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Research ArticleSimulated Annealing Algorithm Combined with Chaos forTask Allocation in Real-Time Distributed Systems

Wenbo Wu Jiahong Liang Xinyu Yao and Baohong Liu

College of Information System and Management National University of Defense Technology Changsha Hunan 410073 China

Correspondence should be addressed to Wenbo Wu just4thingmailcom

Received 17 May 2014 Revised 24 July 2014 Accepted 3 August 2014 Published 14 August 2014

Academic Editor Jun-Juh Yan

Copyright copy 2014 Wenbo Wu 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

This paper addresses the problem of task allocation in real-time distributed systems with the goal of maximizing the systemreliability which has been shown to be NP-hard We take account of the deadline constraint to formulate this problem and thenpropose an algorithm called chaotic adaptive simulated annealing (XASA) to solve the problem Firstly XASA begins with chaoticoptimization which takes a chaotic walk in the solution space and generates several local minima secondly XASA improves SAalgorithm via several adaptive schemes and continues to search the optimal based on the results of chaotic optimization Theeffectiveness of XASA is evaluated by comparing with traditional SA algorithm and improved SA algorithm The results showthat XASA can achieve a satisfactory performance of speedup without loss of solution quality

1 Introduction

In many application domains (eg astronomy genetic engi-neering and military systems) increased complexity andscale has led to the need for more powerful computa-tion resources distributed systems (DS) have emerged asa powerful platform for addressing this issue alternatingto traditional high performance computing systems DSconsists of a set of cooperating nodes (either homogeneousor heterogeneous) communicating over the communicationlinks An application running in a DS could be divided intoa number of tasks and executed concurrently on differentnodes in the system referred to as the task allocation problem(TAP) To improve the performance of DS several studieshave been devoted to the TAP with the main concern onthe performance measures such as minimizing the executionand communication cost [1ndash3] minimizing the applicationturnaround time [4 5] and achieving better fault tolerance[6 7]

On the other hand the real-time property is required inmany DS (eg military systems) In such system the appli-cation should complete its work before deadline not onlypromising the logical correctnessWhile the complexity ofDScould increase the potential of system failure because in such

a large and complex system the nodes and communicationlinks failures are inevitable Hence the reliability is a crucialrequirement for DS especially for the real-time DS (RTDS)

Distributed system reliability (DSR) has been definedby Kumar et al [8] as the probability for the successfulcompletion of distributed programs which requires that allthe allocated processors and involved communication linksare operational during the execution lifetime Redundancyand diversity is the traditional technique to attain betterreliability [6 7 9ndash14]They process hardware andor softwareredundancy hence impose extra cost Moreover in manysituations the system configuration is fixed and we have nofreedom to introduce system redundancy Task allocation isthe alternative way to improve DS reliability and this methoddoes not require additional resources neither hardware norsoftware

The TAP with the goal of maximizing the DSR is atypical combinatorial optimization problem unfortunatelythis has been shown to be NP-hard in strong sense andthe computational complexity of optimal algorithms (egbranch and bound technique) is exponential in nature Wecannot obtain the optimal results in reasonable time for largescale problems Hence several heuristic and metaheuristicalgorithmshave been implemented such as genetic algorithm

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 151394 13 pageshttpdxdoiorg1011552014151394

2 Mathematical Problems in Engineering

(GA) [7 14 15] simulated annealing algorithm (SA) [16]particle swarm optimization (PSO) [17] honeybee matingoptimization (HMO) [18] cat swarm optimization (CSO)[19] and iterated greedy algorithm (IG) [20] These algo-rithms may obtain suboptimal results but they can sharplyreduce the calculation time

A common thread among these algorithms is that they allstart from a randomly chosen initial solution or a set of solu-tions in the solution space and then repeat the exploration-decision procedure until convergence and obtaining goodenough solutions (maybe suboptimal results) [21] Hereexplorationmeans obtaining new solutions based on the cur-rent solution in SA algorithm for instance new solutions arechosen from the neighbors of the current solution Decisionis made after exploration a new solution is either acceptedor rejected according to some rules if the new solution isaccepted then it becomes new current solution and moveon otherwise drop it start new exploration According toa series of the exploration-decision procedures the qualityof the obtained solutions becomes better and better untilmeeting the termination condition which is often defined assome convergent situations Hence the convergence speed ofalgorithms is affected by the choice of initial solutions andrules that are applied in the exploration-decision procedures

Simulated annealing algorithm is one of the earliest andmost wildly used optimization approaches it introducesrandom factor in searching process andmodels the annealingof solids as Metropolis process [22] SA algorithm acceptsldquoworserdquo solution with some probabilities related to the cur-rent temperature so it can escape from the local optima andfind the global optimal solution The convergence speed ofSA algorithm is depending on its initial solution and coolingschedule

Chaos is a bounded unstable dynamic behavior thatexhibits sensitive dependence on initial conditions andincludes infinite unstable periodic motions [23] Althoughit appears to be stochastic it occurs in a deterministicnonlinear system under deterministic conditions In recentyears chaotic optimization algorithm (COA) has arousedintense interests due to its ergodicity easy implementationand ability to escape local optima [24] However COA islack of heuristic mostly needs a large number of iterationsto reach the global optimum which means its convergencespeed is slow

In this paper we propose a combinational algorithmcalledXASA (Chaotic Adaptive SimulatedAnnealing) whereX alludes to the Greek spelling of chaos (120594120572119900120589) and whichis proposed to solve the TAP in RTDS with the goal ofmaximizing the DSR We take into account several kinds ofconstraints including deadline XASA starts from COA andobtains several optima via its ergodicity then SA algorithmwill operate based on these optima in relative smaller rangesto find the best solution This method can overcome theslow convergence of SA algorithm and COA without loss ofsolution quality

The rest of this paper is organized as follows Section 2presents the related work in the application of SA algorithmand Chaotic Optimization to the TAP and the contributionsof this paper are stated Section 3 describes the formulation

of the TAP with the goal of maximizing reliability and thesolution approach is presented in Section 4 with somedetails of implementation Section 5 discusses the perfor-mance evaluation of proposed algorithm according to sev-eral experiments and analyses And Section 6 concludesthis work

2 Related Works

The idea of simulated annealing algorithm was proposed byMetropolis et al [22] in 1953 and was applied to optimizationproblems by Kirkpatrick et al [25] in 1983 To our knowledgethe first application of SA algorithm to the TAP was made byvan Laarhoven et al [26] in 1992 which applied SA algorithmto a job shop scheduling problem From then on severalworks [27ndash29] have been done that compare SA algorithmto other optimization algorithms for problems related tothe TAP Attiya and Hamam applied SA algorithm to solvethe TAP and compared it with branch-and-bound techniquein 2006 in terms of maximizing the reliability of DS [16]extending this work Faragardi et al proposed improved SAalgorithms to solve this problem [30 31] using the hybrid ofSA and tabu search with a nonmonotonic cooling schedulein 2012 and adding systematic search of neighborhood andmemory to SA algorithm in 2013

COA often combines with other optimization algorithmto overcome its drawbacks and take advantage of its beneficialproperty such as ergodicity for example chaotic simulatedannealing (CSA) [32] chaotic particle swarm optimization(CPSO) [23] and chaotic improved imperialist competitivealgorithm (CICA) [33] et al CSA was proposed by Chen andAihara [32] to solve combinatorial optimization problems in1995 which used Hopfield neural networks Then severalother papers have expanded this work [34ndash37] Mingjunand Huanwen have proposed another version of CSA tosolve the optimization problem of continuous functions [38]Most of these works focus on the application of continuousfunctions optimization while the method proposed by Chenand Aihara is actually based on artificial neural network notSA algorithm Ferens and Cook [39] adapted CSA developedby Mingjun and Huanwen into the TAP in 2013 wherechaos was infused into a solution by setting the numberof perturbations made by the value of a chaotic variableHowever this method does notmake full use of the beneficialproperty of COA and does not improve the convergencespeed either

