comparison of fuzzy numbers based on the probability measure of fuzzy events abstract most approaches for ranking fuzzy numbers proposed in the literature are based on fuzzy sets theory only and suffer from lack of discrimination and occasionally conflict with intuition it is true that fuzzy numbers are frequently partial order and cannot be compared however this does not alleviate the need for comparison in practical applications in this paper the order of fuzzy numbers are determined based on the concept of probability measure of fuzzy events due to zadeh it considers both the mean and dispersion of alternatives and gives two groups of indices based on the uniform and the proportional probability distributions the approach is also extended to the comparison of random fuzzy numbers by means of a mean fuzzy number it is shown that several comparison indices in the literature can be obtained based on the probability present measure approach finally some typical examples are used to compare the various different approaches the different interpretations of the dispersion index under different physical situations are emphasized

a repairing technique for the local search of the job shop problem abstract the local search technique has become a widely used tool for solving many combinatorial optimization problems in the case of the job shop the implementation of such a technique is not straightforward at all due to the existence of the technological constraints among the operations that belong to the same job their presence renders a certain set of schedules infeasible consequently special attention is required when defining optimization algorithms to prevent the possibility of reaching an infeasible schedule during execution traditionally the problem is tackled on the neighborhood level by using only a limited set of moves for which feasibility inherently holds this paper proposes an alternative way to avoid infeasibility by incorporating a repairing technique into the mechanism for applying moves to a schedule whenever an infeasible move is being applied a repairing mechanism rearranges the underlying schedule in such a way that the feasibility of the move is restored the possibility of reaching infeasible solutions is therefore eliminated on the lowest possible conceptual level consequently neighborhood functions need not to be constrained to a limited set of feasible moves any more

solving the flexible job shop scheduling problem with sequence dependent setup times abstract this paper addresses the flexible job shop scheduling problem with sequence dependent setup times and where the objective is to minimize the makespan we first present a mathematical model which can solve small instances to optimality and also serves as a problem representation after studying structural properties of the problem using a disjunctive graph model we develop a tabu search algorithm with specific neighborhood functions and a diversification structure extensive experiments are conducted on benchmark instances test results first show that our algorithm outperforms most existing approaches for the classical flexible job shop scheduling problem additional experiments also confirm the significant improvement achieved by integrating the propositions introduced in this study

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

the average amount of information lost in multiplication we show that if x and y are integers independently and uniformly distributed in the set 1 n then the information lost in forming their product which is given by the equivocation h x y xmiddoty is theta loglogn we also prove two extremal results regarding cases in which x and y are not necessarily independently or uniformly distributed first we note that the information lost in multiplication can of course be 0 we show that the condition h x y xmiddoty 0 implies 2log2n h x y omega loglogn furthermore if x and y are independent and uniformly distributed on disjoint sets of primes it is possible to have h x y xmiddoty 0 with log2n h x and log2n h y each o loglogn second we show that no matter how x and y are distributed h x y xmiddoty o logn loglogn furthermore there are distributions in which x and y are independent and uniformly distributed over sets of numbers having only small and distinct prime factors for which we have h x y xmiddoty omega logn loglogn

fuzzy job shop scheduling based on ranking level spl lambda 1 interval valued fuzzy numbers job shop scheduling is a difficult problem both theoretically and practically the theoretical problems stem from the search for optimal schedules subject to a limited number of constraints while the complexity of practical problems is due to the number and variety of constraints that are not rigid in the actual situations indeed in real world descriptions there are many vaguely formulated relations and imprecise data although the job shop scheduling problem has often been investigated very little of this research is concerned with the uncertainty characterized by the imprecision in problem variables in this paper we consider a fuzzy job shop scheduling problem with imprecise processing times we use fuzzy numbers and level spl lambda 1 interval valued fuzzy numbers for the representation of vague processing times this problem is similar to a fuzzy multiple criteria optimization problem the primary results obtained from this research are 1 signed distance ranking fuzzy numbers used to obtain a job shop scheduling problem in the fuzzy sense and 2 signed distance ranking level spl lambda 1 interval valued fuzzy numbers used to obtain another such problem

feasible insertions in job shop scheduling short cycles and stable sets insertion problems arise in scheduling when additional activities have to be inserted into a given schedule this paper investigates insertion problems in a general disjunctive scheduling framework capturing a variety of job shop scheduling problems and insertion types first a class of scheduling problems is introduced characterized by disjunctive graphs with the so called short cycle property and it is shown that in such problems the feasible selections correspond to the stable sets of maximum cardinality in an associated conflict graph two types of insertion problems are then identified where the underlying disjunctive graph is through or bi connected for these cases it is shown that the short cycle property holds and the conflict graph is bipartite allowing to derive a polyhedral characterization of all feasible insertions an efficient method for deciding whether there exists a feasible insertion and a lower and upper bound procedure for the minimum makespan insertion problem are developed for bi connected graphs this procedure solves the insertion problem to optimality the obtained results are applied to three extensions of the classical job shop the multi processor task blocking and no wait job shop and two types of insertions job and block insertion

