Multiobjective Simulated Annealing: Principles and...

Review ArticleMultiobjective Simulated Annealing Principles andAlgorithm Variants

Khalil Amine

MOAD6 Research Team MASI Laboratory E3S Research Centre Mohammadia School of EngineeringMohammed V University of Rabat Morocco

Correspondence should be addressed to Khalil Amine khaminesbaigmailcom

Received 9 November 2018 Revised 7 March 2019 Accepted 6 May 2019 Published 23 May 2019

Academic Editor Imed Kacem

Copyright copy 2019 Khalil AmineThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Simulated annealing is a stochastic local search method initially introduced for global combinatorial mono-objective optimisationproblems allowing gradual convergence to a near-optimal solution An extended version for multiobjective optimisation has beenintroduced to allow a construction of near-Pareto optimal solutions by means of an archive that catches nondominated solutionswhile exploring the feasible domain Although simulated annealing provides a balance between the exploration and the exploitationmultiobjective optimisation problems require a special design to achieve this balance due to many factors including the number ofobjective functions Accordingly many variants of multiobjective simulated annealing have been introduced in the literature Thispaper reviews the state of the art of simulated annealing algorithm with a focus upon multiobjective optimisation field

1 Introduction

Simulated annealing is a probabilistic local search methodfor global combinatorial optimisation problems that allowsgradual convergence to a near-optimal solution It consistsof a sequence of moves from a current solution to a betterone according to certain transition rules while acceptingoccasionally some uphill solutions in order to guarantee adiversity in the domain exploration and to avoid gettingcaught at local optima The process is managed by a certaincooling schedule that controls the number of iterationsSimulated annealing has shown robustness and success formany applications and has received significant attentionsince its introduction Consequently multiobjective simu-lated annealing has been introduced with the aim to allow aconstruction of near-Pareto optimal solutions bymeans of anarchive in which nondominated solutions are gradually gath-eredwhile exploring the feasible domain Although simulatedannealing provides a balance between the exploration and theexploitation multiobjective optimisation problems require aspecial design to achieve this balance due to many factorsincluding the number of objective functions Accordinglymany variants of multiobjective simulated annealing havebeen introduced in the literature

This paper reviews and discusses simulated annealingalgorithm for multiobjective optimisation and delineatessome algorithm variants

2 Overview on Mono-ObjectiveSimulated Annealing

Simulated annealing is a meta-heuristic that dates back tothe works of Kirkpatrick et al [1] and Cerny [2] havingshown that the Metropolis algorithm [3] (an algorithm ofstatistical physics that consists in constructing a sequence ofMarkov chains for sampling from a probability distributionThe algorithm is often used under an extended versioncalled Metropolis-Hastings algorithm) can be used to obtainoptimal solution of optimisation problems by consideringthe energetic state as objective function and thermodynamicequilibrium states as local optima Simulated annealingmimics the metallurgical process of careful annealing thatconsists in cooling a heated metal or alloy until reaching themost solid state called the ground state Simulated annealingis considered as an extension of the hill climbing algorithmwhich consists of a sequence of transitions across solutionswhile improving a certain energetic objective function at

HindawiAdvances in Operations ResearchVolume 2019 Article ID 8134674 13 pageshttpsdoiorg10115520198134674

2 Advances in Operations Research

Start

Current solution

Neighbouringsolution

generation

Candidate solution

Energy evaluation Probabilisticacceptance

Regular acceptance Equilibriumcondition

Temperatureupdate

Cooling condition

Stop

yes

no

yes

no

yes

no

Figure 1 Flowchart of the mono-objective simulated annealing algorithm

each iteration until reaching the global optimum Simulatedannealing has introduced an occasional acceptance mech-anism of uphill solutions in order to explore the feasibledomain intensively and to avoid getting caught at localoptimaThe process is managed by a certain cooling schedulethat controls the temperature variation during the algorithmprocess Figure 1 presents the generic simulated annealingalgorithm flowchart

The generic simulated annealing algorithm consists oftwo nested loops Given a current solution and a fixedtemperature the inner loop consists at each iterationin generating a candidate neighbouring solution that willundergo an energy evaluation to decide whether to acceptit as current Occasionally some nonimproving solutionsare accepted according to a certain probabilistic rule Theloop is controlled by an equilibrium condition that limits thenumber of iterations at every temperature level It refers to athermal equilibrium when the algorithm is not expected tofind more improving solutions within the current exploredneighbourhood The outer loop consists in decreasing thetemperature level according to a certain cooling schemeThe temperature is supposed to be iteratively decreased untilreaching a cooling condition that often corresponds to a final

temperature due The algorithm involves therefore two mainconsiderations namely the neighbourhood structure and thecooling schedule

Neighbourhood structure refers to a set of conceptualconsiderations regarding both the representation and thegeneration mechanism of neighbouring solutions It is sup-posed to be proper to the problem under considerationand to provide a restricted number of possible transitionsFor ease handling solutions of optimisation problems areusually considered in a certain indirect representation thatconsists of a finite number of components within a certainconfiguration Thus two solutions are neighbours if theirconfigurations are similar that is the two configurationsdiffer in few components only or may have close energyvalues Neighbourhood structure consists rather in definingthe candidate solutions that the algorithm may move to ateach iteration Even often underestimated in early studies[4 5] the neighbourhood structure choice has a significantimpact on the neighbourhood size and even on the simulatedannealing performance Moreover it may affect the conceptof local optimality itself [6 7] thereby identifying a local opti-mum for every neighbourhood Accordingly the algorithm

Advances in Operations Research 3

Table 1 Summary of classical cooling schemes

Denotation General functional form Iterative formula AuthorsLinear 119879119896 = 1198790 minus 119896120573 119879119896 = 119879119896minus1 minus 120573 Strenski and Kirkpatrick [9]Geometric 119879119896 = 1205721198961198790 119879119896 = 120572119879119896minus1 Kirkpatrick et al [1]

Logarithmic 119879119896 = 1198790ln (119896 + 1) 119879119896 = ln (119896)

ln (119896 + 1)119879119896minus1 Geman and Geman [10]

Hybrid 119879119896 = 11987901 + 119896 if 119896 le 120573

1205721198961198790 if 119896 gt 120573 119879119896 = 119896119896 + 1119879119896minus1 if 119896 le 120573

120572119879119896minus1 if 119896 gt 120573 Sekihara et al [11]

Exponential 119879119896 = 11987901 + 1198961205731198790 119879119896 = 119879119896minus11 + 120573119879119896minus1 Lundy and Mees [12]

Adaptive 119879119896 = 120601 (119878119888) 119879119896minus1 Ingber [13]The table notations are summarised as follows1198790 is the initial temperature value119879119896 is the temperature value for the 119896th iteration120572 and 120573 are two constants such that 120572 isin [0 1] and 120573 gt 0120601 is a function depending on the candidate solution 119878119888 that takes a value less than 1 if 119878119888 improves the objective function in comparison to the best solutionfound so far and greater than 1 otherwise

purpose is to hopefully reach at each temperature level thelocal optimum before reaching the equilibrium condition

The cooling schedule controls the temperature decreasein the course of the algorithm It consists of four enti-ties namely an initial temperature a final temperature anequilibrium condition and a cooling scheme that consistsof a decreasing sequence of temperature Whilst the finaltemperature is supposed to be preset to zero or aroundthe choice of the other parameters is rather challenging andstrongly related to the case under investigation no specificrule does exist for all situations and preliminary experimentsare often required The equilibrium condition is a parameterthat limits the number of iterations at every temperaturelevel It could be merely preset to a certain upper bound onthat number plausibly proportional to the current solutionneighbourhood cardinality In order to utterly mimic thecareful annealing thereby receiving satisfactory solutionsany parameters choice should ensure that the process startsat a sufficiently high temperature value and terminates atsufficiently low temperature value according to a sufficientlyslow cooling scheme

