bottom Bottom BottomMetapopulation

Subpopulation MGoal makespan

Subpopulation EGoal energy

Subpopulation LGoal link

Subpopulation BGoal balance

Top 10Top 10Top 10Top 10

DEA-ready pool

Top 10 cross efficiency

Figure 2 The Subpopulation updating process

an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level

In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2

52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895

The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows

Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step

Step 2 (evaluation) Performance of each individual in thepool is evaluated


Table 1 Two randomly generated chromosomes

Chromosome Index1 2 3 4 5 6 7 8 9

chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3

Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool

Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size

Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size

Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process

For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3

In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3

In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4

53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool

First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule

chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap

Figure 3 The crossover operation

chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33

SwapArbitraryposition 1

Arbitraryposition 2

Mutation

Figure 4 The mutation operation

The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm

Pseudocode 1 shows cross evaluation process in ouralgorithm

6 Computational Experiments and Discussion

61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency

CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross


BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574

119897in the DEA-ready pool Γ DO

(11) Calculate 119864119897119897using FDH DEA

(2) END FOR(3) Regroup efficient schedule (119864

119897119897= 1) as Γ

119890

(4) FOR each schedule 120574119895in Γ119890

(41) FOR each schedule 120574119894= 120574119895in Γ119890DO

(411) Calculate the peer-appraisal FDH 119864119895119894

(42) END FOR(43) Calculate the average cross efficiency 119864

119895of 120574119895

(5) END FOR(6) Sorting the cross efficiency of schedules in Γ

119890

(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION

Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool

efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]

Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo

Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI

119894= (119864119894119894minus 119864119894)119864119894119894

MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU

From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies

ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared

in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric

As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53

Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This


Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash


Table 3 The average value of metrics of each subpopulation

Subpopulation Averagemakespan

Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727

phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric

62 Simulation Results of CrosFDH-GA Scheduling Algorithm

621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated

The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows

fitnessMD = 119872minus1

sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861

fitnessWES = 119908119872119872minus2

+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612

fitnessEWC = (119890119908119872 minus 1) sdot 119890

119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908

119861are the weight coefficients of the

corresponding metric In our implementation 119908119872 119908119864 119908119871

and119908119861are all set to 025 which means there is no preference

between four metricsMoreover the sel ratio cros ratio and mut ratio of GA

are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals

All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG

622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator

As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics

To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4

From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics

Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced

Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation


0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(c) Average link load

76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(d) Workload balance

4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time

Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA

Table 4The average value of metrics of each scheduling algorithm

Algorithms Averagemakespan

Averageenergy

Averagelink load

Averagebalance

CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483

the data in Figure 5(e) are preprocessed by applying log10to

the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such

long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time

In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)

The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem


40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA

(a) MD- WS- WES- and EWC-GA

40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA

Figure 6 The relation of solving time and task number

but to the number of efficient schedules in the DEA-readypool in each generation

Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time

Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric

10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

Number of efficient schedules

Aver

age s

olvi

ng ti

me (

seco

nd)

(a) Solving time and efficient number

Number of efficient schedules10 20 30 40 50 60 70 80 90

0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05

(b) Efficient number distribution

Figure 7 Efficient schedule number in CROSFDH-GA

which both converge in the same pace And the 119861 metric isthe last converged metric

7 Conclusion

In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs


0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02

MakespanEnergy Workload balance

Average link load

Figure 8 The trend of convergence of four metrics

Conflict of Interests

The authors certify that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)

References

[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960

[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966

[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970

[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988

[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003

[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005

[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001

[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010

[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International

Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011

[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010

[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011

[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013

[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011

[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004

[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011

[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004

[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002

[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007

[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013

[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006

[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994

[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978


[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001

[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006

[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-

ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008

[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998

[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990

[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988

[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004

[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008

[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009

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


R R R R

R

R

RRRR

R

R R R

RR

PE

PE PE PE

PEPE PE

PE

PE

PE PE

PEPE PE

PE PE

(0 0) (1 0) (2 0) (3 0)

(1 1) (2 1) (3 1)

(1 2) (2 2) (3 2)

(0 1)

(0 2)

(0 3) (1 3) (2 3) (3 3)