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

constructing a fuzzy controller from data abstract fuzzy control at the executive level can be interpreted as an approximation technique for a control function based on typical imprecisely specified input output tuples that are represented by fuzzy sets the imprecision is characterized by similarity relations that are induced by transformations of the canonical distance function between real numbers taking this interpretation of fuzzy controllers into account in order to derive a fuzzy controller from observed data typical input output tuples have to be identified in addition a concept of similarity based on a transformations of the canonical distance is needed in order to characterize the typical input output tuples by suitable fuzzy sets a variety of fuzzy clustering algorithms that are exactly working in this spirit exists they identify prototypes and assign fuzzy sets to the prototypes on the basis of a suitable transformed distance in this paper we discuss how such fuzzy clustering techniques can be applied to construct a fuzzy controller from data and introduce special clustering algorithms that are tailored for this problem

criteria for evaluating fuzzy ranking methods abstract in this paper four criteria for evaluating fuzzy ranking methods are investigated fuzzy preference representation rationality of fuzzy ordering distinguishability and robustness based on these criteria two existing ranking methods baas and kwakernaak s and nakamura s are evaluated and a new improved ranking method is suggested

fuzzy decision modeling for supply chain management managing a supply chain sc is very difficult since various sources of uncertainty and complex interrelationships between various entities exist in the sc moreover the reducing product life cycle and the heightened expectations of customers have also made the sc even harder to manage especially for innovative products although the innovative products can enable a company to achieve higher profit margins they make demands for them unpredictable because no historical data is available this paper develops a fuzzy decision methodology that provides an alternative framework to handle sc uncertainties and to determine sc inventory strategies while there is lack of certainty in data or even lack of available historical data fuzzy set theory is used to model sc uncertainty a fuzzy sc model based on possibility theory is developed to evaluate sc performances based on the proposed fuzzy sc model a genetic algorithm approach is developed to determine the order up to levels of stock keeping units in the sc to minimize the sc inventory cost subject to the restriction of fulfilling the target fill rate of the finished product the proposed model allows decision makers to express their risk attitudes and to analyze the trade off between customer service level and inventory investment in the sc and better sc inventory strategies can be made a simulation approach is used to validate the concept developed

a general approach for optimizing regular criteria in the job shop scheduling problem even though a very large number of solution methods has been developed for the job shop scheduling problem a majority has been designed for the makespan criterion in this paper we propose a general approach for optimizing any regular criterion in the job shop scheduling problem the approach is a local search method that uses a disjunctive graph model and neighborhoods generated by swapping critical arcs the connectivity property of the neighborhood structure is proved and a novel efficient method for evaluating moves is presented besides its generality another prominent advantage of the proposed approach is its simple implementation that only requires to tune the range of one parameter extensive computational experiments carried out on various criteria makespan total weighted flow time total weighted tardiness weighted sum of tardy jobs maximum tardiness show the efficiency of the proposed approach best results were obtained for some problem instances taken from the literature

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

alternating cluster estimation a new tool for clustering and function approximation many clustering models define good clusters as extrema of objective functions optimization of these models is often done using an alternating optimization ao algorithm driven by necessary conditions for local extrema we abandon the objective function model in favor of a generalized model called alternating cluster estimation ace ace uses an alternating iteration architecture but membership and prototype functions are selected directly by the user virtually every clustering model can be realized as an instance of ace out of a large variety of possible instances of non ao models we present two examples 1 an algorithm with a dynamically changing prototype function that extracts representative data and 2 a computationally efficient algorithm with hyperconic membership functions that allows easy extraction of membership functions we illustrate these non ao instances on three problems a simple clustering of plane data where we show that creating an unmatched ace algorithm overcomes some problems of fuzzy c means fcm ao and possibilistic c means pcm ao b functional approximation by clustering on a simple artificial data set and c functional approximation on a 12 input 1 output real world data set ace models work pretty well in all three cases

the permutation flow shop with buffers a tabu search approach the paper deals with the criteria of makespan minimisation for the permutation flow shop problem with buffers of limited capacity located between successive machines as the problem is np hard an approximation algorithm is proposed which has been designed using a non trivial generalisation of the block elimination properties known for the classic flow shop problem this algorithm is able to achieve excellent results for problems up to 200 jobs and 20 machines as it exploits the structural properties defined in the problem and combines these with a local search technique controlled by a tabu search strategy

a fuzzy robust scheduling approach for product development projects abstract efficient scheduling of a product development project is difficult since a development project is usually unique in nature and high level of design imprecision exists at the early stages of product development moreover risk averse project managers are often more interested in estimating the risk of a schedule being late over all potential realizations the objective of this research is to develop a robust scheduling methodology based on fuzzy set theory for uncertain product development projects the imprecise temporal parameters involved in the project are represented by fuzzy sets a measure of schedule robustness based on qualitative possibility theory is proposed to guide the search process to determine the robust schedule i e the schedule with the best worst case performance a genetic algorithm approach is developed for solving the problem with acceptable performance an example of electronic product development project is used to illustrate the concept developed

