stability of cauchy jensen type functional equation in generalized fuzzy normed spaces in this paper we establish some stability results concerning the cauchy jensen functional equation in generalized fuzzy normed spaces the results of the present paper improve and extend some recent results

stability of homomorphisms for a 3d cauchy jensen type functional equation on c ternary algebras abstract in this paper we investigate homomorphisms between c ternary algebras and derivations on c ternary algebras associated with the following cauchy jensen type additive functional equation f x y 2 z f x z 2 y f y z 2 x 2 f x f y f z

stability of a general mixed additive cubic functional equation in non archimedean fuzzy normed spaces we establish some stability results concerning the general mixed additive cubic functional equation in non archimedean fuzzy normed spaces in addition we establish some results of approximately general mixed additive cubic mappings in non archimedean fuzzy normed spaces the results improve and extend some recent results

generalization of ulam stability problem for euler lagrange quadratic mappings in 1968 s m ulam proposed the problem when is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true in 1978 p m gruber proposed the ulam type problem suppose a mathematical object satisfies a certain property approximately is it then possible to approximate this object by objects satisfying the property exactly in this paper we solve the generalized ulam stability problem for non linear euler lagrange quadratic mappings satisfying approximately a mean equation and an euler lagrange type functional equations in quasi banach spaces and p banach spaces

a multi objective simulated annealing algorithm for scheduling in flowshops to minimize the makespan and total flowtime of jobs in this paper we consider the problem of permutation flowshop scheduling with the objectives of minimizing the makespan and total flowtime of jobs and present a multi objective simulated annealing algorithm mosa two initial sequences are obtained by using simple and fast existing heuristics supplemented by the implementation of three improvement schemes each of the two resultant sequences corresponds to a possible non dominated solution containing the minimum value of one objective function these sequences taken one at a time are given as the starting sequences to the mosa the mosa seeks to obtain non dominated solutions through the implementation of a simple probability function that attempts to generate solutions on the pareto optimal front the probability function selects probabilistically a particular objective function considering which the algorithm uncovers non dominated solutions moreover the probability function is varied in such a way that the entire objective function space is covered uniformly so as to obtain as many non dominated and well dispersed solutions as possible the parameters in the proposed mosa are determined after conducting a pilot study two variants of the proposed algorithm called mosa i and mosa ii with different parameter settings with respect to the temperature and epoch length are considered in the performance evaluation of algorithms in order to evaluate mosa i and mosa ii we have made use of 90 benchmark problems provided by taillard eur j operation res 64 1993 278 after an extensive literature survey the following flowshop multi objective scheduling algorithms have been identified as benchmark procedures a mogls multi objective genetic local search by ishibuchi and murata ieee trans syst man cybernet c appl rev 28 1998 392 b elitist non dominated sorting genetic algorithm enga by bagchi multi objective scheduling by genetic algorithms kluwer academic publishers 1999 c gpw gradual priority weighting approach by chang hsieh and lin int j prod econ 79 2002 171 and d a posteriori approach based heuristic by framinan leisten and ruiz usano eur j operation res 141 2002 559 the non dominated sets obtained from each of the existing benchmark algorithms and the proposed mosa i and mosa ii are compared and subsequently combined to obtain a net non dominated front it is found that most of the solutions in the net non dominated front are yielded by mosa i and mosa ii in addition it is noteworthy that both mosa i and mosa ii require less computational effort than the mogls enga and gpw

on the ulam stability of mixed type mappings on restricted domains in 1941 d h hyers solved the well known ulam stability problem for linear mappings in 1951 d g bourgin was the second author to treat the ulam problem for additive mappings in 1982 1998 we established the hyers ulam stability for the ulam problem of linear and nonlinear mappings in 1983 f skof was the first author to solve the ulam problem for additive mappings on a restricted domain in 1998 s m jung investigated the hyers ulam stability of additive and quadratic mappings on restricted domains in this paper we improve the bounds and thus the results obtained by s m jung in 1998 besides we establish the ulam stability of mixed type mappings on restricted domains finally we apply our recent results to the asymptotic behavior of functional equations of different types

