extended vikor method in comparison with outranking methods the vikor method was developed to solve mcdm problems with conflicting and noncommensurable different units criteria assuming that compromising is acceptable for conflict resolution the decision maker wants a solution that is the closest to the ideal and the alternatives are evaluated according to all established criteria this method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria and on proposing compromise solution one or more the vikor method is extended with a stability analysis determining the weight stability intervals and with trade offs analysis the extended vikor method is compared with three multicriteria decision making methods topsis promethee and electre a numerical example illustrates an application of the vikor method and the results by all four considered methods are compared

vector coherent state constructions of u 3 symmetric tensors and their su 3 su 2 u 1 wigner coefficients generalized vector coherent state constructions of totally symmetric u 3 tensors are used to gain new expressions for the su 3 su 2 u 1 wigner coefficients for the coupling 1 1 20 3 3 these expressions show how the extremely simple formulas of le blanc and biedenharn involving a single 9 j coefficient arise as special cases of a general result that involves 12 j coefficients a simpler general result involving only 9 j coefficients and k normalization factors is derived in a way that can in principle be generalized to the generic coupling with multiplicity

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

lp metric sensitivity analysis for single and multi attribute decision analysis abstract analyzing the sensitivity of decisions to probability estimation error in single and multi attribute problems and to errors in estimating additive multi attribute value models in multi attribute problems is an integral part of decision analysis this paper presents an intuitive and tractable approach to this sensitivity analysis here a decision is considered insensitive if 1 the probabilities or multi attribute weights required for any other alternative to become preferred are not close to the original estimated probabilities and weights and 2 the rank order of states implied by the probabilities or the rank order of attributes implied by the additive multi attribute weights must change for any other alternative to become preferred the sensitivity analysis is conducted using straight forward linear programming models an example is used to demonstrate their application

proportional hesitant fuzzy linguistic term set for multiple criteria group decision making we propose the general concept of phflts we define the negation union and intersection operations on phfltss we present the phflwa and phflowa operators a transformation algorithm is proposed to convert the proportional comparative linguistic pairs into phfltss we develop a proportional hesitant fuzzy linguistic mcgdm model the theory of hesitant fuzzy linguistic term sets hfltss is a powerful technique used to describe hesitant situations which are typically assessed by experts using several possible linguistic values or rich expressions instead of a single term the union of hfltss with respect to each expert that is an extended hflts ehflts further facilitates the elicitation of linguistic assessments for addressing group decision making problems because ehfltss can deal with generalized either consecutive or non consecutive linguistic terms in this study we propose proportional hfltss phfltss which include the proportional information of each generalized linguistic term the mathematical form for a phflts is consistent with that for a linguistic distribution assessment however the underlying meanings of the proportions associated with generalized linguistic terms are different phfltss can be viewed as a special method for performing linguistic distribution assessments phfltss are recognized as a useful extension of hfltss and a possibility distribution for hfltss under different assumptions we define the basic operations with closed properties among phfltss on the basis of t norms and t conorms we then propose a probability theory based outranking method for phfltss by providing possibility degree formulas we also study two fundamental aggregation operators for phfltss namely the proportional hesitant fuzzy linguistic weighted averaging operator and the proportional hesitant fuzzy linguistic ordered weighted averaging operator several important properties of these aggregation operators are investigated finally we use the proposed multiple criteria group decision making model in practical applications

consistency measures for hesitant fuzzy linguistic preference relations hesitant fuzzy linguistic term sets hfltss are used to deal with situations in which the decision makers dms think of several possible linguistic values or richer expressions than a single term for an indicator alternative variable etc compared with fuzzy linguistic approaches they are more convenient and flexible to reflect the dms preferences in decision making for further applications of hfltss to decision making we develop a concept of hesitant fuzzy linguistic preference relations hflprs as a tool to collect and present the dms preferences due to the importance of the consistency measures using preference relations in decision making we develop some consistency measures for hflprs to ensure that the dms are being neither random nor illogical a consistency index is defined to establish the consistency thresholds of hflprs to measure whether an hflpr is of acceptable consistency for hflprs with unacceptable consistency we develop two optimization methods to improve the consistency until they are acceptable several illustrative examples are given to validate the consistency measures and the optimization methods

operators and comparisons of hesitant fuzzy linguistic term sets the theory of hesitant fuzzy linguistic term sets hfltss is very useful in objectively dealing with situations in which people are hesitant in providing linguistic assessments the purpose of this paper is to develop comparison methods and study the aggregation theory for hfltss we first define operations on hfltss and give possibility degree formulas for comparing hfltss we then define two aggregation operators for hfltss a hesitant fuzzy lwa operator and a hesitant fuzzy lowa operator in actual application we use these operators and the comparison methods to deal with multicriteria decision making problems with different situations in which importance weights of criteria or experts are known or unknown

clebsch gordan coefficients of su 3 with simple symmetry properties using an appropriate labelling operator constructed from representation generators su 3 clebsch gordan coefficients are introduced whose symmetry properties are similar to those of their su 2 counterparts an algebraic algorithm for computing the coefficients is presented

group decision making based on acceptable multiplicative consistency and consensus of hesitant fuzzy linguistic preference relations abstract in this paper we propose a new group decision making gdm method based on the acceptable multiplicative consistency and consensus of hesitant fuzzy linguistic preference relations hflprs first an approach for improving the consistency of hflprs is proposed to generate an acceptable multiplicative consistent hflpr then a consensus index of hflprs is defined and an optimization model is presented to meet an acceptable consensus requirement under the premise of the acceptable multiplicative consistency and the smallest information loss where it yields adjusted hflprs with an acceptable consistency and consensus then the weights of decision makers dms are calculated based on the obtained adjusted hflprs moreover we propose a new gdm method based on hflprs finally the proposed gdm method is illustrated by an application example and comparative analyses are conducted to show the performance and the superiority of the proposed gdm method