a contribution to convergence theory of fuzzy c means and derivatives in this paper we revisit the convergence and optimization properties of fuzzy clustering algorithms in general and the fuzzy c means fcm algorithm in particular our investigation includes probabilistic and a slightly modified implementation of possibilistic memberships which will be discussed under a unified view we give a convergence proof for the axis parallel variant of the algorithm by gustafson and kessel that can be generalized to other algorithms more easily than in the usual approach using reformulated fuzzy clustering algorithms we apply banach s classical contraction principle and establish a relationship between saddle points and attractive fixed points for the special case of fcm we derive a sufficient condition for fixed points to be attractive allowing identification of them as local minima of the objective function excluding the possibility of a saddle point

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

about the use of fuzzy clustering techniques for fuzzy model identification abstract in this work we present an alternative approach to generate fuzzy rules with a functional consequent associated to the tsk fuzzy model in our case using fuzzy clustering algorithms that look for linear behaviours in the product space of the input output data we analyse different methods to generate the associated fuzzy rules using in some cases multidimensional reference fuzzy sets in the product space of the input variables and in other cases fuzzy sets in each of the different dimensions in any case the rules being generated correspond to a tsk fuzzy model

embedding of linear programming in a simulated annealing algorithm for solving a mixed integer production planning problem the present paper is concerned with the grouping of book covers on offset plates in order to minimize the total production cost the mathematical formulation of the problem involves both binary and continuous variables as exact methods are unable to provide solutions in reasonable time a heuristic algorithm of the simulated annealing type is proposed at each iteration the values of the current solution binary variables are altered in order to yield a neighboring solution to compute the corresponding values of the continuous variables and the value of the objective function a linear programming routine is called at each iteration this constitutes the main originality of the present approach and is in principle applicable in mixed integer programming problems the procedure is tested on several examples

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

job sequencing with fuzzy processing times abstract in practice processing times can be more accurately represented as intervals with the most probable completion time somewhere near the middle of the interval a fuzzy number which is essentially a generalized interval can represent this processing time interval exactly and naturally in this work triangular and trapezoidal fuzzy numbers are used to represent those vague job processing times in job shop production systems the job sequencing algorithms of johnson and ignall and schrage are modified to accept fuzzy job processing times fuzzy makespans and fuzzy mean flow times are then calculated for greater decision making information numerous examples are used to illustrate the approach

tasc two attribute set clustering through decision tree construction clustering is the process of grouping a set of objects into classes of similar objects in the past clustering algorithms had a common problem that they use only one set of attributes for both partitioning the data space and measuring the similarity between objects this problem has limited the use of the existing algorithms on some practical situation hence this paper introduces a new clustering algorithm which partitions data space by constructing a decision tree using one attribute set and measures the degree of similarity using another three different partitioning methods are presented the algorithm is explained with illustration the performance and accuracy of the four partitioning methods are evaluated and compared

robust tsk fuzzy modeling for function approximation with outliers the takagi sugeno kang tsk type of fuzzy models has attracted a great attention of the fuzzy modeling community due to their good performance in various applications most approaches for modeling tsk fuzzy rules define their fuzzy subspaces based on the idea of training data being close enough instead of having similar functions besides training data sets algorithms often contain outliers which seriously affect least square error minimization clustering and learning algorithms a robust tsk fuzzy modeling approach is presented in the approach a clustering algorithm termed as robust fuzzy regression agglomeration rfra is proposed to define fuzzy subspaces in a fuzzy regression manner with robust capability against outliers to obtain a more precision model a robust fine tuning algorithm is then employed various examples are used to verify the effectiveness of the proposed approach from the simulation results the proposed robust tsk fuzzy modeling indeed showed superior performance over other approaches

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

on not first not last conditions in disjunctive scheduling abstract this paper is concerned with the development of constraint propagation techniques for the characterization of feasible solutions in disjunctive scheduling in disjunctive scheduling a set of uninterruptible tasks is to be performed on a set of resources each task has a release date a deadline and a fixed processing time each resource can handle only one task at a time some of these propagation techniques are implemented by rules that deduce either mandatory or forbidden sequences between tasks or sets of tasks for instance certain rules indicate whether a given task must or cannot be performed before or after a set of other competing tasks we focus our attention on the latter problem known as the not first not last nf nl problem the genericity of propagation rules is a question of major importance it induces that the result of the overall propagation must not depend on the order in which the inference rules are applied hence one must search for completeness in the time windows narrowing in order to ensure the convergence of the propagation towards a unique fix point an efficient algorithm is proposed it guarantees the completeness of time windows narrowing due to nf nl conditions it has been integrated in a branch and bound procedure to solve job shop instances it has also been tested within several lower bounding procedures computational results are reported and the power and complementarity of nf nl rules with other classical inference rules are discussed