(a) A 4 times 4 2D mesh NoC

East Inport

FIFO

Dec

oder

Req

ReqAck

Ack

East Outport

Arbiter

Sel

WestOutport

North Outport

North Inport

South Outport

South Inport

Local OutportLocal Inport

MU

X

West Inport

full in

east in

full out

west out

(b) Microstructure of NoC node

Figure 1 Architecture of NoC-based MPSoC

simulations results and discussion are given in Section 6Section 7 concludes the paper

2 Related Works

The multicriterion scheduling algorithms for CMPs havebeen widely researched in recent years In [12] a schedulingalgorithm is proposed for multicore processors to avoidresource contention aswell as to reduce energy consumptionA modified genetic algorithm which incorporates bacterio-logical algorithm is proposed in [13] to maximize the systemreliability and reduce makespan In [14] the optimization ofmakespan and workload balance is addressed and a NSGA-II based schedule algorithm is proposed for multicore-basedgrid Amultiobjective evolutionary algorithm (MOEA) basedschedule heuristic is proposed for the joint optimization ofperformance energy and temperature on multicore proces-sors in [15]

As for the DEArsquos application in the field of task schedulingonmulticore a FDH-based evaluationmethod for the assess-ment of schedule heuristics is proposed in [16] Althoughboth [16] and ourwork adoptDEAFDHmodel as the analytictool ourwork introduces the concept of cross efficiency to theFDHmodel and uses FDH cross efficiency to rank schedulesMoreover the incorporation of DEA evaluation method intometaheuristic also distinguishes itself from the work in [16]

3 Problem Formulation

31 Task Model In this paper tasks are modeled usingdirected acyclic graphs (DAGs) A DAG 119866 = (119881 119864) is anacyclic graph where 119881 is the set of nodes which represent thetasks and 119864 is the set of edges in which an element 119890

119894119895denotes

the communication from task 119894 to task 119895 The edge indicatesthe precedent relation between two tasks

Each node V119894and edge 119890

119894119895are associated with a weight

denoted by119862119894and119879119894119895 respectivelyWeight119862

119894is the computa-

tional load required by a processing Element (PE) to executetask 119894 and119879

119894119895is the data transmission load between task 119894 and

task 119895 In our work both 119862119894and 119879

119894119895are presented using time

unit (cycles)

32 Network-on-Chip Hardware The target hardware is a 2Dmesh NoC-based MPSoC as illustrated in Figure 1(a) EachPE is connected to a router and routers are interconnectedwith each other through bidirection links Data is transferredthrough NoC in the form of packets

PEs are homogenous processor cores with local datacache If two consequential tasks are scheduled to the samePE the successor task reads the predecessorrsquos data directlyfrom the data cache of the PE without routing in NoC

The microstructure of a NoC router is shown in Fig-ure 1(b) The router has five Inports and Outports corre-sponding to five directions of East West North South andLocal The decoder in the Inport scans the first flit of theFIFO for any incoming packet If the decoder detects thehead flit of a packet it performs XY routing algorithm andsends request signal to the arbiter of the correspondingOutport If the arbiter receives multiple request signals thecontention is solved using Round-Robin arbitration Thegranted Inport then forwards the packet to the downstreamrouter Wormhole routing is adopted to minimize the bufferrequirement as well as the packet latency [17] The backpressure mechanism is also employed to further reduce end-to-end delay [18]

The energy model of a NoC is presented by the BitEnergy proposed in [19] Analytically the average energyconsumption of transmitting one bit from node 119894 to node 119895is calculated by

119864119894119895

bit = 119899hops sdot 119864119873bit + (119899hops minus 1) sdot 119864119871bit (1)


where 119864119873bit

and 119864119871bit








2

)

12

(2)



max 119885 = 119906119879

sdot 119910119894

st

V sdot 119909119894= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899

V 119906 ge 0

(3)

where 119909119894= (119909

11989411199091198942

sdot sdot sdot 119909119894119898)119879

isin 119877119898 and 119910

119894=

(1199101198941

1199101198942

sdot sdot sdot 119910119894119904)119879



V2


119906 = (11990611199062




min120579120582

119885 = 120579

st

120579119909119894minus 119883 sdot 120582 ge 0

119910119894minus 119884 sdot 120582 le 0

120582 ge 0

(4)

where 119883 = (11990911199092


(11991011199102


120582 = (12058211205822

sdot sdot sdot 120582119899)119879









119894119894 then the


max 119864119895119894= 119906119879

sdot 119910119895

st

V sdot 119909119895= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899


119894= 0

V 119906 ge 0

(5)