The choice of an appropriate cooling scheme is crucialin order to ensure success of the algorithm A wide rangeof cooling schemes have been introduced in the literaturethat are either monotonic or adaptive A cooling scheme isa function of iteration index that depends more often onthe previous temperature value and the initial temperatureMonotonic schemes consist of static decrease of temperatureat every iteration independently of the quality of the foundsolutions nor the current explored neighbourhood structurewhereas adaptive cooling schemes involve a mechanism ofdecreasing the temperature in accordance to the qualityof transitions Therefore more satisfactory moves occurmore the decrease hop is larger This would imply somereannealing occurrences allowing more diversification andintensification A summary of classical schemes is presentedin Table 1 Nevertheless Triki et al [8] have demonstrated

based on an empirical study that almost all classical coolingschemes are equivalent that is they can be tuned to result insimilar decrease of the temperature during the course of thealgorithm

During the course of the algorithm two kinds of accep-tance are involved namely a regular acceptance correspond-ing to the case when an improving solution is found and aprobabilistic acceptance corresponding to the case when anonimproving solution is found that would imply diversifica-tion and escape from local optimaGiven a candidate solution119878119888 the acceptance of the candidate move to 119878119888 is expressed bymeans of the Metropolis rule given by the following formula

P (Accept 119878119888) = min(1 exp (minusΔ119864119896119879 )) (1)

where Δ119864 is the move energetic variation and 119879 is thecurrent temperature 119896 is called Boltzmann constant andoften assumed to be equal to 1 This rule tolerates fre-quent acceptance occurrences of uphill solutions at hightemperature levels That is almost all candidate solutionswould be accepted if the temperature is high whereas onlyimproving solutions would be accepted if the temperatureis low Furthermore given that the exponential function isincreasing the acceptance rule favours small deteriorationsthan large ones The algorithm is supposed to stop whenone of the two following cases is due the temperature levelattains the final temperature or either the process results ona solution for which no improving solution could be foundwithin a reasonable number of annealing cases A criterionresponsible for dealing with both cases is called a coolingcondition

Simulated annealing has shown significant success formany applications and has received significant attention sinceits introduction Moreover it is problem-dependent and thechoice of appropriate parameters is rather a challenging taskMore detailed review can be found in [14ndash19]


3 Multiobjective Simulated Annealing

31 Multiobjective Optimisation and Metaheuristics Multi-objective optimisation also called multicriteria or multiat-tribute optimisation deals with problems involving morethan one objective function to be simultaneously optimisedThe objective functions under consideration are often in con-tradiction (or conflicting) in such a way no improvement onany objective function is possible without any deteriorationin any one of the othersmdashotherwise the problem is said tobe trivial and is reduced to a mono-objective optimisationproblem Given 119898 objective functions 119864119894 Rn 997888rarr R for119894 = 1 119898 a multiobjective optimisation problem (oftensaid to be bi-objective if119898 = 2 and many-objective if119898 ge 4)can generally be formulated as

minimise (1198641 (119909) 119864119898 (119909))subject to 119909 isin 119863 (MOP)

where 119863 sub Rn called the feasible domain is a set ofequalities and inequalities of the decision vector variable119909 The optimality concept in multiobjective optimisation isbased on the dominance binary relation which serves as acomparison function between feasible solutions A solutionis said to be dominating another one if the former improvesat least one objective function value compared to the latterwithout any deterioration in the other objective functionvalues Given 119909 and 119910 two feasible solutions of (MOP) that is119909 119910 isin 119863 the following alternatives are possible

(i) 119909 ≺ 119910 that is 119909 dominates 119910 if 119864119894(119909) le 119864119894(119909) forall 119894 = 1 119898 and (1198641(119909) 119864119898(119909))=(1198641(119910) 119864119898(119910))

(ii) 119909 sim 119910 that is 119909 and 119910 are equivalent if 119864119894(119909) = 119864119894(119910)for all 119894 = 1 119898

(iii) 119909 119910 that is 119909 and 119910 are noncomparable if thereexist 119894 and 119895 such that 119864119894(119909) lt 119864119894(119910) and 119864119895(119909) gt119864119895(119910)

A nondominated solution is defined then as one forwhich no feasible solution exists that dominates itmdashit corre-sponds to the best trade-off between all considered objectivefunctions Compared to mono-objective optimisation theoptimality is typically attained by means of a set of non-dominated solutions called Pareto set rather than a singlesolution Pareto set is usually referred to by means of itsimage called Pareto front or Pareto frontier which serves as agraphical representation of dimension119898 minus 1 of optimality inthe objective space that is the space constituted by objectivefunctions Each Pareto optimal solution is said to refer to apreference structure or either a utility function Furthermorethe Pareto solutions are said to be noncomparable that is nopreference can be stated between any pair of them Despitethe mathematical significance of this concept of optimality awide range of investigations have been deemed in economicsgiving rise to many advances in the utility theory that dealwith preference structures in accordance with the users(customers) satisfaction Nevertheless finding all the optimalsolutions for a multiobjective optimisation problem is rather

challenging and difficult at least in comparison with mono-objective optimisation case

A variety of techniques have been addressed in theliterature to deal with multiobjective optimisation that canbe classified into three approaches a priori a posteriori andprogressive An a priori approach consists of a transformationof the mathematical multiobjective model into a mono-objective one Many methods have been introduced fromthis perspective namely goal programming aggregationor scalarisation and lexicographical which return typicallyone (Pareto) optimal solution of the original model An aposteriori approach consists of a construction of the Paretoset Yet such a construction is usually difficult and compu-tationally consuming Thus approximation algorithms areplausibly suggested in this case They consist in constructinga subset of the Pareto set or otherwise a set of efficientlynondominated (or near-Pareto optimal) solutions that isfeasible solutions that are not dominated by any solution thatcan be alternatively computed in reasonable computationalcost The construction of the actual or an approximatingPareto set is sometimes referred to as Pareto optimisationAs to progressive approach also called interactive approachit is a hybridisation of the two aforementioned approacheswhere the decision-maker provides a kind of guidance tothe algorithm execution It consists actually of an a prioriapproach that involves an a posteriori partial result learningand preference defining as long as the algorithm progressesMore details about progressive optimisation can be found in[20 21]

Due to the success of metaheuristics in mono-objectiveoptimisation especially for combinatorial problems whereinformation about the problem (in terms of the feasibledomain called in this context the search space) is oftenaccessible metaheuristics have been widely adapted to copewith multiobjective combinatorial problems as well Con-trary to the constructive techniques metaheuristics aresolution-based allowing therefore a manageable compro-mise between the quality of solution (efficiency) and the exe-cution time (number of iterations) Moreover metaheuristicsare expected to bring a generic and adaptable frameworkfor a wide range of problems Vector evaluated geneticalgorithm [22] has been the firstmultiobjectivemetaheuristicproposed in the literature consisting of an adaptation of thegenetic algorithm to the multiobjective optimisation caseThenceforth many other multiobjective metaheuristics havebeen developed that include multiobjective Tabu search [23]multiple ant colony system for vehicle routing problems withtime windows [24] multi-objective simulated annealing [25]and multiobjective scatter search [26]

Metaheuristics adaptations have been developed into twomain paradigms one run and multiruns One-run-basedtechniques consist in adapting the original metaheuristic toreturn a set of solutions at only one execution One of theimportant advantages of this paradigm is that it follows themain principle of metaheuristics consisting of dealing with asolution at each iteration rather than a solution componentwhich refers in the multiobjective optimisation case to a setof solutions with a certain level of efficiency Yet the numberof returned solutions is rather uncontrollable Indeed if the


potential near-optimal solutions are archived progressivelyeach solution returned should undergo an evaluation phasewhere it is compared to the previous archived potentialnear-optimal solutions in order to eliminate any dominatedone If the near-optimal solutions are returned unsteadilyat the algorithm termination for example for population-based meta-heuristics like genetic algorithms the solutionsare compared to each other at this stage

Inspired from the a priori multiobjective approachmultiruns-based techniques consist in applying the originalmetaheuristic on a certain aggregation of the objective func-tions under consideration At each run the metaheuristic isexpected to reach one near-optimal solution of the problemthat satisfies a certain preference structure The algorithmis assumed to be executed as many times as the requirednumber of near-optimal solutions Unfortunately there isno guarantee that two different preference structures leadto two distinct near-optimal solutions Once again thegathered solutions are compared to each other to eliminateany dominated one Nevertheless the near-optimal solutionsfound should be diverse that is dispersed on the Pareto frontDas and Dennis [27] showed that evenly spread solutions arenot guaranteed evenwith an evenly distributed set of weightsThus although the concept that multiruns-based techniquesfollow is quite simple determining effective and pertinentpreference structures in advance is not straightforward atleast for a whole range of multiobjective problems

