reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

logarithmic deformations of the rational superpotential landau ginzburg construction of solutions of the wdvv equations the superpotential in the landau ginzburg construction of solutions to the witten dijkgraaf verlinde verlinde or wdvv equations is modified to include logarithmic terms this results in deformations quadratic in the deformation parameters of the normal prepotential solutions of the wdvv equations such solutions satisfy various pseudo quasi homogeneity conditions on assigning a notional weight to the deformation parameters these solutions originate in the so called water bag reductions of the dispersionless kp hierarchy this construction includes as a special case deformations which are polynomial in the flat coordinates resulting in a new class of polynomial solutions of the wdvv equations

feedforward artificial neural networks for solving discrete multiple criteria decision making problems decision making involves choosing some course of action among various alternatives in almost all decision making problems there are several criteria for judging possible alternatives the main concern of the decision maker dm is to fulfill his conflicting goals while satisfying the constraints of the system in this paper we present an adaptive feedforward artificial neural network af ann approach to solve discrete multiple criteria decision making mcdm problems the af ann is used to capture and represent the dm s preferences and then to select the most desirable alternative the af ann can adjust and improve its representation as more information from the dm becomes available we begin with the assumption that an af ann topology is given i e specific numbers of nodes and links are predetermined to adjust the parameters of the af ann we present an iterative learning algorithm consisting of two steps a generating a direction and b a one dimensional search along that direction we then present a methodology to obtain the most appropriate af ann topology and set its parameters the procedure starts with a small number of nodes and links and then adaptively increases the number of nodes and links until the proper topology is obtained furthermore when the set of training patterns alternatives with their associated evaluations by the dm changes the af ann model can adapt itself by re training or expanding the existing model some illustrative examples are presented to solve discrete mcdm problems by an af ann we show how to incorporate basic properties of efficiency concavity and convexity into the af ann we formulate the mcdm problems and use the af ann to rank the set of discrete alternatives where each alternative is associated with a set of conflicting and noncommensurate criteria we present a method for solving discrete mcdm problems through af anns which consists of a formulating and assessing the utility function by eliciting information from the dm and then training the af ann and b ranking a rating alternatives by using the trained af ann model some computational experiments are presented to show the effectiveness of the method

interactive fuzzy decision making for multiobjective nonlinear programming usingaugmented minimax problems an interactive fuzzy decision making method for solving multiobjective nonlinear programming problems is presented in this paper by assuming that the decision maker dm has fuzzy goals for each of the objective functions the fuzzy goals of the dm are quantified by eliciting corresponding membership functions through the interaction with the dm having determined the membership functions if the dm specifies his reference membership values the augmented minimax problem is solved and the dm is supplied with the corresponding pareto optimal solution together with the trade off rates between the membership functions then by considering the current values of the membership functions as well as the trade off rates the dm responds by updating his reference membership values in this way the compromise or satisficing solution for the dm can be derived efficiently from among a pareto optimal solution set on the basis of the proposed method a time sharing computer program is written and an illustrative numerical example is demonstrated along with the computer outputs

decision making with multiple objectives using gai networks this paper deals with preference representation on combinatorial domains and preference based recommendation in the context of multicriteria or multiagent decision making the alternatives of the decision problem are seen as elements of a product set of attributes and preferences over solutions are represented by generalized additive decomposable gai utility functions modeling individual preferences or criteria thanks to decomposability utility vectors attached to solutions can be compiled into a graphical structure closely related to junction trees the so called gai network using this structure we present preference based search algorithms for multicriteria or multiagent decision making although such models are often non decomposable over attributes we actually show that gai networks are still useful to determine the most preferred alternatives provided preferences are compatible with pareto dominance we first present two algorithms for the determination of pareto optimal elements then the second of these algorithms is adapted so as to directly focus on the preferred solutions we also provide results of numerical tests showing the practical efficiency of our procedures in various contexts such as compromise search and fair optimization in multicriteria or multiagent problems

the light beam search approach an overview of methodology applications the paper gives a general overview of the light beam search lbs methodology and applications lbs enables an interactive analysis of multiple objective decision problems due to presentation of samples of a large set of non dominated points to the decision maker dm in each iteration a local preference model in the form of an outranking relation is used to define the neighborhood of a current non dominated point the sample comes from the first current point is obtained by projection of an aspiration point onto the non dominated set in the direction of a reservation point the dm can control the search by either modifying the aspiration and reservation points or by shifting the current point to a selected better point from its neighborhood the paper describes applications of the approach to several real life problems and discusses observations made while working on these problems the lbs approach is compared to other existing methods and the class of problems suitable to this methodology is defined