1 120574

119894 120574

119899 = (119909

1 1199101) (119909

119894 119910119894)



119894= (119872119894119864119894119871119894)119879






(i) 120574119895= (119909119895 119910119895) isin Γ and 119909

119894= (119872119894119864119894119871119894) ge 119909119895or

(ii) 119910119894= 119861119894le 119910119895








min120579120582119904+119904minus


st

120579119909119894minus 119883 sdot 120582 minus 119904

+

= 0119910119894minus 119884 sdot 120582 + 119904

minus

= 0

1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899

119904+

119904minus

ge 0

(6)

where 119909119894= (119872

119894119864119894119871119894)119879 and 119910



119883 = (11990911199092


11199102



sdot sdot sdot 120582119899)119879


119894


119894and all




119903is found with






119894inefficient

120579119909119894ge 119909119897

119910119894le 119910119897 119897 = 1 2 119899

(7)




119898in the







119886119864119886119871119886)119879 is not



119887








dominates it





min120579120582119904+119904minus

119885 = 120579 minus 120576 sdot (119904+

119872+ 119904+

119864+ 119904+

119871+ 119904minus

119861)

st


119899) sdot 120582 minus 119904

+

119872= 0


+

119864= 0


119899) sdot 120582 minus 119904

+

119871= 0


minus

119861= 0

sum120582119897= 1 120582

119897isin 0 1 119897 = 1 2 119899

(8)


to

119872119894minus119872119895minus 119904+

119872= 0

119864119894minus 119864119895minus 119904+

119864= 0

119871119894minus 119871119895minus 119904+

119871= 0

119861119894minus 119861119895+ 119904minus

119861= 0

(9)


119895

119864119894ge 119864119895 119871119894ge 119871119895 and 119861



119904+

119871 and 119904minus



119864+ 119904+

119871+ 119904minus

119861) lt 1





have

120579lowast

sdot 119872119894= 119872119895+ 119904+lowast

119872

120579lowast

sdot 119864119894= 119864119895+ 119904+lowast

119864

120579lowast

sdot 119871119894= 119871119895+ 119904+lowast

119871


119861

(10)


119864+ 119904+lowast


119861) lt 1



119895



(1) 119872119894lt 119872119897 or

(2) 119864119894lt 119864119897 or

(3) 119871119894lt 119871119897 or

(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)




119894is efficient



119890



119890 Thus Γ






min120579119897120582119904+119904minus

119885 = sum

119897

120579119897+ 120576 sdot sum

119897

(1 sdot 119904+119897+ 119904minus

119897)

st

120579119897119909119894minus 119909119897sdot 120582119897minus 119904+

119897= 0 119897 = 0 1 119899

(119910119894minus 119910119897) sdot 120582119897+ 119904minus

119897= 0 119897 = 0 1 119899

1 sdot 120582 = 1120582 119904+

119897 119904minus

119897ge 0

(11)


max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(12)





(119906119897sdot 119910119894minus 119911 sdot V

119897sdot 119909119894) minus (119906





119894is 119864119894119894 the FDH



of 119911 in

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899



119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(14)











max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899



119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(15)



119894)


(16)


119894119894



119895119895



using (12)

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(17)





119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894




119894= sum119899





119895


119895119895


119895119894of schedule 120574


119894is




119895119895




120572= (119909

120572 119910120572) be the most




120572be


119894


119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are





120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=



119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1



119897sdot 119909119897+ 119911120573



119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)




120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast





120572 That










120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=




max119906119897V119897119911120573

119911120573

st


119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)



= 119906119897sdot (119910119897minus 119910119894)


+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864


119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max


119897 1199061015840119897 and

1199111015840








120573=










119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119864119873bit

and 119864119871bit








2

)

12

(2)



max 119885 = 119906119879

sdot 119910119894

st

V sdot 119909119894= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899

V 119906 ge 0

(3)

where 119909119894= (119909

11989411199091198942

sdot sdot sdot 119909119894119898)119879

isin 119877119898 and 119910

119894=

(1199101198941

1199101198942

sdot sdot sdot 119910119894119904)119879



V2


119906 = (11990611199062




min120579120582

119885 = 120579

st

120579119909119894minus 119883 sdot 120582 ge 0

119910119894minus 119884 sdot 120582 le 0

120582 ge 0

(4)

where 119883 = (11990911199092


(11991011199102


120582 = (12058211205822

sdot sdot sdot 120582119899)119879









119894119894 then the


max 119864119895119894= 119906119879

sdot 119910119895

st

V sdot 119909119895= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899


119894= 0

V 119906 ge 0

(5)