stability of an alternative functional equation abstract let f s x map an abelian semigroup s into a banach space x we deal with stability of the following alternative functional equation f x y f x f y 0 f x y f x f y we assume that f x y f x f y 1 x y f x y f x f y 2 x y for all x y s where 1 2 s r are given functions and prove that under some additional assumptions on 1 2 there exists a unique additive mapping a s x such that f x a x x for x s where s r is a function which can be explicitly computed starting from 1 and 2

stability of multi jensen mappings in non archimedean normed spaces in this paper we study the stability of the multi jensen mappings in non archimedean normed spaces the results improve and extend some recent results

on approximation of approximate solutions of dhombres equation abstract let f s x map an abelian semigroup s into a banach space x we deal with stability of dhombres equation f x f y 0 f x y f x f y we assume that f x f y 1 x y f x y f x f y 2 x y for all x y s where 1 2 s r are given functions and prove that under some additional assumptions on 1 2 there exists a solution of dhombres equation a s x such that f x a x x for x s where s r is a function which can be explicitly computed starting from 1 and 2

pacemakers in large arrays of oscillators with nonlocal coupling we model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling we assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation we concentrate on the case of heterogeneities with positive average and show that steady solutions to the nonlocal problem exist in particular we show that these heterogeneities act as a wave source this effect is not possible in 3 dimensional systems such as the complex ginzburg landau equation where the wavenumber of weak sources decays at infinity to obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation we then use fredholm properties of the laplace operator in kondratiev spaces to obtain solutions to the eikonal equation and by extension to the nonlocal problem

efficient heuristics for flowshop sequencing with the objectives of makespan and flowtime minimisation in this paper we analyse the performance of flowshop sequencing heuristics with respect to the objectives of makespan and flowtime minimisation for flowtime minimisation we propose the strategy employed by the neh heuristic to construct partial solutions results show that this approach outperforms the common fast heuristics for flowtime minimisation while performing similarly or slightly worse than others which on reward prove to be much more cpu time consuming additionally the suggested approach is well balanced with respect to makespan and flowtime minimisation based on the previous results two algorithms are proposed for the sequencing problem with multiple objectives makespan and flowtime minimisation these algorithms provide the decision maker with a set of heuristically efficient solutions such that he she may choose the most suitable sequence for a given ratio between costs associated with makespan and those assigned to flowtime computational experience shows both algorithms to perform better than the current heuristics designed for the two criteria problem

on the stability of a quartic functional equation abstract in this paper we prove the generalized hyers ulam stability for the following quartic functional equation f 2 x y f 2 x y 4 f x y 4 f x y 24 f x 6 f y

pacemaker guided noise induced spatial periodicity in excitable media we study the impact of subthreshold periodic pacemaker activity and internal noise on the spatial dynamics of excitable media for this purpose we examine two systems that both consist of diffusively coupled units in the first case the local dynamics of the units is driven by a simple one dimensional model of excitability with a piece wise linear potential in the second case a more realistic biological system is studied and the local dynamics is driven by a model for calcium oscillations internal noise is introduced via the leap stochastic integration procedure and its intensity is determined by the finite size of each constitutive system unit we show that there exists an intermediate level of internal stochasticity for which the localized pacemaker activity maps best into coherent periodic waves whose spatial frequency is uniquely determined by the local subthreshold forcing via an analytical treatment of the simple minimal model for the excitable spatially extended system we explicitly link the pacemaker activity with the spatial dynamics and determine necessary conditions that warrant the observation of the phenomenon in excitable media our results could prove useful for the understanding of interplay between local and global agonists affecting the functioning of tissue and organs

intuitionistic fuzzy stability of a general mixed additive cubic equation we establish some stability results concerning the general mixed additive cubic functional equation f kx y f kx y kf x y kf x y 2f kx 2kf x in intuitionistic fuzzy normed spaces in addition we show under some suitable conditions that an approximately mixed additive cubic function can be approximated by a mixed additive and cubic mapping

stability of functional equations on restricted domains in a group and their asymptotic behaviors in this paper we consider hyers ulam stability problems for the pexider equation the cauchy equation and the jensen equation in general restricted domains in a group the main purpose of this paper is to find restricted domains such that the functional inequality satisfied in those domains extends to the inequality for the whole domain and such that the hyers ulam stability theorem holds for the inequalities as it does when the inequality holds globally we also consider a distributional version of the hyers ulam stability of the pexider equation in restricted domains and its asymptotic behaviors