The present paper differs from the above mentionedresearches because it can combine the advantages of both twoalgorithms Firstly with the ergodicity property COA can getthe skeleton of solution space by chaoticwalking in it therebypreventing the result from falling into local optima Secondlybased on the results of COA we can easily determine thecooling schedule of SA algorithm which is very important tothe performance of SA algorithmbut hard to deal with Lastlyseveral adaptive schemes are used in SA algorithm all theseschemes including COA preliminary results can increase theconvergence speed significantly

Mathematical Problems in Engineering 3

Node1

Processor

Processor

Processor

Memory

Memory

Memory

Node2

Node3

S1

S2

S3

M1

M2

M3

(a) A distributed system

10 16 11 14 5 30

[1 2] [2 2]

1217 25

11 8 12[5 3]

[4 1]

[6 4]

4 21 13

t1 t2

t3t4

t5

1

2

2

3

45

(b) A task graph

Figure 1 Results of the evaluation

3 Problem Statement

We consider a heterogeneous DS that runs a real-time appli-cation Each node of DSmay have different processing speedmemory size and failure rate Moreover the communicationlinks may also have different bandwidth and failure rate Thestate of nodes and communication links is either operationalor failed and the failure events are statistically independentWe also assume that the failure rates of nodes and communi-cation links are constant

There are119872 tasks to be executed on a DS with119873 nodesand 119872 gt 119873 for most cases Tasks executed on the noderequire resources including processing load resources andmemory space Additionally two tasks executed on differentnodes require communication bandwidth to communicatewith each other Figure 1 illustrates a simple case consistingof 5 tasks and 3 nodes An application running on DS can berepresented by a task interaction graph 119866(119881 119864) as shown inFigure 1(b) where119881 represents a set of tasks and119864 representsthe interactions between two tasks Each task 119894 isin 119881 isassociated with two properties 119902

1198941 1199021198942 119902

119894119873 represents

the execution time of the task at different node and [119904119894 119898119894]

represents the processing load and memory requirements ofthe task And the label of each edge (119894 119895) isin 119864 represents thecommunication requirements among tasks

The purpose of this work is to find a task assignmentthat all119872 tasks are assigned to 119873 nodes (note that one taskshould be assigned to one and only one node while onenode can execute multitasks or none) so that the overallsystem reliability is maximized the deadline and otherrequirements of tasks are satisfied and the capacities of thesystem resources are not violated

31 Notations The notations that are used to formulate theproblem are listed in Abbreviation Section

32 System Reliability Reliability of a distributed systemmaybe defined as the probability that the system can run the entireapplication successfully [7 14 40] Due to the independenceof the failures of the node andpath the system reliability is theproduct of the components reliabilities Which is the productof the reliabilities of all nodes and communication links

We assume that all components of the node exceptprocessor are perfect which means the reliability of the nodeequals the reliability of its processor The reliability of theprocessor at time 119905 is 119890minus120582119896119905 [16] under a task assignment 119883the total execution time of tasks that are assigned to 119901

119896is

sum119872

119894=1119909119894119896119890119894119896 so the reliability of the node 119901

119896is

119877119896 (119883) = 119890

minus120582119896 sum119872

119894=1119909119894119896119890119894119896 (1)

Similarly the reliability of the communication link 119897119896119887

at time 119905 is 119890minus120593119896119887119905 under a task assignment 119883 the totalcommunication time via 119897

119896119887issum119872119894=1sum119894 =119895119909119894119896119909119895119887(119888119894119895120596119896119887) so the

reliability of the communication link 119897119896119887is

119877119896119887 (119883) = 119890

minus120593119896119887 sum119872

119894=1sum119894 =119895119909119894119896119909119895119887(119888119894119895120596119896119887) (2)

Hence we obtain the reliability of the system

119877 (119883) =

119873

prod

119896=1

119877119896 (119883)

119873minus1

prod

119896=1

119873

prod

119887=119896+1

119877119896119887 (119883) = 119890

minus119884(119883) (3)

where

119884 (119883) =

119873

sum

119896=1

119872

sum

119894=1

120582119896119909119894119896119890119894119896

+

119873

sum

119896=1

119873

sum

119887=119896+1

119872

sum

119894=1

sum

119894 =119895

120593119896119887119909119894119896119909119895119887(119888119894119895

120596119896119887

)

(4)

33 Constraints In order to achieve a satisfactory allocationthere are several basic constraints of the TAP in RTDSthat should be met Traditionally the aim of allocationconstraints is devoted to not violate the availability of thesystem resources like memory capacity While the deadlinerequirement should be met for real-time property as well

(i) Memory ConstraintThe total amount of memory require-ments of tasks assigned to a node should not exceed thecapacity of the node That is

119872

sum

119894=1

119898119894119909119894119896le 119872119896 forall119896 isin [1119873] (5)


(ii) Computation Resource Constraints The total amount ofcomputation resource requirements of tasks assigned to anode should not exceed the capacity of the node That is

119872

sum

119894=1

119904119894119909119894119896le 119878119896 forall119896 isin [1119873] (6)

(iii) Communication Resource Constraints The total amountof communication resource requirements of tasks via acommunication link should not exceed the capacity of thelink That is

119872

sum

119894=1

sum

119895 =119894

119888119894119895119909119894119896119909119895119887le 119862119896119887 1 le 119896 lt 119887 le 119873 (7)

(iv)DeadlineConstraintsAll tasks should complete executionbefore their deadline Since there is no priority of tasks alltasks that are assigned to a node can execute in any order Weshould take account of the worst case that is considering taskalways executed at last Hence the deadline constraints is

119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896le 119889119894 forall119894 isin [1119872] (8)

34 Problem Formulation According to the above discus-sion we can tell that maximizing the reliability of RTDS isequivalent to minimizing the object function 119884(119883) with allconstraints mentioned before Hence we can formulate theTAP by the following combinatorial optimization problem

min 119884 (119883)

subject to (5) sim (8)

(9)

4 Task Allocation Solution

This section describes basic SA algorithm briefly at first withthe discussion of the cooling schedule which has a significanteffect on its convergence speed then presents the XASA andexplains the details of how it can be applied into the TAP interms of the statement proposed in this paper

41 Basic Simulated Annealing Algorithm The SA algorithmstarts from a randomly chosen initial solution and generatesa series of Markov chains according to the descent ofthe control parameter (ie temperature) In these Markovchains a new solution is chosen by making a small randomperturbation of the solution and if the new solution is betterthen it is kept but if it is worse it is kept with some probabilityrelated to the current temperature and the difference betweenthe new solution and the previous solution According to aseries of iteration of solutions an optimal one was foundTheSA algorithm applied in the TAP is listed as follows

Step 1 Choose an initial task arrangement (119883119904) at random

Step 2 Calculate the cost (119891119904) of119883

119904

Step 3 Set the initial solution as the optimal 119891lowast larr 119891119904119883lowast larr

119883119904

Step 4 Initialize the temperature (119879 = 1198790)

Step 5 Select a neighbor (119883119899) of119883

119904


119899

Step 7 If 119891119899le 119891119904 then 119883

119904larr 119883119899and 119891

119904larr 119891119899 otherwise go

to Step 9

Step 8 If119891119899lt 119891lowast then119883lowast larr 119883

119899and119891lowast larr 119891

119899 go to Step 10

Step 9 If random(0 1) lt 119890minus(119891119899minus119891119904)119879 then119883119904larr 119883119899and119891119904larr