32 Principles of Multiobjective Simulated Annealing Mul-tiobjective simulated annealing originates from the worksof Serafini [28] where many probabilistic acceptance ruleshave been designed and discussed with the aim at increasingthe probability of accepting nondominated solutions that issolutions nondominated by any generated solution so farTwo alternative approaches have been proposed in conse-quence Given119898 objective functions 119864119894 assigned to119898 scalar-valued weights 120596119894 for 119894 = 1 119898 and a candidate solution119878119888 the first approach consists in accepting with certaintyonly improving solutions thereby accepting with probabilityless than 1 any other solution This strong acceptance ruleallows deep exploration of the feasible domain and has beenexpressed as follows

P (Accept 119878119888) = 119898prod119894=1

min(1 exp(120596119894Δ119864119894119879 )) (2)

The second approach consists in accepting with certaintyany either dominating or noncomparable solution allowingdiversity in the exploration of the feasible domain This weakacceptance rule is expressed in the following form

P (Accept 119878119888) = min(1 max119894=1119898

(exp(120596119894Δ119864119894119879 ))) (3)

The two alternatives lead to quite differentMarkov chainsfor which nondominated solutions were proved to have

higher stationary probability A composite acceptance rulehas been proposed consisting of the following formula

P (Accept 119878119888)= 120572 119898prod119894=1

min(1 exp(120596119894Δ119864119894119879 ))+ (1 minus 120572)min(1 max

119894=1119898(exp(120596119894Δ119864119894119879 )))

(4)

where 120572 isin [0 1] and 119879120596119894 is considered as a temperature119879119894 related to each objective function 119864119894 that is like eachobjective function has its own annealing scheme Besideseach considered occurrence of weights (120596119894)119894=1119898 is assumedto correspond to a certain preference structure In order toguarantee entire exploration of the Pareto set a slow randomvariation of the weights has been proposed in consequence atevery iteration

33 MOSA Methods Multiobjective Simulated Annealingmethod (MOSA) is a class of simulated annealing extensionsto multiobjective optimisation exploiting the idea of con-structing an estimated Pareto front by gathering nondomi-nated solutions found while exploring the feasible domainAn archive is considered that serves for maintenance ofthose efficient solutions Accordingly many multiobjectivesimulated annealing paradigms have been proposed in theliterature [25 29ndash34] that have in common the suggestion ofconsidering a variation of a certain composite energy oftenas a linear combination of the considered objective functionsUnder certain choice of acceptance probabilities asymptoticconvergence to the Pareto solutions set has been proved [35]

Ulungu et al [25 36] have developed a MOSA methodthat involves a potentially efficient solutions list that catchesnon-dominated solutions it contains all the generated solu-tions which are not dominated by any other generatedsolution so far The method uses weighted functions formeasuring the quality of transitions Each scalarising func-tion would induce a privileged search direction towardsa relative optimal solution If that optimal solution is notreached some near-optimal solutions would be consideredThe use of a wide diversified set of weights has been proposedwith the aim to cover all the efficient front The potentiallyefficient solutions list is assumed to be updated after everyoccurrence of a new solution acceptance at which anydominated solution is removed from the list If no dominatedsolution exists the potentially efficient solutions list increasesin cardinality

Czyzak and Jaszkiewicz [30] have proposed a Pareto Sim-ulated Annealing (PSA) procedure using a weak acceptancecriterion where any solution not dominated by the currentsolution could be accepted Inspired from genetic algorithmsPSA is a population-based metaheuristic that considers ateach temperature a set of (sample) generated solutions tobe hopefully improved Each solution from the sample isimproved in such a way the new accepted solution should bedistant from the closest solution to the former solution Thisapproach is performed by increasing weights of objectivesaccording to which the closest solution is better than the


Input A cooling schedule 120591 a starting temperature 119879 larr997888 1198790 a starting sample of generated solutions Sand an initial memory Mlarr997888 S

Output The archive M representing an approximation of the Pareto solutions set1 repeat2 for each 119878119888 isin S do3 repeat4 Construct a neighbouring solution 1198781198991198901199085 if 119878119899119890119908 is not dominated by 119878119888 then6 Update M with 1198781198991198901199087 Select 119878119888119897 (if exists) the closest solution to 1198781198888 Update weights of objectives in accordance with 119878119888 and 119878119899119890119908 partial dominance9 else10 Accept 119878119899119890119908 with certain probability11 end if12 until Equilibrium condition13 end for14 Decrease temperature 11987915 until Cooling condition16 returnM

Algorithm 1 PSA procedure

current solution thereby decreasing the weights of objectivesaccording to which the closest solution is better than thecurrent solution The new weights combination will be usedfor the evaluation step in the next iteration as well as for theprobabilistic acceptance The PSA procedure is summarisedin Algorithm 1

PSA attempts to keep a kind of uniformity while improv-ing the sample of generated solutions That is each sequenceof improving solutions related to a certain solution fromthe starting sample should be moved away from the othersequences of improving solutions related to the other ele-ments of the starting sample This would hopefully suggestthe sample cardinality conservation and thus allowing tocontrol the number of the returned efficient solutions Fur-thermore the fact that PSA deals at each temperature with asample of potential solutions allows parallelisationAs shownin Figure 2 when PSA is parallel all the sample elementsare updated at the same time with the improving solutionsof all parallel annealing sequences Banos et al [37] haveinvestigated four alternative parallelisations of PSA in termsof quality of the solutions and execution time Moreoverextensions to the fuzzy multiobjective combinatorial opti-misation [38] and stochastic multiobjective combinatorialoptimisation [39] have been addressed in the literature

With the same idea of using a population of solutionsas a starting archive Li and Landa-Silva [40 41] haveproposed an adaptive evolutionary multiobjective approachcombinedwith simulated annealing (EMOSA) It is a hybridi-sation of the multiobjective evolutionary algorithm basedon decomposition (MOEAD) proposed by Zhang and Li[42] and simulated annealing metaheuristic The proposedmethod consists in considering a set of starting solutionsthat is assumed to be evenly spread and where each solutioncorresponds to an initial solution of a certain subproblemthat is a problem corresponding to an aggregation objectivefunction Solutions evaluation is performed with respect

E2

Starting sample

Feasible domain

Pareto front

E1

Figure 2 Hypothetical improvement pattern of parallel annealingchains in parallel PSA procedure

to both weighted aggregation and Pareto dominance basedon the concept of 120598-dominance introduced by Laumannset al [43] the regular acceptance of candidate solutionsis performed in accordance to the improvement of thecorresponding aggregation function while the update of thenondominated solutions archive is performed using the 120598-dominance Besides for each aggregation function weightsare adaptively modified at the lowest temperature in order toassure search diversify

Suppapitnarm et al [32] proposed an extension to aprevious work conducted by Engrand [44] which consists inusing a logarithmic composite objective function

sum119894

ln (119864119894) = ln(prod119894

119864119894) (5)

for a given number of objective functions 119864119894 and anacceptance probability formulation for a generated candidate


E2

Feasible domainFeasible domainPareto set

Pareto front

E1

y

x

Objective spaceVariable space

y1

x1

E2(x1 yS

S

1)

E1(x1 y1)

Figure 3 Hypothetical correspondence between variable space and objective space for a bi-objective minimisation problem

solution 119878119888 in reference to a given current solution 119878119888119906119903defined as follows

P (Accept 119878119888) = exp(minus 1119879sum119894

ln( 119864119894 (119878119888)119864119894 (119878119888119906119903))) (6)

The lack of any weight assigned to objectives in additionto the multiplicative scheme of the considered compositeobjective would suggest that all objectives have the samemagnitude This implies an acceptance issue in both regularand probabilistic modes Indeed any weak dominating ornoncomparable solution is supposed to be accepted How-ever the magnitude order of some objectives could affectthe acceptance in a way that favours some objectives overothers Similarly the probabilistic acceptance depends on therelative change of all the objectives thereby influenced by theobjectives with high magnitude order Suppapitnarm et alhave proposed to not use any aggregation objective functionfor candidate solutions evaluation so that any decrease in anyobjective will systematically result in regular acceptance Asto the probabilistic acceptance a probability rule that involvesa temperature parameter for each objective function has beenintroduced as follows