a fixed point method for perturbation of bimultipliers and jordan bimultipliers in c ternary algebras let a be a c ternary algebra a c bilinear t a a a is called a c ternary algebra bi multiplier if it satisfies t abc d t a b cd t a bcd abt c d for all a b c d a also the mapping t a a a is a called c ternary algebra jordan bimultiplier if it satisfies t aaa a t a a aa t a aaa aat a a for all a a using the fixed point method we investigate the generalized hyers ulam rassias stability of bimultipliers and jordan bimultipliers in c ternary algebras the concept of hyers ulam rassias stability originated from the th m rassias stability theorem that appeared in his paper th m rassias proc am math soc 72 297 1978

fuzzy approximately cubic mappings we establish some stability results concerning the cubic functional equationf 2x y f 2x y 2f x y 2f x y 12f x in fuzzy normed spaces we discuss the fuzzy continuity of the cubic mappings and show that the existence of a solution for any approximately cubic mapping guarantees the completeness of the fuzzy normed space

heuristics for scheduling in flowshop with multiple objectives abstract most of the heuristics for flowshop scheduling aim at minimizing makespan however scheduling with multiple objectives such as that of minimizing makespan total flowtime and machine idletime is more effective in reducing the total scheduling cost in this article we first address the problem of scheduling to minimize makespan and total flowtime and propose a new heuristic algorithm a heuristic preference relation is developed as the basis for the heuristic so that only the potential job interchanges are checked for possible improvement with respect to these two objectives the proposed as well as the existing heuristics are evaluated in a large number of randomly generated large sized problems the proposed heuristic algorithm is then extended to cover the problem of scheduling to minimize makespan total flowtime and machine idletime the results of the experimental investigation of the evaluation of this heuristic algorithm and the existing heuristic in meeting these objectives are also reported it is found that both the proposed heuristics are more effective than the existing one for scheduling with multiple objectives

approximation of radical functional equations related to quadratic and quartic mappings abstract in this paper we introduce and solve of the radical quadratic and radical quartic functional equations f a x 2 b y 2 a f x b f y f a x 2 b y 2 f a x 2 b y 2 2 a 2 f x 2 b 2 f y we also establish some stability results in 2 normed spaces and then the stability by using subadditive and subquadratic functions in p 2 normed spaces for these functional equations

on the ulam stability of jensen and jensen type mappings on restricted domains in 1941 hyers solved the well known ulam stability problem for linear mappings in 1951 bourgin was the second author to treat this problem for additive mappings in 1982 1998 rassias established the hyers ulam stability of linear and nonlinear mappings in 1983 skof was the first author to solve the same problem on a restricted domain in 1998 jung investigated the hyers ulam stability of more general mappings on restricted domains in this paper we introduce additive mappings of two forms of jensen and jensen type and achieve the ulam stability of these mappings on restricted domains finally we apply our results to the asymptotic behavior of the functional equations of these types

fuzzy bounded linear operators in this paper a notion of boundedness of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space is introduced and two types strong and weak of fuzzy bounded linear operators are defined relation between fuzzy continuity and fuzzy boundedness is studied definitions of fuzzy bounded linear functionals are given and the notions of fuzzy dual spaces are developed the hahn banach theorem the open mapping theorem the closed graph theorem and the uniform boundedness principle theorem are established

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

fuzzy versions of hyers ulam rassias theorem we introduce three reasonable versions of fuzzy approximately additive functions in fuzzy normed spaces more precisely we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in a fuzzy sense

isomorphisms between c ternary algebras in this paper we prove the hyers ulam rassias stability of homomorphisms in c ternary algebras and of derivations on c ternary algebras for the following generalized cauchy jensen additive mapping 2f j 1pxj2 j 1dyj j 1pf xj 2 j 1df yj this is applied to investigate isomorphisms between c ternary algebras the concept of hyers ulam rassias stability originated from the rassias stability theorem that appeared in his paper on the stability of the linear mapping in banach spaces see proc amr math soc 72 297 300 1978

