on estimating the distribution of optimal traveling salesman tour lengths using heuristics the traveling salesman problem is an important combinatorial optimization problem due to its significance in academic research and its real world applications the problem has been extensively studied and much is known about its polyhedral structure and algorithms for exact and heuristic solutions while most work is concentrated on solving the deterministic version of the problem there also has been some research on the stochastic tsp research on the stochastic tsp has concentrated on asymptotic properties and estimation of the tsp constant not much is however known about the probability distribution of the optimal tour length in this paper we present some empirical results based on monte carlo simulations for the symmetric euclidean and rectilinear tsps we derive regression equations for predicting the first four moments of the distribution of estimated tsp tour lengths using heuristics we then show that a beta distribution gives excellent fits for small to moderate sized tsp problems we derive regression equations for predicting the parameters of the beta distribution finally we predict the tsp constant using two alternative approaches

a note on the assignment problem with seniority and job priority constraints consider an assignment problem in which persons are qualified for a subset of the jobs assume the persons to belong to given seniority classes seniority constraints impose that the solution is such that no free person can be given a job unless an assigned person with the same or higher seniority becomes free similarly jobs can belong to priority classes for which priority constraints must hold it is shown that the assignment problem with both types of constraints can be solved by successively reoptimizing a rectangular linear assignment problem of increasing size the related complexity is lower than for a known coefficient scaling approach a further advantage of this approach is that it is easy to modify to solve the problem under the bottleneck criterion

improving heuristics for the frequency assignment problem abstract lower bounds for the frequency assignment problem can be found from maximal cliques and subgraphs related to cliques in this paper we show that for many types of problem optimal assignments can be found by a process of assigning these subgraphs first fixing the assignment and then extending the assignment to the full problem we demonstrate the advantages of the method for some typical examples in particular we give the first optimal assignments of several variants of the philadelphia problems these problems have been used by several authors to assess assignment methods and lower bounds

an evolutionary approach to construction of outranking models for multicriteria classification the case of the electre tri method outranking methods constitute an important class of multicriteria classification models often however their implementation is cumbersome due to the large number of parameters that the decision maker must specify past studies tried to address this issue using linear and nonlinear programming to elicit the necessary preferential information from assignment examples in this study an evolutionary approach based on the differential evolution algorithm is proposed in the context of the electre tri method computational results are given to test the effectiveness of the methodology and the quality of the obtained models

multiple criteria hierarchy process for electre tri methods abstract sorting methods in particular electre tri methods are widely used in multiple criteria decision aiding to deal with ordinal classification problems problems of this kind encountered in practice involve the evaluation of different alternatives actions on several evaluation criteria that are structured in a hierarchical way in order to deal with a hierarchical structure of criteria in decision problems multiple criteria hierarchy process mchp has been recently proposed in this paper we apply the mchp to the electre tri methods in particular we extend electre tri b electre tri c and electre tri nc methods we also adapt the mchp concept to the case where interaction among evaluation criteria has either strengthening or weakening or antagonistic effect finally we present an extension of the srf method to determine the weights of criteria in case they are hierarchically structured

rough set approach to multiple criteria classification with imprecise evaluations and assignments dominance based rough set approach drsa has been introduced to deal with multiple criteria classification also called multiple criteria sorting or ordinal classification with monotonicity constraints where assignments of objects may be inconsistent with respect to dominance principle in this paper we consider an extension of drsa to the context of imprecise evaluations of objects on condition criteria and imprecise assignments of objects to decision classes the imprecisions are given in the form of intervals of possible values in order to solve the problem we reformulate the dominance principle and introduce second order rough approximations the presented methodology preserves well known properties of rough approximations such as rough inclusion complementarity identity of boundaries and precisiation moreover the meaning of the precisiation property is extended to the considered case the paper presents also a way to reduce decision tables and to induce decision rules from rough approximations

using logical surrogate information in lagrangean relaxation an application to symmetric traveling salesman problems abstract the traveling salesman problem tsp is a classical combinatorial optimization problem which has been intensively studied the lagrangean relaxation was first applied to the tsp in 1970 the lagrangean relaxation limit approximates what is known today as held and karp hk bound a very good bound less than 1 from optimal for a large class of symmetric instances it became a reference bound for new heuristics mainly for the very large scale instances where the use of exact methods is prohibitive a known problem for the lagrangean relaxation application is the definition of a convenient step size control in subgradient like methods even preserving theoretical convergence properties a wrong defined control can affect the performance and increase computational times we show in this work how to accelerate a classical subgradient method while conserving good approximations to the hk bounds the surrogate and lagrangean relaxations are combined using the local information of the relaxed constraints it results in a one dimensional search that corrects the possibly wrong step size and is independent of the used step size control comparing with the ordinary subgradient method and beginning with the same initial multiplier the computational times are almost twice as fast for medium instances and greatly improved for some large scale tsplib instances