119891119899

Step 10 Repeat Step 5 to Step 9 for a given number ofiterations

Step 11 Reduce the temperature via some cooling function119879 = 119891(119879)

Step 12 If the termination condition is satisfied (eg 119879 le 119879119891)

then go to Step 13 otherwise go to Step 5

Step 13 Output the solution

Note that the neighborhood defines the procedure tomove from a solution point to another solution point [16]In this paper a neighbor is obtained by randomly choosing atask 119894 among119872 tasks and replacing its current assigned nodewith another randomly selected one

42 Cooling Schedule It has been shown that the SA algo-rithm converges to the global optimal with probability 1 [41]which needs a sufficiently slow cooling schedule (ie suffi-cient hot initial temperature sufficient low final temperatureand sufficient slow cooling speed) However the requiredslow cooling schedule may lead to an unacceptably longsolution convergence time which can be exponential

The cooling schedule is a set of parameters whichcontrols the procedure of SA algorithm so that it can beasymptotic converge to a suboptimal in reasonable time Thecooling schedule is made up of these parameters

(i) The Initial Value of the Control Parameter (ie Tem-perature) 119879

0 The initial temperature represents one of the

most important parameters in SA algorithm If the initialtemperature is very high it will take very long time to beconvergent On the other hand poor solutions are obtainedif the initial temperature is low A basic principle of choosinginitial temperature is that the acceptance probability ofworse solutions is close to 1 Which means the exchangingof neighboring solutions should be almost freely at firstHence we can determine the initial temperature 119879

0via initial

acceptance probability of worse solution 1198750 In this paper 119875

0

is set to be 09 1198750= 119890minusΔ1198790 where Δ is the difference between

the neighboring solutions so 1198790= minusΔ ln119875

0 We can use the


Δ = 119891max minus 119891min as the estimation of Δ where 119891max and 119891minis the maximum and minimum of energy function amongrandomly chosen 119870 solutions So the initial temperature 119879

0

is determined by the following formula

1198790=119891min minus 119891max

ln1198750

(10)

(ii) The Cooling Function 119865(119879) The cooling function definesthe coolingmethod of temperature119879 A commonly used typeof cooling function is exponential descent function 119865(119879) =120572119879 where 120572 is constant and 120572 lt 1 So that the cooling speeddepends on the parameter 120572 and we call it the cooling factorA large value of 120572 represents slow cooling which yields goodsolution but expensive in time 120572 is set to be slightly less than1 in the most reported literatures and it is chosen to be 095in this paper

(iii) The Final Value of Control Parameter 119879119891 Termination

Condition in AnotherWordThe criterion for termination canbe either final temperature or steady state of the system Theformer one can control the total calculation time but notthe solution quality a given final temperature may introducecalculation redundancy for small scale problem but obtainpoor solution for large scale problem On the other hand thelatter one can take into account both time and quality In thispaper the SA algorithmwill terminate if the solution remainsunchanged (neither upgrading nor downgrading) for a givennumber of iterations The number of iterations that solutionremains unchanged is chosen to be 119872 times 119873 Furthermorethe validation of the final solution should be satisfied as wellwhich will be discussed later

(iv) The Length of Markov Chain 119871119896 It is the number of inner

loop repetitionsThis parameter is chosen to be119872times (119873minus 1)which is the size of solution neighborhood because each taskcan be assigned to another119873 minus 1 nodes

The cooling schedule has a significant effect on theresults of the algorithm especially on the convergence speedBesides the initial solution of SA algorithm can affect theconvergence speed as well

43 Chaotic Optimization Algorithm The chaotic variablesare produced by the following well-known one-dimensionallogistic map

119911119896+1

= 120583119911119896(1 minus 119911

119896) 119896 = 0 1 (11)

where 120583 = 4 119911119896isin [0 1] The logistic map has special

characters such as the ergodicity stochastic property andsensitivity dependence on initial conditions The chaoticoptimization algorithm applied in this paper is listed asfollows

Step 1 Initialize the chaotic vector Z at random note thatthe value of chaotic variables cannot be 0 025 05 075 and10 which are the fixed points of logistic map and all chaoticvariables are different from each other

Start

Generate K chaotic vectors at randomeach vector contains M variables

Calculate corresponding K solutions A and their costs

Search the solution space via COA

The new solution s is better Nothan the worst one of A

YesReplace the worst solution of A by s

Termination of COA No

YesDetermine the cooling schedule of SA

according to K optimized solutions

Choose the best one of K optimizedsolutions to be the initial solution of SA

Search the optimal solution by SA

End

Figure 2 The flowchart of XASA

Step 2 Generate the solution vector A via Z then generatethe task assignment 119883 via A and calculate the cost function119891

Step 3 Set the initial solution as the optimal 119891lowast = 119891 Zlowast = Z

Step 4 Calculate new chaotic vector Z via formula [9]

Step 5 Generate119883 as Step 2 calculate the cost function 119891

Step 6 If 119891 lt 119891lowast then 119891lowast = 119891 Zlowast = Z

Step 7 Repeat Steps 4 5 and 6 until 119891lowast remains unchangedfor a given number of iterations

Note that the iteration number in Step 7 is chosen to beas same as SA algorithm which is discussed before while thevalidation of the solution is not required in COA since it isnot heuristic and it will take very long time to get convergenceif there are few valid solutions in large scale cases

44 Simulated Annealing Algorithm Combined with ChaosThebasic idea of our proposed algorithm is simulated anneal-ing combined with chaotic search and adding some adaptiveschemes to the cooling schedule so that we can improvethe convergence speed without loss of solution quality Theflowchart of XASA is shown in Figure 2


There are 4 schemes to speed up the convergence of SAalgorithm

First we apply COA to find 119870 optimized solutions which isa preliminary search in the solution space and the solutiondistribution can be found according to the ergodicity ofchaos system Hence we can search optimal solution via SAalgorithm in a relative smaller range with optimized initialsolution

Second the initial temperature 1198790can also be smaller based

on the results of COA because we can replace the 119891max and119891min with that of 119870 optimized solutions

Third the length of Markov chain 119871119896is constant in SA

algorithm while it is adaptive in XASA The algorithm willjump out of the inner loop if the rejections of new solutionexceed a given threshold 120579 The threshold is given by theformula 120579 = min(119871

119896times1205751 120579times1205752) at each temperature Because

1198750is set to be 09 we can set the initial value of 120579 to be

lceil119871119896times005rceil 120575

1represents themaximum threshold and120575

1lt 1

in this paper it is set to be 06 1205752represents the increasing

speed of 120579 this is similar to the cooling factor 120572 so it is set tobe 105

Fourth the cooling factor 120572 is adaptive in XASA as well themore solutions are accepted (both better andworse solutions)at the current temperature the smaller the cooling factoris and vice versa The rationale is that high temperature atthe beginning of the algorithm generates numerous solutionacceptances thus a rapid reduction of temperature can bemade while fewer solutions will be accepted as the temper-ature is cooling down thus a slow cooling speed should beapplied since we need carefully a solution search Hence thecooling factor of XASA is 1205721015840 = 120572 times 119890

minus120581(120581+4times119899) where 120581 isthe acceptance number of last inner loop and 119899 is the actuallength of Markov chains Note that 120581 isin [0 119899] so 119890minus120581(120581+4times119899) isin[08187 1] and 1205721015840 isin [07778 095]

To implement the algorithm some details should bepresented as follows

(1) Solution Representation In this paper solutions are pre-sented with a vector A(M1) each element represents a taskits value is between 1 and 119873 denoting which node this taskis assigned to In order to apply COA we use a chaotic vectorZ related to A(M 1) where A = [Z times (119873 minus 1) + 1] A taskassignment119883 of the TAP is generated by 119860 as follows