experiments with classification based scalarizing functions in interactive multiobjective optimization in multiobjective optimization methods the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways we compare both theoretically and numerically the performance of three classification based scalarizing functions and pay attention to how well they obey the classification information in particular we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing pareto optimality it turns out that scalarizing functions with or without so called augmentation terms have significant differences in this respect we also collect a set of mostly nonlinear benchmark test problems that we use in the numerical comparisons

a combined tchebycheff aspiration criterion vector interactive multiobjective programming procedure in this paper we combine the tchebycheff method of steuer and choo with wierzbicki s aspiration criterion vector method in order to form an improved procedure for interactive multiple objective programming the combined procedure is sensible because the tchebycheff and aspiration criterion vector methods possess complementary distinguishing characteristics solve similar optimization problems to probe the nondominated set and share a similar computer user interface in the early iterations tchebycheff probes of the nondominated set might be conducted to locate promising neighborhoods of search in later iterations aspiration criterion vector probes might be used to pinpoint a final solution computational experience is reported showing the improved effectiveness of the combined procedure when employed in this fashion when compared against the tchebycheff and aspiration criterion vector methods run separately

the weak link approach to network dea for two stage processes abstract we introduce in this paper a new definition for the overall system efficiency in network dea which is inspired by the weak link notion in supply chains and the maximum flow minimum cut problem in networks we use a two phase max min optimization technique in a multi objective programming framework to estimate the individual stage efficiencies and the overall system efficiency in two stage processes of varying complexity this enables us to estimate unique and unbiased efficiency scores and if required to drive the efficiency assessments effectively in line with specific priorities given to the stages a comparison with the multiplicative decomposition approach on data drawn from the literature brings into light the advantages of our method

multiplicative decision rules for multiobjective decision problems abstract properties of consistency of general decision rules for multiobjective decision problems are discussed decision rules based on the family of weighted geometric averages are introduced their properties are analysed and possible implementations are discussed

an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters abstract this paper presents an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy numbers on the basis of the level sets of the fuzzy numbers the concept of multiobjective nonlinear programming and local m pareto optimality is introduced through interaction with the decision maker dm the fuzzy goals of the dm for each of the objective functions in multiobjective nonlinear programming are quantified by eliciting the corresponding membership functions after determining the membership functions in order to generate a candidate for the satisficing solution which is also locally m pareto optimal if the dm specified the degree of the level sets and the reference membership values the augmented minimax problem is solved and the dm is supplied with the corresponding local m pareto optimal solution together with the trade off rates among the values of the membership functions and the degree then by considering the current values of the membership functions and as well as the trade off rates the dm responds by updating his reference membership values and or the degree in this way the local satisfying solution for the dm can be derived efficiently from among an m pareto optimal solution set based on the proposed method an interactive computer program has been written and an illustrative numerical example is given along with the corresponding computer outputs

zero one programming with multiple criteria abstract the study of multiple criteria problems is not new but during the last seventeen years more and more study has been done in this area the study of 0 1 multiple criteria problems started much later than the study of continuous multiple criteria problems some of the earliest works on 0 1 problems with multiple criteria was first presented in 1973 quite a few solution methods have been developed since then especially during the last seven years in this paper most of these methods and some applications will be treated

quantitative comparison of approximate solution sets for multicriteria optimization problems with weighted tchebycheff preference function we consider the problem of evaluating the quality of solution sets generated by heuristics for multiple objective combinatorial optimization problems we extend previous research on the integrated preference functional ipf which assigns a scalar value to a given discrete set of nondominated points so that the weighted tchebycheff function can be used as the underlying implicit value function this extension is useful because modeling the decision maker s value function with the weighted tchebycheff function reflects the impact of unsupported points when evaluating sets of nondominated points we present an exact calculation method for the ipf measure in this case for an arbitrary number of criteria we show that every nondominated point has its optimal weight interval for the weighted tchebycheff function accordingly all nondominated points and not only the supported points in a set contribute to the value of the ipf measure when using the weighted tchebycheff function two and three criteria numerical examples illustrate the desirable properties of the weighted tchebycheff function providing a richer measure than the original ipf based on a convex combination of objectives