P (Accept 119878119888) = prod119894

exp(minusΔ119864119894119879119894 ) = exp(sum119894

minus Δ119864119894119879119894 ) (7)

where Δ119864119894 is the energy variation of the objective function119864119894 between the candidate and the current solution and 119879119894 isthe current temperature assigned to119864119894 which serves as weightfor the objective From this perspective some objectivescould be favoured independently of their magnitude orderThe method enables a return to base strategy that consistsin recommencing the search periodically from an archivednondominated solution This strategy allows diversity in theexploration of the feasible domain Besides and in order toallow intensification in less explored regions Suppapitnarmet al proposed to restart the search from the most isolatedsolution from the archive An isolation measurement hasbeen introduced in addition to a monitoring strategy for therate at which the return to base could be activated

34 Domination-Based MOSA Methods In the above meth-ods the quality of solutions at hand is measured in accor-dance to the last accepted solution as a unique referencecriteria An alternative to this approach would consist incomparing generated solutions to the actual Pareto front aswell Yet it is crucial to determine a certain performance indi-cator to characterise the closeness to the Pareto front Thusmany performance indicators have been introduced in theliterature that consist of either a certain distance to the Paretofront or the ratio of the volume dominated by the resultantarchive to that dominated by the Pareto front Neverthelessthis approach is faced to the prevailing lack of advancedinformation about the actual Pareto front Hence algorithmsperformance is often investigated using test problems that areaverage computing cost optimisation problems with knownproperties Discussions on and construction of test problemshave been widely addressed in the literature within differentcontexts of multiobjective optimisation (eg [45ndash50])

In concept solutions are mainly configurations of deci-sion variables however they could be considered as con-figurations of objective functions as well This gives rise totwo different but equivalent representations of the feasibledomain Figure 3 illustrates the correspondence betweenthe two representations in the decision space which isconstituted by variables and in the objective space which isconstituted by objective functions (Note that the dimensionof the decision space and that of the objective space are notnecessary the same) Given a Pareto front P of a certainoptimisation problem and an archive under evaluation Athat are assumed without loss of generality to be finite thedistance between any solution 119878A from A to P denoteddist(119878AP) is measured in the objective space by a Euclideandistance based formula

dist (119878AP) = min119878PisinP

d (119878A 119878P) (8)

where 119889 is the classical Euclidean distance Yet the naiveformulation of the distance between A and P

dist (AP) = min119878AisinA

dist (119878AP) = min119878AisinA119878PisinP

d (119878A 119878P) (9)

returns the distance between the nearest pairs fromA andPwhich does not reflect the closeness between the two sets as


E2E2

Feasibledomain

E1E1

Objective space Objective space

Archive Archive

Pareto front

Pareto sample

Archive dominated area

Volume ratio

Figure 4 Hypothetical volume domination versus practical volume domination

a matter of course indeed if only one solution is very closeto the Pareto set the distance would be lower providing aconfusing closeness measurement Therefore a sophisticateddistance formulation would consist in taking into accountthe distances between all pairs from A and P (Either theconsidered optimisation problem is discrete or continuousonly a discrete sample of the actual Pareto front would beavailable and yet a discrete approximating Pareto frontwouldbe solicited) Accordingly Smith [51] has proposed to use themedian of the Euclidean distance between pairs from the twosets

M (AP) = median119878Aisin119860

dist (119878AP) (10)

Besides two metrics that have been widely used in theliterature are the generational distance [52] 119866 and theinverted generational distance [53] 119868119866 that consist of thefollowing

G (AP) = 1|A|radic sum119878AisinA

dist2 (119878AP) (11)

IG (AP) = 1|P|radic sum119878PisinP

dist2 (119878PA) (12)

where |A| and |P| are the cardinality ofA andP respectivelyAnother example of metrics used in the literature is thedistance of Hausdorff consisting of the following

H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)

that has been used in an extended version in the works ofSchutze et al [50] and Bogoya et al [54]

It is straightforward that a multiobjective optimisationshould result in a set that is sufficiently covering as well as effi-ciently close to the actual Pareto set Deb et al [55] proposeda spread metric parameter to characterise the covering Yetno performance indicator from the aforementioned has gotthe general approval as an efficient and accurate indicator forboth closeness and spreadmeasurement Volumedomination

has been proposed in the literature [56ndash58] as a promisingalternative towards efficient evaluation of the constructedarchive

Hypervolume is a useful metric for characterising thecoverage of an archive as well as its closeness to the Paretofront This metric which dates back to the works of Zitzlerand Thiele [58] consists in calculating in the objectivespace the size of the area covered by the archive elementsThe considered optimisation problem would be reduced tomaximising such measurement Similarly a certain ratioof the volume dominated by the resultant archive to thatdominated by the Pareto front would be an alternativeperformance indicator Smith [51] has considered the areadominated by the Pareto front and not dominated by thearchive under evaluation Figure 4 illustrates the conceptualareas delimited by the Pareto front and the archive Of coursethe domination is related to the acceptance rule (weak orstrong) Yet in practice three areas can be differentiatednamely the area dominated by both the available Paretosample and the archive under evaluation (hatched grey zone)the area dominated by the Pareto sample only (hatched darkgrey zone) and the unexplored area (grey zone)

Many MOSA methods have used one or moredominance-oriented performance indicators in order toprovide an advanced metaheuristic An early discussion hasbeen conducted by Nam and Park [59] where six acceptancecriteria have been suggested and evaluated Besides Smithet al [60] have proposed a composite energy function basedon domination as an alternative to the classical weightedcombination of objective functionsWhile the latter approachallows a priori biasing guidance towards a certain region ofthe Pareto front thereby making the algorithm sensitive tothe weighting given to the competitive objective functionsthe proposed approach aims at eschewing any limitingproposal on the feasible domain exploration in favour ofdominance concept

Sankararao andYoo [61] proposed a robustmultiobjectivesimulated annealing (rMOSA) with the aim to speed upthe convergence process and to get a uniform Pareto setapproximation The method is developed with application tochemical engineering problems

A domination-based method called Archived Multiob-jective Simulated Annealing method (AMOSA) has been


proposed [62] using a parameter called amount of domi-nation For a given number of objective functions 119864119894 andconsidering an archived solution 119878119886 and a candidate solution119878119888 the amount of domination between the two solutions isgiven by the following

Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)

where 119877119894 is the range of the 119894119905ℎ objective function Thisdomination amount represents a hypervolume for whicheach side in the Euclidean plane constituted by certain twoobjective functions 119864119894 and 119864119895 is a rectangle delimited by thepoints 119878119888 (119864119894(119878119886) 119864119895(119878119888)) 119878119886 and (119864119894(119878119888) 119864119895(119878119886)) At eachiteration AMOSA constructs a candidate solution from acurrent solution at hand that will undergo an evaluation inrespect of the archive as well as the current solutionmdashanaverage amount of domination Δ119889119900119898119886V119903 is computed inregard to either the solutions dominated by the candidatesolution or those dominating it Any candidate solution thatis nondominated by the current solution or any other solutionfrom the archive is accepted as current and added to thearchiveMeanwhile any dominated solution from the archiveis removed If the candidate solution is dominated by acertain number of solutions from the archive the consideredacceptance rule consists of the following

P (Accept 119878119888) = 11 + exp (Δ119889119900119898119886V119903 (119878119888) times 119879) (15)

where 119879 is the current value of the temperature parameterIf the candidate solution dominates the current solution(which does not necessary belong to the archive) while itis dominated by a certain number of archived solutions atransition to an archived solution is performed with certainprobability Overall the possible transitions consist in accept-ing the candidate solution keeping the current solution ormoving to a solution from the archive In order to decideon the resultant archive size AMOSA defines two limitingparameters called hard limit and soft limit The archive sizeis allowed to increase up to the soft limit after which thenumber of solutions is reduced to the hard limit by meansof a certain clustering process

