Research Article List-Based Simulated Annealing Algorithm ...Research Article List-Based Simulated Annealing Algorithm for Traveling Salesman Problem Shi-huaZhan, 1,2 JuanLin, 1 Ze-junZhang,

Research ArticleList-Based Simulated Annealing Algorithm forTraveling Salesman Problem

Shi-hua Zhan12 Juan Lin1 Ze-jun Zhang1 and Yi-wen Zhong12

1College of Computer and Information Science Fujian Agriculture and Forestry University Fuzhou 350002 China2Center of Modern Education Technology and Information Management Fujian Agriculture and Forestry UniversityFuzhou 350002 China

Correspondence should be addressed to Yi-wen Zhong yiwzhongfafueducn

Received 16 August 2015 Revised 21 January 2016 Accepted 10 February 2016

Academic Editor Leonardo Franco

Copyright copy 2016 Shi-hua Zhan 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

Simulated annealing (SA) algorithm is a popular intelligent optimization algorithm which has been successfully applied in manyfields Parametersrsquo setting is a key factor for its performance but it is also a tediouswork To simplify parameters setting we present alist-based simulated annealing (LBSA) algorithm to solve traveling salesman problem (TSP) LBSA algorithmuses a novel list-basedcooling schedule to control the decrease of temperature Specifically a list of temperatures is created first and then the maximumtemperature in list is used byMetropolis acceptance criterion to decide whether to accept a candidate solutionThe temperature listis adapted iteratively according to the topology of the solution space of the problemThe effectiveness and the parameter sensitivityof the list-based cooling schedule are illustrated through benchmark TSP problems The LBSA algorithm whose performance isrobust on a wide range of parameter values shows competitive performance compared with some other state-of-the-art algorithms

1 Introduction

Simulated annealing (SA) algorithm which was first inde-pendently presented as a search algorithm for combinato-rial optimization problems in [1 2] is a popular iterativemetaheuristic algorithm widely used to address discrete andcontinuous optimization problems The key feature of SAalgorithm lies in means to escape from local optima byallowing hill-climbing moves to find a global optimum Oneof the major shortages of SA is that it has several parametersto be tuned and its performance is sensitive to the values ofthose control parameters There are two special strategies forparameter tuning the online parameter tuning and the off-line parameter tuning In the online approach the param-eters are controlled and updated dynamically or adaptivelythroughout the execution of the SA algorithm [3ndash7] whereasin the off-line approach the values of different parametersare tuned before the execution of the SA algorithm and arefixed during its execution Fine-tuning parameter values isnot trivial and those parameters are quite often very poorlymade by trial and error So SA algorithm which has less

parameter or is less sensitive to parameter setting is veryattractive for practical users

Recently a metaheuristic algorithm called the list-basedthreshold-accepting (LBTA) algorithm has been developedand has shown significant performance for combinatorialoptimization problems that are NP-complete The advantageof LBTA over the majority of other neighbourhood search-based metaheuristic methods is that it has fewer controllingparameters that have to be tuned in order to producesatisfactory solutions Since its appearance LBTA has beensuccessfully applied to many combinatorial optimizationproblems [8ndash14]

In this work we are motivated by the success of LBTAin simplifying parameter tuning to study how list-basedparameter control strategy can be applied to SA algorithmTowards by this goal our paper presents a novel list-basedcooling schedule for SA algorithm to solve travelling sales-man problem (TSP) and we call our proposed algorithmas list-based simulated annealing (LBSA) algorithm In list-based cooling schedule all temperatures are stored in a listwhich is organized as a priority queue A higher temperature

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 1712630 12 pageshttpdxdoiorg10115520161712630

2 Computational Intelligence and Neuroscience

has higher priority In LBSA a list of temperatures is createdfirst and then in each generation the maximum value inthe list is used as current temperature to calculate acceptanceprobability for candidate solution The temperature list isupdated adaptively according to the topology of the solutionspace of the problem Using the list-based cooling scheduleSA algorithm has good performance on a wide range ofparameter values and it also has competitive performancecompared with some other state-of-the-art algorithms Theparameter robustness of list-based cooling schedule cangreatly simplify the design and implementation of LBSAalgorithm for practical applications

The remainder of this paper is organized as followsSection 2 provides a short description of TSP problem andSA algorithm Section 3 presents our proposed list-basedSA algorithm Section 4 gives the experimental approachand results of experiments carried out on benchmark TSPproblems Finally in Section 5 we summarize our study

2 Preliminaries

21 Traveling Salesman Problem TSP problem is one ofthe most famous hard combinatorial optimization problemsIt belongs to the class of NP-hard optimization problemsThis means that no polynomial time algorithm is known toguarantee its global optimal solution Consider a salesmanwho has to visit 119899 cities The TSP problem consists of findingthe shortest tour through all the cities such that no city isvisited twice and the salesman returns back to the starting cityat the end of the tour It can be defined as follows For 119899 citesproblem we can use a distance matrix 119863 = (119889

119894119895)119899times119899

to storedistances between all the pair of cites where each element 119889

119894119895

of matrix 119863 represents the distance between cities V119894and V119895

And we use a set of permutations 120587 of the integers from 1 to119899 which contains all the possible tours of the problem Thegoal is to find a permutation 120587 = (120587(1) 120587(2) 120587(119899)) thatminimizes

119891 (120587) =

119899minus1

sum

119894=1

119889120587(119894)120587(119894+1)

+ 119889120587(119899)120587(1) (1)

TSP problem may be symmetric or asymmetric In thesymmetric TSP the distance between two cities is the same ineach opposite direction forming an undirected graph Thissymmetry halves the number of possible solutions In theasymmetric TSP paths may not exist in both directions orthe distances might be different forming a directed graphTraffic collisions one-way streets and airfares for cities withdifferent departure and arrival fees are examples of how thissymmetry could break down

22 Simulated Annealing Algorithm SA algorithm is com-monly said to be the oldest among the metaheuristics andsurely one of the few algorithms that have explicit strategiesto avoid local minima The origins of SA are in statisti-cal mechanics and it was first presented for combinatorialoptimization problems The fundamental idea is to acceptmoves resulting in solutions of worse quality than thecurrent solution in order to escape from local minima Theprobability of accepting such a move is decreased during the

Start

Yes No

Yes

No

Stop condition of outer loop is met

Yes

Yes

No

No

End

Generate an initial solution x randomly

Generate a candidate solution y randomlybased on current solution x and a specified

neighbourhood structure

y is better than x

Generate r in [0 1) randomly

r lt p

x = y

Stop condition of inner loop is met

Decrease the temperature t

Output the solution x

p = exp(minus(f(y) minus f(x))t

)