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

lexicographic maximin optimisation for fair bandwidth allocation in computer networks this paper addresses the problem of bandwidth allocation in multi application computer network environments allocations are determined from the solution of a multiple objective optimisation problem under network constraints where the lexicographic maximin criterion is applied to solve the problem and guarantees fairness and efficiency properties to the solution an algorithm based on a series of maximum concurrent multicommodity flow subproblems is proposed numerical results show the advantage of the approach compared to other traditional bandwidth allocation solutions

an interactive approach for placing alternatives in preference classes abstract in this paper we consider the multiple criteria decision making problem of partitioning alternatives into preference classes we develop an interactive procedure based on the assumption that the decision maker dm has a linear utility function the approach requires the dm to place an alternative in a preference class from time to time based on the preference information derived from the dm s placement as well as from dominance and linearity we try to place other alternatives we present results on the performance of the algorithm applied to four different sets of alternatives

lmi stability conditions for fractional order systems after an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives also denoted fractional order systems this paper deals with linear matrix inequality lmi stability conditions for fractional order systems under commensurate order hypothesis it is shown that a direct extension of the second lyapunov s method is a tedious task if the fractional order n is such that 0 n 1 the stability domain is not a convex region of the complex plane however through a direct stability domain characterization three lmi stability analysis conditions are proposed the first one is based on the stability domain deformation and the second one on a characterization of the instability domain which is convex the third one is based on generalized lmi framework these conditions are applied to the gain margin computation of a crone suspension

probabilistic analysis of the held and karp lower bound for the euclidean traveling salesman problem we analyze probabilistically the classical held karp lower bound derived from the 1 tree relaxation for the euclidean traveling salesman problem etsp we prove that if n points are identically and independently distributed according to a distribution with bounded support and absolutely continuous part fxdx over the d cube the held karp lower bound on these n points is almost surely asymptotic to hkdn d 1 d fx d 1 d dx where hkd is a constant independent of n the result suggests a probabilistic explanation of the observation that the lower bound is very close to the length of the optimal tour in practice since the etsp is almost surely asymptotic to tspdn d 1 d fx d 1 d dx the techniques we use exploit the polyhedral description of the held karp lower bound and the theory of subadditive euclidean functionals

electre tri nb a new multiple criteria ordinal classification method abstract this paper presents a new method for multiple criteria ordinal classification sorting problems in an ordinal classification problem the categories also called classes in related literature are pre defined and ordered from the best to the worst or from the worst to the best the actions not necessarily known a priori are assigned to the different ordered categories several electre type methods were designed to deal with this kind of problem however none of them proposes a characterization of the categories through a set of limiting profiles this is the novelty of the current method which may be considered as an extension of electre tri b it fulfills a set of structural requirements uniqueness of the assignments independence monotonicity homogeneity conformity and stability with respect to merging and splitting operations all these features will be presented in the current paper and illustrated through three examples

robust stability of fractional order system with general interval uncertainties abstract this paper focuses on the analysis of robust stability of fractional order system with general interval uncertainties the concept of general interval uncertainties means that the interval uncertainties exist both in the coefficients and orders of the fractional order system necessary and sufficient conditions are proposed to check the robust stability of general interval fractional order system according to the proposed stability criterion it is interesting to find that the edge theorem which was initially proposed for integer order system cannot be directly extended to test the stability of fractional order system with general interval uncertainties examples are followed to verify the validity of the proposed method

an algorithm for the ranking of shortest paths an efficient computational implementation of a path deletion k shortest paths algorithm and a new algorithm for the same problem are presented in a path deletion k shortest paths algorithm a sequence g1 g2 gk of networks is is defined such that g1 is given network and its k th shortes path is trivially determined from the shortest path in gk in essence as soon as the shortest path in gk is determined it is excluded from gk in such a way that no new paths are formed and no more paths are deleted so far each gk two procedures are executed a shortest path algorithm and a path deletion algorithm in the presented computational implementation all the information resulting from the determination of the k th shortest path is carried throughout gk 1 gk 2 gk the new algorithm not only keeps this characteristics but also avoids the last k 1 executions of a shortest path algorithm which results in a suprising and very substantial reduction in the execution time in fact for randomly generated networks with 104 nodes and 105 arcs once the shortest the new algorithm computes the next 100 shortest paths in times of the order of 10 1 seconds to illustrate the efficiency of this algorithm comparative computational experiments are reported

an improved transformation of the symmetric multiple traveling salesman problem we improve the standard transformation of the symmetric single depot multiple traveling salesman problem mtsp to one on a sparser edge configuration the improvement tends to suppress to a large extent the degeneracy of the resulting symmetric traveling salesman problem tsp as a result a 1 tree based tsp algorithm solves large mtsps of the type we consider as easily as it solves standard tsps we demonstrate the improvement by presenting computational results that are currently the best available

the partial sum criterion for steiner trees in graphs and shortest paths many optimization problems are defined in a weighted graph generally the criterion is min sum i e the total weight of all edges in the solution here we consider a different criterion

the efficiency of ballstep subgradient level methods for convex optimization we study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we establish convergence and efficiency estimates for simple ballstep level controls without requiring that the feasible set be compact our framework may handle accelerations based on cheap projections surrogate constraints and conjugate subgradient techniques

on dual solutions of the linear assignment problem abstract for the linear assignment problem we describe how to obtain different dual solutions it turns out that a shortest path algorithm can be used to compute such solutions with several interesting properties that enable to do better post optimality analysis two examples illustrate how different dual solutions can be used in the context of the traveling salesman problem

brief paper pseudo state feedback stabilization of commensurate fractional order systems this paper addresses the problem of pseudo state feedback stabilization of commensurate fractional order systems fos in the proposed approach linear matrix inequalities lmi formalism is used to check if the pseudo state matrix eigenvalues belong to the fos stability region of the complex plane a review of lmi stability conditions is first proposed for fractional order 0 n 1 and 1 n 2 the paper then focuses particularly on the case 0 n 1 as the stability region is non convex and associated lmi condition is not as straightforward to obtain as in the case 1 n 2 a new lmi stability condition is thus proposed based on this condition a necessary and sufficient lmi method for the design of stabilizing controllers is given this method paves the way for extension to fos of various lmi based results among these possible extensions a first result on robust control of polytopic fractional order systems is given in this paper