119883 (119894 119896) = 1 119896 = 119860 (119894) 119894 = 1 2 119872

0 119896 = 119860 (119894) 119896 = 1 2 119873(12)

(2) Energy Function We integrate the object function 119884(119883)and all constraints into a cost function to fit the SA algorithmframework And the cost function is used as the energyfunction

All constraints are formulated as penalty functions asfollows

119864119872=

119873

sum

119896=1

max(0119872

sum

119894=1

119898119894119909119894119896minus119872119896)

119864119878=

119873

sum

119896=1

max(0119872

sum

119894=1

119904119894119909119894119896minus 119878119896)

119864119862=

119873

sum

119896=1

119873

sum

119887=119896+1

max(0119872

sum

119894=1

sum

119894 =119895

119888119894119895119909119894119896119909119895119887minus 119862119896119887)

119864119863=

119872

sum

119894=1

max(0119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896minus 119889119894)

(13)

As all constraints are of the same importance we use acommon coefficient 120574 for all penalty function Hence theenergy function is

119891 (119883) = 119884 (119883) + 120574 (119864119872 + 119864119878 + 119864119862 + 119864119863) (14)

The criterion of choosing penalty function coefficient 120574 isthat it should scale the values of the penalty functions to thecomparable values as that of object function 119884(119883) such thatthe procedures of the algorithm will be toward the directionof penalty avoiding hence the valid solution can be foundwith high probability Besides the validation of a solution canbe represented as 119891(119883) = 119884(119883)

5 Performance Evaluation

To evaluate the performance of the proposed algorithm bothSA and XASA are coded in Matlab and tested for numerousrandomly generated task sets that are allocated onto a RTDSThere are two variations of SA algorithm implemented in thispaper the traditional one (SA1) and the improved one (SA2)SA2 applies the last two adaptive schemes of XASA thatis adaptive length of Markov chains and cooling factor Allother components of SA2 are as same as SA1 including initialsolution initial temperature and termination conditionTheused computation system is Matlab 7110 with Intel Core i7-2600 340GHz and 16Gbmain memory under aWindows7 environment

51 Experiment Parameters Settings All DS parameters arefollowed by the former researches [16 17 40] The failurerates of processors and communication links are given inthe ranges [000005ndash000010] and [000015ndash000030] respec-tively The time of processing a task at different processors isgiven in the range [15ndash25] The memory requirement of eachtask and node memory capacity is given in the range [1ndash10]and [100ndash200] respectively The task processing load versusnode processing capacity is given in the ranges [1ndash50] and[100ndash300] The value of data to be communicated betweentasks is given in the range [5ndash10] The bandwidth and loadcapacity of communication links are given in the ranges [1ndash4]and [100ndash200] The range of task deadline value is [10ndash200]


Table 1 Simulation results for the case of 12 nodes

Cases XASA SA1 SA2Δ1199051

Δ1199052

Δ1199053

Δ1198771

Δ1198772

119873 119872 119877avg 119905avg 119877avg 119905avg 119877avg 119905avg

12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027


Cases XASA SA1 SA2119873 119872 119877std 119905std 119881

11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653

The network topology is star 119873 is set to be 12 and 25and 119872 is set to be [16 18 20 22 25] and [40 45 50 55 60]for two cases respectivelyThe coefficient of penalty function120574 is set to be 1 The number of randomly chosen solutionsat the beginning of the algorithm is set to be 119870 = 10 Andthe initial solution of the two SA algorithms is one of these119870 solutions which is chosen at random Because SA is astochastic algorithm each independent run of the algorithmon a same applicationmay yield different result we thus runall of the three algorithms on an application 10 times andobtain the average values

52 Experiment Results Table 1 summarizes the reliabilityand calculation time of all TAPs by deploying the XASAand other two SA algorithms The title 119877 with suffix avgrepresentsReliabilitywith the average value of 10 independentruns and 119879 represents Time where the unit is second Δ119905

1

is the acceleration ratio of XASA versus SA1 where Δ1199051=

(119905SA1 minus 119905XASA)119905SA1 times 100 Δ1199052and Δ119905

3is the acceleration

ratio of XASA versus SA2 and SA2 versus SA1 respectivelyΔ119877 represents the average deviation in percentage betweenXASA and SA algorithms in terms of reliability where Δ119877

119894=

(119877SAi minus 119877XASA)119877SAi times 100 119894 = 1 2

The comparative results from Table 1 show that XASAcan sharply reduce the convergence time against the othertwo SA algorithms while the solution quality (ie reliability)is slightly worse (less than 001 in terms of value and 1in percentage) Note that the value of Δ119905

3is steady in the

range of 72 sim 75 with small variation which means thethird and fourth adaptive schemes take a fixed effect on theSA algorithm Furthermore Δ119905

1is steady in all cases except

the last one (119873 = 25 119872 = 60) as well which has anaverage value of 9266 with standard deviation 01036 Inour preliminary experiments the value of function 119884(119883) isalways below 2 during the whole algorithm while the valueof the penalty function can be hundreds Besides accordingto the experiment parameters set in this paper there is nosignificant difference between nodes nor do the tasks Hencethere are lots of local minima in the solution space withoutconsidering the validation of solutions Constraints are easyto be satisfied when the problem scale is small so COA canobtain several valid solutions and the initial temperature forthe SA algorithm in the next step of XASA can be relativelysmall therefore it is fast to get convergence On the otherhand COA can hardly obtain valid solutions in the case oflarge scale (eg 119873 = 25 119872 = 60 in this paper) if thereare insufficient valid solutions (less than 119870) a large initial


0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104

Figure 3 Energy function at each iteration for119873 = 12119872 = 20

temperature will be set because of the giant value of penaltyfunction this will affect the convergence speed significantlyAdditionally constraints are not so easy to be satisfied whenproblem scale is large so there are much less local minima inthe solution space this will slow down the convergence speedas well All these factors weaken the speedup effect based onCOA so the performance in the last case is not so good

Table 2 shows some overall statics characters of all threealgorithms Where the 119877std and 119905std represent the standarddeviation of reliability and calculation time 119881

1represents

the valid solutions in percentage of COA in XASA and 1198812

represents that of inSA (inSA represents the SA algorithm inXASA) the other two119881 columns have the same meaning Aswe can see XASA is the best in terms of mean value of timestandard deviation and there is no case that SA1 excels XASAin this criterion119881

2gets the best result compared to other two

algorithms aswell Note that1198811shows poor results in the large

cases this is an evidence for our analysis of the large scaleissue presented before

53 Time Series Analysis Figure 3 shows the values of theenergy function which are calculated at each iteration of thealgorithms for the case119873 = 12119872 = 20 Note that the valuesof the invalid solutions are set to be 022 because the realvalues of the invalid solutions are too large and they mayconceal the details of the valid ones

As we can see COA cannot guarantee the validation ofthe solutions it is completely stochastic without heuristicHowever it can generate a good start for the next step ofXASA which is shown in the results of inSA where allsolutions are valid and the convergence speed is quite fastThe inSA begins to reach to the good enough solutions before2000 iterations The other two SA algorithms start with theworse condition and spend much more iterations to reachto the good enough solutions They both need thousandsof iterations to explore the solution space which generateslots of invalid solutions as COA but the cost is much moreexpensive Because of the adaptive schemes SA2 can quicklypass through invalid solutions as can be seen in Figure 3Hence these schemes are effective