multi objective sequence dependent setup times permutation flowshop a new algorithm and a comprehensive study abstract the permutation flowshop scheduling problem has been thoroughly studied in recent decades both from single objective as well as from multi objective perspectives to the best of our knowledge little has been done regarding the multi objective flowshop with pareto approach when sequence dependent setup times are considered as setup times and multi criteria problems are important in industry we must focus on this area we propose a simple yet powerful algorithm for the sequence dependent setup times flowshop problem with several criteria the presented method is referred to as restarted iterated pareto greedy or ripg and is compared against the best performing approaches from the relevant literature comprehensive computational and statistical analyses are carried out in order to demonstrate that the proposed ripg method clearly outperforms all other algorithms and as a consequence it is a state of art method for this important and practical scheduling problem

target patterns in two dimensional heterogeneous oscillatory reaction diffusion systems properties of target patterns created by pacemakers representing local regions with the modified oscillation frequency are studied for twodimensional oscillatory reaction diffusion systems described by the complex ginzburg landau equation an approximate analytical solution based on the phase dynamics approximation is constructed for a circular core and compared with numerical results for circular and square cores the dependence of the wavenumber and frequency of generated waves on the size and frequency shift of the pacemaker is discussed instabilities of target patterns involving repeated creations of ring shaped amplitude defects are further considered c 2006 elsevier b v all rights reserved

stability of n lie homomorphisms and jordan n lie homomorphisms on n lie algebras the motivation of this paper is to apply the hypers ulam stability problems of some kinds of functional equations to the classes of n lie homomorphisms and n lie algebras by using the structures of n lie homomorphisms and n lie algebras in this paper the generalized hyers ulam rassias stability of n lie homomorphisms and jordan n lie homomorphisms on n lie algebras associated to the generalized cauchy jensen rassias additive functional equation are investigated using the fixed point methods

on the stability of a pexiderized gol ab schinzel equation in this paper we establish the stability of the pexiderized email protected b schinzel functional equation f x yf x g x h y under the condition that lim t 0h tx exists and is non zero

theory and methodology a bicriteria approach to the two machine flow shop scheduling problem in this paper we address the problem of minimizing makespan and sum of completion times simultaneously in a two machine flow shop environment we formulate the problem as a bicriteria scheduling problem and develop a branch and bound procedure that iteratively solves restricted single objective scheduling problems until the set of efficient solutions is completely enumerated we report computational results and explore certain properties of the set of efficient solutions we then discuss their implications for the decision maker

multiple and bicriteria scheduling a literature survey abstract real life scheduling problems require the decision maker to consider a number of criteria before arriving at any decision a solution which is optimal with respect to a given criterion might be a poor candidate for some other the trade offs involved in considering several different criteria provide useful insights to the decision maker thus considering problems with more than one criterion is more relebant in the context of real life scheduling problems surprisingly research in this important field has been scarce when compared to research in single criterion scheduling in this paper we provide a detailed literature survey of multiple and bicriteria problems in scheduling we also provide a broad classification scheme for scheduling problems

solution of the ulam stability problem for euler lagrange quadratic mappings abstract in 1940 s m ulam proposed at the university of wisconsin theproblem give conditions in order for a linear mapping near an approximately linear mapping to exist in 1968 s u ulam proposed the moregeneral problem when is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true in 1978 p m gruber proposed theulam type problem suppose a mathematical object satisfies a certain property approximately is it then possible to approximate this object by objects satisfying the property exactly according to p m gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types in 1982 1996 we solved the above ulam problem or equivalently the ulam type problem for linear mappings and established analogous stability problems in this paper we first introduce new quadratic weighted means and fundamental functional equations and then solve the ulam stability problem for non linear euler lagrange quadratic mappings q x y satisfying a mean equation and functional equation formula for all 2 dimensional vectors x1 x2 x2 withxa normed linear space y a real complete normed linear space and any fixed pair a1 a2 of realsaiand any fixed pair m1 m2 of positive realsmi i 1 2 formula

separation of fuzzy normed linear spaces abstract this paper deals with hausdorff separation axiom for fuzzy topological spaces the necessary and sufficient condition obtained in the fuzzy vector topology induced by a fuzzy norm is separated properties of separated and non separated fuzzy norms along with examples are studied separation of a fuzzy norm in terms of the ordinary norm it induces and its consequences are characterised separation of quotient fuzzy semi norm is studied

fuzzy topological vector spaces ii abstract some of the properties of fuzzy topological vector spaces are investigated also there are given necessary and sufficient conditions for a family of fuzzy sets in a vector space e to be the family of all neighborhoods of zero for a fuzzy linear topology