robust stability and stabilization of fractional order interval systems with the fractional order alpha the 0 alpha 1 case this technical note firstly presents a sufficient and necessary condition for the robust asymptotical stability of fractional order interval systems with the fractional order satisfying 0 1 and then a sufficient condition for the robust asymptotical stabilization of such fractional order interval systems is derived all the results are obtained in terms of linear matrix inequalities finally two illustrative examples are given to show that our results are effective for checking the robust stability and designing the robust stabilizing controller for fractional order interval systems

a multiple criteria sorting method where each category is characterized by several reference actions the electre tri nc method this paper presents electre tri nc a new sorting method which takes into account several reference actions for characterizing each category this new method gives a particular freedom to the decision maker in the co construction decision aiding process with the analyst to characterize the set of categories while there is no constraint for introducing only one reference action as typical of each category like in electre tri c almeida dias et al 2010 as in such a sorting method this new sorting method is composed of two joint rules electre tri nc also fulfills a certain number of natural requirements additional results on the behavior of the new method are also provided in this paper namely the ones with respect to the addition or removal of the reference actions used for characterizing a certain category a numerical example illustrates the manner in which electre tri nc can be used by a decision maker a comparison with some related sorting procedures is presented and it allows to conclude that the new method is appropriate to deal with sorting problems

lin kernighan heuristic adaptations for the generalized traveling salesman problem abstract the lin kernighan heuristic is known to be one of the most successful heuristics for the traveling salesman problem tsp it has also proven its efficiency in application to some other problems in this paper we discuss possible adaptations of tsp heuristics for the generalized traveling salesman problem gtsp and focus on the case of the lin kernighan algorithm at first we provide an easy to understand description of the original lin kernighan heuristic then we propose several adaptations both trivial and complicated finally we conduct a fair competition between all the variations of the lin kernighan adaptation and some other gtsp heuristics it appears that our adaptation of the lin kernighan algorithm for the gtsp reproduces the success of the original heuristic different variations of our adaptation outperform all other heuristics in a wide range of trade offs between solution quality and running time making lin kernighan the state of the art gtsp local search

assignment problems a golden anniversary survey abstract having reached the 50th golden anniversary of the publication of kuhn s seminal article on the solution of the classic assignment problem it seems useful to take a look at the variety of models to which it has given birth this paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years the intention here is not to identify every such paper of which there have been hundreds nor to identify the best solution procedure for each variation rather the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature

an approximation algorithm for the traveling salesman problem with backhauls the traveling salesman problem with backhauls tspb is defined on a graph g v e the vertex set is partitioned into v v1 l b where v1 is a depot l is a set of linehaul customers and b is a set of backhaul customers a cost matrix satisfying the triangle inequality is defined on the edge set e the tspb consists of determining a least cost hamiltonian cycle on g such that all vertices of l are visited contiguously after v1 followed by all vertices of b following a result by christofides for the traveling salesman problem we propose an approximation algorithm with worst case performance ratio of 3 2 for the tspb

a new approach to multi criteria sorting based on fuzzy outranking relations the theseus method abstract in this paper we propose the theseus method a new approach based on fuzzy outranking relations to multi criteria sorting problems compared with other outranking based methods theseus is inspired by another view of multi criteria classification problems it utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined this way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects the appropriate assignment is determined by solving a simple selection problem the capacity of a reference set for making appropriate assignments is related to a good characterization of the categories a single reference action characterizing a category may be insufficient to achieve well determined assignments in this paper the reference set capacity to perform appropriate assignments is characterized by some new concepts this capacity may be increased when more objects are added to the reference set theseus is a method for handling the preference information contained in such larger reference sets

embedding learning capability in lagrangean relaxation an application to the travelling salesman problem abstract this paper presents an effective procedure that finds lower bounds for the travelling salesman problem based on the 1 tree using a learning based lagrangian relaxation technique the procedure can dynamically alter its step size depending upon its previous iterations along with having the capability of expansion contraction the procedure performs a learning process in which lagrange multipliers are influenced by a weighted cost function of their neighbouring nodes in analogy with simulated annealing paradigm here a learning process is equivalent to escaping local optimality via exploiting the structure of the problem computational results conducted on euclidean benchmarks from the tsplib library show that the procedure is very effective

the symmetric clustered traveling salesman problem abstract the clustered traveling salesman problem is the well known standard problem with additional constraints the cities are divided into groups and each group of cities cluster must be visited contiguously we describe an extensive adaptation of an existing traveling salesman algorithm based on the 1 tree relaxation in the lagrangean approach a new multiplier is introduced to improve the lower bounds furthermore we describe a heuristic to find satisfactory upper bounds and discuss the elimination of nonoptimal edges the computational results are satisfying and include random euclidean problems up to 150 cities

bounds for maximin latin hypercube designs latin hypercube designs lhds play an important role when approximating computer simulation models to obtain good space filling properties the maximin criterion is frequently used unfortunately constructing maximin lhds can be quite time consuming when the number of dimensions and design points increase in these cases we can use heuristical maximin lhds in this paper we construct bounds for the separation distance of certain classes of maximin lhds these bounds are useful for assessing the quality of heuristical maximin lhds until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs i e for maximin designs without a latin hypercube structure the separation distance of maximin lhds also satisfies these unrestricted bounds by using some of the special properties of lhds we are able to find new and tighter bounds for maximin lhds within the different methods used to determine the upper bounds a variety of combinatorial optimization techniques are employed mixed integer programming the traveling salesman problem and the graph covering problem are among the formulations used to obtain the bounds besides these bounds also a construction method is described for generating lhds that meet baer s bound for the l distance measure for certain values of n