an interactive weight space reduction procedure for nonlinear multiple objective mathematical programming to make a decision that is defined by multiple conflicting objectives it is necessary to know the relative importance of the different objectives in this paper we present an interactive method and the underlying theory for solving multiple objective mathematical programming problems defined by a convex feasible region and concave continuously differentiable objective functions the relative importance of the different objectives for a decision maker is elicited by using binary comparisons of objective function vectors the method is cognitively easy to use and in test problems has rapidly converged to an optimal solution

a comparison of two reference point methods in multiple objective mathematical programming abstract when making decisions with multiple criteria a decision maker often thinks in terms of an aspiration point or levels of achievement for the criteria in multiple objective mathematical programming solution methods based on aspiration points can generate nondominated solutions using a variety of scalarizing functions these reference point solution methods commonly use a scalarizing function that reaches down from the ideal solution in a direction specified by the aspiration point conversely a similar scalarizing function can push out from the nadir point toward a specified aspiration point these scalarizing functions are similar in structure but diametrically opposed in their reference points in this paper we examine how these approaches behave from a technical point of view and conduct an experiment to understand better the human behaviour of users of these approaches insights we gain on the evenness of dispersion are relevant when attempting to construct a representation of the nondominated set further the technical characteristics of the two formulations solutions combined with behavioural tendencies allow us to comment on the implications for their use in interactive multiple objective methods

choquet based optimisation in multiobjective shortest path and spanning tree problems this paper is devoted to the search of choquet optimal solutions in finite graph problems with multiple objectives the choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives we first present a condition on preferences name hereafter preference for interior points that characterizes preferences favouring compromise solutions a natural attitude in various contexts such as multicriteria optimisation robust optimisation and optimisation with multiple agents within choquet expected utility theory this condition amounts to using a submodular capacity and a convex utility function under these assumptions we focus on the fast determination of choquet optimal paths and spanning trees after investigating the complexity of these problems we introduce a lower bound for the choquet integral computable in polynomial time then we propose different algorithms using this bound either based on a controlled enumeration of solutions ranking approach or an implicit enumeration scheme branch and bound finally we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes

nonlinear multicriteria clustering based on multiple dissimilarity matrices we present a new algorithm capable of partitioning sets of objects by taking simultaneously into account their relational descriptions given by multiple dissimilarity matrices the novelty of the algorithm is that it is based on a nonlinear aggregation criterion weighted tchebycheff distances more appropriate than linear combinations such as weighted averages for the construction of compromise solutions we obtain a hard partition of the set of objects the prototype of each cluster and a weight vector that indicates the relevance of each matrix in each cluster since this is a clustering algorithm for relational data it is compatible with any distance function used to measure the dissimilarity between objects results obtained in experiments with data sets synthetic and real show the usefulness of the proposed algorithm

a multiobjective discrete optimization model for land allocation a multiobjective integer programming model is presented for allocating an area of land for development the objectives considered in the allocation are cost proximity to desirable and undesirable land features and the shape of the area an interactive multiobjective optimization algorithm is presented and applied to the model the algorithm generates a subset of efficient solutions with some guidance from the decision maker at each iteration as to what constitutes a preferred efficient point the algorithm calls for the frequent solution of subproblems which constrain all but one of the objectives while optimizing the remaining one in the land allocation model the subproblems are integer programs solved efficiently by specialized enumeration techniques for some of the subproblems namely those using proximity as the single objective the first feasible solution we enumerate is guaranteed optimal for the other subproblems we show that an algorithm with this fortunate property would require the solution of an np hard problem at each step of the enumeration the model and algorithm were tested in locating potential sites for a 13 acre residential development within a 2250 acre study area near norris tennessee

a new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems abstract most real life decision making activities require more than one objective to be considered therefore several studies have been presented in the literature that use multiple objectives in decision models in a mathematical programming context the majority of these studies deal with two objective functions known as bicriteria optimization while few of them consider more than two objective functions in this study a new algorithm is proposed to generate all nondominated solutions for multiobjective discrete optimization problems with any number of objective functions in this algorithm the search is managed over p 1 dimensional rectangles where p represents the number of objectives in the problem and for each rectangle two stage optimization problems are solved the algorithm is motivated by the well known e constraint scalarization and its contribution lies in the way rectangles are defined and tracked the algorithm is compared with former studies on multiobjective knapsack and multiobjective assignment problem instances the method is highly competitive in terms of solution time and the number of optimization models solved