1 120574

119894 120574

119899 = (119909

1 1199101) (119909

119894 119910119894)



119894= (119872119894119864119894119871119894)119879






(i) 120574119895= (119909119895 119910119895) isin Γ and 119909

119894= (119872119894119864119894119871119894) ge 119909119895or

(ii) 119910119894= 119861119894le 119910119895








min120579120582119904+119904minus


st

120579119909119894minus 119883 sdot 120582 minus 119904

+

= 0119910119894minus 119884 sdot 120582 + 119904

minus

= 0

1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899

119904+

119904minus

ge 0

(6)

where 119909119894= (119872

119894119864119894119871119894)119879 and 119910



119883 = (11990911199092


11199102



sdot sdot sdot 120582119899)119879


119894


119894and all




119903is found with






119894inefficient

120579119909119894ge 119909119897

119910119894le 119910119897 119897 = 1 2 119899

(7)




119898in the







119886119864119886119871119886)119879 is not



119887








dominates it





min120579120582119904+119904minus

119885 = 120579 minus 120576 sdot (119904+

119872+ 119904+

119864+ 119904+

119871+ 119904minus

119861)

st


119899) sdot 120582 minus 119904

+

119872= 0


+

119864= 0


119899) sdot 120582 minus 119904

+

119871= 0


minus

119861= 0

sum120582119897= 1 120582

119897isin 0 1 119897 = 1 2 119899

(8)


to

119872119894minus119872119895minus 119904+

119872= 0

119864119894minus 119864119895minus 119904+

119864= 0

119871119894minus 119871119895minus 119904+

119871= 0

119861119894minus 119861119895+ 119904minus

119861= 0

(9)


119895

119864119894ge 119864119895 119871119894ge 119871119895 and 119861



119904+

119871 and 119904minus



119864+ 119904+

119871+ 119904minus

119861) lt 1





have

120579lowast

sdot 119872119894= 119872119895+ 119904+lowast

119872

120579lowast

sdot 119864119894= 119864119895+ 119904+lowast

119864

120579lowast

sdot 119871119894= 119871119895+ 119904+lowast

119871


119861

(10)


119864+ 119904+lowast


119861) lt 1



119895



(1) 119872119894lt 119872119897 or

(2) 119864119894lt 119864119897 or

(3) 119871119894lt 119871119897 or

(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)




119894is efficient



119890



119890 Thus Γ






min120579119897120582119904+119904minus

119885 = sum

119897

120579119897+ 120576 sdot sum

119897

(1 sdot 119904+119897+ 119904minus

119897)

st

120579119897119909119894minus 119909119897sdot 120582119897minus 119904+

119897= 0 119897 = 0 1 119899

(119910119894minus 119910119897) sdot 120582119897+ 119904minus

119897= 0 119897 = 0 1 119899

1 sdot 120582 = 1120582 119904+

119897 119904minus

119897ge 0

(11)


max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(12)





(119906119897sdot 119910119894minus 119911 sdot V

119897sdot 119909119894) minus (119906





119894is 119864119894119894 the FDH



of 119911 in

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899



119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(14)











max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899



119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(15)



119894)


(16)


119894119894



119895119895



using (12)

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(17)





119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894




119894= sum119899





119895


119895119895


119895119894of schedule 120574


119894is




119895119895




120572= (119909

120572 119910120572) be the most




120572be


119894


119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are





120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=



119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1



119897sdot 119909119897+ 119911120573



119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)




120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast





120572 That










120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=




max119906119897V119897119911120573

119911120573

st


119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)



= 119906119897sdot (119910119897minus 119910119894)


+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864


119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max


119897 1199061015840119897 and

1199111015840








120573=










119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119894 then the


max 119864119895119894= 119906119879

sdot 119910119895

st

V sdot 119909119895= 1

V sdot 119909119897minus 119906 sdot 119910

119897ge 0 119897 = 1 2 119899


119894= 0

V 119906 ge 0

(5)







1 120574

119894 120574

119899 = (119909

1 1199101) (119909

119894 119910119894)