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

on the denominator function for canonical su 3 tensor operators the definition of a canonical unit su 3 tensor operator is given in terms of its characteristic null space as determined by group theoretic properties of the intertwining number this definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique up to phases construction of all su 3 wcg coefficients wigner clebsch gordan using this construction an explicit su 3 invariant denominator function characterizing completely the canonically defined wcg coefficients is obtained it is shown that this denominator function squared is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials these polynomials denoted gtq are defined over three shift parameters and three barycentric coordinates the properties of these polynomials hence of the corresponding invariant denominator function are developed in detail these include a derivation of their degree symmetries and zeros the

integration of indifference zone with multi objective computing budget allocation in this paper we consider how to address the issues of having designs with close performance in the multi objective ranking and selection mors problem to resolve this issue we propose integrating the indifference zone iz concept into the multi objective computing budget allocation mocba framework in particular when iz is introduced into the mocba framework we address how to determine the probability of non dominance how to define the pareto set and how to derive allocation rules for the simulation replications empirical results show that the mocba framework with iz can significantly save simulation budget when designs to be compared have close performance

how to select and how to rank projects the promethee method abstract in this paper we present the promethee methods a new class of outranking methods in multicriteria analysis their main features are simplicity clearness and stability the notion of generalized criterion is used to construct a valued outranking relation all the parameters to be defined have an economic signification so that the decision maker can easily fix them two ways of treatment are proposed it is possible to obtain either a partial preorder promethee i or a complete one promethee ii both on a finite set of feasible actions a comparison is made with the electre iii method the stability of the results given by the two methods is analysed numerical applications are given in order to illustrate the properties of the new methods and some further problems are discussed

a modified topsis with a different ranking index abstract as a tool for decision analysis the topsis technique for order preference by similarity to ideal solution attempts to choose alternative that should simultaneously have the closest distance from the positive ideal solution pis and the farthest distance from the negative ideal solution nis although the ranking index of topsis is reasonable it contains a flaw that is this ranking index is irrespective of the weights of separations of an alternative from the pis and the nis in other words no matter what weights the decision maker assigns to these two separations the ranking results would not differ as if he has no preference for these two separations this flaw will certainly limit the applicability of topsis by treating the separations of an alternative from the pis and the nis as a cost criterion and a benefit criterion respectively we reduced the original mcdm multi criteria decision making problem to a new mcdm problem with these two criteria only by proposing w and w as the weights of the cost criterion and the benefit criterion respectively we defined a new ranking index experimental results showed that if the number of alternatives exceeds two or if the relative importance of the two separations should be considered the proposed ranking index would be a better choice finally two numerical examples of a real life case are given for illustration in summary the proposed ranking index is intelligible and intrinsically superior to the original ranking index in seeking compromised solutions

novel basic operational laws for linguistic terms hesitant fuzzy linguistic term sets and probabilistic linguistic term sets in the process of decision making people sometimes may feel more comfortable to express their preferences by linguistic terms instead of the quantitative form however as the basic premise of operations the existing operational laws of linguistic terms and the extended linguistic term sets are very unreasonable in order to overcome this issue in this paper we redefine some more logical operational laws for linguistic terms hesitant fuzzy linguistic elements hfles and probabilistic linguistic term sets pltss based on two equivalent transformation functions these novel operational laws can not only avoid the operation values exceeding the bounds of ltss but also keep the operation results more reasonable in decision making with linguistic information furthermore the operational laws can keep the probability information complete when computing with pltss additionally lots of properties of the operational laws are discussed and some three dimensional figures are drawn to show the regions of different operational laws of linguistic terms more vividly

optimal incentive strategy for leader follower games an optimal incentive strategy by which the leader suffers the least losses in punishing the follower s deviation from the decision desired by the leader is defined sufficient conditions for the existence of optimal incentive strategy are given static and dynamic leader follower games with quadratic cost functionals are investigated it is shown that leader follower games with quadratic cost functionals admit optimal incentive strategies under the follower s decision variable being scalar there exists a unique linear optimal incentive strategy such an incentive strategy can be explicitly determined

an experimental comparison of different approaches to determining weights in additive utility models several studies this past decade have examined differences between holistic and decomposed approaches to determining weights in additive utility models some have argued that it matters little which procedure is used whereas others strongly favored particular methods in this paper we address this controversy experimentally by comparing five conceptually different approaches in terms of their weights and predictive ability the five methods are 1 multiple linear and non linear regression analyses of ten and fifteen holistic assessments 2 direct decomposed tradeoffs as proposed by keeney and raiffa keeney r l h raiffa 1977 decisions with multiple objectives wiley new york 3 a recent eigen vector technique of saaty saaty t l 1977 a scaling method for priorities in hierarchical structures j math psych 15 3 234 281 involving redundant pairwise comparisons of attributes 4 a straightforward allocation of hundred importance points and 5 unit weighting i e equal weighting after standardizing the attributes the decision task involved college admissions subjects were asked to evaluate hypothetical college applicants on the basis of verbal sat quantitative sat high school grade point average and a measure of extra curricular activity linear as well as nonlinear attribute utility functions were used in constructing the additive models the nonlinear functions were specified graphically by the subjects through selection from five different shapes i e one per attribute to test the predictive ability of the various models each subject made twenty separate pairwise comparisons of alternatives including direction and strength of preference the prediction criteria were percentage correct predictions as well as correlations using these twenty pairs seventy subjects were tested using an order controlled within subject design in comparing the different methods of weight determination monetary incentives were used to enhance motivation in terms of findings the methods generally differed systematically concerning the weights given to the various attributes as well as the variances of the resulting predictions on average however the methods predicted about equally well except for unit weighting which was clearly inferior the findings differ in this regard from the general literature furthermore nonlinear models were found to be inferior to linear ones finally subjects judged the methods to differ significantly in difficulty and trustworthiness which were found to correlate inversely the overall results raise various applied and theoretical issues which are discussed