Suman et al [63] have presented an extension of AMOSAmethod on the basis of the orthogonal experiment design(OED) called Orthogonal Simulated Annealing (OSA) Themain contribution of this method lies in the use of the OEDfor guiding the selection of good neighbouring solutionsby means of an orthogonal table and a fractional factorialanalysis OED aims at allowing effective search while reduc-ing computational cost Figure 5 presents the OSA methodflowchart The equilibrium condition is set to impose anupper bound on the number of iterations that should beperformed for the same current solution at each temperaturelevel

On the basis of AMOSA method Sengupta and Saha[64] proposed a reference point based many-objective sim-ulated annealing algorithm referred to as RSA Rather than

moving from one solution to a neighbouring one the workintroduced the archive-to-archive concept that consists inmoving from an archive of solutions to another one The ideais analogous to the use of population of solutions in geneticalgorithms The method involves in addition a switchingstrategy called mutation switching which consists in peri-odically switching between different mutation techniquesBesides the method uses the reference point based clusteringalgorithm in order to get a uniform spread of the returnedarchive

Due to its computational efficiency and simplicity simu-lated annealing has been widely used to solve multiobjectiveoptimisation problems as well as mono-objective ones invarious fields including clustering [65ndash67] jobshop problems[68] and scheduling [69] Besides genetic algorithms andparticle swarm optimisation methods have received muchattention in the recent years and many commercial solverspermit simple implementation of suchmethods Accordinglyfew hybridisations of multiobjective simulated annealingwith one of the aforementioned metaheuristics have beenintroduced in the literature [70 71] in addition tomany appli-cations in supply chain [72 73] distribution networks [7475] facility layout design [76 77] design optimisation[66]and scheduling [69 78ndash80]

4 Summary and Concluding Remarks

Simulated annealing is a metaheuristic belonging to the localsearch methods family It guarantees gradual convergence toa near-optimal solution and provides the ability of escapinglocal optima and beingmemoryless Due to its adaptability tomany combinatorial and continuous optimisation problemsa wide range of adaptations have hitherto emerged Such asuccess has motivated an extension of the simulated anneal-ing meta-heuristic to the multiobjective optimisation contextthat consists in approximating the optimal solutions set bymeans of an archive that holds the nondominated solutionsfound while exploring the feasible domain Accordinglymany variants of multiobjective simulated annealing havebeen introduced in the literature Nevertheless the followingremarks are observed

(i) Compared to othermetaheuristics simulated anneal-ing is a powerful tool for many applications due toits asymptotic convergence However the algorithmprocess is rather slow especially for some coolingschemes and using another heuristic seems to givebetter result for problems with few optima

(ii) The use of an archive within a sequential storageprocess has emerged in almost all Pareto optimisa-tion including swarm optimisation and evolutionaryalgorithms (eg [81 82]) The process involves anupdating mechanism that becomes restrictive whenthe nondominated solutions number gets largerA discussion has been conducted on the use ofarchive regarding the performance of metaheuristicsin Pareto optimisation [83]

(iii) Controlling the number of returned solutions keepschallenging in multiobjective simulated annealing


Start

current_Solution

Pertubation

new_Solution

Fractionalfactorialanalysis

OED procedure

new_Solution

new_Solution isfeasible

Penaltyfunctionapproach

new_Solutionimproves

current_Solution

Accept new_Solutionwith certainprobability

Accept new_Solutionas current_Solution

and update Pareto set

Equilibriumcondition

Update temperature

Cooling conditionRandom

selection fromPareto set

Stop

no

yes

yes

no

yes

no

yes

no

Figure 5 Orthogonal simulated annealing flowchart

Generally all metaheuristics involving an archive torepresent an approximation of the actual Pareto solu-tions set should strive for this archive to be boundedand of limited size [84] Yet it may be difficult todecide on the archive cardinality in practice andthe same metaheuristic may lead to a very concisearchive for a certain problem occurrence and to avery large one for others Knowles and Corne [84]

have addressed a theoretical study of metaheuristicsarchiving as a sequential storage process leadingto the no significant performance of any proposedapproach so far to control the archive size Never-theless Gaspar-Cunha et al [85] have proposed asolution clusteringmechanismwith the aim to reducethe returned near-Pareto front when it is of largesize


(iv) Tuning simulated annealing parameters is challeng-ing in both single and multiple criteria optimisationAs a wise choice of parameters would result on fairlyrepresentative approximation of the Pareto-optimalsolutions set many works [86ndash88] have requiredpreliminary experiments Yet experiments wouldrequire some knowledge on the actual Pareto front interms of range or cardinality which is not often thecase The situation becomes more problematic if theexperiments would involve comparison with othermetaheuristics

Conflicts of Interest

The author declares that they have no conflicts of interest

References

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

[2] 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

[3] 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

[4] S Alizamir S Rebennack and P M Pardalos ldquoImproving theneighborhood selection strategy in simulated annealing usingthe optimal stopping problemrdquo in Simulated Annealing C MTan Ed pp 363ndash382 InTech Rijeka Croatia 2008

[5] L Goldstein andMWaterman ldquoNeighborhood size in the sim-ulated annealing algorithmrdquo American Journal of Mathematicaland Management Sciences vol 8 no 3-4 pp 409ndash423 1988

[6] E Aarts J Korst and W Michiels ldquoSimulated annealingrdquo inSearch Methodologies Introductory Tutorials in Optimizationand Decision Support Techniques K Burke and G Kendall Edspp 91ndash120 Springer US Boston Mass USA 2nd edition 2014

[7] E-G TalbiMetaheuristics fromDesign to Implementation JohnWiley amp Sons Hoboken New Jersey NJ USA 2009

[8] E Triki Y Collette and P Siarry ldquoA theoretical study onthe behavior of simulated annealing leading to a new coolingschedulerdquo European Journal of Operational Research vol 166no 1 pp 77ndash92 2005

[9] P N Strenski and S Kirkpatrick ldquoAnalysis of finite lengthannealing schedulesrdquo Algorithmica vol 6 no 1 pp 346ndash3661991

[10] S Geman and D Geman ldquoStochastic relaxation Gibbs distri-butions and the Bayesian restoration of imagesrdquo IEEE Transac-tions on Pattern Analysis and Machine Intelligence vol 6 no 6pp 721ndash741 1984

[11] K Sekihara H Haneishi and N Ohyama ldquoDetails of simulatedannealing algorithm to estimate parameters of multiple currentdipoles using biomagnetic datardquo IEEE Transactions on MedicalImaging vol 11 no 2 pp 293ndash299 1992

[12] M Lundy and A Mees ldquoConvergence of an annealing algo-rithmrdquo Mathematical Programming vol 34 no 1 pp 111ndash1241986

[13] L Ingber ldquoVery fast simulated re-annealingrdquoMathematical andComputer Modelling vol 12 no 8 pp 967ndash973 1989

[14] K Amine ldquoInsights into simulated annealingrdquo in Handbookof Research on Modeling Analysis and Application of Nature-Inspired Metaheuristic Algorithms S Dash B K Tripathy andA Rahman Eds Advances in Computational Intelligence andRobotics pp 121ndash139 IGI Global Hershey Pa USA 2018

[15] K A Dowsland and J M Thompson ldquoSimulated annealingrdquo inHandbook of Natural Computing G Rozenberg T Back and JN Kok Eds pp 1623ndash1655 Springer-Verlag Berlin Germany2012

[16] D Fouskakis andD Draper ldquoStochastic optimization a reviewrdquoInternational Statistical Review vol 70 no 3 pp 315ndash349 2002

[17] A Franzin and T Stutzle ldquoRevisiting simulated annealing acomponent-based analysisrdquo Computers amp Operations Researchvol 104 pp 191ndash206 2019

[18] N Siddique and H Adeli ldquoSimulated annealing its variantsand engineering applicationsrdquo International Journal onArtificialIntelligence Tools vol 25 no 6 Article ID 1630001 2016

[19] B Suman N Hoda and S Jha ldquoOrthogonal simulated anneal-ing for multiobjective optimizationrdquo Computers amp ChemicalEngineering vol 34 no 10 pp 1618ndash1631 2010