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

some inverse optimization problems under the hamming distance given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal the aim is to optimize the difference between the initial cost vector and the disturbed one this difference can be measured in several ways we consider the hamming distance measuring in how many components two vectors are different where weights are associated to the components general algorithms for the bottleneck or minimax criterion are described and after modification applied to the inverse minimum spanning tree problem the inverse shortest path tree problem and the linear assignment problem

multicriteria sorting using a valued preference closeness relation abstract to define a metric or a similarity relation with good properties is a crucial issue in most analogy and rule induction oriented methods for multicriteria sorting here we propose a valued indifference closeness relation which is inspired on concordance and discordance measures the criterion weights are obtained from the preferential information embedded in a reference set this proposal performs very well in several practical problems

improved extremal optimization for the asymmetric traveling salesman problem abstract this paper presents an improved extremal optimization ieo algorithm for solving the asymmetric traveling salesman problem atsp at each update step the ieo algorithm proceeds through two main steps extremal dynamics and cooperative optimization as an improvement of extremal optimization eo the ieo provides a general combinatorial optimization framework by emphasizing the step of cooperative optimization in the paper an effective cooperative optimization strategy with combination of greedy search and random walk is designed in terms of the microscopic characteristics of the atsp solutions simulation results on a set of benchmark atsp instances show that the proposed ieo algorithm provides satisfactory performance on computational effectiveness and efficiency

a branch and bound algorithm for the singly constrained assignment problem the singly constrained assignment problem scap is a linear assignment problem lap with one extra side constraint e g due to a time restriction the scap is in contrast to the lap difficult to solve a branch and bound algorithm is presented to solve the scap to optimality lower bounds are obtained by lagrangean relaxation computational results show that the algorithm is able to solve different types of scap instances up to size n 1000 within short running times on a standard personal computer

resolving inconsistencies among constraints on the parameters of an mcda model we consider a framework where decision makers dms interactively define a multicriteria evaluation model by providing imprecise information i e a linear system of constraints to the model s parameters and by analyzing the consequences of the information provided dms may introduce new constraints explicitly or implicitly results that the model should yield if a new constraint is incompatible with the previous ones then the system becomes inconsistent and the dms must choose between removing the new constraint or removing some of the older ones we address the problem of identifying subsets of constraints which when removed lead to a consistent system identifying such subsets would indicate the reason for the inconsistent information given by dms there may exist several possibilities for the dms to resolve the inconsistency we present two algorithms to identify such possibilities one using 0 1 mixed integer linear programming and the other one using linear programming both approaches are based on the knowledge that the system was consistent prior to introducing the last constraint the output of these algorithms helps the dm to identify the conflicting pieces of information in a set of statements he she asserted the relevance of these algorithms for mcda is illustrated by an application to an aggregation disaggregation procedure for the electre tri method

the multi vehicle traveling purchaser problem with pairwise incompatibility constraints and unitary demands a branch and price approach abstract in this work we study a supplier selection and routing problem where a fleet of homogeneous vehicles with a predefined capacity is available for procuring different products from different suppliers with the aim to satisfy demand at the minimum traveling and purchasing cost decisions are further complicated by the presence of pairwise incompatibility constraints among products implying the impossibility of loading two incompatible products on the same vehicle the problem is known as the multi vehicle traveling purchaser problem with pairwise incompatibility constraints we study the special case in which the demand for each product is unitary and propose a column generation approach based on a dantzig wolfe reformulation of the problem where each column represents a feasible vehicle route associated with a compatible purchasing plan to solve the pricing problem we propose a hybrid strategy exploiting the advantages of two alternative exact methods a labeling algorithm solving a resource constrained elementary shortest path problem on an expanded graph and a tailored branch and cut algorithm due to the integrality request on variables we embed the column generation in a branch and bound framework and propose different branching rules extensive tests carried out on a large set of instances show that our branch and price method performs well improving on average both in quality and in computational time solutions obtained by a state of art branch and cut approach applied to a three index connectivity constraints based formulation

a double loop hybrid algorithm for the traveling salesman problem with arbitrary neighbourhoods abstract this paper addresses the traveling salesman problem with arbitrary neighbourhoods which is an np hard problem in combinatorial optimization and is important in operations research and theoretical computer science existing methods based on neighbourhood predigestion and discretization are impractical in cases in which the neighbourhoods are arbitrary in this paper the neighbourhoods of the traveling salesman problem are generalized to arbitrarily connected regions in planar euclidean space a novel approach to solving this problem is proposed including a boundary based encoding scheme and a double loop hybrid algorithm based on particle swarm optimization and genetic algorithm in the hybrid algorithm linear descending inertia weight particle swarm optimization is adopted to search continuous visiting positions in the outer loop and the genetic algorithm is used to optimize the discrete visiting sequence in the inner loop the boundary based encoding scheme can reduce the search space significantly without degrading the solution quality and the hybrid algorithm can find a high quality solution in a reasonable time the computational results on both small and large instances demonstrate that the proposed approach can guarantee a high quality solution in a reasonable time compared with three other state of the art algorithms iterative algorithm branch and bound algorithm and upper and lower bound algorithm moreover the proposed approach works efficiently in a real world application that cannot be solved by existing algorithms