Figure 1 Flowchart of simulated annealing algorithm

search through parameter temperature SA algorithm startswith an initial solution 119909 and candidate solution 119910 is thengenerated (either randomly or using some prespecified rule)from the neighbourhood of 119909 The Metropolis acceptancecriterion [15] which models how a thermodynamic systemmoves from one state to another state in which the energy isbeingminimized is used to decidewhether to accept119910 or notThe candidate solution 119910 is accepted as the current solution 119909based on the acceptance probability

119901 =

1 if 119891 (119910) le 119891 (119909)

119890minus(119891(119910)minus119891(119909))119905

otherwise(2)

where 119905 is the parameter temperature The SA algorithm canbe described by Figure 1

In order to apply the SA algorithm to a specific problemone must specify the neighbourhood structure and coolingschedule These choices and their corresponding parametersetting can have a significant impact on the SArsquos performanceUnfortunately there are no choices of these strategies thatwill be good for all problems and there is no general easyway to find the optimal parameter sets for a given problem

Computational Intelligence and Neuroscience 3

A cooling schedule should consist of starting temperaturetemperature decrement function Markov chain length andtermination condition Geometric cooling schedule whichcan be described by the temperature-update formula 119905

119896+1=

119886119905119896 is probably the most commonly used in the SA literature

and acts as a base line for comparison with other moreelaborate schemes [16] Typical values of 119886 for moderatelyslow cooling rates are 08 through 099

For practical application where computation complexityof objective function is high SA algorithm may run withconstant Markov chain length and use fixed iteration timesas termination condition As a result initial temperatureand cooling coefficient 119886 are the two key parameters thatimpact the performance of SA algorithm Even in this simplesituation it is not an easy task to find optimal values forthose two parameters because SArsquos performance is sensitiveto those parameters One way to tackle the parameter settingproblem of SA algorithm is to use adaptive cooling strategy[3ndash7] Adaptive cooling strategy is definitely efficient and itmakes SA algorithm less sensitive to user defined parametersthan canonical SA but it also makes SA algorithm lose thefeature of simplicity Another way to tackle this problem is tofind new cooling schedule which has fewer parameters or theparameters are more robust

23 Simulated Annealing Algorithm for TSP Problems Inorder to use SA algorithm for TSP problem we can representsolution 119909 as a permutation 120587 Then a large set of operatorssuch as the famous 2-Opt 3-Opt inverse insert and swapoperators can be used to generate candidate solution 119910 Sinceits appearance SA algorithm has been widely and deeplystudied on TSP problems [17ndash21] Many cooling schedulessuch as geometric exponential logarithmic and arithmeticones and their variants have been used in literature

The theory convergence conditions of SA make it verytime consuming inmost cases [22]Wang et al [23] proposeda two-stage SA algorithm which was tested on 23 TSPbenchmark instanceswith scale from51 to 783Thenumericalresults show that it is difficult for SA algorithm to solve TSPbenchmark instances with scale exceeding 1000 cities basedon time complexity Geng et al [24] proposed an adaptivesimulated annealing algorithm with greedy search (ASA-GS) where greedy search technique is used to speed up theconvergence rate The ASA-GS achieves a reasonable trade-off among computation time solution quality and complexityof implementation It has good scalability and has goodperformance even for TSP benchmark instances with scaleexceeding 1000 cities Wang et al [25] proposed a multiagentSA algorithm with instance-based sampling (MSA-IBS) byexploiting learning ability of instance-based search algo-rithm The learning-based sampling can effectively improvethe SArsquos sampling efficiency Simulation results showed thatthe performance of MSA-IBS is far better than ASA-GS interms of solution accuracy andCPU time In this paperMSA-IBS is used as basis to use list-based cooling schedule

3 List-Based Simulated Annealing Algorithm

31 The Neighbourhood Structure In this paper we use thegreedy hybrid operator proposed by Wang et al [25] to

produce candidate solution This is a kind of multiple moveoperators which select the best one from three neighboursSpecifically after two positions 119894 and 119895 are selected ituses inverse operator insert operator and swap operatorto produce three neighbour solutions And the best one isused as the candidate solution The inverse insert and swapoperators can be defined as follows

Define 1 (inverse operator inverse(120587 119894 119895)) Itmeans to inversethe cities between positions 119894 and 119895 The inverse(120587 119894 119895) willgenerate a new solution 1205871015840 such that 1205871015840(119894) = 120587(119895) 1205871015840(119894 + 1) =120587(119895minus1) 1205871015840(119895) = 120587(119894) where 1 le 119894 119895 le 119899and1 le 119895minus119894 lt 119899minus1in addition if 119895 minus 119894 = 119899 minus 1 it means 119894 = 1 and 119895 = 119899 and then1205871015840(119894) = 120587(119895) and 1205871015840(119895) = 120587(119894) Two edges will be replaced by

inverse operator for symmetric TSP problems

Define 2 (insert operator insert(120587 119894 119895)) It means tomove thecity in position 119895 to position 119894 The insert(120587 119894 119895) operator willgenerate a new solution 1205871015840 such that 1205871015840(119894) = 120587(119895) 1205871015840(119894 + 1) =120587(119894) 1205871015840(119895) = 120587(119895 minus 1) in the case of 119894 lt 119895 or 1205871015840(119895) =1205871015840(119895+1) 1205871015840(119894minus1) = 120587(119894) 1205871015840(119894) = 120587(119895) in the case of 119894 gt 119895

In general three edges will be replaced by insert operator

Define 3 (swap operator swap(120587 119894 119895)) It means to swap thecity in position 119895 and city in position 119894 The swap(120587 119894 119895)operator will generate a new solution1205871015840 such that1205871015840(119894) = 120587(119895)and 1205871015840(119895) = 120587(119894) In general four edges will be replaced byswap operator

Using the above three operators the used hybrid greedyoperator can be defined as follows

1205871015840= min (inverse (120587 119894 + 1 119895) insert (120587 119894 + 1 119895)

swap (120587 119894 + 1 119895)) (3)

where min returns the best one among its parameters

32 Produce Initial Temperature List As in LBTA algorithms[8ndash14] list-based parameter controlling strategy needs toproduce an initial list of parameters Because temperature inSA is used to calculate acceptance probability of candidatesolution we use initial acceptance probability 119901

0to produce

temperature 119905 as follows Suppose 119909 is current solution and119910 is candidate solution which is randomly selected from 119909rsquosneighbours Their objective function values are 119891(119909) and119891(119910) respectively If 119910 is worse than 119909 then the acceptanceprobability 119901 of 119910 can be calculated using formula (2)Conversely if we know the acceptance probability 119901

0 then

we can calculate the corresponding temperature 119905 as follows

119905 =

minus (119891 (119910) minus 119891 (119909))

ln (1199010)

(4)

Figure 2 is the flowchart of producing initial temperaturelist In Figure 2 after a feasible solution 119909 is produced a can-didate solution 119910 is randomly selected from 119909rsquos neighbours If119910 is better than 119909 then 119909 is replaced by 119910 And using formula(4) the absolute value of 119891(119910) minus 119891(119909) is used to calculate aninitial temperature value 119905 Then 119905 will be inserted into initial