[20] J Korhonen and Y Wang ldquoEffect of packet size on loss rateand delay in wireless linksrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNCrsquo05) vol3 pp 1608ndash1613 New Orleans Louisiana La USA 2005

[21] K Sorensen and J Springael ldquoProgressive multi-objectiveoptimizationrdquo International Journal of Information Technologyamp Decision Making vol 13 no 5 pp 917ndash936 2014

[22] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms J J Grefenstette Ed pp 93ndash100 Lawrence Erlbaum Associates Inc Hillsdale NJ USA1985

[23] P M Hansen ldquoTabu search for multiobjective optimizationMOTSrdquo in Proceedings of the 13th International Conference onMultiple Criteria Decision Making (MCDM 97) Cape TownSouth Africa 1996

[24] LMGambardella E Taillard andGAgazzi ldquoMACS-VRPTWa multiple ant colony system for vehicle routing problems withtime windowsrdquo in New Ideas in Optimization D Corne MDorigo F Glover et al Eds pp 63ndash76 McGraw-Hill LtdMaidenhead UK 1999

[25] E L Ulungu J Teghem P H Fortemps and D TuyttensldquoMOSA method a tool for solving multiobjective combinato-rial optimization problemsrdquo Journal of Multi-Criteria DecisionAnalysis vol 8 no 4 pp 221ndash236 1999

[26] R P Beausoleil ldquolsquoMOSSrsquo multiobjective scatter search appliedto non-linear multiple criteria optimizationrdquo European Journalof Operational Research vol 169 no 2 pp 426ndash449 2006

[27] I Das and J E Dennis ldquoA closer look at drawbacks of mini-mizing weighted sums of objectives for Pareto set generationin multicriteria optimization problemsrdquo Journal of StructuralOptimization vol 14 no 1 pp 63ndash69 1997

[28] P Serafini ldquoSimulated annealing for multi objective optimiza-tion problemsrdquo in Multiple Criteria Decision Making Proceed-ings of the Tenth International Conference Expand and Enrichthe Domains of Thinking and Application G H Tzeng H FWang U P Wen and P L Yu Eds pp 283ndash292 Springer-Verlag New York NY USA 1994


[29] M H Alrefaei and A H Diabat ldquoA simulated annealingtechnique formulti-objective simulation optimizationrdquoAppliedMathematics and Computation vol 215 no 8 pp 3029ndash30352009

[30] P Czyzak and A Jaszkiewicz ldquoPareto simulated annealingametaheuristic technique for multiple-objective combinatorialoptimizationrdquo Journal of Multi-Criteria Decision Analysis vol7 no 1 pp 34ndash47 1998

[31] B Suman ldquoSimulated annealing-based multiobjective algo-rithms and their application for system reliabilityrdquo EngineeringOptimization vol 35 no 4 pp 391ndash416 2003

[32] A Suppapitnarm K A Seffen G T Parks and P J ClarksonldquoA Simulated annealing algorithm for multiobjective optimiza-tionrdquo Engineering Optimization vol 33 no 1 pp 59ndash85 2000

[33] J TeghemD Tuyttens and E LUlungu ldquoAn interactive heuris-tic method for multi-objective combinatorial optimizationrdquoComputers amp Operations Research vol 27 no 7-8 pp 621ndash6342000

[34] O Tekinalp and G Karsli ldquoA new multiobjective simulatedannealing algorithmrdquo Journal of Global Optimization vol 39no 1 pp 49ndash77 2007

[35] M Villalobos-Arias C A Coello Coello and O Hernandez-Lerma ldquoAsymptotic convergence of a simulated annealing algo-rithm for multiobjective optimization problemsrdquoMathematicalMethods of Operations Research vol 64 no 2 pp 353ndash3622006

[36] E L Ulungu J Teghem and P Fortemps ldquoHeuristic for multi-objective combinatorial optimization problems by simulatedannealingrdquo inMCDMTheory and Applications J Gu G ChenQ Wei and S Wang Eds pp 229ndash238 Sci-Tech 1995

[37] R Banos C Gil B Paechter and J Ortega ldquoParallelization ofpopulation-basedmulti-objectivemeta-heuristics an empiricalstudyrdquo Applied Mathematical Modelling vol 30 no 7 pp 578ndash592 2006

[38] M Hapke A Jaszkiewicz and R Słowinski ldquoPareto simulatedannealing for fuzzy multi-objective combinatorial optimiza-tionrdquo Journal of Heuristics vol 6 no 3 pp 329ndash345 2000

[39] W J Gutjahr ldquoTwometaheuristics for multiobjective stochasticcombinatorial optimizationrdquo in Stochastic Algorithms Founda-tions and Applications O B Lupanov O M Kasim-Zade A VChaskin and K Steinhofel Eds pp 116ndash125 Springer BerlinGermany 2005

[40] H Li and D Landa-Silva ldquoAn adaptive evolutionary multi-objective approach based on simulated annealingrdquo EvolutionaryComputation vol 19 no 4 pp 561ndash595 2011

[41] H Li and D Landa-Silva ldquoEvolutionary multi-objective simu-lated annealingwith adaptive and competitive search directionrdquoin Proceedings of the 2008 IEEE Congress on Evolutionary Com-putation (IEEE World Congress on Computational Intelligence)IEEE Hong Kong June 2008

[42] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[43] M Laumanns L Thiele K Deb and E Zitzler ldquoCombiningconvergence and diversity in evolutionary multiobjective opti-mizationrdquoEvolutionary Computation vol 10 no 3 pp 263ndash2822002

[44] P Engrand ldquoA multi-objective approach based on simulatedannealing and its application to nuclear fuel managementrdquo inProceedings of the 5th International Conference on Nuclear Engi-neering ICONE-5 pp 416ndash423 American Society of Mechani-cal Engineers New York NY USA 1997

[45] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[46] K Deb ldquoMulti-objective genetic algorithms problem difficul-ties and construction of test problemsrdquo Evolutionary Computa-tion vol 7 no 3 pp 205ndash230 1999

[47] S Huband P Hingston L Barone and L While ldquoA reviewof multiobjective test problems and a scalable test problemtoolkitrdquo IEEE Transactions on Evolutionary Computation vol10 no 5 pp 477ndash506 2006

[48] Y Jin and B Sendhoff ldquoConstructing dynamic optimization testproblems using the multi-objective optimization conceptrdquo inApplications of Evolutionary Computing G R Raidl S Cagnoniand J Branke Eds pp 525ndash536 Springer Berlin Germany2004

[49] G Rudolph O Schutze C Grimme C Domınguez-Medinaand H Trautmann ldquoOptimal averaged Hausdorff archivesfor bi-objective problems theoretical and numerical resultsrdquoComputational Optimization and Applications vol 64 no 2 pp589ndash618 2016

[50] O Schutze X Esquivel A Lara and C A C Coello ldquoUsingthe averaged Hausdorff distance as a performance measure inevolutionary multiobjective optimizationrdquo IEEE Transactionson Evolutionary Computation vol 16 no 4 pp 504ndash522 2012

[51] K I Smith A Study of Simulated Annealing Techniquesfor Multi-Objective Optimisation [PhD thesis] University ofExeter England UK 2006

[52] D A van Veldhuizen and G B Lamont ldquoEvolutionary com-putation and convergence to a pareto frontrdquo in Proceedings ofthe Late Breaking Papers at the 1998 Genetic Programming Con-ference J R Koza Ed Omni Press University of WisconsinMadison Wis USA July 1998

[53] H Ishibuchi H Masuda Y Tanigaki and Y Nojima ldquoModifieddistance calculation in generational distance and invertedgenerational distancerdquo in Evolutionary Multi-Criterion Opti-mization A Gaspar-Cunha C H Antunes and C C CoelloEds vol 9019 of Lecture Notes in Computer Science pp 110ndash125Springer International Publishing Cham Switzerland 2015

[54] J M Bogoya A Vargas O Cuate and O Schutze ldquoA (pq)-averaged Hausdorff distance for arbitrary measurable setsrdquoMathematical amp Computational Applications vol 23 no 3Article ID 51 2018

[55] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms Wiley Chichester UK 2001

[56] A Auger J Bader D Brockhoff and E Zitzler ldquoHypervolume-based multiobjective optimization theoretical foundations andpractical implicationsrdquo Theoretical Computer Science vol 425pp 75ndash103 2012