design of a near optimal wide range fuzzy logic controller the paper describes a novel method for the design of a fuzzy logic controller with near optimal performance in the wide range of operating conditions the approach is based on the analysis of the system behaviour in the error state space the final control structure in a form of a dual fuzzy logic controller is arrived at in two stages the first stage encompasses design and tuning of the pi like fuzzy controller the second stage consists of placing an additional fuzzy controller of a structure similar to that of the first one in parallel with the pi like fuzzy controller designed in the first stage the final structure of the dual fuzzy controller is obtained by synthesis and tuning of those segments of the second controller that are different the resulting dual controller is characterised with high performance in the wide range of operating conditions and with small number of parameters that can be adjusted using simple optimisation methods the controller is developed and tested for a plant comprising a variable speed dc motor drive

estimating the concentration of optically active constituents of sea water by takagi sugeno models with quadratic rule consequents determining the concentrations of chlorophyll suspended particulate matter and coloured dissolved organic matter in the sea water is basic to support the monitoring of upwelling phenomena algae blooms and changes in the marine ecosystem since these concentrations affect the spectral distribution of the solar light back scattered by the water body their estimation can be computed by using a set of remotely sensed multispectral measurements of the reflected sunlight in this paper the relation between the concentrations of interest and the average subsurface reflectances is modelled by means of a set of second order takagi sugeno ts fuzzy rules unlike first order ts rules which adopt linear functions as consequent second order ts rules exploit quadratic functions thus improving the modelling capability of the rule in the subspace determined by the antecedent first we show how we can build a second order ts model through a simple transformation which allows estimating the consequent parameters using standard linear least squares algorithms and by adopting one of the most used methods proposed in the literature to generate first order ts models then we compare first order and second order ts models against mean square error and interpretability of rules we highlight how second order ts models allow us to achieve better approximation than first order ts models though maintaining interpretability of the rules finally we show how second order ts models perform considerably better the mean square error is lower by two orders of magnitude than the specific implementations of radial basis function networks and multi layer perceptron networks used in previous papers for the same application domain

fuzzy scheduling modelling flexible constraints vs coping with incomplete knowledge abstract an overview of some fuzzy set based approaches to scheduling is proposed emphasizing two distinct uses of fuzzy sets representing preference profiles and modelling uncertainty distributions the first setting leads to a valued non compensatory generalization of constraint directed scheduling the other setting yields a possibility theoretic counterpart of pert where probability distributions of activity durations are changed into possibility distributions for the purpose of modelling incomplete information it is pointed out that a special case of the latter interval valued pert is a difficult ill known problem regarding the determination of critical activities latest starting times and floats lastly when flexible constraints and uncertain processing times are to be jointly considered the use of possibilistic decision theory leads to the computation of robust schedules

two simulated annealing based heuristics for the job shop scheduling problem abstract in this paper we present two simulated annealing based algorithms for the classical general job shop scheduling problem where the objective is to minimize the makespan we consider sets of jobs consisting of tasks and sets of machines which can handle at most one task at a time to represent the problem we employ the model of disjunctive graphs simulated annealing has been applied to this problem earlier e g by van laarhoven et al where the neighborhood function is based on reversing a single arc of a longest path of the underlying graph in our approach we analyze a neighborhood function which involves a non uniform generation probability to obtain the neighbors of a schedule we reverse more than a single arc of longest paths and perform a control on the increase of the makespan the selection of the arcs depends on the number of longest paths to which a particular arc belongs furthermore we have designed two cooling schedules which employ a detailed analysis of the objective function depending on the specified neighborhood relation the expected run times can be either o n2 e or o n3 e m for the first cooling schedule and o n5 2 e m1 2 or o n7 2 e m1 2 for the second cooling schedule where n is the number of tasks m the number of machines and e represents o ln ln n ln n our computational experiments on small to large benchmark problems have shown that within short series of consecutive trials relatively stable results equal or close to optimal solutions are repeatedly obtained including the well known benchmark problems ft10 and la38 we could improve five upper bounds for the yn1 yn4 swv12 swv13 and swv15 benchmark problems e g for swv13 the gap between the lower and the former upper bound has been shortened by about 57 in our approach we rely only on basic information about the given problem instance

a study about dimensional change of industrial parts using fuzzy rules abstract this work presents the development of a fuzzy model aimed to predict the dimensional changes of actual industrial parts made in a new emerging material called austempered ductile iron adi the dimensional change is an important factor that must be taken into account to improve the fabrication process of any actual part of adi this change is produced during a heat treatment cycle and in the factory practice its prediction is difficult due to the high number of variables affecting it in the present case it was considered that the dimensional change is strongly influenced by 11 operative variables of the fabrication process associated to the chemical composition and heat treatment temperatures and holding times the data employed to build the model corresponds to measurements done by experts on different actual industrial parts the knowledge acquisition about the origin of the data was necessary to understand the uncertainty factors and to adjust the parameters for obtaining a proper model the model was developed with a takagi sugeno type structure and its parameters adjusted with the anfis learning algorithm and introduced into an expert system previously developed it allows the accurate prediction of dimensional changes and helps in the efficient design of new industrial adi parts