Begin

End

Yes

No

No

Yes

Create an initial solution x

Create an empty temperature list LDefining maximum list length Lmax

Defining initial acceptance probability p0i = 0

Create neighbour solution y from x randomly

f(y) lt f(x)

x = y

Insert t into temperature list Li = i + 1

i lt Lmax

t =minus|f(y) minus f(x)|

ln(p0)

Figure 2 Flowchart of producing initial temperature list

temperature list The temperature list is a priority list wherehigher temperature has higher priority The same procedureis repeated until the list of temperature values is filled

33 Temperature Controlling Procedure In each iteration119896 the maximum temperature in list is used as currenttemperature 119905max If Markov chain length is119872 then 119905max maybe used at best 119872 times for the calculation of acceptanceprobability of candidate solution Suppose there are 119899 timesthat bad solution is accepted as current solutionwe use119889

119894and

119901119894to represent the difference of objective function values and

acceptance probability respectively where 119894 = 1 sdot sdot sdot 119899 and therelation between 119889

119894and 119901

119894can be described as the following

equation

119901119894= 119890minus119889119894119905max (5)

According to the Metropolis acceptance criterion foreach time a bad candidate solution is met a random number119903 is created If 119903 is less than the acceptance probability thenthe bad candidate solution will be accepted So for each pairof 119889119894and 119901

119894 there is a random number 119903

119894and 119903119894is less than

119901119894 We use 119889

119894and 119903119894to calculate a new temperature 119905

119894as the

following formula described

119905119894=

minus119889119894

ln (119903119894)

(6)

We use the average of 119905119894to update the temperature list

Specifically 119905max will be removed from list and average of 119905119894

will be inserted into list Because 119905119894is always less than 119905max the

average of 119905119894is less than 119905max also In this way the temperature

decreases always as the search proceeds

34 Description of LBSA Algorithm Because the main pur-pose of this paper is to study the effectiveness of list-basedcooling schedule we use a simple SA framework which usesfixed iteration times for outer loop and fixed Markov chainlength in each temperature The detailed flowchart of themain optimization procedure is shown in Figure 3 Besidesthe creation of initial temperature list the main differencebetween the flowchart of LBSA and the flowchart of canonicalSA is about the temperature control strategy In canonicalSA the temperature controlling procedure is independent ofthe topology of solution space of the problem Converselythe temperature controlling procedure of LBSA is adaptiveaccording to the topology of solution space of the problemIn Figure 3 119870 is maximum iteration times of outer loopwhich is termination condition of LBSA119872 is Markov chainlength which is termination condition of inner loop Counter119896 is used to record the current iteration times of outer loop119898 is used to record current sampling times of inner loopand 119888 is used to record how many times bad solution isaccepted in each temperature Variable 119905 is used to store thetotal temperature calculated by formula (6) and the averagetemperature 119905119888 will be used to update the temperature listWe can use a maximum heap to implement the temperaturelist As the time complexity of storage and retrieval from heapis logarithmic the use of temperature list will not increase thetime complexity of SA algorithm

4 Simulation Results

In order to observe and analyse the effect of list-basedcooling schedule and the performance of LBSA algorithmfive kinds of experiments were carried out on benchmarkTSP problems The first kind of experiments was used toanalyse the effectiveness of the list-based cooling scheduleThe second kind was carried out to analyse the parametersensitivity of LBSA algorithmThe third kind was carried outto compare the performance of different ways to update thetemperature list The fourth kind was carried out to comparethe performance of different neighbourhood structures Andthe fifth kind was carried out to compare LBSArsquos performancewith some of the state-of-the-art algorithms

41 The Effectiveness of List-Based Cooling Schedule Toillustrate the effectiveness of list-based cooling schedule wecompare the temperature varying process and search processof LBSA algorithm with SA algorithms based on the classicalgeometric cooling schedule and arithmetic cooling schedule


Begin

End

Yes No

No

Yes

Yes No

NoYes

temperature listUsed to update

temperature listUpdateNo

Yes

Produce initial temperature list Lk = 0

Fetch the maximum temperature tmax from list Lk = k + 1 t = 0 c = 0 m = 0

Create neighbour solutiony from x randomlym = m + 1

f(y) lt f(x)


r lt p

c = c + 1

x = y

m le M

Pop the maximum value from list L

k le K

Insert tc into list L

p = exp(minus(f(y) lt f(x))

tmax)

c == 0

t =t minus (f(y) minus f(x))

ln(r)

Figure 3 Flowchart of list-based SA algorithm

Those experiments were carried out on BCL380 XQL662XIT1083 and XSC6880 problems from VLSI data sets Thebest known integer solutions of those problems are 1621 25133558 and 21537 respectivelyThe iteration times of outer loopare 1000 and the Markov chain length in each temperature isthe city number of the problem which is 380 662 1083 and6880 respectively

Figure 4 compares the temperature varying process ofdifferent cooling schedules on the four benchmark problemsList-based cooling schedule can be viewed as a kind ofgeometric cooling schedule with variable cooling coefficientCompared with the temperature varying of geometric cool-ing schedule temperature of list-based cooling scheduledecreases quicker in the early stage but slower in the laterstage As indicated by Abramson et al [16] geometric cooling

schedule always uses constant cooling coefficient regardlessof the stages of search However at high temperaturesalmost all candidate solutions are accepted even thoughmany of them could be nonproductive To use variablecooling coefficients which depend on the stages would allowSA algorithm to spend less time in the high temperaturestages Consequently more time would be spent in the lowtemperature stages thus reducing the total amount of timerequired to solve the problem

Figure 5 compares the search process of different coolingschedules on the four benchmark problems Compared withgeometric cooling schedule and arithmetic cooling schedulelist-based cooling schedule has quicker convergence speedand has good long-term behaviour also This good perfor-mance may be due to the variable cooling coefficient featureof list-based cooling schedule The temperature is updatedadaptively according to the topology of solution space of theproblem As a result LBSA algorithm can spend more timeto search on promising area finely

42 Sensitivity Analysis of Temperature List Length and InitialTemperature Values We observe the sensitivity of list lengthon BCL380 XQL662 XIT1083 and XSC6880 problems Foreach problem we test 30 different list lengths from 10 to300 with a step 10 For each list length we run LBSAalgorithm 50 times and calculate the average tour lengthand the percentage error of the mean tour length to thebest known tour length Figure 6 is the relation betweenpercentage error and list length It shows the following (1)for all the problems there is a wide range of list length forLBSA to have similar promising performance (2) list lengthis more robust on small problems than on large problems(3) list length should not be too big for large problems Forbig problems the used Markov chain length which is thecity number of the problem is not big enough for LBSA toreach equilibrium on each temperature Big list length meansthe temperature decreases more slowly so the algorithm willspend too much time on high temperature and accept toomuch nonproductive solutions As a result a big list lengthwill dramatically deteriorate LBSArsquos performance for largeproblems Because of the robust temperature list length weuse fixed temperature list length in the following simulationswhich is 120 for all instances