0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581

(a) Inner iteration characters of inSA

0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581

(b) Inner iteration characters of SA2

0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or

(c) Cooling factor of inSA

0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or

(d) Cooling factor of SA2

0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA

(e) Temperature of inSA

0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA

(f) Temperature of SA2

Figure 4 Cooling schedule characters for119873 = 12119872 = 20


0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104


Figure 4 shows the details of the adaptive cooling sched-ule schemes which are calculated at each cooling step ofthe algorithms for the case 119873 = 12 119872 = 20 The leftthree figures are the results of inSA and right ones are theresults of SA2 Note that the temperature in Figures 4(c) and4(f) is set to be unitary At the beginning of the coolingsteps the acceptance rates of the new solutions are both highin two algorithms hence the cooling factor is small andthe temperature reduces rapidly while the actual length ofMarkov chains 119899 is much smaller in inSA than SA2 andthe cooling factor increases faster as well It is caused bythe differences of the initial temperature and initial solutionsince inSA and SA2 are actually the same Hence COA cantruly improve the convergence speed

Figure 5 shows the details of the case 119873 = 25 119872 = 60

where the invalid solution values are set to be 17 The resultof COA is quite bad only three valid solutions are foundtherefore inSA cannot get a good start and it has to explorewide solution space at the beginning which generates lots ofinvalid solutions

Figure 6 shows the results of the case119873 = 25119872 = 60 asFigure 4The cooling factor and temperature curve are not sodifferent between inSA and SA2 and the character 119899 cannotbring as much benefit as before These cause the inefficientsituation of the largest case

6 Conclusions

In this paper we consider a heterogeneousDS that runs a real-time application to achieve maximization of system reliabil-ity with task allocation technique By formulating the reliabil-ity and constraints wemodel this problem as a combinatorialproblem To solve this problem with fast convergence speedwe improve the well-known simulated annealing algorithmbased on the analysis of the cooling schedule of SA algorithmwhich has a significant effect on its convergence speedThen we propose the algorithm XASA which combines SAalgorithm with chaotic optimization algorithm with severaladaptive schemes The experimental results show that the


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




proposed algorithm can achieve a satisfactory performanceof speedup while the solution quality is just slightly worse

Notations

119872 Number of tasks119873 Number of nodes119901119896 Node 119896

119905119894 Task 119894119897119896119887 Communication link between 119901

119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896

119890119894119896 Execution time of 119905

119894on 119901119896

119888119894119895 Communication cost between 119905

119894and 119905119895

119862119896119887 Communication capacity of 119897

119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887

119898119894 Memory required by 119905

119894

119872119896 Memory capacity of 119901

119896

119904119894 Processing load of 119905

119894

119878119896 Processing capacity of 119901

119896

119889119894 Deadline of 119905

119894

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous referees and the editor fortheir valuable comments and suggestions This work is sup-ported by the National Natural Science Foundation of China(Grant no 61374185)

References

[1] C H Lee and K G Shin ldquoOptimal task assignment in homo-geneous networksrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 8 no 2 pp 119ndash129 1997

[2] T P Ajith and C S R Murthy ldquoOptimal task allocation indistributed systems by graph matching and state space searchrdquoJournal of Systems and Software vol 46 no 1 pp 59ndash75 1999

[3] G Attiya and Y Hamam ldquoStatic task assignment in distributedcomputing systemsrdquo in Proceedings of the 21st IFIP TC7 Confer-ence on System Modeling and Optimization 2003

[4] M Kafil and I Ahmad ldquoOptimal task assignment in heteroge-neous distributed computing systemsrdquo IEEE Concurrency vol6 no 3 pp 42ndash51 1998

[5] G Attiya and Y Hamam ldquoOptimal allocation of tasks ontonetworked heterogeneous computers using minimax criterionrdquoin Proceedings of the International Network Optimization Con-ference (INOC rsquo03) Paris France 2003

[6] S Kartik and C S R Murthy ldquoImproved task-allocation algo-rithms tomaximize reliability of redundant distributed comput-ing systemsrdquo IEEE Transactions on Reliability vol 44 no 4 pp575ndash586 1995

[7] C Hsieh ldquoOptimal task allocation and hardware redundancypolicies in distributed computing systemsrdquo European Journal ofOperational Research vol 147 no 2 pp 430ndash447 2003

[8] V K P Kumar S Hariri and C S Raghavendra ldquoDistributedprogram reliability analysisrdquo IEEE Transactions on SoftwareEngineering vol 12 no 1 pp 42ndash50 1986

[9] S M Shatz and J Wang ldquoModels and algorithms for reliability-oriented task-allocation in redundant distributed-computersystemsrdquo IEEE Transactions on Reliability vol 38 no 1 pp 16ndash27 1989

[10] A Kumar and D P Agrawal ldquoGeneralized algorithm for eval-uating distributed-program reliabilityrdquo IEEE Transactions onReliability vol 42 no 4 pp 416ndash426 1993

[11] P A Tom and C S R Murthy ldquoAlgorithms for reliability-oriented module allocation in distributed computing systemsrdquoJournal of Systems and Software vol 40 no 2 pp 125ndash138 1998

[12] C C Chiu Y S Yeh and J S Chou ldquoA fast algorithm for reli-ability-oriented task assignment in a distributed systemrdquo Com-puter Communications vol 25 no 17 pp 1622ndash1630 2002

[13] A O Charles Elegbede C Chu K H Adjallah and F YalaouildquoReliability allocation through cost minimizationrdquo IEEE Trans-actions on Reliability vol 52 no 1 pp 106ndash111 2003

[14] C Hsieh and Y Hsieh ldquoReliability and cost optimizationin distributed computing systemsrdquo Computers amp OperationsResearch vol 30 no 8 pp 1103ndash1119 2003

[15] D P Vidyarthi and A K Tripathi ldquoMaximizing reliability ofdistributed computing system with task allocation using simplegenetic algorithmrdquo Journal of Systems Architecture vol 47 no 7pp 549ndash554 2001

[16] G Attiya and Y Hamam ldquoTask allocation for maximizing reli-ability of distributed systems a simulated annealing approachrdquoJournal of Parallel andDistributed Computing vol 66 no 10 pp1259ndash1266 2006

[17] P Yin S S Yu P P Wang and Y T Wang ldquoTask allocationfor maximizing reliability of a distributed system using hybridparticle swarm optimizationrdquo Journal of Systems and Softwarevol 80 no 5 pp 724ndash735 2007

[18] Q M Kang H He H M Song and R Deng ldquoTask allocationfor maximizing reliability of distributed computing systemsusing honeybee mating optimizationrdquo Journal of Systems andSoftware vol 83 no 11 pp 2165ndash2174 2010

[19] R Shojaee H R Faragardi S Alaee andN Yazdani ldquoA newCatSwarm Optimization based algorithm for reliability-orientedtask allocation in distributed systemsrdquo in Proceedings of the 6thInternational Symposium on Telecommunications (IST rsquo12) pp861ndash866 IEEE Tehran Iran November 2012

[20] Q M Kang H He and J Wei ldquoAn effective iterated greedyalgorithm for reliability-oriented task allocation in distributedcomputing systemsrdquo Journal of Parallel and Distributed Com-puting vol 73 no 8 pp 1106ndash1115 2013

[21] C BlumandARoli ldquoMetaheuristics incombinatorial optimiza-tion overview and conceptual comparisionrdquo IEEE Transactionson Evolutionary Computation vol 32 no 1 pp 48ndash77 2003

[22] N Metropolis A W Rosenbluth M N Rosenbluth A HTeller and E Teller ldquoEquation of state calculations by fastcomputing machinesrdquo The Journal of Chemical Physics vol 21no 6 pp 1087ndash1092 1953