[57] J Wu and S Azarm ldquoMetrics for quality assessment of amultiobjective design optimization solution setrdquo Journal ofMechanical Design vol 123 no 1 pp 18ndash25 2001

[58] E Zitzler and L Thiele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEETransactions onEvolutionaryComputation vol3 no 4 pp 257ndash271 1999

[59] D Nam and C Park ldquoMultiobjective simulated annealing acomparative study to evolutionary algorithmsrdquo InternationalJournal Fuzzy Systems vol 2 no 2 pp 87ndash97 2000

[60] K I Smith R M Everson J E Fieldsend C Murphy and RMisra ldquoDominance-basedmultiobjective simulated annealingrdquo


IEEE Transactions on Evolutionary Computation vol 12 no 3pp 323ndash342 2008

[61] B Sankararao and C K Yoo ldquoDevelopment of a robustmultiobjective simulated annealing algorithm for solving mul-tiobjective optimization problemsrdquo Industrial amp EngineeringChemistry Research vol 50 no 11 pp 6728ndash6742 2011

[62] S Bandyopadhyay S Saha U Maulik and K Deb ldquoA simu-lated annealing-based multiobjective optimization algorithmAMOSArdquo IEEE Transactions on Evolutionary Computation vol12 no 3 pp 269ndash283 2008

[63] B Suman and P Kumar ldquoA survey of simulated annealing as atool for single and multiobjective optimizationrdquo Journal of theOperational Research Society vol 57 no 10 pp 1143ndash1160 2006

[64] R Sengupta and S Saha ldquoReference point based archived manyobjective simulated annealingrdquo Information Sciences vol 467pp 725ndash749 2018

[65] S G Devi and M Sabrigiriraj ldquoA hybrid multi-objective fireflyand simulated annealing based algorithm for big data classifica-tionrdquo Concurrency and Computation Practice and ExperienceArticle ID e4985 2018

[66] S L Ho S Yang H CWong andGNi ldquoA simulated annealingalgorithm for multiobjective optimizations of electromagneticdevicesrdquo IEEE Transactions on Magnetics vol 39 no 3 pp1285ndash1288 2003

[67] S Saha and S Bandyopadhyay ldquoA newmultiobjective clusteringtechnique based on the concepts of stability and symmetryrdquoKnowledge and Information Systems vol 23 no 1 pp 1ndash27 2010

[68] R K Suresh and K M Mohanasundaram ldquoPareto archivedsimulated annealing for job shop scheduling with multipleobjectivesrdquo The International Journal of Advanced Manufactur-ing Technology vol 29 no 1-2 pp 184ndash196 2006

[69] V Yannibelli and A Amandi ldquoHybridizing a multi-objectivesimulated annealing algorithm with a multi-objective evolu-tionary algorithm to solve a multi-objective project schedulingproblemrdquo Expert Systems with Applications vol 40 no 7 pp2421ndash2434 2013

[70] A Abubaker A Baharum and M Alrefaei ldquoMulti-objectiveparticle swarm optimization and simulated annealing in prac-ticerdquo Applied Mathematical Sciences vol 10 no 42 pp 2087ndash2103 2016

[71] P Vasant ldquoHybrid simulated annealing and genetic algorithmsfor industrial productionmanagement problemsrdquo InternationalJournal of Computational Methods vol 7 no 2 pp 279ndash2972010

[72] V Babaveisi M M Paydar and A S Safaei ldquoOptimizinga multi-product closed-loop supply chain using NSGA-IIMOSA and MOPSO meta-heuristic algorithmsrdquo Journal ofIndustrial Engineering International vol 14 no 2 pp 305ndash3262017

[73] T Dereli and G Sena Das ldquoA hybrid simulated annealing algo-rithm for solving multi-objective container-loading problemsrdquoApplied Artificial Intelligence vol 24 no 5 pp 463ndash486 2010

[74] C H Antunes P Lima E Oliveira and D F Pires ldquoAmulti-objective simulated annealing approach to reactive powercompensationrdquo Engineering Optimization vol 43 no 10 pp1063ndash1077 2011

[75] J Marques M Cunha and D Savic ldquoMany-objective opti-mization model for the flexible design of water distributionnetworksrdquo Journal of Environmental Management vol 226 pp308ndash319 2018

[76] S Turgay ldquoMulti objective simulated annealing approach forfacility layout designrdquo International Journal of MathematicalEngineering andManagement Sciences vol 3 no 4 pp 365ndash3802018

[77] U R Tuzkaya T Ertay and D Ruan ldquoSimulated annealingapproach for the multi-objective facility layout problemrdquo inIntelligent Data Mining D Ruan G Chen E E Kerre and GWets Eds Studies in Computational Intelligence pp 401ndash418Springer Berlin Germany 2005

[78] C Akkan and A Gulcu ldquoA bi-criteria hybrid genetic algorithmwith robustness objective for the course timetabling problemrdquoComputers amp Operations Research vol 90 pp 22ndash32 2018

[79] H J Fraire Huacuja J Frausto-Solıs J D Teran-Villanueva JC Soto-Monterrubio J J Gonzalez Barbosa and G Castilla-Valdez ldquoAMOSA with analytical tuning parameters for het-erogeneous computing scheduling problemrdquo inNature-InspiredDesign of Hybrid Intelligent Systems P Melin O Castillo and JKacprzyk Eds vol 667 of Studies in Computational Intelligencepp 701ndash711 Springer International Publishing Cham Switzer-land 2017

[80] B Shahul Hamid Khan and K Govindan ldquoA multi-objectivesimulated annealing algorithm for permutation flow shopscheduling problemrdquo International Journal of Advanced Oper-ations Management vol 3 no 1 pp 88ndash100 2011

[81] S Agrawal B K Panigrahi and M K Tiwari ldquoMultiobjectiveparticle swarm algorithm with fuzzy clustering for electricalpower dispatchrdquo IEEE Transactions on Evolutionary Computa-tion vol 12 no 5 pp 529ndash541 2008

[82] X Cai Y Li Z Fan and Q Zhang ldquoAn external archive guidedmultiobjective evolutionary algorithm based on decompositionfor combinatorial optimizationrdquo IEEE Transactions on Evolu-tionary Computation vol 19 no 4 pp 508ndash523 2015

[83] A Jaszkiewicz and T Lust ldquoND-tree-based update a fastalgorithm for the dynamic nondominance problemrdquo IEEETransactions on Evolutionary Computation vol 22 no 5 pp778ndash791 2018

[84] J Knowles and D Corne ldquoBounded pareto archiving theoryand practicerdquo inMetaheuristics for Multiobjective OptimisationX Gandibleux M Sevaux K Sorensen and V Trsquokindt Edspp 39ndash64 Springer Berlin Germany 2004

[85] A Gaspar-Cunha P Oliveira and J A Covas ldquoUse of geneticalgorithms in multicriteria optimization to solve industrialproblemsrdquo in Proceedings of the Seventh International Con-ference on Genetic Algorithms Michigan State University EastLansing MI July 19-23 1997 T Back Ed pp 682ndash688 MorganKaufmann Publishers Inc San Francisco Calif USA 1997

[86] P Cabrera-Guerrero G Guerrero J Vega and F JohnsonldquoImproving simulated annealing performance by means ofautomatic parameter tuningrdquo Studies in Informatics and Con-trol vol 24 no 4 pp 419ndash426 2015

[87] W G Jackson E Ozcan and R I John ldquoTuning a simulatedannealing metaheuristic for cross-domain searchrdquo in Proceed-ings of the 2017 IEEE Congress on Evolutionary Computation(CEC) pp 1055ndash1062 San Sebastian Spain June 2017

[88] S M Sait and H Youssef Iterative Computer Algorithms withApplications in Engineering Solving Combinatorial Optimiza-tion Problems IEEE Computer Society Press Los AlamitosCalif USA 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Start