In order to observe the sensitivity of initial temperaturevalues we carried out experiments with different initialtemperature values on BCL380 XQL662 XIT1083 andXSC6880 problems Because the initial temperature valuesare produced according to initial acceptance probability 119901

0

we use 30 different 1199010 which range from 10minus20 to 09 to create

initial temperature values Specifically the set of 1199010is the

union of geometric sequence from 10minus20 to 10minus2 with commonratio 10 and arithmetic sequence from01 to 09with commondifference 01 For each initial acceptance probability we runLBSA algorithm 50 times and calculate the average tourlength and the percentage error of the mean tour length tothe best known tour length Figure 7 is the relation betweenpercentage error and index of initial acceptance probabilityIt shows the following (1) the performance of LBSA is not


BCL380

GeometricArithmeticList

100 200 300 400 500 600 700 800 900 10000Iteration times

0

2

4

6

8

10

12

14

16

18Te

mpe

ratu

re

(a)


XQL662

100 200 300 400 500 600 700 800 900 10000Iteration times

02468

1012141618202224

Tem

pera

ture

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

0

4

8

12

16

20

24

28

32

Tem

pera

ture

(c)


XSC6880

100 200 300 400 500 600 700 800 900 10000Iteration times

05

101520253035404550556065

Tem

pera

ture

(d)

Figure 4 Compare the temperature varying process of different cooling schedule

sensitive to the initial temperature values (2) there is adifferent varying direction among different problems Forsmall problems the initial temperature should not be too lowbut for big problems the initial temperature should not betoo high This difference is due to the limited computationresource used and the nonlinear computation complexityof TSP problem The LBSA algorithm can have similarpromising performance on awide range of initial temperaturevalues this high robustness of initial temperature is due to theadaptive nature of list-based cooling schedule

43 Comparing Different Temperature Updating Schemes Inour algorithm described in Section 34 we use the average oftemperature 119905

119894to update temperature list There are several

variants such as using maximum 119905119894or minimum 119905

119894 To show

the advantage of using average of 119905119894 we compare the results

and the decreasing process of temperature using those threeupdating schemes Table 1 is the simulation results it is clearthat using average temperature to update temperature listhas far better performance than the other methods Thetemperature decreasing process of different strategies onBCL380 which is showed in Figure 8 can explain why usingaverage temperature is the best option If we use maximumtemperature to update the temperature list the temperaturewill decrease very slowly As a result the acceptation proba-bility of bad solution is always high and SA algorithm willsearch randomly in the solution space Conversely if weuse minimum temperature to update the temperature listthe temperature will decrease sharply As a result the SAalgorithm will be trapped into local minimum quickly and


BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)

Figure 5 Compare the search process of different cooling schedule

Table 1 Comparison of different temperature list updating schemes

Instance OPT Average Maximum MinimumBCL380 1621 16302 23574 16519XQL662 2513 25272 41178 25658XIT1083 3558 35929 66630 36746XSC6880 21537 219200 634722 226498

lose the advantage of escaping from local minimum by usingMetropolis acceptance criterion

44 Comparing Different Neighbourhood Structures In ouralgorithm we use a greedy hybrid neighbour operator pro-posed byWang et al [25] to produce candidate solutionThis

Table 2 Comparison of different neighbourhood structures

Instance Inverse Insert Swap HybridBCL380 163196 177624 266644 163084XQL662 25352 283652 44372 25272XIT1083 360348 411872 676084 359208XSC6880 221156 29048 642634 219306

hybrid operator uses the best one produced by inverse insertand swap operators To show its advantage we comparethe results produced by inverse insert swap and hybridoperators And we compare the percentage of inverse insertand swap operators accepted when we use hybrid operatorTable 2 is the simulation results it is clear that hybrid


BCL380XQL662

XIT1083XSC6880

20 40 60 80 100 120 140 160 180 200 220 240 260 280 3000Temperature list length

0

10

20

30

40

50

60

70

Erro

r (

)

Figure 6 Sensitivity of parameter temperature list length

BCL380XQL662

XIT1083XSC6880

2 4 6 8 10 12 14 16 18 20 22 24 26 28 300Index of different initial acceptance probability

04

06

08

10

12

14

16

18

20

22

Erro

r (

)

Figure 7 Sensitivity of parameter initial temperature values

operator has the best performance Among the three basicoperators inverse is the best If we compare the percentagesof different basic operators accepted when we use hybridoperator we found that inverse operator is accepted mostFigure 9 is the percentages of different operators accepted oninstance BCL380Thepercentages of inverse insert and swapoperators are 65 31 and 4 respectively The relativepercentages of the three operators accepted are similar onother instances as on BCL380 The good performance ofinverse operator is due to its ability to produce more fine-grained neighbourhood structure because it changes twoedges only The hybrid operator which uses inverse insertand swap operators at the same time has a bigger neighbour-hood structure So it has higher probability to find promisingsolutions

AverageMaximumMinimum

minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times

Figure 8 Temperature decreasing process of different temperaturelist updating schemes on instance BCL380

Inverse65

Insert31

Swap4

Figure 9 Percentage of inverse insert and swap operators acceptedin hybrid operator on instance BCL380

45 Competitiveness of LBSA Algorithm We compare LBSAalgorithm with genetic simulated annealing ant colonysystem (GSAACS) [26] and MSA-IBS on 24 benchmarkinstances with cities from 51 to 1655The GSAACS is a hybridalgorithmwhich uses the ant colony system (ACS) to generatethe initial solutions of the genetic algorithms Then it usesgenetic algorithm (GA) which uses SA as mutation operatorto generate offspring solutions based on the initial solutionsIf the solutions searched by GA are better than the initialsolutions GSAACS will use these better solutions to updatethe pheromone for the ACS After a predefined numberof generations GSAACS uses particle swarm optimization(PSO) to exchange the pheromone information betweengroups


Table 3 Compare LBSA with GSAACS and MSA-IBS on 24 benchmark instances from TSPLIB

Number Instance OPT GSAACS MSA-IBS LBSABest Mean PEav Best Mean PEav Best Mean PEav