a fast tabu search algorithm for the permutation flow shop problem abstract a fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in the permutation flow shop is presented the algorithm is based on a tabu search technique with a specific neighborhood definition which employs a block of jobs notion computational experiments up to 500 jobs and 20 machines show its excellent numerical properties

optimal design of radial basis function neural networks for fuzzy rule extraction in high dimensional data abstract the design of an optimal radial basis function neural network rbfnf is not a straightforward procedure in this paper we take advantage of the functional equivalence between rbfn and fuzzy inference systems to propose a novel efficient approach to rbfn design for fuzzy rule extraction the method is based on advanced fuzzy clustering techniques solutions to practical problems are proposed by combining these different solutions a general methodology is derived the efficiency of our method is demonstrated on challenging synthetic and real world data sets

kernel based fuzzy clustering and fuzzy clustering a comparative experimental study in this study we present a comprehensive comparative analysis of kernel based fuzzy clustering and fuzzy clustering kernel based clustering has emerged as an interesting and quite visible alternative in fuzzy clustering however the effectiveness of this extension vis a vis some generic methods of fuzzy clustering has neither been discussed in a complete manner nor the performance of clustering quantified through a convincing comparative analysis our focal objective is to understand the performance gains and the importance of parameter selection for kernelized fuzzy clustering generic fuzzy c means fcm and gustafson kessel gk fcm are compared with two typical generalizations of kernel based fuzzy clustering one with prototypes located in the feature space kfcm f and the other where the prototypes are distributed in the kernel space kfcm k both generalizations are studied when dealing with the gaussian kernel while kfcm k is also studied with the polynomial kernel two criteria are used in evaluating the performance of the clustering method and the resulting clusters namely classification rate and reconstruction error through carefully selected experiments involving synthetic and machine learning repository http archive ics uci edu beta data sets we demonstrate that the kernel based fcm algorithms produce a marginal improvement over standard fcm and gk for most of the analyzed data sets it has been observed that the kernel based fcm algorithms are in a number of cases highly sensitive to the selection of specific values of the kernel parameters

robust scheduling of parallel machines with sequence dependent set up costs in this paper we propose a robust approach for solving the scheduling problem of parallel machines with sequence dependent set up costs in the literature several mathematical models and solution methods have been proposed to solve such scheduling problems but most of which are based on the strong assumption that input data are known in a deterministic way in this paper a fuzzy mathematical programming model is formulated by taking into account the uncertainty in processing times to provide the optimal solution as a trade off between total set up cost and robustness in demand satisfaction the proposed approach requires the solution of a non linear mixed integer programming nlmip that can be formulated as an equivalent mixed integer linear programming milp model the resulting milp model in real applications could be intractable due to its np hardness therefore we propose a solution method technique based on the solution of an approximated model whose dimension is remarkably reduced with respect to the original counterpart numerical experiments conducted on the basis of data taken from a real application show that the average deviation of the reduced model solution over the optimum is less than 1 5

constraint satisfaction problems algorithms and applications a constraint satisfaction problem csp requires a value selected from a given finite domain to be assigned to each variable in the problem so that all constraints relating the variables are satisfied many combinatorial problems in operational research such as scheduling and timetabling can be formulated as csps researchers in artificial intelligence ai usually adopt a constraint satisfaction approach as their preferred method when tackling such problems however constraint satisfaction approaches are not widely known amongst operational researchers the aim of this paper is to introduce constraint satisfaction to the operational researcher we start by defining csps and describing the basic techniques for solving them we then show how various combinatorial optimization problems are solved using a constraint satisfaction approach based on computational experience in the literature constraint satisfaction approaches are compared with well known operational research or techniques such as integer programming branch and bound and simulated annealing

jobshop scheduling with imprecise durations a fuzzy approach jobshop scheduling problems are np hard problems the durations in the reality of manufacturing are often imprecise and the imprecision in data is very critical for the scheduling procedures therefore the fuzzy approach in the framework of the dempster shafer theory commands attention the fuzzy numbers are considered as sets of possible probabilistic distributions after a review of some issues concerning fuzzy numbers we discuss the determination of a unique optimal solution of the problem and then we cast a meta heuristic simulated annealing sa to this particular framework for optimization it should be stressed that the obtained schedule remains feasible for all realizations of the operations durations