[23] B Liu L Wang Y Jin F Tang and D Huang ldquoImprovedparticle swarm optimization combined with chaosrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1261ndash1271 2005

[24] L Wnag D Z Zheng and Q S Lin ldquoSurvey on chaotic opti-mization methodsrdquo Computing Technology and Automationvol 20 no 1 pp 1ndash5 2001 (Chinese)

[25] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983


[26] P J M van Laarhoven E H L Aarts and J K Lenstra ldquoJobshop scheduling by simulated annealingrdquo Operations Researchvol 40 no 1 pp 113ndash125 1992

[27] P Eles Z Peng K Kuchcinski and A Doboli ldquoSystem levelhardwaresoftware partitioning based on simulated annealingand tabu searchrdquoDesign Automation for Embedded Systems vol2 no 1 pp 5ndash32 1997

[28] T Wiangtong P K Cheung and W Luk ldquoComparing threeheuristic search methods for functional partitioning in hard-ware-software codesignrdquo Design Automation for EmbeddedSystems vol 6 no 4 pp 425ndash449 2002

[29] F Ferrandi P L Lanzi C Pilato D Sciuto and A TumeoldquoAnt colony heuristic for mapping and scheduling tasks andcommunications on heterogeneous embedded systemsrdquo IEEETransactions on Computer-Aided Design of Integrated Circuitsand Systems vol 29 no 6 pp 911ndash924 2010

[30] H R Faragardi R Shojaee and N Yazdani ldquoReliability-awaretask allocation in distributed computing systems using hybridsimulated annealing and tabu searchrdquo in Proceedings of the IEEE9th International Conference on High Performance Computingand Communication pp 1088ndash1095 IEEE Liverpool UK 2012

[31] H R Faragardi R Shojaee M A Keshtkar and H TabanildquoOptimal task allocation for maximizing reliability in dis-tributed real-time systemsrdquo inProceedings of the IEEEACIS 12thInternational Conference on Computer and Information Science(ICIS rsquo13) pp 513ndash519 IEEE Niigata Japan 2013

[32] L Chen and K Aihara ldquoChaotic simulated annealing by aneural network model with transient chaosrdquo Neural Networksvol 8 no 6 pp 915ndash930 1995

[33] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[34] L Chen and K Aihara ldquoCombinatorial optimization by chaoticdynamicsrdquo in Proceedings of the IEEE International Confer-ence on Systems Man and Cybernetics pp 2921ndash2926 IEEEOrlando Fla USA October 1997

[35] L Wang and K Smith ldquoOn chaotic simulated annealingrdquo IEEETransactions on Neural Networks vol 9 no 4 pp 716ndash718 1998

[36] L Wang and F Tian ldquoNoisy chaotic neural networks forsolving combinatorial optimization problemsrdquo in Proceedings ofthe IEEE-INNS-ENNS International Joint Conference on NeuralNetworks (IJCNN rsquo00) vol 4 pp 37ndash40 IEEE Como Italy2000

[37] KMasuda and E Aiyoshi ldquoSolution to combinatorial problemsby using chaotic global optimization method on a simplexrdquo inProceedings of the 41st SICE Annual Conference pp 1313ndash1318IEEE 2002

[38] J Mingjun and T Huanwen ldquoApplication of chaos in simulatedannealingrdquo Chaos Solitons and Fractals vol 21 no 4 pp 933ndash941 2004

[39] K Ferens and D Cook ldquoChaotic walk in simulated annealingsearch space for task allocation in a multiprocessing systemrdquoInternational Journal of Cognitive Informatics and Natural Intel-ligence vol 7 no 3 pp 58ndash79 2013

[40] S Kartik and C S R Murthy ldquoTask allocation algorithms formaximizing reliability of distributed computing systemsrdquo IEEETransactions on Computers vol 46 no 6 pp 719ndash724 1997

[41] BHajek ldquoCooling schedules for optimal annealingrdquoMathemat-ics of Operations Research vol 13 no 2 pp 311ndash329 1988

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


(GA) [7 14 15] simulated annealing algorithm (SA) [16]particle swarm optimization (PSO) [17] honeybee matingoptimization (HMO) [18] cat swarm optimization (CSO)[19] and iterated greedy algorithm (IG) [20] These algo-rithms may obtain suboptimal results but they can sharplyreduce the calculation time

A common thread among these algorithms is that they allstart from a randomly chosen initial solution or a set of solu-tions in the solution space and then repeat the exploration-decision procedure until convergence and obtaining goodenough solutions (maybe suboptimal results) [21] Hereexplorationmeans obtaining new solutions based on the cur-rent solution in SA algorithm for instance new solutions arechosen from the neighbors of the current solution Decisionis made after exploration a new solution is either acceptedor rejected according to some rules if the new solution isaccepted then it becomes new current solution and moveon otherwise drop it start new exploration According toa series of the exploration-decision procedures the qualityof the obtained solutions becomes better and better untilmeeting the termination condition which is often defined assome convergent situations Hence the convergence speed ofalgorithms is affected by the choice of initial solutions andrules that are applied in the exploration-decision procedures

Simulated annealing algorithm is one of the earliest andmost wildly used optimization approaches it introducesrandom factor in searching process andmodels the annealingof solids as Metropolis process [22] SA algorithm acceptsldquoworserdquo solution with some probabilities related to the cur-rent temperature so it can escape from the local optima andfind the global optimal solution The convergence speed ofSA algorithm is depending on its initial solution and coolingschedule

Chaos is a bounded unstable dynamic behavior thatexhibits sensitive dependence on initial conditions andincludes infinite unstable periodic motions [23] Althoughit appears to be stochastic it occurs in a deterministicnonlinear system under deterministic conditions In recentyears chaotic optimization algorithm (COA) has arousedintense interests due to its ergodicity easy implementationand ability to escape local optima [24] However COA islack of heuristic mostly needs a large number of iterationsto reach the global optimum which means its convergencespeed is slow

In this paper we propose a combinational algorithmcalledXASA (Chaotic Adaptive SimulatedAnnealing) whereX alludes to the Greek spelling of chaos (120594120572119900120589) and whichis proposed to solve the TAP in RTDS with the goal ofmaximizing the DSR We take into account several kinds ofconstraints including deadline XASA starts from COA andobtains several optima via its ergodicity then SA algorithmwill operate based on these optima in relative smaller rangesto find the best solution This method can overcome theslow convergence of SA algorithm and COA without loss ofsolution quality

The rest of this paper is organized as follows Section 2presents the related work in the application of SA algorithmand Chaotic Optimization to the TAP and the contributionsof this paper are stated Section 3 describes the formulation

of the TAP with the goal of maximizing reliability and thesolution approach is presented in Section 4 with somedetails of implementation Section 5 discusses the perfor-mance evaluation of proposed algorithm according to sev-eral experiments and analyses And Section 6 concludesthis work

2 Related Works

The idea of simulated annealing algorithm was proposed byMetropolis et al [22] in 1953 and was applied to optimizationproblems by Kirkpatrick et al [25] in 1983 To our knowledgethe first application of SA algorithm to the TAP was made byvan Laarhoven et al [26] in 1992 which applied SA algorithmto a job shop scheduling problem From then on severalworks [27ndash29] have been done that compare SA algorithmto other optimization algorithms for problems related tothe TAP Attiya and Hamam applied SA algorithm to solvethe TAP and compared it with branch-and-bound techniquein 2006 in terms of maximizing the reliability of DS [16]extending this work Faragardi et al proposed improved SAalgorithms to solve this problem [30 31] using the hybrid ofSA and tabu search with a nonmonotonic cooling schedulein 2012 and adding systematic search of neighborhood andmemory to SA algorithm in 2013