multicriteria decision support using rules that represent rough graded preference relations the approach described in this paper aims to support multicriteria choice and ranking of actions when the input preference information acquired from the decision maker is a graded comprehensive pairwise comparison or ranking of reference actions it is based on decision rule preference model induced from a rough approximation of the graded comprehensive preference relation among the reference actions the set of decision rules applied to a new set of actions provides a graded fuzzy preference relation which can be exploited by weighted fuzzy net flow score or lexicographic fuzzy net flow score procedure to obtain a final recommendation in terms of the best choice or of the ranking

on the choice of step size in subgradient optimization abstract this paper recommends some procedures for the selection of step sizes in the context of subgradient optimization the first of these procedures is developed in detail in this study and is a theoretically convergent scheme this method has two phases the first phase is designed to accelerate the solution procedure towards an optimal solution while the second phase helps to close in on an optimal solution a second technique recommended is a simple minded scheme which although not theoretically convergent seems to be computationally very efficient these two methods are shown to compare favorably with held wolfe and crowder s scheme for prescribing step sizes we also suggest some modifications of the latter scheme to make it computationally more efficient

the symmetric traveling salesman problem and edge exchanges in minimal 1 trees abstract in this article the effect of exchanging edges inside a minimal 1 tree with edges outside is analysed in combination with an upper bound this analysis enables the elimination of variables in the symmetric traveling salesman problem after discussion on a number of improvements for this analysis the implementation is described in a traveling salesman algorithm based on the 1 tree relaxation computational results show the advantages of the edges exchanges for euclidean problems up to 120 cities as well as for random table problems up to 200 cities

a quantitative analysis of the simulated annealing algorithm a case study for the traveling salesman problem a quantitative study is presented of the typical behavior of the simulated annealing algorithm based on a cooling schedule presented previously by the authors the study is based on the analysis of numerical results obtained by systematically applying the algorithm to a 100 city traveling salesman problem the expectation and the variance of the cost are analyzed as a function of the control parameter of the cooling schedule a semiempirical average case performance analysis is presented from which estimates are obtained on the expectation of the average final result obtained by the simulated annealing algorithm as a function of the distance parameter which determines the decrement of the control parameter

new lower bounds for the symmetric travelling salesman problem in this paper new lower bounds for the symmetric travelling salesman problem are proposed and combined in additive bounding procedures efficient implementations of the algorithms are given in particular fast procedures for computing the linear programming reduced costs of the shortest spanning tree sst problem and for finding all ther sst of a given graph are presented computational results on randomly generated test problems are reported

the assignment problem with seniority and job priority constraints consider an assignment problem in which persons are qualified for some but usually not all of the jobs moreover assume persons belong to given seniority classes and jobs have given priority levels seniority constraints impose that the solution be such that no unassigned person can be given a job unless an assigned person with the same or higher seniority becomes unassigned priority constraints specify that the solution must be such that no unassigned job can become assigned without a job with the same or higher priority becoming unassigned it is shown that i adding such constraints does not reduce and may even increase the number of assigned persons in the optimal solution ii using a greedy heuristic for constrained assignment as often done in practice may reduce the number of assigned persons by half and iii an optimal solution to the assignment problem with both types of constraints can be obtained by solving a classical assignment problem with adequately modified coefficients

an aggregation disaggregation approach to obtain robust conclusions with electre tri electre tri is a well known method to assign a set of alternatives to a set of predefined categories considering multiple criteria using this method requires setting many parameters which is often a difficult task we consider the case where the decision makers dms in the decision process are unsure of which values should each parameter take which may result from uncertain imprecise or inaccurately determined information as well as from lack of consensus among them this paper discusses the synergy between two approaches developed independently to deal with this difficulty the first approach infers the value of parameters from assignment examples provided by the dms as an elicitation aid each assignment example originates mathematical constraints that the parameter values should satisfy the second approach considers a set of constraints on the parameter values reflecting the imprecise information that the dms are able to provide then it computes the best and worst categories for each alternative compatible with constraints in order to present robust conclusions both approaches avoid asking for precise values for the parameters rather they proceed to solve the problem in a way that requires from the dms much less effort by integrating these two approaches this paper proposes a new interactive approach where the insight obtained during robustness analyses guides the dms during the elicitation phase

a branch and bound algorithm for the symmetric traveling salesman problem based on the 1 tree relaxation abstract in 1970 held and karp introduced the lagrangean approach to the symmetric traveling salesman problem we use this 1 tree relaxation in a new branch and bound algorithm it differs from other algorithms not only in the branching scheme but also in the ascent method to calculate the 1 tree bounds urthermore we determine heuristic solutions throughout the computations to provide upperbounds we present computational results for both a depth first and a breadth first version of our algorithm on the average our results on a number of euclidean problems from the literature are obtained in about 60 less 1 trees than the best known algorithm based on the 1 tree relaxation for random table problems up to 100 cities the average results are also satisfactory

a lagrangean approach to the degree constrained minimum spanning tree problem abstract a known branch and bound algorithm for the degree constrained minimum spanning tree problem is adapted for the use of lagrangean multipliers they improve the bounds from which the edge elimination analysis in the algorithm benefits in particular computational results are given for problems up to 150 nodes both random drawn coordinate problems as well drawn table problems these results are much better than other results reported in literature and they show the lagrangean approach to be more effective when the problems are more difficult to solve