novel types of sensitivity analysis for additive mcdm methods three novel types of sensitivity analysis are presented they determine the following 1 the sensitivity of a ranking to specific changes in the evaluations of all alternatives on certain criteria 2 the influence of specific changes in certain criterion scores of an alternative 3 the minimum modification of the weights required to make an alternative ranked first the use of these instruments is demonstrated by the results of a simulation experiment results show that the first and second type of sensitivity analysis enable to apply multiple criteria decision making mcdm methods in dynamic circumstances the third type of sensitivity analysis is demonstrated to be a tool to analyze the total weight space

hesitant fuzzy linguistic vikor method and its application in qualitative multiple criteria decision making the hesitant fuzzy linguistic term set hflts has turned out to be a powerful and flexible technique in representing decision makers qualitative assessments in the processes of decision making the aim of this paper is to develop a method to solve the multicriteria decision making mcdm problem within the context of hflts in which the criteria conflict with each other to do so the concepts of ideal solutions for a hfl mcdm problem have been introduced in addition in order to represent the closeness of one solution to the ideal one we propose a sort of hesitant fuzzy linguistic measures such as the hesitant fuzzy linguistic group utility measure the hesitant fuzzy linguistic individual regret measure and the hesitant fuzzy linguistic compromise measure based on these measures we develop a hesitant fuzzy linguistic vikor hfl vikor method which is motivated by the traditional vikor method the general procedures for the hfl vikor method are given some numerical examples are provided to demonstrate the advantages and practicality of our method finally we make some discussions on the advantages of the hfl vikor method as well as future work

local stability intervals in the analytic hierarchy process abstract this paper provides a sensitivity analysis of the judgements used in the analytic hierarchy process ahp in relation to the rank reversal produced in two different situations the selection of the best alternative p problem and the ranking of all the alternatives p problem in both cases under the supposition that we employ the row geometric mean method rgmm to determine the local priorities we obtain in the case of a single criterion a local stability interval for each judgement for each alternative and for the paired comparisons matrix which allow us to guarantee the best alternative and the ranking of all of them the p and p problems respectively with respect to these three situations judgement alternative and matrix and two problems p and p we also calculate a local stability index to detect the critical values of the resolution process both the local stability intervals and indexes are used as management tools in the final stage of the decision making process that is to say the exploitation phase especially in the negotiation process and the search for consensus between the actors

modeling subjective evaluation for fuzzy group multicriteria decision making this paper presents a new fuzzy multicriteria decision making mcdm approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers a pairwise comparison process is used to help individual decision makers make comparative judgements and a linguistic rating method is used for making absolute judgements a hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons to reflect the inherent imprecision of subjective judgements individual assessments are aggregated as a group assessment using triangular fuzzy numbers to obtain a cardinal preference value for each decision alternative a new fuzzy mcdm algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement an empirical study of aircraft selection is presented to illustrate the effectiveness of the approach

explicit canonical tensor operators and orthonormal coupling coefficients of su 3 the canonical unit su 3 tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators with the renormalization factors expressed in terms of the denominator functions of biedenharn gustafson lohe louck and milne the matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of asherova and smirnov the self conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique the corresponding extreme isoscalar factors of the clebsch gordan wigner coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of su 3 with its considerable simplication for the boundary values of parameters the general isofactors are also expanded in the different ways in terms of their boundary values the new classes of the generaliz

using the fuzzy multi criteria decision making approach for measuring the possibility of successful knowledge management this study develops a forecasting framework based on the fuzzy multi criteria decision making fmcdm approach to help organizations build awareness of the critical influential factors on the success of knowledge management km implementation measure the success possibility of knowledge management projects as well as identify the necessary actions prior to embarking on conducting knowledge management subjectivity uncertainty and vagueness within the prediction process are dealt with using linguistic variables parameterized by triangular fuzzy numbers by multiplying the importance weights of influential factors and the possible rating of success failure predicted success failure values are determined to enable organizations to decide whether to initiate knowledge management inhibit adoption or undertake remedial improvements to increase the possibility of successful knowledge management project the proposed approach is demonstrated using a real case involving a taiwanese semiconductor engineering corporation

a new version of 2 tuple fuzzy linguistic representation model for computing with words in this paper we provide a new proportional 2 tuple fuzzy linguistic representation model for computing with words cw which is based on the concept of symbolic proportion this concept motivates us to represent the linguistic information by means of 2 tuples which are composed by two proportional linguistic terms for clarity and generality we first study proportional 2 tuples under ordinal contexts then under linguistic contexts and based on canonical characteristic values ccvs of linguistic labels we define many aggregation operators to handle proportional 2 tuple linguistic information in a computational stage for cw without any loss of information our approach for this proportional 2 tuple fuzzy linguistic representation model deals with linguistic labels which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having equal distance between them moreover this new model not only provides a space to allow a continuous interpolation of a sequence of ordered linguistic labels but also provides an opportunity to describe the initial linguistic information by members of a continuous linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant meanwhile under the assumption of equally informative which is defined by a condition based on the concept of ccv we show that our model reduces to herrera and mart spl inodot spl acute nez s translational 2 tuple fuzzy linguistic representation model