COA often combines with other optimization algorithmto overcome its drawbacks and take advantage of its beneficialproperty such as ergodicity for example chaotic simulatedannealing (CSA) [32] chaotic particle swarm optimization(CPSO) [23] and chaotic improved imperialist competitivealgorithm (CICA) [33] et al CSA was proposed by Chen andAihara [32] to solve combinatorial optimization problems in1995 which used Hopfield neural networks Then severalother papers have expanded this work [34ndash37] Mingjunand Huanwen have proposed another version of CSA tosolve the optimization problem of continuous functions [38]Most of these works focus on the application of continuousfunctions optimization while the method proposed by Chenand Aihara is actually based on artificial neural network notSA algorithm Ferens and Cook [39] adapted CSA developedby Mingjun and Huanwen into the TAP in 2013 wherechaos was infused into a solution by setting the numberof perturbations made by the value of a chaotic variableHowever this method does notmake full use of the beneficialproperty of COA and does not improve the convergencespeed either

The present paper differs from the above mentionedresearches because it can combine the advantages of both twoalgorithms Firstly with the ergodicity property COA can getthe skeleton of solution space by chaoticwalking in it therebypreventing the result from falling into local optima Secondlybased on the results of COA we can easily determine thecooling schedule of SA algorithm which is very important tothe performance of SA algorithmbut hard to deal with Lastlyseveral adaptive schemes are used in SA algorithm all theseschemes including COA preliminary results can increase theconvergence speed significantly


Node1

Processor

Processor

Processor

Memory

Memory

Memory

Node2

Node3

S1

S2

S3

M1

M2

M3


10 16 11 14 5 30

[1 2] [2 2]

1217 25

11 8 12[5 3]

[4 1]

[6 4]

4 21 13

t1 t2

t3t4

t5

1

2

2

3

45

(b) A task graph


3 Problem Statement



1198941 1199021198942 119902








119896is

sum119872


119896is

119877119896 (119883) = 119890

minus120582119896 sum119872

119894=1119909119894119896119890119894119896 (1)



119896119887issum119872119894=1sum119894 =119895119909119894119896119909119895119887(119888119894119895120596119896119887) so the


119877119896119887 (119883) = 119890

minus120593119896119887 sum119872

119894=1sum119894 =119895119909119894119896119909119895119887(119888119894119895120596119896119887) (2)


119877 (119883) =

119873

prod

119896=1

119877119896 (119883)

119873minus1

prod

119896=1

119873

prod

119887=119896+1

119877119896119887 (119883) = 119890

minus119884(119883) (3)

where

119884 (119883) =

119873

sum

119896=1

119872

sum

119894=1

120582119896119909119894119896119890119894119896

+

119873

sum

119896=1

119873

sum

119887=119896+1

119872

sum

119894=1

sum

119894 =119895

120593119896119887119909119894119896119909119895119887(119888119894119895

120596119896119887

)

(4)



119872

sum

119894=1

119898119894119909119894119896le 119872119896 forall119896 isin [1119873] (5)



119872

sum

119894=1

119904119894119909119894119896le 119878119896 forall119896 isin [1119873] (6)


119872

sum

119894=1

sum

119895 =119894

119888119894119895119909119894119896119909119895119887le 119862119896119887 1 le 119896 lt 119887 le 119873 (7)


119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896le 119889119894 forall119894 isin [1119872] (8)


min 119884 (119883)


(9)






119904


119883119904



119904


119899

Step 7 If 119891119899le 119891119904 then 119883

119904larr 119883119899and 119891


to Step 9


119899and119891lowast larr 119891



119891119899












0via initial


0



0 We can use the



0


1198790=119891min minus 119891max

ln1198750

(10)








119911119896+1

= 120583119911119896(1 minus 119911

119896) 119896 = 0 1 (11)




Start











End





















1lt 1







119883 (119894 119896) = 1 119896 = 119860 (119894) 119894 = 1 2 119872

0 119896 = 119860 (119894) 119896 = 1 2 119873(12)



119864119872=

119873

sum

119896=1

max(0119872

sum

119894=1

119898119894119909119894119896minus119872119896)

119864119878=

119873

sum

119896=1

max(0119872

sum

119894=1

119904119894119909119894119896minus 119878119896)

119864119862=

119873

sum

119896=1

119873

sum

119887=119896+1

max(0119872

sum

119894=1

sum

119894 =119895

119888119894119895119909119894119896119909119895119887minus 119862119896119887)

119864119863=

119872

sum

119894=1

max(0119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896minus 119889119894)

(13)


119891 (119883) = 119884 (119883) + 120574 (119864119872 + 119864119878 + 119864119862 + 119864119863) (14)








Δ1199052

Δ1199053

Δ1198771

Δ1198772


12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027



11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653



1





119894=



3is steady in the





0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Node1

Processor

Processor

Processor

Memory

Memory

Memory

Node2

Node3

S1

S2

S3

M1

M2

M3


10 16 11 14 5 30

[1 2] [2 2]

1217 25

11 8 12[5 3]

[4 1]

[6 4]

4 21 13

t1 t2

t3t4

t5

1

2

2

3

45

(b) A task graph


3 Problem Statement



1198941 1199021198942 119902








119896is

sum119872


119896is

119877119896 (119883) = 119890

minus120582119896 sum119872

119894=1119909119894119896119890119894119896 (1)



119896119887issum119872119894=1sum119894 =119895119909119894119896119909119895119887(119888119894119895120596119896119887) so the


119877119896119887 (119883) = 119890

minus120593119896119887 sum119872

119894=1sum119894 =119895119909119894119896119909119895119887(119888119894119895120596119896119887) (2)


119877 (119883) =

119873

prod

119896=1

119877119896 (119883)

119873minus1

prod

119896=1

119873

prod

119887=119896+1

119877119896119887 (119883) = 119890

minus119884(119883) (3)

where

119884 (119883) =

119873

sum

119896=1

119872

sum

119894=1

120582119896119909119894119896119890119894119896

+

119873

sum

119896=1

119873

sum

119887=119896+1

119872

sum

119894=1

sum

119894 =119895

120593119896119887119909119894119896119909119895119887(119888119894119895

120596119896119887

)

(4)



119872

sum

119894=1

119898119894119909119894119896le 119872119896 forall119896 isin [1119873] (5)



119872

sum

119894=1

119904119894119909119894119896le 119878119896 forall119896 isin [1119873] (6)


119872

sum

119894=1

sum

119895 =119894

119888119894119895119909119894119896119909119895119887le 119862119896119887 1 le 119896 lt 119887 le 119873 (7)


119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896le 119889119894 forall119894 isin [1119872] (8)


min 119884 (119883)


(9)






119904


119883119904



119904


119899

Step 7 If 119891119899le 119891119904 then 119883

119904larr 119883119899and 119891


to Step 9


119899and119891lowast larr 119891



119891119899












0via initial


0



0 We can use the



0


1198790=119891min minus 119891max

ln1198750

(10)








119911119896+1

= 120583119911119896(1 minus 119911

119896) 119896 = 0 1 (11)




Start











End





















1lt 1







119883 (119894 119896) = 1 119896 = 119860 (119894) 119894 = 1 2 119872

0 119896 = 119860 (119894) 119896 = 1 2 119873(12)



119864119872=

119873

sum

119896=1

max(0119872

sum

119894=1

119898119894119909119894119896minus119872119896)

119864119878=

119873

sum

119896=1

max(0119872

sum

119894=1

119904119894119909119894119896minus 119878119896)

119864119862=

119873

sum

119896=1

119873

sum

119887=119896+1

max(0119872

sum

119894=1

sum

119894 =119895

119888119894119895119909119894119896119909119895119887minus 119862119896119887)