119894= (119872119894119864119894119871119894)119879






(i) 120574119895= (119909119895 119910119895) isin Γ and 119909

119894= (119872119894119864119894119871119894) ge 119909119895or

(ii) 119910119894= 119861119894le 119910119895








min120579120582119904+119904minus


st

120579119909119894minus 119883 sdot 120582 minus 119904

+

= 0119910119894minus 119884 sdot 120582 + 119904

minus

= 0

1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899

119904+

119904minus

ge 0

(6)

where 119909119894= (119872

119894119864119894119871119894)119879 and 119910



119883 = (11990911199092


11199102



sdot sdot sdot 120582119899)119879


119894


119894and all




119903is found with






119894inefficient

120579119909119894ge 119909119897

119910119894le 119910119897 119897 = 1 2 119899

(7)




119898in the







119886119864119886119871119886)119879 is not



119887








dominates it





min120579120582119904+119904minus

119885 = 120579 minus 120576 sdot (119904+

119872+ 119904+

119864+ 119904+

119871+ 119904minus

119861)

st


119899) sdot 120582 minus 119904

+

119872= 0


+

119864= 0


119899) sdot 120582 minus 119904

+

119871= 0


minus

119861= 0

sum120582119897= 1 120582

119897isin 0 1 119897 = 1 2 119899

(8)


to

119872119894minus119872119895minus 119904+

119872= 0

119864119894minus 119864119895minus 119904+

119864= 0

119871119894minus 119871119895minus 119904+

119871= 0

119861119894minus 119861119895+ 119904minus

119861= 0

(9)


119895

119864119894ge 119864119895 119871119894ge 119871119895 and 119861



119904+

119871 and 119904minus



119864+ 119904+

119871+ 119904minus

119861) lt 1





have

120579lowast

sdot 119872119894= 119872119895+ 119904+lowast

119872

120579lowast

sdot 119864119894= 119864119895+ 119904+lowast

119864

120579lowast

sdot 119871119894= 119871119895+ 119904+lowast

119871


119861

(10)


119864+ 119904+lowast


119861) lt 1



119895



(1) 119872119894lt 119872119897 or

(2) 119864119894lt 119864119897 or

(3) 119871119894lt 119871119897 or

(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)




119894is efficient



119890



119890 Thus Γ






min120579119897120582119904+119904minus

119885 = sum

119897

120579119897+ 120576 sdot sum

119897

(1 sdot 119904+119897+ 119904minus

119897)

st

120579119897119909119894minus 119909119897sdot 120582119897minus 119904+

119897= 0 119897 = 0 1 119899

(119910119894minus 119910119897) sdot 120582119897+ 119904minus

119897= 0 119897 = 0 1 119899

1 sdot 120582 = 1120582 119904+

119897 119904minus

119897ge 0

(11)


max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(12)





(119906119897sdot 119910119894minus 119911 sdot V

119897sdot 119909119894) minus (119906





119894is 119864119894119894 the FDH



of 119911 in

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899



119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(14)











max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899



119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(15)



119894)


(16)


119894119894



119895119895



using (12)

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(17)





119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894




119894= sum119899





119895


119895119895


119895119894of schedule 120574


119894is




119895119895




120572= (119909

120572 119910120572) be the most




120572be


119894


119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are





120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=



119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1



119897sdot 119909119897+ 119911120573



119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)




120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast





120572 That










120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=




max119906119897V119897119911120573

119911120573

st


119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)



= 119906119897sdot (119910119897minus 119910119894)


+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864


119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max


119897 1199061015840119897 and

1199111015840








120573=










119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119886119864119886119871119886)119879 is not



119887








dominates it





min120579120582119904+119904minus

119885 = 120579 minus 120576 sdot (119904+

119872+ 119904+

119864+ 119904+

119871+ 119904minus

119861)

st


119899) sdot 120582 minus 119904

+

119872= 0


+

119864= 0


119899) sdot 120582 minus 119904

+

119871= 0


minus

119861= 0

sum120582119897= 1 120582

119897isin 0 1 119897 = 1 2 119899

(8)


to

119872119894minus119872119895minus 119904+

119872= 0

119864119894minus 119864119895minus 119904+

119864= 0

119871119894minus 119871119895minus 119904+

119871= 0

119861119894minus 119861119895+ 119904minus

119861= 0

(9)


119895

119864119894ge 119864119895 119871119894ge 119871119895 and 119861



119904+

119871 and 119904minus



119864+ 119904+

119871+ 119904minus

119861) lt 1





have

120579lowast

sdot 119872119894= 119872119895+ 119904+lowast

119872

120579lowast

sdot 119864119894= 119864119895+ 119904+lowast

119864

120579lowast

sdot 119871119894= 119871119895+ 119904+lowast

119871


119861

(10)