1 eil51 426 427 42727 030 426 426 0 426 426 02 eil76 538 538 54020 041 538 538 0 538 538 03 eil101 629 630 63523 099 629 629 0 629 629 04 berlin52 7542 7542 754200 0 7542 7542 0 7542 7542 05 bier127 118282 118282 11942183 096 118282 1182912 001 118282 118282 06 ch130 6110 6141 620563 157 6110 6110 0 6110 61106 0017 ch150 6528 6528 656370 055 6528 65444 025 6528 65375 0158 rd100 7910 7910 798757 098 7910 7910 0 7910 79101 09 lin105 14379 14379 1440637 019 14379 14379 0 14379 14379 010 lin318 42029 42487 4300290 232 42040 421709 034 42029 4213855 02611 kroA100 21282 21282 2137047 042 21282 21282 0 21282 2128415 00112 kroA150 26524 26524 2689920 141 26524 2652415 0 26524 2652405 013 kroA200 29368 29383 2973873 126 29368 2938345 005 29368 293719 00114 kroB100 22141 22141 2228287 064 22141 221742 015 22141 221842 0215 kroB150 26130 26130 2644833 122 26130 2613405 002 26130 261379 00316 kroB200 29437 29541 3003523 203 29438 294394 001 29437 2943865 00117 kroC100 20749 20749 2087897 063 20749 20749 0 20749 20749 018 kroD100 21294 21309 2162047 153 21294 2134275 023 21294 2129455 019 kroE100 22068 22068 2218347 052 22068 221144 021 22068 220926 01120 rat575 6773 6891 693387 238 6813 682465 076 6789 681555 06321 rat783 8806 8988 907923 310 8845 88697 072 8846 88666 06922 rl1323 270199 277642 28018147 369 270893 2719723 066 270475 271415 04523 fl1400 20127 20593 2134963 607 20299 203924 132 20140 201822 02724 d1655 62128 64151 6562113 562 62786 629663 135 62454 626101 078

Average 162 025 015

The results of GSAACS are from [26] GSAACS uses 120ants and 1000 generations In each generation of GSAACSGA runs 100 generations In MSA-IBS and LBSA algorithmwe set the population size to 30 and the iteration times ofouter loop to 1000 and the Markov chain length in eachtemperature is two times the number of cities The endcondition is that either it finds the optimal solution or theiteration times of outer loop reach 1000 The algorithm wasexecuted 25 times on each TSP problem and the results arelisted in Table 3

In Table 3 the columns Instance OPT Best Mean andPEav denote the name of the TSP problem the optimal tourlength from TSPLIB the shortest tour length found theaverage tour length among the 25 trials and the percentageerror of the mean tour length to the OPT respectively Ascan be seen in Table 3 both MSA-IBS and LBSA have betterperformance than GSAACS on all 24 instances LBSA is alittle better than MSA-IBS in terms of average percentageerror

We compare LBSA algorithm with GA-PSO-ACO [27]and MSA-IBS on 35 benchmark instances with cities from48 to 33810 GA-PSO-ACO combines the evolution ideas ofthe genetic algorithms particle swarm optimization and ant

colony optimization In the GA-PSO-ACO algorithm thewhole process is divided into two stages In the first stage GAand PSO are used to obtain a series of suboptimal solutionsto adjust the initial allocation of pheromone for ACO In thesecond stage ACO is used to search the optimal solutionTheresults of GA-PSO-ACO are from [27] GA-PSO-ACO uses100 individuals and 1000 generations In LBSA andMSA-IBSalgorithm we set the number of population size to 10 and theiteration times of outer loop to 1000 and the Markov chainlength in each temperature is the number of cities The endcondition of LBSA is that either it finds the optimal solutionor the iteration times of outer loop reach 1000 Like GA-PSO-ACOandMSA-IBS LBSAwas executed 20 times on eachTSPinstance and the results are listed in Table 4

As can be seen in Table 4 both MSA-IBS and LBSA havebetter performance than GA-PSO-ACO on all 35 instancesLBSA is a little bit better than MSA-IBS in terms of averagepercentage error

We compare LBSA algorithm with ASA-GS [24] andMSA-IBS algorithm on 40 benchmark instances with citiesfrom 151 to 85900 The experiments were run on a 28GHzPC and the ASA-GS was run on a 283GHz PC For allinstances we set the iteration times of outer loop to 1000


Table 4 Compare LBSA with GA-PSO-ACO and MSA-IBS on 35 benchmark instances from TSPLIB

Instance OPT GA-PSO-ACO MSA-IBS LBSABest Mean PEav Best Mean PE Best Mean PEav

Att48 33522 33524 33662 042 33522 3355464 010 33522 335366 004Eil51 426 426 43184 137 426 42648 011 426 4265 012Berlin52 7542 754437 754437 003 7542 7542 0 7542 7542 0St70 675 67960 69460 290 675 67716 032 675 67555 008Eil76 538 54539 55016 226 538 5382 004 538 538 0Pr76 108159 109206 110023 172 108159 108288 012 108159 1082683 010Rat99 1211 1218 1275 528 1211 121104 000 1211 12111 001Rad100 7910 7936 8039 163 7910 791456 006 7910 79147 006KroD100 21294 21394 21484 089 21294 2134064 022 21294 213142 009Eil101 629 63307 63793 142 629 6296 010 629 629 0Lin105 14379 14397 14521 099 14379 1438048 001 14379 14379 0Pr107 44303 44316 44589 065 44303 4437988 017 44303 4439225 020Pr124 59030 59051 60157 191 59030 5903288 000 59030 590318 000Bier127 118282 118476 120301 171 118282 1183346 004 118282 1183512 006Ch130 6110 612115 620347 153 6110 612196 020 6110 612795 029Pr144 58537 58595 58662 021 58537 5854972 002 58537 5857015 006KroA150 26524 26676 26803 105 26524 265382 005 26524 265426 007Pr152 73682 73861 73989 042 73682 7372796 006 73682 736888 001U159 42080 42395 42506 101 42080 4218232 024 42080 4219885 028Rat195 2323 2341 2362 168 2328 23342 048 2328 2331 034RroA200 29368 29731 31015 561 29368 2942264 019 29368 2940535 013Gil262 2378 2399 2439 257 2379 238356 023 2379 238245 019Pr299 48191 48662 48763 119 48192 4826308 015 48191 48250 012Lin318 42029 42633 42771 177 42076 4229204 063 42070 4226435 056Rd400 15281 15464 15503 145 15324 1537756 063 15311 1537375 061Pcb442 50778 51414 51494 141 50879 5105012 054 50832 510417 052Rat575 6773 6912 6952 264 6819 685464 121 6829 684795 111U724 41910 42657 42713 192 42150 4230212 094 42205 423578 107Rat783 8806 9030 9126 363 8897 891828 128 8887 891325 122Pr1002 259045 265987 266774 298 261463 2622117 122 261490 2622025 122D1291 50801 52378 52443 323 51098 5134084 106 51032 513587 110D1655 62128 64401 65241 501 62784 6301196 142 62779 6299465 139Nl4461 182566 18933 192574 548 185377 1856311 168 185290 1855017 161Brd14051 469385 490432 503560 728 478040 4786188 197 477226 4776127 175Pla33810 66048945 70299195 72420147 965 67868250 680388331 301 67754877 678485351 272