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

a fast taboo search algorithm for the job shop problem a fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in a job shop is presented the algorithm is based on a taboo search technique with a specific neighborhood definition which employs a critical path and blocks of operations notions computational experiments up to 2 000 operations show that the algorithm not only finds shorter makespans than the best approximation approaches but also runs in shorter time it solves the well known 10 10 hard benchmark problem within 30 seconds on a personal computer

scatter search with path relinking for the flexible job shop scheduling problem abstract the flexible job shop scheduling is a challenging problem due to its high complexity and the huge number of applications it has in real production environments in this paper we propose effective neighborhood structures for this problem including feasibility and non improving conditions as well as procedures for fast estimation of the neighbors quality these neighborhoods are embedded into a scatter search algorithm which uses tabu search and path relinking in its core to develop these metaheuristics we define a novel dissimilarity measure which deals with flexibility we conducted an experimental study to analyze the proposed algorithm and to compare it with the state of the art on standard benchmarks in this study our algorithm compared favorably to other methods and established new upper bounds for a number of instances

computing improved optimal solutions to max min flexible constraint satisfaction problems abstract the formal framework for decision making in a fuzzy environment is based on a general max min bottleneck like optimization problem proposed by zadeh it is also the basis for extending the constraint satisfaction paradigm of artificial intelligence to accommodating flexible or prioritized constraints this paper surveys refinements of the ordering of solutions supplied by the max min formulation namely the discrimin partial ordering and the leximin complete preordering a general algorithm is given which computes all maximal solutions in the sense of these relations it also sheds light on the structure of the set of best solutions moreover classes of problems for which there is a unique best discrimin and leximin solution are exhibited namely continuous problems with convex domains and so called isotonic problems noticeable examples of such problems are fuzzy linear programming problems and fuzzy pert like scheduling problems

measuring the satisfaction of constraints in fuzzy linear programming the paper proposes a new kind of method for solving fuzzy linear programming problems based on the satisfaction or fulfillment degree of the constraints using a new ranking method of fuzzy numbers the fulfillment of the constraints can be measured then the properties of the ranking index are discussed with this ranking index the decision maker can make the constraints tight or loose based on his optimistic or pessimistic attitude and get the optimal solution from the fuzzy constraint space the corresponding value of objective distribution function can be obtained a numerical example illustrates the merits of the approach

new optimization heuristics in a former paper we introduced a very effective new general purpose optimization principle we compared this method which we called threshold accepting ta with the well known simulated annealing sa method for discrete optimization the empirical results demonstrated the superiority of the ta algorithm in further experiments with the ta principle we discovered two new powerful optimization heuristics the great deluge algorithm gda and the record to record travel rrt these algorithms resemble in their structure the formerly studied ta and the sa method the differences lie in their acceptance rules for worse intermediate solutions both the gda and the rrt are essentially one parameter algorithms i e for the achievement of best possible performance a good choice is necessary only for a single parameter this is in contrast for instance to the classical as algorithm where it is necessary to choose carefully a certain sequence of parameters the so called annealing schedule the quality of the computational results obtained so far by rrt and gda shows that the new algorithms behave equally well as ta and thus a fortiori better than sa

general local search methods abstract this paper is a tutorial introduction to three recent yet widely used general heuristics simulated annealing tabu search and genetic algorithms a relatively precise description and an example of application are provided for each of the methods as well as a tentative evaluation and comparison from a pragmatic point of view

an alternative framework to lagrangian relaxation approach for job shop scheduling abstract a new lagrangian relaxation lr approach is developed for job shop scheduling problems in the approach operation precedence constraints rather than machine capacity constraints are relaxed the relaxed problem is decomposed into single or parallel machine scheduling subproblems these subproblems which are np complete in general are approximately solved by using fast heuristic algorithms the dual problem is solved by using a recently developed surrogate subgradient method that allows approximate optimization of the subproblems since the algorithms for subproblems do not depend on the time horizon of the scheduling problems and are very fast our new lr approach is efficient particularly for large problems with long time horizons for these problems the machine decomposition based lr approach requires much less memory and computation time as compared to a part decomposition based approach as demonstrated by numerical testing

particle swarm optimization combined with genetic operators for job shop scheduling problem with fuzzy processing time job shop scheduling problem is a np hard problem the processing time for each job is often imprecise in many real world applications and the imprecision in the data is critical for the scheduling procedures therefore job shop scheduling problem with fuzzy processing time is addressed in the paper the processing time is described by triangular fuzzy numbers the objective is to find a job sequence that minimizes the makespan and the uncertainty of the makespan by using an approach for ranking fuzzy numbers the particle swarm optimization pso is a randomized population based optimization method that was inspired by the flocking behavior of birds and human social interactions pso has been successfully applied to various real world applications but there is a little literature reported regarding application to scheduling problems as it was unsuitable for them in this paper pso is redefined and modified by introducing genetic operators such as crossover and mutation operator to update the particles we call this particle swarm optimization combined with genetic operators gpso this is successfully employed to solve the formulated problem ten benchmarks with fuzzy processing time are used to test gpso the feasibility as well as the efficiency of the proposed method is assessed in comparison with genetic algorithm ga