electregkms robust ordinal regression for outranking methods we present a new method called electregkms which employs robust ordinal regression to construct a set of outranking models compatible with preference information the preference information supplied by the decision maker dm is composed of pairwise comparisons stating the truth or falsity of the outranking relation for some real or fictitious reference alternatives moreover the dm specifies some ranges of variation of comparison thresholds on considered pseudo criteria using robust ordinal regression the method builds a set of values of concordance indices concordance thresholds indifference preference and veto thresholds for which all specified pairwise comparisons can be restored such sets are called compatible outranking models using these models two outranking relations are defined necessary and possible whether for an ordered pair of alternatives there is necessary or possible outranking depends on the truth of outranking relation for all or at least one compatible model respectively distinguishing the most certain recommendation worked out by the necessary outranking and a possible recommendation worked out by the possible outranking electregkms answers questions of robustness concern the method is intended to be used interactively with incremental specification of pairwise comparisons possibly with decreasing confidence levels in this way the necessary and possible outranking relations can be respectively enriched or impoverished with the growth of the number of pairwise comparisons furthermore the method is able to identify troublesome pieces of preference information which are responsible for incompatibility the necessary and possible outranking relations are to be exploited as usual outranking relations to work out recommendation in choice or ranking problems the introduced approach is illustrated by a didactic example showing how electregkms can support real world decision problems

an algebraic algorithm for determining the desired gain of multivariable feedback systems in this paper an algebraic method is presented whereby one can obtain the relevant values of the gain k of a multivariable multi input multi output feedback system to be stable or relatively stable the basis of the method involves the algebraic solution of the two equations obtained from the real and imaginary parts of the characteristic equation of the system the proposed method offers an alternative procedure for determining stability relative stability to the graphical methods based on the generalized nyquist criterion and the multivariable root locus method this method is analogous to the analytical form of the root locus combined with the routh hurwitz criterion as compared to the nyquist or graphical root locus methods when applied to single input single output feedback systems several examples are presented illustrating the applications of the method and compared with the graphical methods

brief analysis of fractional delay systems of retarded and neutral type this paper analyses several properties linked to the robust control of fractional differential systems with delays the bibo stability of both retarded and neutral fractional delay systems is related to the location of their poles in the particular case of retarded systems we give some properties of the poles of the system and the singular values of its hankel operator

transformation of multidepot multisalesmen problem to the standard travelling salesman problem abstract one type of multidepot multisalesmen problem can be transformed into a standard travelling salesman problem by cooying depots and introducing imaginary depots

electre tri c a multiple criteria sorting method based on characteristic reference actions in this paper a new sorting method following a decision aiding constructive approach is proposed this method is called electre tri c as a sorting method a set of categories must be defined to represent the way in which the actions that are going to be assigned to each of them should further be processed this method is appropriate to deal with decision aiding contexts where the categories are completely ordered and each of them is defined through a single characteristic reference action the set of characteristic actions should be co constructed through an interactive process between the analyst and the decision maker electre tri c has been conceived to verify a set of natural structural requirements conformity homogeneity monotonicity and stability which can be viewed as its fundamental properties this method is composed of two joint rules called descending rule and ascending rule which must be used conjointly and not separately each one of these rules selects only one category or a range of possible categories for a possible assignment of an action this assignment depends on the comparison of such an action to the characteristic actions according to a chosen credibility level numerical examples are also presented in order to illustrate the main theoretical results provided by the method

exact stability test and stabilization for fractional systems in this paper we point out a connection between regular chains and the stability of fractional order systems this observation leads to an elementary test for the stability of commensurate fractional systems

an empirical study of a new metaheuristic for the traveling salesman problem abstract we present a deceptively simple yet empirically powerful metaheuristic called jump search for solving traveling salesman problems that has been found to be more effective than tabu search on both random and benchmark test problems from the literature while the underlying philosophy of jump search applying local search from different starting points has been discussed in the literature previously using random starting points the use of construction based heuristic solutions has heretofore not been considered extensive empirical analysis shows the method to be surprisingly effective vis a vis a randomized strategy and in comparison with tabu search the approach is quite robust and suggests that a systematic search among neighborhoods of good not random solutions provides distinct advantages this suggests that further research be focused on better construction heuristics and hybrid schemes that incorporate jump search in for example tabu search

multiple criteria sorting with a set of additive value functions we present a new multiple criteria sorting method that aims at assigning actions evaluated on multiple criteria to p pre defined and ordered classes the preference information supplied by the decision maker dm is a set of assignment examples on a subset of actions relatively well known to the dm these actions are called reference actions each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes the set of assignment examples is used to build a preference model of the dm represented by a set of general additive value functions compatible with the assignment examples for each action a the method computes two kinds of assignments to classes concordant with the dm s preference model the necessary assignment and the possible assignment the necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously the possible assignment specifies in turn the range of classes to which the action can be assigned considering any compatible value function individually the compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs

equivalent weights for lexicographic multi objective programs characterizations and computations abstract a recent paper 19 demonstrated the existence of a set of equivalent weights for which the optimal solution set of a preemptive priority multi objective program is precisely equal to the set of optimal solutions to the resulting nonpreemptive program with the objective function given as a linear weighting of the multiple objectives this paper addresses two further issues firstly for some important special cases or applications it is demonstrated that not only is the computation of a set of equivalent weights feasible but it is also highly desirable two algorithms are presented to compute a set of equivalent weights one method is a direct specialization of the approach adopted in 19 whereas the second approach is an alternative technique the latter method is shown to yield weights of uniformly smaller values than the former method while being of the same computational complexity and is hence preferable secondly as opposed to constructing one vector of equivalent weights a characterization is provided for the entire set of equivalent weights