Average 243 053 049

and set the Markov chain length in each temperature tothe number of cities As in MSA-IBS algorithm a suitablepopulation size is selected for each instance such that theCPUtime of LBSA is less than that of ASA-GS The end conditionof LBSA and MSA-IBS is either when it finds the optimalsolution or when the iteration times of outer loop reach 1000The algorithm is executed 25 trials on each instance and theresults are listed in Table 5

As can be seen in Table 5 the average PEav of LBSA for allinstances is 075 which is better than 187 of ASA-GS and the

average CPU time of LBSA for all instances is 28249 which isfar less than 65031 of ASA-GS LBSA is a little bit better thanMSA-IBS in terms of average PEav

5 Conclusion

This paper presents a list-based SA algorithm for TSPproblems The LBSA algorithm uses novel list-based coolingschedule to control the decrease of temperature parameterThe list-based cooling schedule can be viewed as a special


Table 5 Compare LBSA with ASA-GS and MSA-IBS on 40 benchmark instances from TSPLIB

Number Instance OPT ASA-GS MSA-IBS LBSAMean PEav Time Mean PEav Time Mean PEav Time

1 Ch150 6528 65398 016 1091 6529 002 086 65298 003 1292 Kroa150 26524 265386 005 109 26524 0 082 26524 0 0983 Krob150 26130 261781 018 109 26135 002 151 26137 003 1654 Pr152 73682 736947 001 1085 73682 0 084 73682 0 0875 U159 42080 423989 075 1149 42080 0 079 42080 0 0916 Rat195 2323 234805 107 1437 23302 031 186 2328 022 1937 D198 15780 158454 041 146 15780 0 139 15780 0 1538 Kroa200 29368 294384 023 1426 29378 003 174 293738 002 1679 Krob200 29437 295131 025 1424 294398 001 195 294422 002 2110 Ts225 126643 126646 000 1605 126643 0 13 126643 0 15411 Pr226 80369 806874 039 167 80369 0 193 803698 000 21612 Gil262 2378 239861 086 1943 23788 003 239 23792 005 27213 Pr264 49135 491389 000 1909 49135 0 143 49135 0 14914 Pr299 48191 483264 028 2194 482264 007 267 482212 006 29315 Lin318 42029 423837 084 2335 421844 037 24 421956 040 25816 Rd400 15281 154298 097 304 153472 043 32 153504 045 34617 Fl417 11861 120438 154 3202 118756 012 372 118678 006 40118 Pr439 107217 110226 280 3492 1074072 018 36 1074652 023 39519 Pcb442 50778 512692 096 3575 50970 038 368 50877 019 43120 U574 36905 373698 125 4847 371558 068 521 371646 070 61321 Rat575 6773 690482 194 521 68398 099 527 68374 095 59922 U724 41910 424704 133 6683 422122 072 811 42252 082 83423 Rat783 8806 898219 200 789 88934 099 899 88882 093 8924 Pr1002 259045 264274 201 16442 2614818 094 1271 2618052 107 129625 Pcb1173 56892 578205 163 19308 575616 118 89 574318 095 96126 D1291 50801 522523 285 21464 513438 107 1059 511988 078 117727 Rl1323 270199 273444 120 21016 2718184 06 1153 2717144 056 126428 Fl1400 20127 207822 325 23202 203748 123 1772 202494 061 154329 D1655 62128 641559 326 28188 62893 123 1618 630014 141 164530 Vm1748 336556 343911 218 27698 3396178 091 197 3397108 094 190531 U2319 234256 236744 106 41097 235236 042 1702 235975 073 189432 Pcb3038 137694 141242 257 55428 1397062 146 2764 1396352 141 290533 Fnl4461 182566 187409 265 8309 1855354 163 3043 1855094 161 296734 Rl5934 556045 575437 348 104395 5661668 182 5076 566053 180 52535 Pla7397 23260728 24166453 389 124522 238E+07 248 10069 238E+07 224 896136 Usa13509 19982859 20811106 414 201605 204E+07 206 36512 204E+07 189 3267637 Brd14051 469385 486197 358 20805 4786096 197 37528 478010 184 3698638 D18512 645238 669445 375 259397 6581492 200 65485 6574572 189 6291439 Pla33810 66048945 69533166 527 419988 68075607 307 195968 680292264 300 19981940 Pla85900 142382641 156083025 963 885513 1464955156 289 759618 1455265426 221 75866

Average 187 65031 081 28352 075 28249

geometric cooling schedule with variable cooling coefficientThe variance of cooling coefficient is adaptive according tothe topology of solution space which may be more robustto problems at hand Simulated results show that this novelcooling schedule is insensitive to the parameter values andthe proposed LBSA algorithm is very effective simple andeasy to implement The advantage of insensitive parameteris very attractive allowing LBSA algorithm to be applied in

diverse problems without much effort on tuning parametersto produce promising results

Conflict of Interests

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


Acknowledgment

This work was supported by Nature Science Foundation ofFujian Province of China (no 2014J01219 no 2013J01216)

References

[1] V Cerny ldquoThermodynamical approach to the traveling sales-man problem an efficient simulation algorithmrdquo Journal ofOptimization Theory and Applications vol 45 no 1 pp 41ndash511985

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

[3] L Ingber ldquoAdaptive simulated annealingrdquo Lester Ingber vol 52no 5 pp 491ndash500 1993

[4] L Ingber ldquoAdaptive simulated annealing (ASA) lessonslearnedrdquo Control and Cybernetics vol 25 no 1 pp 32ndash54 1996

[5] S-J Jeong K-S Kim and Y-H Lee ldquoThe efficient searchmethod of simulated annealing using fuzzy logic controllerrdquoExpert Systems with Applications vol 36 no 3 pp 7099ndash71032009

[6] H A E Oliveira Jr and A Petraglia ldquoSolving nonlinear systemsof functional equations with fuzzy adaptive simulated anneal-ingrdquo Applied Soft Computing Journal vol 13 no 11 pp 4349ndash4357 2013

[7] H Oliveira Jr and A Petraglia ldquoEstablishing nash equilibria ofstrategic games amultistart fuzzy adaptive simulated annealingapproachrdquo Applied Soft Computing Journal vol 19 pp 188ndash1972014

[8] C D Tarantilis and C T Kiranoudis ldquoA list-based thresholdaccepting method for job shop scheduling problemsrdquo Interna-tional Journal of Production Economics vol 77 no 2 pp 159ndash1712002

[9] C D Tarantilis C T Kiranoudis and V S Vassiliadis ldquoA listbased threshold accepting algorithm for the capacitated vehiclerouting problemrdquo International Journal of ComputerMathemat-ics vol 79 no 5 pp 537ndash553 2002

[10] C D Tarantilis C T Kiranoudis and V S Vassiliadis ldquoA listbased threshold accepting metaheuristic for the heterogeneousfixed fleet vehicle routing problemrdquo Journal of the OperationalResearch Society vol 54 no 1 pp 65ndash71 2003