problem difficulty for tabu search in job shop scheduling tabu search algorithms are among the most effective approaches for solving the job shop scheduling problem jsp yet we have little understanding of why these algorithms work so well and under what conditions we develop a model of problem difficulty for tabu search in the jsp borrowing from similar models developed for sat and other np complete problems we show that the mean distance between random local optima and the nearest optimal solution is highly correlated with the cost of locating optimal solutions to typical random jsps additionally this model accounts for the cost of locating sub optimal solutions and provides an explanation for differences in the relative difficulty of square versus rectangular jsps we also identify two important limitations of our model first model accuracy is inversely correlated with problem difficulty and is exceptionally poor for rare very high cost problem instances second the model is significantly less accurate for structured non random jsps our results are also likely to be useful in future research on difficulty models of local search in sat as local search cost in both sat and the jsp is largely dictated by the same search space features similarly our research represents the first attempt to quantitatively model the cost of tabu search for any np complete problem and may possibly be leveraged in an effort to understand tabu search in problems other than job shop scheduling

fuzzy job sequencing for a flow shop abstract in job sequencing for a flow shop processing times are frequently not known exactly and only estimated intervals are given fuzzy numbers are ideally suited to represent these intervals in this work the campbell dudek and smith cds job sequencing algorithm is modified to accept trapezoidal fuzzy processing times deterministic sequences result but the sequence performance measurements of makespan and job mean flow time are fuzzy having been calculated using fuzzy arithmetic the use of possibility theory and the fuzzy integral enables the schedular to meaningfully interpret these fuzzy results deterministic approximations to this fuzzy approach are also investigated

disproof of the mertens conjecture the mertens conjecture states that m x 1 where m x n x n and n is the mo bius function this conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the riemann hypothesis this paper disproves the mertens conjecture by showing that x lim sup m x x 2 1 06 the disproof relies on extensive computations with the zeros of the zeta function and does not provide an explicit counterexample disproof of the mertens conjecture a m odlyzko at t bell laboratories murray hill new jersey 07974 usa and h j j te riele centre for mathematics and computer science kruislaan 413 1098 sj amsterdam the netherlands

extended grasp for the job shop scheduling problem with total weighted tardiness objective abstract the paper proposes a heuristic for the job shop scheduling problem with minimizing the total weighted tardiness of jobs as objective it is built upon the well known metaheuristic grasp and strengthened with an inclusion of specific local search components the design is based on an advanced disjunctive graph model which enables capturing solution schedules through a tree graph called critical tree the tree graph allows for effectively steering a first descent search algorithm which further incorporates powerful neighborhood operators and a fast move evaluation procedure based on heads updating additionally amplifying and path relinking is adaptively applied to the best schedules discovered we present computational results of the new heuristic on two famous sets of benchmark instances we identify ten new best solutions and we demonstrate the high potential of the approach through a comparison with state of the art methods

optimal fuzzy counterparts of scheduling rules the optimality of a fuzzy logic alternative to the usual treatment of uncertainties in a scheduling system using probability theory is examined formally fuzzy scheduling techniques proposed in the literature either fuzzify directly the existing scheduling rules or solve mathematical programming problems to determine the optimal schedules in the former method the fuzzy optimality for the optimal scheduling rules is usually not justified but still assumed in this paper the necessary conditions for fuzzy optimality are defined and fuzzy counterparts of some of the well known scheduling rules such as shortest processing time spt and earliest due date edd are developed

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

handling flexibility in a generalised job shop with a fuzzy approach abstract in this paper we deal with a particular scheduling problem inspired by a practical case coming from a belgian pharmaceutical company this problem has some particularities with respect to the classical job shop problem the principal one is the existence of a delay between the end of an operation and the start of the next operation of the same job this delay is not fixed but belongs generally to an interval range of possible values a resolution method the horizon method has been proposed in the deterministic case fixed data and strict constraints in this method the horizon time of each machine is discretised by unit times and at each unit time a binary value is associated all the operations applied on the machines horizons are logic operations or and right or left shifting and so on unfortunately this method does not take into account several aspects of the problem the existence of a preference relation on the possible values of the delay between successive operations and of flexible due dates in this work we modelise such flexible parameters using the fuzzy logic we propose a new method generalizing the horizon method this new method try to find for the realisation of the jobs a compromise between the choice of good values for the delay between couples of successive operations of the same job and good values for the completion time of the jobs