a decision support system for multiple criteria alternative ranking using topsis and vikor in fuzzy and nonfuzzy environments abstract topsis and vikor are two well known and widely used multiple attribute decision making methods many extensions of these methods have been proposed that either use different techniques to rank alternatives or utilize fuzzy logic to handle alternatives and criteria that are unquantifiable and or incomplete in this paper we present the implementation of a web based decision support system that incorporates topsis and vikor to solve multicriteria decision making problems either in a nonfuzzy or in a fuzzy environment the aim of this paper is to present a tool that will be used by decision makers to compare various alternative solutions and understand how robust a decision will be the proposed system can be used both in single and in group decision making problems in addition we review several variations for each step of these methods and implement different techniques when applicable hence decision makers can experiment with different techniques in each method we implement ten normalization techniques three methods for the calculation of the ideal and anti ideal solutions fifteen distance metrics five fuzzy distance metrics and eight defuzzification techniques an illustrative example is presented to highlight the key features of the implemented system and the different scenarios that can be built using the proposed dss

a subjective and objective integrated approach to determine attribute weights this paper proposes an integrated approach to determine attribute weights in the multiple attribute decision making madm problems the approach makes use of the subjective information provided by a decision maker and the objective information to form a two objective programming model thus the resultant attribute weights and rankings of alternatives reflect both the subjective considerations of a decision maker dm and the objective information an example is used to illustrate the applicability of the proposed approach

robustness analysis in multi objective mathematical programming using monte carlo simulation abstract in most multi objective optimization problems we aim at selecting the most preferred among the generated pareto optimal solutions a subjective selection among objectively determined solutions in this paper we consider the robustness of the selected pareto optimal solution in relation to perturbations within weights of the objective functions for this task we design an integrated approach that can be used in multi objective discrete and continuous problems using a combination of monte carlo simulation and optimization in the proposed method we introduce measures of robustness for pareto optimal solutions in this way we can compare them according to their robustness introducing one more characteristic for the pareto optimal solution quality in addition especially in multi objective discrete problems we can detect the most robust pareto optimal solution among neighboring ones a computational experiment is designed in order to illustrate the method and its advantages it is noteworthy that the augmented weighted tchebycheff proved to be much more reliable than the conventional weighted sum method in discrete problems due to the existence of unsupported pareto optimal solutions

on the problem of weights in multiple criteria decision making the noncompensatory approach abstract this paper shows how the notion of relative importance of attributes can be defined within the framework of the noncompensatory approach to multiple criteria decision making the problem of weights then appears as a problem of functional representation of relations we state some theoretical results concerning this problem and outline a practical decision aid tactic based on the ideas introduced in the paper

rim reference ideal method in multicriteria decision making abstract multiple criteria decision making methods provide a useful tool to support decision making to date different methods have been developed using different approaches in particular the conception used by the topsis and vikor methods identifies the best alternative based on the positive and the negative ideal solutions in this paper a new method based on the concept of ideal solution is presented as a possible variant of the aforementioned however in our case these values can vary between the maximum value and the minimum value this does not occur in the vikor and topsis methods in which these values are the extreme values moreover our method is characterized to be independent of the type of data moreover rim does not present rank reversal an aspect that is not present in other multiple criteria decision making methods

symmetry properties of matrix elements of canonical su 3 tensor operators the symmetries of the su 3 3 j symbols which are defined as symmetrized matrix elements of the canonical su 3 tensor operators are investigated the symmetries considered are those which in su 2 correspond to the interchange of columns of the 3 j symbol as well as the symmetry under conjugation it is found that for each tensor operator in a multiplicity set the matrix elements for a fixed operator pattern carry a one dimensional representation of the symmetric group s3

su3 isoscalar factors a summary of the properties of the wigner clebsch gordan coefficients and iso scalar factors for the group su3 in the su2 u1 decomposition is presented the outer degeneracy problem is discussed in detail with a proof of a conjecture braunschweig s which has been the basis of previous work on the su3 coupling coefficients recursion relations obeyed by the su3 isoscalar factors are produced along with an algorithm which allows numerical determination of the factors from the recursion relations the algorithm produces isoscalar factors which share all the symmetry properties under permutation of states and conjugation which are familiar from the su2 case the full set of symmetry properties for the su3 wigner clebsch gordan coefficients and isoscalar factors are displayed

a method for determining the weights of criteria the centralized weights abstract in multicriteria analysis many methods use weights to represent the relative importance of criteria the noncompensatory aggregation procedure used in outranking methods enables comparisons of subsets of criteria from this point of view an interactive method is presented which requests only ordinal comparisons from the decision maker the relation more important than is assumed to be a semiorder therefore the indifference part of this relation is not necessarily transitive the judgements can be formulated as linear inequalities which constrain the set of feasible weights and threshold values all vertices of this polyhedron are determined the threshold and the unknown real weights are estimated by the centroid some characteristics a possible refinement in the estimation and an illustrative example are also given

note a preference ranking organisation method the promethee method for multiple criteria decision making principles for a new family of outranking methods are given the main aim of the proposed promethee approach is to be as easily understood as possible by the decision maker it is based on extensions of the notion of criterion six possible extensions are considered these extensions can easily be identified by the decision maker because the parameters to be defined at most 2 have an economic significance a valued outranking graph is constructed by using a preference index two possibilities are considered to solve the ranking problem by using this valued graph promethee i provides a partial preorder and promethee ii a total preorder on the set of the possible actions

implementation of the u n tensor operator calculus in a vector bargmann hilbert space using a vector coherent state framework and an associated vector bargmann hilbert space 1961 the structure of two classes of u n u n 1 unit projective operators is shown to be intimately related to 6 j and 9 j coefficients of the u n 1 subalgebra explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels

a linear programming method for multiple criteria decision making with probabilistic linguistic information abstract the hesitant fuzzy linguistic term set hflts is applicable in expressing people s hesitant qualitative preference information however it may lose preference information as all the terms in the hflts are assigned the same weight the probabilistic linguistic term set plts can avoid this drawback by allowing the decision makers dms to give the weight of each linguistic term as a probability due to the flexibility of the plts the multiple criteria decision making mcdm with probabilistic linguistic information is now becoming a hot research topic the aim of this paper is to propose a linear programming method to solve the mcdm problems in which the dms preferences over the alternatives with respect to the given criteria are expressed as the pltss we first give the weighted deviation square measure for the normalized probabilistic linguistic positive ideal solution and each normalized probabilistic linguistic evaluation vector of each alternative then we define the inconsistency index and the consistency index according to the objective and subjective evaluations given by the dms based on which a linear programming model is then built to derive the weight of each criterion furthermore a linear programming method is developed to solve the mcdm problems with probabilistic linguistic information and a case study about the evaluation of hospitals is conducted to illustrate the proposed method finally some comparative analyses with the aggregation operator based method and the probabilistic linguistic preference ranking organization method for enrichment evaluation pl promethee method are made to show the advantages of the proposed method

a multiple attribute utility theory approach to ranking and selection managers of large industrial projects often measure performance by multiple attributes for example our paper is motivated by the simulation of a large industrial project called a land seismic survey in which project performance is based on duration cost and resource utilization to address these types of problems we develop a ranking and selection procedure for making comparisons of systems e g project configurations that have multiple performance measures the procedure combines multiple attribute utility theory with statistical ranking and selection to select the best configuration from a set of possible configurations using the indifference zone approach we apply our procedure to results generated by the simulator for a land seismic survey that has six performance measures and describe a particular type of sensitivity analysis that can be used as a robustness check

aggregation of utility based individual preferences for group decision making abstract multi attribute utility theory maut elicits an individual decision maker s preferences for single attributes and develops a utility function by mathematics formulation to add up the preferences of the entire set of attributes when assessing alternatives a common aggregation method of maut for group decisions is the simple additive weighting saw method which does not consider the different preferential levels and preferential ranks for individual decision makers assessments of alternatives in a decision group and thus seems too intuitive in achieving the consensus and commitment for group decision aggregation in this paper the preferential differences denoting the preference degrees among different alternatives and preferential priorities denoting the favorite ranking of the alternatives for each decision maker are both considered and aggregated to construct the utility discriminative values for assessing alternatives in a decision group a comparative analysis is performed to compare the proposed approach to the saw model and a satisfaction index is used to investigate the satisfaction levels of the final two resulting group decisions in addition a feedback interview is conducted to understand the subjective perceptions of decision makers while examining the results obtained from these two approaches for the second practical case both investigation results show that the proposed approach is able to achieve a more satisfying and agreeable group decision than that of the saw method

on biorthogonal and orthonormal clebsch gordan coefficients of su 3 analytical and algebraic approaches minimal biorthogonal systems of the clebsch gordan wigner coefficients of su 3 u 2 are discussed as well as the dual coupled bases the closed system of analytical expressions for the dual isofactors reduced wigner coefficients and the overlaps of coupled states is obtained with the help of analytical inversion symmetry the regge type symmetry of the overlaps and the boundary orthonormal isofactors orthogonalization coefficients is discovered the polynomial structure of the alternative complete algebraic systems of the orthonormal su 3 isofactors characterized by the null spaces symmetries and additional selection rules and obtained by means of the hecht or gram schmidt process is considered the realizations of the external missing label operators of the third and the fourth orders in the minimal coupled bases which lead to preferable algorithms to evaluate the orthonormal su 3 coupling coefficients satisfying different symmetry properties are presented with the help of the 6j coef

preference programming through approximate ratio comparisons abstract in the context of hierarchical weighting this paper operationalizes interval judgments which allow the decision maker to enter ambiguous preference statements by indicating the relative importance of factors as intervals of values on a ratio scale through such judgments the decision maker can capture the subjective uncertainty in his preferences and thus avoid the often cumbersome elicitation of exact ratio estimates after each new statement the interval judgments are synthesized into dominance relations on the alternatives by solving a series of linear programming problems this leads to an interactive process of preference programming which provides more detailed results as the decision maker gradually enters a more specific preference description moreover the overall effort of preference elicitation is smaller than in the analytic hierarchy process because the most preferred alternative can usually be identified before all possible comparisons between pairs of factors have been completed

a calculus for su 3 leading to an algebraic formula for the clebsch gordan coefficients we develop a simple computational tool for su 3 analogous to bargmann s calculus for su 2 crucial new inputs are i explicit representation of the gelfand zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable ii an auxiliary gaussian measure with respect to which the gelfand zetlin states are orthogonal but not normalized iii simple generating functions for generating all basis states and also all invariants as an illustration of our techniques an algebraic formula for the clebsch gordan coefficients is obtained for the first time this involves only gaussian integrations thus su 3 is made as accessible for computations as su 2 is