finite dimensional fuzzy normed linear space abstract the concept of a fuzzy normed linear space is introduced it is proved that in a finite dimensional fuzzy normed linear space fuzzy norms are the same upto fuzzy equivalence finite dimensional fuzzy subspaces of a fuzzy normed linear space are proved to be necessarily complete fuzzy normed linear spaces

an efficient heuristic approach to the scheduling of jobs in a flowshop abstract three heuristic algorithms for flowshop scheduling with the objective of minimizing total flowtime are presented in this paper since the flowshop scheduling problem has been found to be an np complete problem the development of heuristic algorithms that guarantee fairly accurate solutions becomes necessary the proposed algorithms are found to give consistently near optimal solutions and fare better than the existing methods

comment on approximate ternary jordan derivations on banach ternary algebras bavand savadkouhi et al j math phys 50 042303 2009 let a be a banach ternary algebra over c and x a ternary banach a module a c linear mapping d a a x x is called a ternary jordan derivation if d xxx a d x xx x xd x x x xxd x x for all x a bavand savadkouhi et al j math phys 50 042303 2009 investigated ternary jordan derivations on banach ternary algebras associated with the following functional equation f x y z 4 f 3x y 4z 4 f 4x 3z 4 2f x and proved the generalized ulam hyers stability of ternary jordan derivations on banach ternary algebras the mapping f in lemma 2 2 of bavand savadkouhi et al is identically zero and all of the results are trivial in this note we correct the statements of the results and the proofs

isomorphisms between c ternary algebras abstract in this paper we prove the hyers ulam rassias stability of homomorphisms in c ternary algebras and of derivations on c ternary algebras for the following cauchy jensen additive mappings 0 1 f x y 2 z f x y 2 z f x 2 f z 0 2 f x y 2 z f x y 2 z f y 0 3 2 f x y 2 z f x f y 2 f z these are applied to investigate isomorphisms between c ternary algebras the concept of hyers ulam rassias stability originated from the th m rassias stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

fuzzy stability of the jensen functional equation we establish a generalized hyers ulam rassias stability theorem in the fuzzy sense in particular we introduce the notion of fuzzy approximate jensen mapping and prove that if a fuzzy approximate jensen mapping is continuous at a point then we can approximate it by an everywhere continuous jensen mapping as a fuzzy version of a theorem of schwaiger we also show that if every fuzzy approximate jensen type mapping from the natural numbers into a fuzzy normed space can be approximated by an additive mapping then the fuzzy norm is complete

on the stability problem for a mixed type of quartic and quadratic functional equation let e1 and e2 be real linear spaces in this paper we determine the general solution for a mixed type functional equation of a quartic and a quadratic mapping f e1 e2 x2 xnn 1f x1 2n 1 n 2 i 1nf xi 2n 2 1 i j n xjf xi for all n variables x1 xn e1 where n 2 in addition we solve the generalized hyers ulam rassias stability problem for the functional equation

hyers ulam stability on a generalized quadratic functional equation in distributions and hyperfunctions we consider the hyers ulam stability of a generalized quadratic functional equation in the spaces of distributions of schwartz and hyperfunctions of gelfand modulo bounded distributions and hyperfunctions

stability of a mixed additive and cubic functional equation in quasi banach spaces abstract in this paper we establish the general solution and investigate the hyers ulam rassias stability of the following functional equation f 2 x y f 2 x y 2 f x y 2 f x y 2 f 2 x 2 f x in the quasi banach spaces

a bicriteria two machine permutation flowshop problem the problem attacked in this paper is the scheduling of n jobs on two machines which are assumed to be continuously available all jobs are available at the beginning of the scheduling period setup times are included in the processing times no preemption of jobs is allowed the objective is the minimization of a weighted combination of the average job flowtime and the schedule makespan three branch and bound bb and each one of these problems is solved for five different levels of the weighting factor an analysis into the resulting efficient points is also included an improved version of fsh1 called fsh2 is developed and both heuristics are subjected to extensive numerical experimentation

on the generalized hyers ulam stability of module left derivations let a be a unital normed algebra and let m be a unitary banach left a module if f a m is an approximate module left derivation then f a m is a module left derivation moreover if m a is a semiprime unital banach algebra and f tx is continuous in t r for each fixed x in a then every approximately linear left derivation f a a is a linear derivation which maps a into the intersection of its center z a and its jacobson radical rad a in particular if a is semisimple then f is identically zero