119864119863=

119872

sum

119894=1

max(0119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896minus 119889119894)

(13)


119891 (119883) = 119884 (119883) + 120574 (119864119872 + 119864119878 + 119864119862 + 119864119863) (14)








Δ1199052

Δ1199053

Δ1198771

Δ1198772


12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027



11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653



1





119894=



3is steady in the





0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872

sum

119894=1

119904119894119909119894119896le 119878119896 forall119896 isin [1119873] (6)


119872

sum

119894=1

sum

119895 =119894

119888119894119895119909119894119896119909119895119887le 119862119896119887 1 le 119896 lt 119887 le 119873 (7)


119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896le 119889119894 forall119894 isin [1119872] (8)


min 119884 (119883)


(9)






119904


119883119904



119904


119899

Step 7 If 119891119899le 119891119904 then 119883

119904larr 119883119899and 119891


to Step 9


119899and119891lowast larr 119891



119891119899












0via initial


0



0 We can use the



0


1198790=119891min minus 119891max

ln1198750

(10)








119911119896+1

= 120583119911119896(1 minus 119911

119896) 119896 = 0 1 (11)




Start











End





















1lt 1







119883 (119894 119896) = 1 119896 = 119860 (119894) 119894 = 1 2 119872

0 119896 = 119860 (119894) 119896 = 1 2 119873(12)



119864119872=

119873

sum

119896=1

max(0119872

sum

119894=1

119898119894119909119894119896minus119872119896)

119864119878=

119873

sum

119896=1

max(0119872

sum

119894=1

119904119894119909119894119896minus 119878119896)

119864119862=

119873

sum

119896=1

119873

sum

119887=119896+1

max(0119872

sum

119894=1

sum

119894 =119895

119888119894119895119909119894119896119909119895119887minus 119862119896119887)

119864119863=

119872

sum

119894=1

max(0119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896minus 119889119894)

(13)


119891 (119883) = 119884 (119883) + 120574 (119864119872 + 119864119878 + 119864119862 + 119864119863) (14)








Δ1199052

Δ1199053

Δ1198771

Δ1198772


12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027



11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653



1





119894=



3is steady in the





0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0


1198790=119891min minus 119891max

ln1198750

(10)








119911119896+1

= 120583119911119896(1 minus 119911

119896) 119896 = 0 1 (11)




Start











End





















1lt 1







119883 (119894 119896) = 1 119896 = 119860 (119894) 119894 = 1 2 119872

0 119896 = 119860 (119894) 119896 = 1 2 119873(12)



119864119872=

119873

sum

119896=1

max(0119872

sum

119894=1

119898119894119909119894119896minus119872119896)

119864119878=

119873

sum

119896=1

max(0119872

sum

119894=1

119904119894119909119894119896minus 119878119896)

119864119862=

119873

sum

119896=1

119873

sum

119887=119896+1

max(0119872

sum

119894=1

sum

119894 =119895

119888119894119895119909119894119896119909119895119887minus 119862119896119887)

119864119863=

119872

sum

119894=1

max(0119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896minus 119889119894)

(13)


119891 (119883) = 119884 (119883) + 120574 (119864119872 + 119864119878 + 119864119862 + 119864119863) (14)








Δ1199052

Δ1199053

Δ1198771

Δ1198772


12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027



11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653



1





119894=



3is steady in the





0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















1lt 1







119883 (119894 119896) = 1 119896 = 119860 (119894) 119894 = 1 2 119872

0 119896 = 119860 (119894) 119896 = 1 2 119873(12)



119864119872=

119873

sum

119896=1

max(0119872

sum

119894=1

119898119894119909119894119896minus119872119896)

119864119878=

119873

sum

119896=1

max(0119872

sum

119894=1

119904119894119909119894119896minus 119878119896)

119864119862=

119873

sum

119896=1

119873

sum

119887=119896+1

max(0119872

sum

119894=1

sum

119894 =119895

119888119894119895119909119894119896119909119895119887minus 119862119896119887)

119864119863=

119872

sum

119894=1

max(0119873

sum

119896=1

119909119894119896

119872

sum

119895=1

119890119895119896119909119895119896minus 119889119894)

(13)


119891 (119883) = 119884 (119883) + 120574 (119864119872 + 119864119878 + 119864119862 + 119864119863) (14)








Δ1199052

Δ1199053

Δ1198771

Δ1198772


12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027



11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653



1





119894=



3is steady in the





0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Δ1199052

Δ1199053

Δ1198771

Δ1198772


12

16 09349 3336 09361 39266 09353 10409 9150 6795 7349 013 00418 09135 3914 09152 50188 09151 13237 9220 7043 7362 018 01720 08948 4014 08992 57456 08968 14454 9301 7223 7484 049 02222 08733 4918 08781 55815 08777 14474 9119 6602 7407 055 05025 08299 5325 08373 84469 08367 22463 9370 7630 7341 089 081

25

40 06000 23187 06048 377104 06028 96961 9385 7609 7429 079 04745 05320 36523 05334 484005 05329 123667 9245 7047 7445 025 01650 04537 36300 04553 606927 04565 161550 9402 7753 7338 035 06155 03732 57156 03753 720068 03726 195631 9206 7078 7283 058 minus01460 02866 157777 02886 795438 02860 217155 8016 2734 7270 071 minus019

Average 9142 6751 7371 049 027



11198812

119877std 119905std 119881 119877std 119905std 119881

12

16 00016 03035 9110 10000 00019 08582 9999 00013 06208 1000018 00008 06041 8557 10000 00020 09552 9876 00018 08322 992120 00062 04114 7239 10000 00029 13870 9812 00036 03914 976322 00016 06186 5430 10000 00015 13489 9323 00017 05886 940725 00049 07613 2927 9982 00038 29824 8599 00031 12888 8688

25

40 00042 15797 2167 9987 00026 87429 8962 00027 46770 913945 00014 54829 985 9967 00024 123919 7651 00027 34313 819550 00039 56073 206 9930 00030 114135 7162 00025 110605 797655 00028 81346 016 9658 00019 89093 5805 00023 77128 673360 00022 93194 003 8976 00016 155811 5563 00022 84473 6703

Average 00029 32823 3664 9850 00024 64570 8275 00024 39051 8653



1





119894=



3is steady in the





0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 1000 2000 3000

012

014

016

018

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4 5

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 2000 4000 6000 8000 10000

012

014

016

018

02

022

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times104




1represents









0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 500

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 600

50

100

150

200

250

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 10 20 30 40075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 10 20 30 40 500

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 600

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA




0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2000 4000 6000 8000 10000

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

COA

0 2 4 612

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

inSA

0 1 2 3 4

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA1

0 5 10

12

13

14

15

16

17

Iterations

Valu

es o

f the

ener

gy fu

nctio

n

SA2

times105

times104

times104






6 Conclusions



0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80 1000

500

1000

1500

Cooling steps

Accu

mul

atio

ns

n

120579

120581


0 20 40 60 80075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 100075

08

085

09

095

Cooling steps

Coo

ling

fact

or


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA


0 20 40 60 80 1000

02

04

06

08

1

Cooling steps

Tem

pera

ture

ASASA





Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Notations



119896and 119901

119887

119909119894119896 Whether or not 119905

119894is on 119901

119896


119894on 119901119896


119894and 119905119895


119896119887

120596119896119887 Bandwidth of 119897

119896119887

120582119896 Failure rate of 119901

119896

120593119896119887 Failure rate of 119897

119896119887


119894


119896


119894


119896

119889119894 Deadline of 119905

119894



Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article simulated annealing algorithm combined with … · 2019. 7. 31. · simulated...

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