119864+ 119904+lowast


119861) lt 1



119895



(1) 119872119894lt 119872119897 or

(2) 119864119894lt 119864119897 or

(3) 119871119894lt 119871119897 or

(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)




119894is efficient



119890



119890 Thus Γ






min120579119897120582119904+119904minus

119885 = sum

119897

120579119897+ 120576 sdot sum

119897

(1 sdot 119904+119897+ 119904minus

119897)

st

120579119897119909119894minus 119909119897sdot 120582119897minus 119904+

119897= 0 119897 = 0 1 119899

(119910119894minus 119910119897) sdot 120582119897+ 119904minus

119897= 0 119897 = 0 1 119899

1 sdot 120582 = 1120582 119904+

119897 119904minus

119897ge 0

(11)


max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(12)





(119906119897sdot 119910119894minus 119911 sdot V

119897sdot 119909119894) minus (119906





119894is 119864119894119894 the FDH



of 119911 in

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899



119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(14)











max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899



119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(15)



119894)


(16)


119894119894



119895119895



using (12)

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(17)





119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894




119894= sum119899





119895


119895119895


119895119894of schedule 120574


119894is




119895119895




120572= (119909

120572 119910120572) be the most




120572be


119894


119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are





120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=



119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1



119897sdot 119909119897+ 119911120573



119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)




120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast





120572 That










120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=




max119906119897V119897119911120573

119911120573

st


119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)



= 119906119897sdot (119910119897minus 119910119894)


+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864


119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max


119897 1199061015840119897 and

1199111015840








120573=










119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min120579119897120582119904+119904minus

119885 = sum

119897

120579119897+ 120576 sdot sum

119897

(1 sdot 119904+119897+ 119904minus

119897)

st

120579119897119909119894minus 119909119897sdot 120582119897minus 119904+

119897= 0 119897 = 0 1 119899

(119910119894minus 119910119897) sdot 120582119897+ 119904minus

119897= 0 119897 = 0 1 119899

1 sdot 120582 = 1120582 119904+

119897 119904minus

119897ge 0

(11)


max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(12)





(119906119897sdot 119910119894minus 119911 sdot V

119897sdot 119909119894) minus (119906





119894is 119864119894119894 the FDH



of 119911 in

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899



119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(14)











max119906119897V119897119911

119911

st

V119897sdot 119909119894= 1 119897 = 1 2 119899



119894) le 0

119897 = 1 2 119899

119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(15)



119894)


(16)


119894119894



119895119895



using (12)

max119906119897V119897119911

119911

st

V119897sdot 119909119895= 1 119897 = 1 2 119899


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(17)





119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894




119894= sum119899





119895


119895119895


119895119894of schedule 120574


119894is




119895119895




120572= (119909

120572 119910120572) be the most




120572be


119894


119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are





120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=



119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1



119897sdot 119909119897+ 119911120573



119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)




120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast





120572 That










120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=




max119906119897V119897119911120573

119911120573

st


119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)



= 119906119897sdot (119910119897minus 119910119894)


+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864


119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max


119897 1199061015840119897 and

1199111015840








120573=










119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119864cross =(

1198641111986412 1198641119899

1198642111986422 1198642119899

11986411989911198641198992 119864119899119899

) (18)

where 119864119895119894




119894= sum119899





119895


119895119895


119895119894of schedule 120574


119894is




119895119895




120572= (119909

120572 119910120572) be the most




120572be


119894


119897119887 Vlowast119897= (Vlowast119897119872

Vlowast119897119864

Vlowast119897119871) 119897 = 1 2 119899 are





120574120573= (119909120573 119910120573) where 119909

120573= 119909120572= (119872 119864 119871)

119879 and 119910120573=



119894as

followsmax119906119897V119897119911120573

119911120573

st

V119897sdot 119909120573= V119897sdot 119909120572= 1



119897sdot 119909119897+ 119911120573



119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(19)




120573= 119911lowast

120572+ max

119897(119906lowast

119897119861sdot Δ119861) 119906lowast





120572 That










120573= (119909120573 119910120573) where 119910

120573= 119861 and 119909

120573=




max119906119897V119897119911120573

119911120573

st


119897119864sdot 119864 + V

119897119871sdot 119871 = 1


= 119906119897119861sdot (119910119897minus 119910120573)