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

a resolution of the su 3 outer multiplicity problem and computation of wigner coefficients for su 3 a simple algorithm is given for the resolution of the su 3 multiplicity problem and the computation of su 3 wigner coefficients using a complete set of u 2 x u 3 bargmann tensors classified by operator patterns null space properties of these tensors are easily derived their structure is such that a direct one to one correspondence is shown to exist between the bargmann tensors and the terms in the clebsch gordan series for su 3 as derived by o reilly 1982 finally the resolution presented herein is shown to be concordant with an alternative resolution advocated long ago by hecht

and like uninorm based consistency analysis and optimized fuzzy weight closed form solution of triangular fuzzy additive preference relations abstract this paper focuses on multiplicative consistency of triangular fuzzy additive preference relations tfaprs and deriving a closed form solution of optimized triangular fuzzy weights tfws from tfaprs and like uninorm based indices are introduced to measure increasing part vagueness decreasing part vagueness and overall vagueness for 0 1 valued triangular fuzzy preferences and tfaprs an index is then defined to measure row vagueness proportionality of tfaprs an and like uninorm based method is further proposed to generate multiplicatively consistent tfaprs from 0 1 valued tfws by discussing equivalency of 0 1 valued tfw vectors the paper presents two frameworks of normalized tfws called multiplicatively modal value normalized tfws and multiplicatively support interval normalized tfws based on crucial properties of consistent tfaprs a logarithmic least square lls model is established to seek multiplicatively support interval normalized tfws from tfaprs by decomposing its goals and constraints the lls model is transformed into two least square models whose closed form solutions are found by the lagrange multiplier method on basis of the obtained closed form solution of tfws an algorithm including acceptable consistency checking is developed for decision making with tfaprs the rationality and advantages of the presented models are illustrated by a numerical example with five tfaprs and a comparative analysis

recommendation using neighborhood methods with preference relation based similarity abstract selecting appropriate items from a list consisting of a large number of items provided by a product website can be difficult and time consuming for the potential customers the development of recommender systems should be an important solution that will help users to select items easily according to their preferences for recommender systems the main aim of the popular collaborative filtering approaches is to recommend items that users with similar preferences have liked in the past because there is a certain degree to which one alternative is not worse than another in decision making it would be interesting to make further use of the preference relation to design a similarity measure by measuring the overall strength of one user s preference over that of another the proposed similarity of one user to another user is therefore dependent on the strength of the preference of the former over the latter in contrast to traditional similarity measures for neighborhood methods in collaborative filtering the proposed preference relation based similarity is not symmetric for any two users experimental results have demonstrated that the generalization ability of the proposed multi criteria neighborhood method performs well in comparison to other single criterion and multi criteria neighborhood methods

increasing selective pressure towards the best compromise in evolutionary multiobjective optimization the extended nosga method most current approaches in the evolutionary multiobjective optimization literature concentrate on adapting an evolutionary algorithm to generate an approximation of the pareto frontier however finding this set does not solve the problem the decision maker still has to choose the best compromise solution out of that set here we introduce a new characterization of the best compromise solution of a multiobjective optimization problem by using a relational system of preferences based on a multicriteria decision aid way of thinking and an outranked based dominance generalization we derive some necessary and sufficient conditions which describe satisfactory approximations to the best compromise such conditions define a lexicographic minimum of a bi objective optimization problem which is a map of the original one the nosga ii method is a nsga ii inspired efficient way of solving the resulting mapped problem

a comparison of the analytic hierarchy process and a simple multi attribute value function abstract the paper reviews the applicability of approaches to multiple criteria decision making to aiding in the selection of a preferred option from a short list of alternatives in the light of a wealth of information about those alternatives it is concluded that saaty s analytic hierarchy process and a simple multi attributed value function are the approaches best suited to this problem and the most widely used in practice the paper goes on to compare these approaches in detail from both a theoretical and a practical standpoint the underlying assumptions of each are examined and considered in the light of practical decision aiding the practical implications of the use of each approach are discussed reference is made to experimental work carried out by the author which has examined both practical and theoretical aspects of the use of each approach in isolation and in comparative studies the paper will be of relevance to anyone facing a decision involving a finite number of alternatives and multiple criteria and considering the use of a formal approach

symmetry properties of s classified su 3 3j 6j and 9j symbols starting from the s classified su 3 clebsch gordan coefficients introduced previously pluhar et al 1985 the s classified su 3 3j 6j and 9j symbols are constructed and examined they satisfy simple symmetry relations similar to those valid for the su 2 case

why triangular membership functions abstract in this note we look at a certain theoretically sound motivation behind the common use of triangular and trapezoidal membership functions the studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing linguistic information it is shown that under some additional mild assumptions these triangular fuzzy sets comply with a request for a uniformly excited codebook in the case of the input interfaces and a satisfaction of a zero error reconstruction criterion being formulated for the output interface

a control theoretic view on incentives the idea of declaring a reward punishment for a decision maker according to his particular choice of action in order to induce certain desired behavior on the part of the decision maker is known as incentive threat this practice is age old however only in recent years have the notions been formalized in the development of a control theoretic view on incentives we first investigate the deterministic version of the incentive problem this reveals the basic simple idea behind the problem it also illustrates the different possibilities introduced by the presence of dynamics and multi follower nature of the problem this is followed by two variants of the stochastic version of the problem where we concentrate on the role of uncertainties relationship to economic literature is also discussed

three level incentive schemes using follower s strategies in differential games we deal with three level incentive differential games in which first and second leaders have access not only to slate information but also to information on follower s strategies we derive sufficient conditions for three level incentive schemes using information on follower s strategies in both linear and non linear differential games and show that three level incentive schemes using information on follower s strategies depend on an initial state value

ranking and selection for multiple performance measures using incomplete preference information abstract this paper presents two new procedures for ranking and selection r s problems where the best system designs are selected from a set of competing ones based on multiple performance measures evaluated through stochastic simulation in the procedures the performance measures are aggregated with a multi attribute utility function and incomplete preference information regarding the weights that reflect the relative importance of the measures is taken into account a set of feasible weights is determined according to preference statements that are linear constraints on the weights given by a decision maker non dominated designs are selected using two dominance relations referred to as pairwise and absolute dominance based on estimates for the expected utilities of the designs over the feasible weights the procedures allocate a limited number of simulation replications among the designs such that the probabilities of correctly selecting the pairwise and absolutely non dominated designs are maximized the new procedures offer ease of eliciting the weights compared with existing r s procedures that aggregate the performance measures using unique weights moreover computational advantages are provided over existing procedures that identify non dominated designs based on the expected values of the performance measures the new procedures allow to obtain a smaller number of non dominated designs they also identify these designs correctly with a higher probability or require a smaller number of replications for correct selection finally the new procedures allocate a larger number of replications to the non dominated designs that are therefore evaluated with greater accuracy these computational advantages are illustrated through several numerical experiments