[11] C D Tarantilis G Ioannou C T Kiranoudis and G P Pras-tacos ldquoSolving the open vehicle routeing problem via a singleparameter metaheuristic algorithmrdquo Journal of the OperationalResearch Society vol 56 no 5 pp 588ndash596 2005

[12] D S Lee V S Vassiliadis and J M Park ldquoList-based threshold-accepting algorithm for zero-wait scheduling of multiproductbatch plantsrdquo Industrial amp Engineering Chemistry Research vol41 no 25 pp 6579ndash6588 2002

[13] D S Lee V S Vassiliadis and J M Park ldquoA novel thresholdaccepting meta-heuristic for the job-shop scheduling problemrdquoComputers amp Operations Research vol 31 no 13 pp 2199ndash22132004

[14] A Nikolakopoulos and H Sarimveis ldquoA threshold acceptingheuristic with intense local search for the solution of specialinstances of the traveling salesman problemrdquo European Journalof Operational Research vol 177 no 3 pp 1911ndash1929 2007

[15] 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

[16] D Abramson M Krishnamoorthy and H Dang ldquoSimulatedannealing cooling schedules for the school timetabling prob-lemrdquo Asia-Pacific Journal of Operational Research vol 16 no1 pp 1ndash22 1999

[17] E H Aarts J H Korst and P J van Laarhoven ldquoA quantitativeanalysis of the simulated annealing algorithm a case study forthe traveling salesman problemrdquo Journal of Statistical Physicsvol 50 no 1-2 pp 187ndash206 1988

[18] J R A Allwright andD B Carpenter ldquoA distributed implemen-tation of simulated annealing for the travelling salesman prob-lemrdquo Parallel Computing vol 10 no 3 pp 335ndash338 1989

[19] C-S Jeong and M-H Kim ldquoFast parallel simulated annealingfor traveling salesman problem on SIMD machines with linearinterconnectionsrdquo Parallel Computing vol 17 no 2-3 pp 221ndash228 1991

[20] M Hasegawa ldquoVerification and rectification of the physicalanalogy of simulated annealing for the solution of the travelingsalesman problemrdquo Physical Review E vol 83 Article ID036708 2011

[21] P Tian and Z Yang ldquoAn improved simulated annealing algo-rithm with genetic characteristics and the traveling salesmanproblemrdquo Journal of Information andOptimization Sciences vol14 no 3 pp 241ndash255 1993

[22] N Sadati T Amraee and A M Ranjbar ldquoA global ParticleSwarm-Based-Simulated Annealing Optimization techniquefor under-voltage load shedding problemrdquo Applied Soft Com-puting Journal vol 9 no 2 pp 652ndash657 2009

[23] Z C Wang X T Geng and Z H Shao ldquoAn effective simulatedannealing algorithm for solving the traveling salesman prob-lemrdquo Journal of Computational andTheoretical Nanoscience vol6 no 7 pp 1680ndash1686 2009

[24] X T Geng Z H Chen W Yang D Q Shi and K ZhaoldquoSolving the traveling salesman problem based on an adaptivesimulated annealing algorithmwith greedy searchrdquoApplied SoftComputing Journal vol 11 no 4 pp 3680ndash3689 2011

[25] C Y Wang M Lin Y W Zhong and H Zhang ldquoSolving trav-elling salesman problem using multiagent simulated annealingalgorithm with instance-based samplingrdquo International Journalof Computing Science and Mathematics vol 6 no 4 pp 336ndash353 2015

[26] S-M Chen and C-Y Chien ldquoSolving the traveling salesmanproblem based on the genetic simulated annealing ant colonysystem with particle swarm optimization techniquesrdquo ExpertSystems with Applications vol 38 no 12 pp 14439ndash14450 2011

[27] WDeng R Chen BHe Y Liu L Yin and J Guo ldquoA novel two-stage hybrid swarm intelligence optimization algorithm andapplicationrdquo Soft Computing vol 16 no 10 pp 1707ndash1722 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



has higher priority In LBSA a list of temperatures is createdfirst and then in each generation the maximum value inthe list is used as current temperature to calculate acceptanceprobability for candidate solution The temperature list isupdated adaptively according to the topology of the solutionspace of the problem Using the list-based cooling scheduleSA algorithm has good performance on a wide range ofparameter values and it also has competitive performancecompared with some other state-of-the-art algorithms Theparameter robustness of list-based cooling schedule cangreatly simplify the design and implementation of LBSAalgorithm for practical applications

The remainder of this paper is organized as followsSection 2 provides a short description of TSP problem andSA algorithm Section 3 presents our proposed list-basedSA algorithm Section 4 gives the experimental approachand results of experiments carried out on benchmark TSPproblems Finally in Section 5 we summarize our study

2 Preliminaries

21 Traveling Salesman Problem TSP problem is one ofthe most famous hard combinatorial optimization problemsIt belongs to the class of NP-hard optimization problemsThis means that no polynomial time algorithm is known toguarantee its global optimal solution Consider a salesmanwho has to visit 119899 cities The TSP problem consists of findingthe shortest tour through all the cities such that no city isvisited twice and the salesman returns back to the starting cityat the end of the tour It can be defined as follows For 119899 citesproblem we can use a distance matrix 119863 = (119889

119894119895)119899times119899

to storedistances between all the pair of cites where each element 119889

119894119895

of matrix 119863 represents the distance between cities V119894and V119895

And we use a set of permutations 120587 of the integers from 1 to119899 which contains all the possible tours of the problem Thegoal is to find a permutation 120587 = (120587(1) 120587(2) 120587(119899)) thatminimizes

119891 (120587) =

119899minus1

sum

119894=1

119889120587(119894)120587(119894+1)

+ 119889120587(119899)120587(1) (1)

TSP problem may be symmetric or asymmetric In thesymmetric TSP the distance between two cities is the same ineach opposite direction forming an undirected graph Thissymmetry halves the number of possible solutions In theasymmetric TSP paths may not exist in both directions orthe distances might be different forming a directed graphTraffic collisions one-way streets and airfares for cities withdifferent departure and arrival fees are examples of how thissymmetry could break down

22 Simulated Annealing Algorithm SA algorithm is com-monly said to be the oldest among the metaheuristics andsurely one of the few algorithms that have explicit strategiesto avoid local minima The origins of SA are in statisti-cal mechanics and it was first presented for combinatorialoptimization problems The fundamental idea is to acceptmoves resulting in solutions of worse quality than thecurrent solution in order to escape from local minima Theprobability of accepting such a move is decreased during the

Start

Yes No

Yes

No

Stop condition of outer loop is met

Yes

Yes

No

No

End

Generate an initial solution x randomly

Generate a candidate solution y randomlybased on current solution x and a specified

neighbourhood structure

y is better than x


r lt p

x = y

Stop condition of inner loop is met

Decrease the temperature t

Output the solution x

p = exp(minus(f(y) minus f(x))t

)