= 119906119897sdot (119910119897minus 119910119894)


+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)


119906119897ge 120576 V

119897ge 120576 sdot 1 119897 = 1 2 119899

(20)

Let V1015840119897

= (V1015840119897119872

V1015840119897119864

V1015840119897119871) = (Vlowast

119897119872(1 + Δ119872 sdot

Vlowast119897119872))(Vlowast119897119872

Vlowast119897119864


119897119872))119906lowast

119897 and

1199111015840

120573= 119911lowast

120572sdot (1 + Δ119872 sdot max


119897 1199061015840119897 and

1199111015840








120573=










119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119909119897= (119872

119897119864119897119871119897)119879 and 119910



119897119872119906lowast

119897119864119906lowast


119897be the



119897= (119872

1015840

1198971198641015840

1198971198711015840

119897)119879

= (120572119872119872119897120572119864119864119897120572119871119871119897)119879


1205741015840

119897= (1199091015840

119897 1199101015840



119897119872120572119872

119906lowast

119897119864120572119864119906lowast

119897119871120572119871) and



ge



119897= (1199061015840lowast

1198971198721199061015840lowast

1198971198641199061015840lowast

119897119871) and V1015840lowast



119897= (1205721198721199061015840lowast

1198971198721205721198641199061015840lowast

1198971198641205721198711199061015840lowast

119897119871) and V

119897= 120573V1015840119897 which


119897 and V








119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000










DEA-ready pool












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3










chr1

chr2

nchr1

nchr2

1 1

11 11 1

2 2 2

22

3 3 3 3

33

1 1 12 2 2

2

2 3 3

1111 3333

Crossover

Swap


chr1

nchr1

1 12 2 223

1 1 2 2 23 3 3 3

33


Arbitraryposition 2

Mutation












119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119897119897= 1) as Γ

119890





119895of 120574119895


119890






119894= (119864119894119894minus 119864119894)119864119894119894








Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2DEA

analysisresults

of40

samples

chedules

Schedu

leSubp

op

Makespan

(inpu

t1)

Energy

(inpu

t2)

Link

(inpu

t3)

Balance

(outpu

t)CC

RBC

CFD

HSuperC

CRCr

osCC

RCr

osFD

H(A

ll)MI(All)

CrosFD

H(Eff)

MI(Eff

)