using objective values to start multiple objective linear programming algorithms we introduce in this paper a new starting mechanism for multiple objective linear programming molp algorithms this makes it possible to start an algorithm from any solution in objective space the original problem is first augmented in such a way that a given starting solution is feasible the augmentation is explicitly or implicitly controlled by one parameter during the search process which verifies the feasibility efficiency of the final solution this starting mechanism can be applied either to traditional algorithms which search the exterior of the constraint polytope or to algorithms moving through the interior of the constraints we provide recommendations on the suitability of an algorithm for the various locations of a starting point in objective space numerical considerations illustrate these ideas

a theorem on nonconvex functions and its application to vector optimization abstract a basic result of convex analysis is the fundamental theorem on convex functions we present a counterpart of this theorem for the case the convexity assumption does not hold and we shall show that some well known necessary conditions for weak efficiency and proper efficiency in nonconvex vector optimization are implicit in our result the significance of this result in vector optimization and in supporting decision making is briefly discussed

a review of interactive methods for multiobjective integer and mixed integer programming this paper makes a review of interactive methods devoted to multiobjective integer and mixed integer programming moip momip problems the basic concepts concerning the characterization of the non dominated solution set are first introduced followed by a remark about non interactive methods vs interactive methods then we focus on interactive moip momip methods including their characterization according to the type of preference information required from the decision maker the computing process used to determine non dominated solutions and the interactive protocol used to communicate with the decision maker we try to draw out some contrasts and similarities of the different types of methods

a unified approach to uncertain optimization abstract in this paper we consider uncertain scalar optimization problems with infinite scenario sets we apply methods from vector optimization in general spaces set valued optimization and scalarization techniques to develop a unified characterization of different concepts of robust optimization and stochastic programming these methods provide new insights on the interrelation between different concepts for handling uncertainties and naturally lead to new concepts of robustness

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

info quantifiers map characterization revisited abstract we highlight the potentiality of a special information theory it approach in order to unravel the intricacies of nonlinear dynamics the methodology being illustrated with reference to the logistic map a rather surprising dynamic feature plane topography map becomes available

minmax combinatorial optimization abstract let e be a finite set and f be a family of subsets of e for each element e e p weights c e i i 1 2 p are prescribed then the minmax combinatorial optimization problem is to minimize s f maximum 1 i p e s c e i several special cases of this general problem viz 3 partition makespan minimization problem bandwidth minimization in matrices and graphs categorized assignment problem etc have been studied in literature in this paper we propose exact and heuristic methods to solve the general problem our algorithms also give some new insights into the application of subgradient optimization in solving the lagrangean dual of combinatorial problems when the lagrange multipliers are required to satisfy additional constraints encouraging computational results are also reported

bicriteria network location bnl problems with criteria dependent lengths and minisum objectives abstract we present a new location model which is a generalization of the bicritera median problem we introduce two sum objectives and criteria dependent edge lengths for this np complete problem a solution method finding all the efficient solutions is presented the method is a two phases approach which can easily be applied as an interactive method in phase 1 the supported solutions are found and in phase 2 the unsupported solutions are found this method can be thought of as a general approach to bi objective combinatorial optimization boco problems

multiple objective minimum cost flow problems a review in this paper theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed for both the continuous and integer case exact and approximation algorithms are presented in addition a section on compromise solutions summarizes corresponding results the reference list consists of all papers known to the authors which deal with the multiple objective minimum cost flow problem

an algorithm for the bi criterion integer programming problem abstract we specify a variation of the weighting method for multi criterion optimization which determines nondominated solutions to the bi criterion integer programming problem the technique makes use of imposed constraints based on nondominated points for the bi criterion case we develop an algorithm which finds all nondominated points by solving a sequence of single criterion integer programming problems we present computational results for the linear 0 1 case and discuss the extension of our algorithm to the general multi criterion integer programming problem