Figure 1 Flowchart of simulated annealing algorithm

search through parameter temperature SA algorithm startswith an initial solution 119909 and candidate solution 119910 is thengenerated (either randomly or using some prespecified rule)from the neighbourhood of 119909 The Metropolis acceptancecriterion [15] which models how a thermodynamic systemmoves from one state to another state in which the energy isbeingminimized is used to decidewhether to accept119910 or notThe candidate solution 119910 is accepted as the current solution 119909based on the acceptance probability

119901 =

1 if 119891 (119910) le 119891 (119909)

119890minus(119891(119910)minus119891(119909))119905

otherwise(2)

where 119905 is the parameter temperature The SA algorithm canbe described by Figure 1

In order to apply the SA algorithm to a specific problemone must specify the neighbourhood structure and coolingschedule These choices and their corresponding parametersetting can have a significant impact on the SArsquos performanceUnfortunately there are no choices of these strategies thatwill be good for all problems and there is no general easyway to find the optimal parameter sets for a given problem



119896+1=
















swap (120587 119894 + 1 119895)) (3)



0to produce


0 then


119905 =

minus (119891 (119910) minus 119891 (119909))

ln (1199010)

(4)



Begin

End

Yes

No

No

Yes





f(y) lt f(x)

x = y


i lt Lmax


ln(p0)




119894and



119894and 119901


equation

119901119894= 119890minus119889119894119905max (5)



119894and 119903119894is less than

119901119894 We use 119889


119894as the


119905119894=

minus119889119894

ln (119903119894)

(6)











Begin

End

Yes No

No

Yes

Yes No

NoYes



Yes




f(y) lt f(x)


r lt p

c = c + 1

x = y

m le M


k le K



tmax)

c == 0


ln(r)








0





BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

0

2

4

6

8

10

12

14

16

18Te

mpe

ratu

re

(a)


XQL662

100 200 300 400 500 600 700 800 900 10000Iteration times

02468

1012141618202224

Tem

pera

ture

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

0

4

8

12

16

20

24

28

32

Tem

pera

ture

(c)


XSC6880

100 200 300 400 500 600 700 800 900 10000Iteration times

05

101520253035404550556065

Tem

pera

ture

(d)






119894 To show




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)










BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119896+1=
















swap (120587 119894 + 1 119895)) (3)



0to produce


0 then


119905 =

minus (119891 (119910) minus 119891 (119909))

ln (1199010)

(4)



Begin

End

Yes

No

No

Yes





f(y) lt f(x)

x = y


i lt Lmax


ln(p0)




119894and



119894and 119901


equation

119901119894= 119890minus119889119894119905max (5)



119894and 119903119894is less than

119901119894 We use 119889


119894as the


119905119894=

minus119889119894

ln (119903119894)

(6)











Begin

End

Yes No

No

Yes

Yes No

NoYes



Yes




f(y) lt f(x)


r lt p

c = c + 1

x = y

m le M


k le K



tmax)

c == 0


ln(r)








0





BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

0

2

4

6

8

10

12

14

16

18Te

mpe

ratu

re

(a)


XQL662

100 200 300 400 500 600 700 800 900 10000Iteration times

02468

1012141618202224

Tem

pera

ture

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

0

4

8

12

16

20

24

28

32

Tem

pera

ture

(c)


XSC6880

100 200 300 400 500 600 700 800 900 10000Iteration times

05

101520253035404550556065

Tem

pera

ture

(d)






119894 To show




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)










BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Begin

End

Yes

No

No

Yes





f(y) lt f(x)

x = y


i lt Lmax


ln(p0)




119894and



119894and 119901


equation

119901119894= 119890minus119889119894119905max (5)



119894and 119903119894is less than

119901119894 We use 119889


119894as the


119905119894=

minus119889119894

ln (119903119894)

(6)











Begin

End

Yes No

No

Yes

Yes No

NoYes



Yes




f(y) lt f(x)


r lt p

c = c + 1

x = y

m le M


k le K



tmax)

c == 0


ln(r)








0





BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

0

2

4

6

8

10

12

14

16

18Te

mpe

ratu

re

(a)


XQL662

100 200 300 400 500 600 700 800 900 10000Iteration times

02468

1012141618202224

Tem

pera

ture

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

0

4

8

12

16

20

24

28

32

Tem

pera

ture

(c)


XSC6880

100 200 300 400 500 600 700 800 900 10000Iteration times

05

101520253035404550556065

Tem

pera

ture

(d)






119894 To show




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)










BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Begin

End

Yes No

No

Yes

Yes No

NoYes



Yes




f(y) lt f(x)


r lt p

c = c + 1

x = y

m le M


k le K



tmax)

c == 0


ln(r)








0





BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

0

2

4

6

8

10

12

14

16

18Te

mpe

ratu

re

(a)


XQL662

100 200 300 400 500 600 700 800 900 10000Iteration times

02468

1012141618202224

Tem

pera

ture

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

0

4

8

12

16

20

24

28

32

Tem

pera

ture

(c)


XSC6880

100 200 300 400 500 600 700 800 900 10000Iteration times

05

101520253035404550556065

Tem

pera

ture

(d)






119894 To show




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)










BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

0

2

4

6

8

10

12

14

16

18Te

mpe

ratu

re

(a)


XQL662

100 200 300 400 500 600 700 800 900 10000Iteration times

02468

1012141618202224

Tem

pera

ture

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

0

4

8

12

16

20

24

28

32

Tem

pera

ture

(c)


XSC6880

100 200 300 400 500 600 700 800 900 10000Iteration times

05

101520253035404550556065

Tem

pera

ture

(d)






119894 To show




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)










BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




BCL380


100 200 300 400 500 600 700 800 900 10000Iteration times

1500175020002250250027503000325035003750400042504500

Tour

leng

th

(a)

XQL662


100 200 300 400 500 600 700 800 900 10000Iteration times

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

Tour

leng

th

(b)


XIT1083

100 200 300 400 500 600 700 800 900 10000Iteration times

4000

5000

6000

7000

8000

9000

10000

11000

12000

Tour

leng

th

(c)

XSC6880


100 200 300 400 500 600 700 800 900 10000Iteration times

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000To

ur le

ngth

(d)










BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




BCL380XQL662

XIT1083XSC6880


0

10

20

30

40

50

60

70

Erro

r (

)


BCL380XQL662

XIT1083XSC6880


04

06

08

10

12

14

16

18

20

22

Erro

r (

)




minus5

0

5

10

15

20

25

30

35

Tem

pera

ture

100 200 300 400 500 600 700 800 900 10000Iteration times


Inverse65

Insert31

Swap4







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Average 162 025 015











Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Average 243 053 049




5 Conclusion






Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Average 187 65031 081 28352 075 28249






Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Acknowledgment


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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