multi criteria classification a new scheme for application of dominance based decision rules abstract we are considering the problem of multi criteria classification in this problem a set of if then decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes given a sample of classification examples called learning data set the rules are induced from dominance based rough approximations of preference ordered decision classes according to the variable consistency dominance based rough set approach vc drsa the main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by i no rule ii exactly one rule iii several rules the proposed classification scheme can be applied to both learning data set to restore the classification known from examples and testing data set to predict classification of new objects a hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals

a note on finding a shortest complete cycle in an undirected graph abstract how to transform a shortest complete cycle problem into a traveling salesman problem is well known although these problems appear ill conditioned a 1 tree relaxation based traveling salesman algorithm solves them rather easily up to 150 nodes provided non optimal edges are eliminated the computational results are considerably better than known results for a linear programming approach

solving some lexicographic multi objective combinatorial problems abstract the lexicographic multi objective optimization problem r lmop with r sum or bottleneck objectives optimizes a first objective then as far as a choice remains a second one then a third one and so on a general solution scheme from literature is based on scaling for the case that only sum criteria are involved the complexity is o rlogn t n m with t n m the complexity of optimization on combinatorial sum problem instances with m arcs and n nodes for cases with at least one bottleneck criterion the complexity is at least o rnlog2n t n m we describe a solution scheme that is suited to solve the shortest path r lmop the minimum spanning tree r lmop and the linear assignment r lmop the complexity is in all cases o r t n m

finding the k shortest paths we give algorithms for finding the k shortest paths not required to be simple connecting a pair of vertices in a digraph our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges in time o m n log n k we can also find the k shortest paths from a given source s to each vertex in the graph in total time o m n log n kn we describe applications to dynamic programming problems including the knapsack problem sequence alignment maximum inscribed polygons and genealogical relationship discovery

ordinal regression revisited multiple criteria ranking using a set of additive value functions abstract we present a new method called utagms for multiple criteria ranking of alternatives from set a using a set of additive value functions which result from an ordinal regression the preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives ar a called reference alternatives the preference model built via ordinal regression is the set of all additive value functions compatible with the preference information using this model one can define two relations in the set a the necessary weak preference relation which holds for any two alternatives a b from set a if and only if for all compatible value functions a is preferred to b and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b these relations establish a necessary and a possible ranking of alternatives from a being respectively a partial preorder and a strongly complete relation the utagms method is intended to be used interactively with an increasing subset ar and a progressive statement of pairwise comparisons when no preference information is provided the necessary weak preference relation is a weak dominance relation and the possible weak preference relation is a complete relation every new pairwise comparison of reference alternatives for which the dominance relation does not hold is enriching the necessary relation and it is impoverishing the possible relation so that they converge with the growth of the preference information distinguishing necessary and possible consequences of preference information on the complete set of actions utagms answers questions of robustness analysis moreover the method can support the decision maker when his her preference statements cannot be represented in terms of an additive value function the method is illustrated by an example solved using the utagms software some extensions of the method are also presented

a branch and bound algorithm for symmetric 2 peripatetic salesman problems abstract the symmetric 2 peripatetic salesman problem 2 psp is an extension of the well known symmetric traveling salesman problem tsp in the sense that 2 edge disjoint hamiltonian cycles of minimum total length are required in a graph g with n vertices lower bound solutions are provided by 2 minimum length edge disjoint 1 trees which can be constructed in at most o n 2 log n operations if g is complete generating upper bound solutions and executing edge elimination tests takes o n 2 time a branch and bound algorithm has been developed based on these procedures to solve symmetric problems computational results are available for euclidean problems up to 60 vertices and problems with randomly generated distance matrices up to 130 vertices

the traveling salesman problem an overview of exact and approximate algorithms in this paper some of the main known algorithms for the traveling salesman problem are surveyed the paper is organized as follows 1 definition 2 applications 3 complexity analysis 4 exact algorithms 5 heuristic algorithms 6 conclusion

an efficient composite heuristic for the symmetric generalized traveling salesman problem the main purpose of this paper is to introduce a new composite heuristic for solving the generalized traveling salesman problem the proposed heuristic is composed of three phases the construction of an initial partial solution the insertion of a node from each non visited node subset and a solution improvement phase we show that the heuristic performs very well on 36 tsplib problems which have been solved to optimality by other researchers we also propose some simple heuristics that can be used as basic blocks to construct more efficient composite heuristics

estimating the held karp lower bound for the geometric tsp the held karp lower bound hk provides a very good problem specific estimate of optimal tour length for the travelling salesman problem tsp this measure which is relatively quick and easy to compute has enormous practical value when evaluating the quality of near optimal solutions for large problems where the true optima are not known although hk can be evaluated exactly by linear programming techniques code for doing this efficiently for problems larger than a few hundred cities is not readily available or easy to produce in addition linear programming implementations even efficient ones do not scale well and rapidly become impractical for problems with many thousands of cities in this study we concentrate on the iterative estimation approach proposed by held and karp in their original papers our paper provides implementation details for two iterative schemes which both use the subgraph speed up technique

on the relations between electre tri b and electre tri c and on a new variant of electre tri b electre tri is a set of methods designed to sort alternatives evaluated on several criteria into ordered categories the original method uses limiting profiles a recently introduced method uses central profiles we study the relations between these two methods we do so by investigating if an ordered partition obtained with one method can also be obtained with the other method after a suitable redefinition of the profiles we also investigate a number of situations in which the original method using limiting profiles gives results that do not fit well our intuition this leads us to propose a variant of electre tri that uses limiting profiles we show that this variant may have some advantages over the original method