on approximation of approximately linear mappings by linear mappings abstract assume a is a normed linear space b is a banach space and f a b is a mapping approximately linear we solve the following ulam problem give conditions in order for a linear mapping near an approximately linear mapping to exist

stability of the multi jensen equation abstract in this paper we study the stability in the sense of th m rassias of multi jensen equation

some efficient heuristic methods for the flow shop sequencing problem abstract in this paper the best heuristic methods known up to now are compared to solve the flow shop sequencing problem and we improve the complexity of the best one next this problem is applied to taboo search a new technique to solve combinatorial optimization problems and computational experiments are reported finally a parallel taboo search algorithm is presented and experimental results show that this heuristic allows very good speed up

approximate ternary jordan derivations on banach ternary algebras let a be a banach ternary algebra over a scalar field r or c and x be a ternary banach a module a linear mapping d a a x x is called a ternary jordan derivation if d xxx a d x xx x xd x x x xxd x x for all x a in this paper we investigate ternary jordan derivations on banach ternary algebras associated with the following functional equation f x y z 4 f 3x y 4z 4 f 4x 3z 4 2f x moreover we prove the generalized ulam hyers stability of ternary jordan derivations on banach ternary algebras

on the stability of the pexiderized trigonometric functional equation abstract the aim of this paper is to investigate the superstability problem of the cosine d alembert the wilson and the sine functional equation from the pexiderized trigonometric functional equation t gh under the condition f x y f x y 2 g x h y x or y and also will investigated the stability of the jensen type functional equation j y under the condition f x y f x y 2 f y x y

homomorphisms and derivations in proper jcq triples we investigate homomorphisms in proper jcq triples and derivations on proper jcq triples associated with the 3 variable jensen functional equation begin array rcl 2f left frac x y z 2 right f x f y f z end array which was introduced and investigated by park cho and han we moreover prove the hyers ulam rassias stability of homomorphisms in proper jcq triples and of derivations on proper jcq triples this is applied to investigate isomorphisms between proper jcq triples

hyers ulam rassias stability of homomorphisms in quasi banach algebras associated to the pexiderized cauchy functional equation abstract in this paper we prove the hyers ulam rassias stability of homomorphisms in quasi banach algebras associated to the pexiderized cauchy functional equation this is applied to investigate homomorphisms between quasi banach algebras the concept of hyers ulam rassias stability originated from th m rassias stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

a particle swarm optimization algorithm for makespan and total flowtime minimization in the permutation flowshop sequencing problem abstract in this paper a particle swarm optimization algorithm pso is presented to solve the permutation flowshop sequencing problem pfsp with the objectives of minimizing makespan and the total flowtime of jobs for this purpose a heuristic rule called the smallest position value spv borrowed from the random key representation of bean j c bean genetic algorithm and random keys for sequencing and optimization orsa journal of computing 6 2 1994 154 160 was developed to enable the continuous particle swarm optimization algorithm to be applied to all classes of sequencing problems in addition a very efficient local search called variable neighborhood search vns was embedded in the pso algorithm to solve the well known benchmark suites in the literature the pso algorithm was applied to both the 90 benchmark instances provided by taillard e taillard benchmarks for basic scheduling problems european journal of operational research 64 1993 278 285 and the 14 000 random narrow random and structured benchmark instances provided by watson et al j p watson l barbulescu l d whitley a e howe contrasting structured and random permutation flowshop scheduling problems search space topology and algorithm performance orsa journal of computing 14 2 2002 98 123 for makespan criterion the solution quality was evaluated according to the best known solutions provided either by taillard or watson et al the total flowtime criterion was evaluated with the best known solutions provided by liu and reeves j liu c r reeves constructive and composite heuristic solutions to the p ci scheduling problem european journal of operational research 132 2001 439 452 and rajendran and ziegler c rajendran h ziegler ant colony algorithms for permutation flowshop scheduling to minimize makespan total flowtime of jobs european journal of operational research 155 2 2004 426 438 for the total flowtime criterion 57 out of the 90 best known solutions reported by liu and reeves and rajendran and ziegler were improved whereas for the makespan criterion 195 out of the 800 best known solutions for the random and narrow random problems reported by watson et al were improved by the vns version of the pso algorithm