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

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

scheduling uncertain orders in the customer subcontractor context within the customer subcontractor negotiation process the first problem of the subcontractor is to provide the customer with a reliable order lead time although his workload is partially uncertain actually a part of the subcontractor workload is composed of orders under negotiation which can be either confirmed or cancelled fuzzy logic and possibility theory have widely been used in scheduling in order to represent the uncertainty or imprecision of processing times but the existence of the manufacturing orders is not usually set into question we suggest a method allowing to take into account the uncertainty of subcontracted orders this method is consistent with list scheduling as a consequence it can be used in many classical schedulers its implementation in a scheduler prototype called tapas is described in this article we focus on the performance of validation tests which show the interest of the method

fuzzy point estimation and its application on fuzzy supply chain analysis this work describes fuzzy point estimation that reflects the subjective measurement of a real number or an interval of real numbers and a new set of definitions for arithmetic operations on fuzzy points in today s changing business world there is a growing need to manage the supply chains of products to allow businesses to offer competitive prices to their customers managing the supply chains is becoming more complex because of market uncertainties this paper provides a new approach to model the uncertainties involved in the supply chain management using the fuzzy point estimation this work provides a clearer description and analysis framework mathematically for supply chain management based on the stationary strategy established by the author the order up to levels may be calculated by means of formulae directly under a certain stationary condition and then the optimal solution may be gotten for a stationary supply chain this paper is intended to open a window toward the complete results for the analysis of non stationary supply chains

a neuro fuzzy method to learn fuzzy classification rules from data abstract neuro fuzzy systems have recently gained a lot of interest in research and application neuro fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules neuro fuzzy techniques have been developed to support the development of e g fuzzy controllers and fuzzy classifiers in this paper we discuss a learning method for fuzzy classification rules the learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly and tunes them by modifying parameters of membership functions our approach is based on nefclass a neuro fuzzy model for pattern classification we also discuss some results obtained by our software implementation of nefclass which is freely available on the internet

membership functions some mathematical programming models and production scheduling abstract the authors have studied in 5 alternative real variable models based on the function d x x x 0 for certain integer or mixed interger programming problems mainly we have shown that there exists a vector 0 such that the solution to the problem min 1 x i 1 n x i ga i x i ax b x 0 is a solution to the problem min x ax b x 0 where x denotes the cardinal of x i e the number of strictly positive components of x thus obtaining a new model for solving in real numbers a generalized lattice point problem cabot 3 the function d x has been introduced by use as a general tool for solving integer or mixed integer problems due to its property of having almost everywhere almost discrete values in the meantime we noticed that this function may represent a membership function of a fuzzy set in this paper we study in detail the features of this membership function and show that cabot s results 3 may be derived in this more general setting using the complementary function s x 1 x x x at the same time in the paper there are some production scheduling models within the framework of fuzzy sets theory to this end a nonconvex production model is presented and it is shown that the value of the objective function 2 1 1 n for a production programming model whose deman and or resource vector components are parametrized may be considered as a grade of membership of the solution of the parametrized model to the feasible set of the nonparametrized production programming model consequently we get a nonconvex production programming model whose convex envelope is linear with coefficients which are in an inverse proportior to the magnitude of the nonparametrized demand or resource vector components this result agrees with the intuitive idea that a high level of demand or resource allows a greater interval of variation in the production process than a lower level of demand or resource

a filter and fan approach to the job shop scheduling problem the job shop scheduling problem jssp is a notoriously difficult problem in combinatorial optimization extensive investigation has been devoted to developing efficient algorithms to find optimal or near optimal solutions this paper proposes a new heuristic algorithm for the jssp that effectively combines the classical shifting bottleneck procedure sbp with a dynamic and adaptive neighborhood search procedure our new search method based on a filter and fan f f procedure uses the sbp as a subroutine to generate a starting solution and to enhance the best schedules produced the f f approach is a local search procedure that generates compound moves by a strategically abbreviated form of tree search computational results carried out on a standard set of 43 benchmark problems show that our f f algorithm performs more robustly and effectively than a number of leading metaheuristic algorithms and rivals the best of these algorithms

towards a generic view of primality through multiset decompositions of natural numbers abstract factorization results in multisets of primes and this mapping can be turned into a bijection between multisets of natural numbers and natural numbers at the same time simpler and more efficient bijections exist that share some interesting properties with the bijection derived from factorization this paper describes mechanisms to emulate properties of prime numbers through isomorphisms connecting them to computationally simpler representations involving bijections from natural numbers to multisets of natural numbers as a result interesting automorphisms of n and emulations of the rad mobius and mertens functions emerge in the world of our much simpler multiset representations finally we generalize the multiset decomposition mechanism derived from the diophantine equation 2 x 2 y 1 z and study emulations of the behavior of the function that counts the number of distinct prime factors of a number as well as the equivalent of b smoothness a property characterizing natural numbers having all their prime factors smaller than b the paper is organized as a self contained literate haskell program the code extracted from the paper is available as a standalone program at http logic cse unt edu tarau research 2012 jprimes hs

job shop scheduling by simulated annealing we describe an approximation algorithm for the problem of finding the minimum makespan in a job shop the algorithm is based on simulated annealing a generalization of the well known iterative improvement approach to combinatorial optimization problems the generalization involves the acceptance of cost increasing transitions with a nonzero probability to avoid getting stuck in local minima we prove that our algorithm asymptotically converges in probability to a globally minimal solution despite the fact that the markov chains generated by the algorithm are generally not irreducible computational experiments show that our algorithm can find shorter makespans than two recent approximation approaches that are more tailored to the job shop scheduling problem this is however at the cost of large running times