weight stability intervals in multicriteria decision aid abstract in this paper we study the stability of the results given by a wide class of mcda methods to changes of the weights of the criteria three aspects of stability are discussed in each case we propose to construct stability intervals for the weights of either single criteria or groups of criteria a necessary and sufficient condition for stability is also considered the use of these instruments is demonstrated with a numerical example

on the structure of tensor operators in su3 a global algebraic formulation of su3 tensor operator structure is achieved a single irreducible unitary representation irrep v ofso 6 2 is constructed which contains every su3 irrep precisely once an algebra of polynomial differential operatorsa acting onv is given the algebraa is shown to consist of linear combinations of all su3 tensor operators with polynomial invariant operators as coefficients by carrying out an analysis ofa the multiplicity problem for su3 tensor operators is resolved

compromise solution by mcdm methods a comparative analysis of vikor and topsis the multiple criteria decision making mcdm methods vikor and topsis are based on an aggregating function representing closeness to the ideal which originated in the compromise programming method in vikor linear normalization and in topsis vector normalization is used to eliminate the units of criterion functions the vikor method of compromise ranking determines a compromise solution providing a maximum group utility for the majority and a minimum of an individual regret for the opponent the topsis method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative ideal solution but it does not consider the relative importance of these distances a comparative analysis of these two methods is illustrated with a numerical example showing their similarity and some differences 2003 elsevier b v all rights reserved

an axiomatic characterization of different majority concepts a lot of decision support systems use some kind of aggregation procedure based on the concept of majority but not always the same one it can be simple majority weak majority or one of the many other kinds of majority this paper attempts to present the main variants of majority and to characterize them in a uniform way consequently it is now easier to compare different kinds of majority and to understand the dissimilarities or similarities between them this should help decision analysts willing to use a majority procedure to choose the right one for their problem and context

explicit orthonormal clebsch gordan coefficients of su 3 the construction of the explicit algebraic polynomial expressions for the nonmultiplicity free orthonormal clebsch gordan wigner coefficients of su 3 u 2 is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the hecht or gram schmidt process the direct and inverse orthogonalization coefficients the first of them being equivalent to the boundary orthonormal isofactors are expressed up to explicitly given multiplicative factors in terms of the numerator and denominator polynomials related with the auxiliary a function of louck biedenharn and lohe that appears as a fragment of the denominator g functions of canonical su 3 tensor operators

an interactive algorithm for large scale multiple objective programming problems with fuzzy parameters through topsis approach in this paper we extend technique for order preference by similarity ideal solution topsis for solving large scale multiple objective programming problems involving fuzzy parameters these fuzzy parameters are characterized as fuzzy numbers for such problems the pareto optimality is introduced by extending the ordinary pareto optimality on the basis of the level sets of fuzzy numbers an interactive fuzzy decision making algorithm for generating pareto optimal solution through topsis approach is provided where the decision maker dm is asked to specify the degree and the relative importance of objectives finally a numerical example is given to clarify the main results developed in the paper

a method for dealing with inconsistencies in pairwise comparisons abstract the pairwise comparison method is a powerful inference tool for assessing the relative importance of a set of items formally its objective is to make compatible decision maker s assignments paired comparison with properties needed for obtaining an overall rank in this paper we propose a distance based framework for analysing this kind of compatibility in this context goal programming is proposed as an attractive and flexible tool

economic analysis of simulation selection problems ranking and selection procedures are standard methods for selecting the best of a finite number of simulated design alternatives based on a desired level of statistical evidence for correct selection but the link between statistical significance and financial significance is indirect and there has been little or no research into it this paper presents a new approach to the simulation selection problem one that maximizes the expected net present value of decisions made when using stochastic simulation we provide a framework for answering these managerial questions when does a proposed system design whose performance is unknown merit the time and money needed to develop a simulation to infer its performance for how long should the simulation analysis continue before a design is approved or rejected we frame the simulation selection problem as a stoppable version of a bayesian bandit problem that treats the ability to simulate as a real option prior to project implementation for a single proposed system we solve a free boundary problem for a heat equation that approximates the solution to a dynamic program that finds optimal simulation project stopping times and that answers the managerial questions for multiple proposed systems we extend previous bayesian selection procedures to account for discounting and simulation tool development costs

complementary group resolution of the su n outer multiplicity problem i the littlewood rules and a complementary u 2n 2 group structure a complementary group to su n is found that realizes all features of the littlewood rules for kronecker products of su n representations this is accomplished by considering a state of su n to be a special gel fand state of the complementary group u 2n 2 with labels of the latter used to distinguish multiple occurrences of irreducible representations of su n irreps in the su n su n su n decomposition that is obtained from the littlewood rules furthermore this realization also helps us to determine su n su n 1 u 1 reduced wigner coefficients rwcs frequently called isoscalar factors and clebsch gordan coefficients cgcs or full nonreduced wigner coefficients of su n using algebraic or numeric methods in either the canonical or a noncanonical basis new explicit formulas for the su 3 and su 4 multiplicities are obtained by using this technique

a closed formula for the product of irreducible representations of su 3 we determine a closed formula in terms of p q r and s for the decomposition of the product p q r s of finite dimensional irreducible representations of su 3 we also determine in terms of p q r s m and n necessary and sufficient conditions that a term m n appears in this decomposition and its multiplicity