modified interactive chebyshev algorithm mica for convex multiobjective programming abstract in this paper we describe an interactive procedural algorithm for convex multiobjective programming based upon the tchebycheff method wierzbicki s reference point approach and the procedure of michalowski and szapiro at each iteration the decision maker dm has the option of expressing his or her objective function aspirations in the form of a reference criterion vector also the dm has the option of expressing minimally acceptable values for each of the objectives in the form of a reservation vector based upon this information a certain region is defined for examination in addition a special set of weights is constructed then with the weights the algorithm of this paper is able to generate a group of efficient solutions that provides for an overall view of the current iteration s certain region by modification of the reference and reservation vectors one can steer the algorithm at each iteration from a theoretical point of view we prove that none of the efficient solutions obtained using this scheme impair any reservation value for convex problems the behavior of the algorithm is illustrated by means of graphical representations and an illustrative numerical example

selecting subsets from the set of nondominated vectors in multiple objective linear programming in this paper we develop methods for selecting certain subsets from the set n of nondominated points for multiple objective linear programming problems one such subset is the set of a points x in n for which the maximum deviation of the objective function values c x from some ideal vector m is as small as possible this subset can be obtained as the set of nondominated points for a multiple objective problem that is considerably smaller than the original problem and the proposed method does not require that the set n be calculated explicitly the method is extended to obtain another subset of n called the trade off compromise set that has some interesting properties and that gives valuable information about possible trade offs amongst the objectives

multiple criteria decision making multiattribute utility theory the next ten years management science and decision science have grown exponentially since midcentury two closely related fields central to this growth are multiple criteria decision making mcdm and multiattribute utility theory maut this paper comments on the history of mcdm and maut and discusses topics we believe are important in their continued development and usefulness to management science over the next decade our aim is to identify exciting directions and promising areas for future research

norm based approximation in multicriteria programming abstract based on new theoretical results on norms heuristic algorithms to approximate the nondominated set of multicriteria programs are proposed by automatically adapting to the problem s structure and scaling the approximation is constructed objectively without interaction with the decision maker as the algorithms extend the results obtained for bicriteria programs difficulties encountered when dealing with more than two criteria are discussed

preference modelling by estimating local utility functions for multiobjective optimization abstract this paper is intended to design goal programming models for capturing the decision maker s dm s preference information and for supporting the search for the best compromise solutions in multiobjective optimization at first a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives the interactive step trade off method istm is employed to generate a typical subset of efficient solutions of a multiobjective problem another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms this technique consisting of the istm method and the newly investigated search process facilitates the identification and elimination of possible inconsistent information which may exist in the dm s preferences it also provides various ways to carry out post optimality analysis to test the robustness of the obtained best solutions a modified nonlinear multiobjective management problem is taken as example to demonstrate the technique

integrating dea oriented performance assessment and target setting using interactive molp methods data envelopment analysis dea and multiple objective linear programming molp are tools that can be used in management control and planning whilst these two types of model are similar in structure dea is directed to assessing past performances as part of management control function and molp to planning future performance targets this paper is devoted to investigating equivalence models and interactive tradeoff analysis procedures in molp such that dea oriented performance assessment and target setting can be integrated in a way that the decision makers preferences can be taken into account in an interactive fashion three equivalence models are investigated between the output oriented dual dea model and the minimax reference point formulations namely the super ideal point model the ideal point model and the shortest distance model these models can be used to support efficiency analysis in the same way as the conventional dea model does and also support tradeoff analysis for setting target values by individuals or groups a case study is conducted to illustrate how dea oriented efficiency analysis can be conducted using the molp methods and how such performance assessment can be integrated into an interactive procedure for setting realistic target values

a class of practical interactive branch and bound algorithms for multicriteria integer programming abstract a practical interactive solution approach to multicriteria integer programming problems is developed the problem is solved by a branch and bound method that employs the zionts and wallenius procedure 23 for solving the multicriteria linear programming problem the development of algorithms for multicriteria decision problems itself is a multicriteria problem which involves the simultaneous minimization of the number of questions asked of the decision maker and the solution time two branch and bound algorithms that follow different search strategies to meet different levels of these criteria have been developed further two families of hybrid algorithms that incorporate a combination of the strategies of the two algorithms have also been developed strategies for the exploration of the decision maker s preference structure are discussed computational experience with the algorithms is presented the class of algorithms represents a collection of viable solution strategies applicable to a variety of decision making styles