an addendum on sensitivity analysis of the optimal assignment we point out that sensitivity results for the linear assignment problem can be produced by a shortest path based approach in a straightforward manner and as efficient as finding an optimal solution

multicriteria assignment method proaftn methodology and medical application abstract this paper presents a new fuzzy multicriteria classification method called proaftn for assigning alternatives to predefined categories this method belongs to the class of supervised learning algorithms and enables to determine the fuzzy indifference relations by generalising the indices concordance and discordance used in the electre iii method then it assigns the fuzzy belonging degree of the alternatives to the categories we also present a clinical application of the proposed method in the cytopathological diagnosis of acute leukaemia

special cases of the traveling salesman problem the paper surveys special cases of the traveling salesman problem that are in the literature the tables include polynomial time recognition algorithms and relationships among special cases as a result of research the most general special cases are identified the paper also contains a table of subclasses of the most general special cases applications and open questions are indicated

a linear time algorithm for the maximum capacity path problem abstract the maximum capacity path problem is to find a path joining two fixed vertices of an edge weighted graph such that the minimum edge weight is maximized the best known algorithm to solve this problem has a worst case complexity of o min m n log n m lognw where m is the number of edges n is the number of vertices and w is the maximum edge capacity we give a simple o m algorithm to solve this problem as a by product of this result we have an o m2 algorithm for a bicriteria path problem which is an improvement over the available o m2log n algorithm

a useful transform of standard input data for a classical np complete problem abstract the archetypal symmetric travelling salesman problem can be seen in a new and interesting way by using first a standard preparatory phase of input data and then by applying a transform from the set d of distances among cities and the set b of loss of optimality the specific form of d b transform is introduced and discussed in order to show in realistic terms the interest of the approach proposed a class of diffusive heuristic procedures operating from b is defined an example of solution by an algorithm included in this class is completely worked out an outline of computational tests done on the same algorithm is also given

solution of large scale symmetric travelling salesman problems in this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality our implementation is based on a fast lp solver ibm s mpsx and makes effective use of polyhedral results on the symmetric travelling salesman polytope we describe the important ingredients of our code and give an extensive documentation of its computational performance

nonoptimal edges for the symmetric traveling salesman problem for the symmetric traveling salesman problem we identify a set of undirected edges that can be eliminated while still retaining at least one optimal solution of the problem the simple identification rules are based on the fact that a solution can be optimal only if it is 2 optimal the rules are less stringent than those formulated for nonoptimal arcs directed applied to the symmetric case so in general they identify more nonoptimal edges application of the theory in a 1 tree based traveling salesman algorithm roughly halves the average computation time for euclidean problems we indicate to what extent the theorems presented here can be adapted for variants of the traveling salesman problem

an optimal solution method for large scale multiple traveling salesmen problems we develop an efficient branch and bound based method for solving the multiple traveling salesman problem and develop lower bounds through a lagrangean relaxation that requires computing a degree constrained minimal spanning tree a subgradient optimization procedure updates the lagrange multipliers we use fast sensitivity analysis techniques to increase the underlying graph sparsity and reduce the problem size the algorithm was tested on 416 problems with up to 500 cities and 10 salesmen we also present computational results on different data sets and parameters in order to identify the major factors that influence the performance of the developed code

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

a stability test for non commensurate fractional order systems this paper presents a necessary and sufficient condition to evaluate non commensurate fractional order systems bounded input bounded output stability this condition is based on an algorithm that relies on a recursively defined closed loop realization of the system and involves cauchy s theorem its efficiency is attested by several numerical examples

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

the black and white traveling salesman problem the black and white traveling salesman problem bwtsp is defined on a graph g whose vertex set is partitioned into black and white vertices the aim is to design a shortest hamiltonian tour on g subject to cardinality and length constraints both the number of white vertices as well as the length of the tour between two consecutive black vertices are bounded above the bwtsp has applications in airline scheduling and in telecommunications this paper proposes an integer linear formulation for the undirected bwtsp as well as several classes of valid inequalities an exact branch and cut algorithm is then developed extensive tests show that it can solve exactly instances involving up to 100 vertices the algorithm can also be applied directly to solve unit demand vehicle routing problems of similar sizes

iterative state space reduction for flexible computation abstract flexible computation is a general framework for decision making under limited computational resources it enables an agent to allocate limited computational resources to maximize its overall performance or utility in this paper we present a strategy for flexible computation which we call iterative state space reduction the main ideas are to reduce a problem space that is difficult to search to one that is relatively easy to explore to use the optimal solution from the reduced space as an approximate solution to the original problem and to iteratively apply multiple reductions to progressively find better solutions the basic operation for state space reduction is heuristic node pruning which excludes the nodes that do not seem to lead to high quality solutions from further examination iterative state space reduction can be combined with a state space search such as best first search or depth first branch and bound dfbnb the resulting algorithm can run as an anytime algorithm which provides a tradeoff between solution quality and computation based on an analytical modal we analyze the probability that one iteration of the iterative process finds a solution furthermore by combining it within dfbnb we apply iterative state space reduction to three combinatorial problems the maximum boolean satisfiability the symmetric tsp and asymmetric tsp our experimental results show that iterative state space reduction is effective and efficient making dfbnb find better solutions with less computation