three level stackelberg decision problems three level stackelberg decision problems are studied by using the inducible region concept through a systematic derivation it is identified that the leader s control has dual purposes which in general are not separable a special class of problems is then considered where explicit results are obtained

orthogonalization of the projected states and isofactors the explicit direct and inverse orthogonalization coefficients ocs are derived for the projected basis states of the two parametric covariant irreducible representations of including elliott s n 3 case and draayer with the alternative choices of the gram schmidt processes began respectively from the lowest vergados or highest absolute value of the intrinsic parameter k the different direct and inverse ocs the latter equivalent to the definite boundary isofactors or the boundary resubducing coefficients are found to be connected by the definite analytical continuation contraction procedures with mutually related ocs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of the orthogonalization coefficients considered are presented as definite algebraic polynomial structures under the square root with the linear factors and numerator denominator polynomials in terms of the partition dependent functions of biedenharn and louck

computing efforts allocation for ordinal optimization and discrete event simulation ordinal optimization has emerged as an efficient technique for simulation and optimization exponential convergence rates can be achieved in many cases in this paper we present a new approach that can further enhance the efficiency of ordinal optimization our approach intelligently determines the optimal number of simulation replications or samples and significantly reduces the total simulation cost numerical illustrations are included the results indicate that our approach can obtain an additional 74 computation time reduction above and beyond the reduction obtained through the use of ordinal optimization for a 10 design example

fuzzy outranking for environmental assessment case study iron and steel making industry recent years have seen a world wide change in the environmental policy towards integrated pollution prevention taking into account all environmental media consequently the environmental assessment of production techniques has to consider multiple criteria which cannot be aggregated to one single index the concepts of multi criteria decision making mcdm seem a suitable means in order to implement life cycle assessment lca into integrated decision processes the outranking methods as a special subgroup of mcdm methods are particularly suitable for integral decision making through the notion of weak preference and incomparability which better represent the real decision situation as demonstrated in this paper especially the outranking method promethee brings together flexibility and simplicity for the user and is therefore chosen for the enhancement towards the evaluation of fuzzy data on preferences scores and weights

the missing link operators for labelling multiplicity in the clebsch gordan series it is indicated that there are invariant operators on the direct product space of irreducible representations of lie groups which distinguish between repeating representations they complete the labelling of the clebsch gordan series

selecting a selection procedure selection procedures are used in a variety of applications to select the best of a finite set of alternatives best is defined with respect to the largest mean but the mean is inferred with statistical sampling as in simulation optimization there are a wide variety of procedures which begs the question of which selection procedure to select the main contribution of this paper is to identify through extensive experimentation the most effective selection procedures when samples are independent and normally distributed we also a summarize the main structural approaches to deriving selection procedures b formalize new sampling allocations and stopping rules c identify strengths and weaknesses of the procedures d identify some theoretical links between them and e present an innovative empirical test bed with the most extensive numerical comparison of selection procedures to date the most efficient and easiest to control procedures allocate samples with a bayesian model for uncertainty about the means and use new adaptive stopping rules proposed here

sensitivity analysis for product design selection with an implicit value function abstract in product design selection the decision maker dm often does not have enough information about the end users needs to state the preferences with precision as is required by many of the existing selection methods we present for the case where the dm gives estimates of the preferences a concept for calculating a robustness index the concept can be used with any iterative selection method that chooses a trial design for each iteration and uses the dm s preference parameters at that trial design to eliminate some design options which have lower value than the trial design such methods like our previously published method are applicable to cases where the dm s value function is implicit our robustness index is a metric of the allowed variation between the actual and estimated preferences for which the set of non eliminated trial designs which could be singleton will not change the dm through experience can use the robustness index and other information generated in calculating the index to determine what action to take make a final selection from the present set of non eliminated designs improve the precision of the preference estimates or otherwise cope with the imprecision we present an algorithm for finding the robustness index and demonstrate and verify the algorithm with an engineering example and a numerical example

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

derivation of necessary and sufficient conditions for single stage stackelberg games via the inducible region concept the inducible region is defined as the collection of all the possible outcomes it is typically a subset of the entire decision space the best the leader can obtain is then the optimal outcome in this inducible region necessary and sufficient conditions are derived by first delineating the inducible region and then obtaining a leader s optimal strategy if it exists if not an e strategy always exists provided that the leader s stackelberg cost is bounded

a fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making decision making is a process common to human beings the uncertainty and fuzziness of problems demand the use of the fuzzy linguistic approach to model qualitative aspects of problems related to decision the recent proposal of hesitant fuzzy linguistic term sets supports the elicitation of comparative linguistic expressions in hesitant situations when experts hesitate among different linguistic terms to provide their assessments the use of linguistic intervals whose results lose their initial fuzzy representation was introduced to facilitate the computing processes in which such expressions are used the aim of this paper is to present a new representation of the hesitant fuzzy linguistic term sets by means of a fuzzy envelope to carry out the computing with words processes this new fuzzy envelope can be directly applied to fuzzy multicriteria decision making models an illustrative example of its application to a supplier selection problem through the use of fuzzy topsis is presented

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

topsis for modm abstract in this study we extend topsis to solve a multiple objective decision making problem the principle of compromise of topsis for multiple criteria decision making is that the chosen solution should have the shortest distance from the positive ideal solution as well as the longest distance from the negative ideal solution thus we reduce a k dimensional objective space to a two dimensional objective space by a first order compromise procedure we then use membership functions of fuzzy set theory to represent the satisfaction level for both criteria we obtain a single objective programming problem by using the max min operator for the second order compromise operation to illustrate the procedure the bow river valley water quality management problem is solved by use of topsis