Current solution

Neighbouringsolution

generation

Candidate solution

Energy evaluation Probabilisticacceptance

Regular acceptance Equilibriumcondition

Temperatureupdate

Cooling condition

Stop

yes

no

yes

no

yes

no

Figure 1 Flowchart of the mono-objective simulated annealing algorithm

each iteration until reaching the global optimum Simulatedannealing has introduced an occasional acceptance mech-anism of uphill solutions in order to explore the feasibledomain intensively and to avoid getting caught at localoptimaThe process is managed by a certain cooling schedulethat controls the temperature variation during the algorithmprocess Figure 1 presents the generic simulated annealingalgorithm flowchart

The generic simulated annealing algorithm consists oftwo nested loops Given a current solution and a fixedtemperature the inner loop consists at each iterationin generating a candidate neighbouring solution that willundergo an energy evaluation to decide whether to acceptit as current Occasionally some nonimproving solutionsare accepted according to a certain probabilistic rule Theloop is controlled by an equilibrium condition that limits thenumber of iterations at every temperature level It refers to athermal equilibrium when the algorithm is not expected tofind more improving solutions within the current exploredneighbourhood The outer loop consists in decreasing thetemperature level according to a certain cooling schemeThe temperature is supposed to be iteratively decreased untilreaching a cooling condition that often corresponds to a final

temperature due The algorithm involves therefore two mainconsiderations namely the neighbourhood structure and thecooling schedule

Neighbourhood structure refers to a set of conceptualconsiderations regarding both the representation and thegeneration mechanism of neighbouring solutions It is sup-posed to be proper to the problem under considerationand to provide a restricted number of possible transitionsFor ease handling solutions of optimisation problems areusually considered in a certain indirect representation thatconsists of a finite number of components within a certainconfiguration Thus two solutions are neighbours if theirconfigurations are similar that is the two configurationsdiffer in few components only or may have close energyvalues Neighbourhood structure consists rather in definingthe candidate solutions that the algorithm may move to ateach iteration Even often underestimated in early studies[4 5] the neighbourhood structure choice has a significantimpact on the neighbourhood size and even on the simulatedannealing performance Moreover it may affect the conceptof local optimality itself [6 7] thereby identifying a local opti-mum for every neighbourhood Accordingly the algorithm






Hybrid 119879119896 = 11987901 + 119896 if 119896 le 120573

1205721198961198790 if 119896 gt 120573 119879119896 = 119896119896 + 1119879119896minus1 if 119896 le 120573





























P (Accept 119878119888) = 119898prod119894=1

min(1 exp(120596119894Δ119864119894119879 )) (2)


P (Accept 119878119888) = min(1 max119894=1119898

(exp(120596119894Δ119864119894119879 ))) (3)



P (Accept 119878119888)= 120572 119898prod119894=1


119894=1119898(exp(120596119894Δ119864119894119879 )))

(4)












E2

Starting sample

Feasible domain

Pareto front

E1




sum119894

ln (119864119894) = ln(prod119894

119864119894) (5)



E2


Pareto front

E1

y

x


y1

x1

E2(x1 yS

S

1)

E1(x1 y1)




ln( 119864119894 (119878119888)119864119894 (119878119888119906119903))) (6)


P (Accept 119878119888) = prod119894


minus Δ119864119894119879119894 ) (7)





d (119878A 119878P) (8)




d (119878A 119878P) (9)



E2E2

Feasibledomain

E1E1


Archive Archive

Pareto front

Pareto sample


Volume ratio




dist (119878AP) (10)



dist2 (119878AP) (11)


dist2 (119878PA) (12)


H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018













Hybrid 119879119896 = 11987901 + 119896 if 119896 le 120573

1205721198961198790 if 119896 gt 120573 119879119896 = 119896119896 + 1119879119896minus1 if 119896 le 120573





























P (Accept 119878119888) = 119898prod119894=1

min(1 exp(120596119894Δ119864119894119879 )) (2)


P (Accept 119878119888) = min(1 max119894=1119898

(exp(120596119894Δ119864119894119879 ))) (3)



P (Accept 119878119888)= 120572 119898prod119894=1


119894=1119898(exp(120596119894Δ119864119894119879 )))

(4)












E2

Starting sample

Feasible domain

Pareto front

E1




sum119894

ln (119864119894) = ln(prod119894

119864119894) (5)



E2


Pareto front

E1

y

x


y1

x1

E2(x1 yS

S

1)

E1(x1 y1)




ln( 119864119894 (119878119888)119864119894 (119878119888119906119903))) (6)


P (Accept 119878119888) = prod119894


minus Δ119864119894119879119894 ) (7)





d (119878A 119878P) (8)




d (119878A 119878P) (9)



E2E2

Feasibledomain

E1E1


Archive Archive

Pareto front

Pareto sample


Volume ratio




dist (119878AP) (10)



dist2 (119878AP) (11)


dist2 (119878PA) (12)


H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018

























P (Accept 119878119888) = 119898prod119894=1

min(1 exp(120596119894Δ119864119894119879 )) (2)


P (Accept 119878119888) = min(1 max119894=1119898

(exp(120596119894Δ119864119894119879 ))) (3)



P (Accept 119878119888)= 120572 119898prod119894=1


119894=1119898(exp(120596119894Δ119864119894119879 )))

(4)












E2

Starting sample

Feasible domain

Pareto front

E1




sum119894

ln (119864119894) = ln(prod119894

119864119894) (5)



E2


Pareto front

E1

y

x


y1

x1

E2(x1 yS

S

1)

E1(x1 y1)




ln( 119864119894 (119878119888)119864119894 (119878119888119906119903))) (6)


P (Accept 119878119888) = prod119894


minus Δ119864119894119879119894 ) (7)





d (119878A 119878P) (8)




d (119878A 119878P) (9)



E2E2

Feasibledomain

E1E1


Archive Archive

Pareto front

Pareto sample


Volume ratio




dist (119878AP) (10)



dist2 (119878AP) (11)


dist2 (119878PA) (12)


H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018















E2

Starting sample

Feasible domain

Pareto front

E1




sum119894

ln (119864119894) = ln(prod119894

119864119894) (5)



E2


Pareto front

E1

y

x


y1

x1

E2(x1 yS

S

1)

E1(x1 y1)




ln( 119864119894 (119878119888)119864119894 (119878119888119906119903))) (6)


P (Accept 119878119888) = prod119894


minus Δ119864119894119879119894 ) (7)





d (119878A 119878P) (8)




d (119878A 119878P) (9)



E2E2

Feasibledomain

E1E1


Archive Archive

Pareto front

Pareto sample


Volume ratio




dist (119878AP) (10)



dist2 (119878AP) (11)


dist2 (119878PA) (12)


H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018









E2


Pareto front

E1

y

x


y1

x1

E2(x1 yS

S

1)

E1(x1 y1)




ln( 119864119894 (119878119888)119864119894 (119878119888119906119903))) (6)


P (Accept 119878119888) = prod119894


minus Δ119864119894119879119894 ) (7)





d (119878A 119878P) (8)




d (119878A 119878P) (9)



E2E2

Feasibledomain

E1E1


Archive Archive

Pareto front

Pareto sample


Volume ratio




dist (119878AP) (10)



dist2 (119878AP) (11)


dist2 (119878PA) (12)


H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018









E2E2

Feasibledomain

E1E1


Archive Archive

Pareto front

Pareto sample


Volume ratio




dist (119878AP) (10)



dist2 (119878AP) (11)


dist2 (119878PA) (12)


H (AP)= max sup

119878AisinA

inf119878PisinP

d (119878A 119878P) sup119878PisinP

inf119878AisinA

d (119878A 119878P) (13)










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018










Δ119889119900119898 (119878119888 119878119886) = prod119894

119864119894(119878119888) =119864119894(119878119886)

1003816100381610038161003816119864119894 (119878119888) minus 119864119894 (119878119886)1003816100381610038161003816119877119894 (14)














Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018









Start

current_Solution

Pertubation

new_Solution


OED procedure

new_Solution




current_Solution





Update temperature



Stop

no

yes

yes

no

yes

no

yes

no








References



































































































Journal of












Journal of







Volume 2018






Volume 2018












References



































































































Journal of












Journal of







Volume 2018






Volume 2018














































































Journal of












Journal of







Volume 2018






Volume 2018








Multiobjective Simulated Annealing: Principles and...

Documents

Transcript of Multiobjective Simulated Annealing: Principles and...