1M

09478

09695

09368

09657

06624

10000

10000

06624

06503

09878

00123

09917

00084

2M

09650

10624

10396

08970

05545

09821

09821

05545

05497

09371

004

80mdash

mdash3

M09683

09993

10154

07146

04522

09788

09788

04522

044

9209484

00320

mdashmdash

4M

09693

11701

11363

14241

08520

09933

1000

008520

08065

09050

01050

09123

00961

5M

09696

10624

10295

09227

05759

09775

09775

05759

05694

09368

00435

mdashmdash

6M

09732

10587

10557

06595

040

1409739

09739

040

1403991

09245

00534

mdashmdash

7M

09732

10736

10738

064

5303862

09739

09739

03862

03847

09184

006

05mdash

mdash8

M09748

11070

10759

09227

05511

09723

09723

05511

05472

09181

00590

mdashmdash

9M

09758

09955

09973

07551

04865

09721

09739

04865

04813

094

8800265

mdashmdash

10M

09758

09250

09429

07274

04957

09953

09963

04957

04889

09822

00144

mdashmdash

11E

1004

809064

09368

060

0204143

09918

09918

04143

040

55096

4900279

mdashmdash

12E

1004

809175

09288

060

0204143

09828

09918

04143

040

55096

4900279

mdashmdash

13E

1004

809138

09469

05826

03989

09839

09841

03989

03900

09611

00239

mdashmdash

14E

10103

09138

09409

05487

03757

09837

09837

03757

03689

09573

00276

mdashmdash

15E

10482

09138

09348

07274

05000

09875

10000

05000

04900

09490

00538

09704

00305

16E

1004

809138

09550

05759

03943

09839

09841

03943

03830

09599

00252

mdashmdash

17E

09745

09138

09187

09227

064

5410

000

1000

0064

5406340

09929

00072

09995

000

0518

E09745

09212

09227

09507

06620

09979

1000

006620

06504

09915

00086

09985

00015

19E

09761

09212

09248

09507

066

0609966

1000

0066

0606504

09915

00086

09985

00015

20E

09722

09212

09227

10153

07070

1000

010

000

07070

06947

09934

000

6609992

000

0821

L10

488

09287

09066

08850

06273

10000

10000

06273

06096

09631

00383

09896

00105

22L

09888

08990

09127

06695

04714

10000

10000

04714

04628

09819

00184

09903

000

9823

L10

087

09584

09147

09657

06784

10000

10000

06784

066

0109755

00251

09927

00074

24L

09729

09287

09147

05965

04191

10000

10000

04191

04106

09821

00183

09937

000

6325

L10

488

09435

09227

08734

06082

09850

09956

06082

05921

09550

004

26mdash

mdash26

L09791

09435

09227

08734

06082

09950

09956

06082

05955

09835

00124

mdashmdash

27L

10133

09175

09227

08412

05858

09924

09960

05858

05739

09701

00266

mdashmdash

28L

10133

09250

09248

09507

066

0609932

09978

066

06064

6809743

00242

mdashmdash

29L

09729

09435

09288

06320

04373

09929

09993

04373

04289

09771

00227

mdashmdash

30L

10495

09510

09288

07209

04988

09785

09892

04988

04857

09414

00507

mdashmdash

31B

09709

10550

10759

16743

10000

10000

10000

10013

09991

09463

00568

09535

004

8832

B09722

10475

10759

16743

10000

10000

10000

10071

09999

09476

00553

09550

00471

33B

10504

1166

411665

15996

08831

09220

09243

08831

08785

08682

006

46mdash

mdash34

B10

103

11590

11665

15996

09181

09586

09610

09181

08843

08848

00861

mdashmdash

35B

10221

11367

11403

15996

09076

09476

09499

09076

08996

08907

006

66mdash

mdash36

B10

279

10698

10859

15996

09465

09747

09907

09465

09408

09196

00774

mdashmdash

37B

10279

10698

10879

15341

09061

09590

09889

09061

09009

09185

00766

mdashmdash

38B

10504

11590

11585

15341

08510

09199

09287

08510

08477

08713

00659

mdashmdash

39B

10504

11293

11202

15341

08800

09314

09604

08800

08730

08893

00800

mdashmdash

40B

10534

10996

10799

15341

09129

09661

09963

09129

09011

09078

00975

mdashmdash




Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Averageenergy

Averagelinkload

Averagebalance

AverageMI (All)

M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727






sdot 119864minus1

sdot 119871minus1

sdot 119861

fitnessWS = 119908119872119872minus1

+ 119908119864119864minus1

+ 119908119871119871minus1

+ 119908119861119861


+ 119908119864119864minus2

+ 119908119871119871minus2

+ 1199081198611198612


119872minus1

+ (119890119908119864 minus 1) sdot 119890

119864minus1

+ (119890119908119871 minus 1) sdot 119890

119871minus1

+ (119890119908119861 minus 1) sdot 119890

119861

(21)

where 119908119872 119908119864 119908119871 and 119908














0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0100020003000400050006000

Mak

espa

n (c

ycle

s)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(a) Makespan

14121

080604020

Ener

gy (J

)

times10minus7

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20

(b) Energy

6005004003002001000

Aver

age l

ink

load

(flits

)

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


76543210W

orkl

oad

bala

nce

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20


4

3

2

1

0

CrosFDHMDWS

WESEWC

LEM

DCtg1

tg2

tg3

tg4

tg5

tg6

tg7

tg8

tg9

tg10

tg11

tg12

tg13

tg14

tg15

tg16

tg17

tg18

tg19

tg20log10

(sol

ving

tim

e)

(e) Solving time




Averageenergy

Averagelink load

Averagebalance








40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

30

Solv

ing

time (

s)

Task number

MD-GAWS-GA

WES-GAEWC-GA


40 50 60 70 80 90 100500

1000

1500

2000

2500

3000

3500

4000

Task number

Solv

ing

time o

f DEA

-GA

(s)

(b) CROSFDH-GA





10 20 30 40 50 60 70 80 900

50

100

150

200

250

300


Aver

age s

olvi

ng ti

me (

seco

nd)



0

1

2

3

4

Prop

ortio

n of

effici

ent n

umbe

r ()

45

35

25

15

05




7 Conclusion



0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40 45 50Generations

Nor

mal

ized

met

rics

16

14

12

1

08

06

04

02


Average link load




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a genetic algorithm for task scheduling...

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